Systems and methods for generating and using projector curve sets for universal calibration for noninvasive blood glucose and other measurements

[0001] This application is a divisional of and claims the benefit of priority under 35 U.S.C. §121 to U.S. patent application Ser. No. 14/869,550 entitled “Systems and Methods for Noninvasive Blood Glucose and Other Analyte Detection and Measurement Using Collision Computing,” filed on Sep. 29, 2015, which claims the benefit of priority to U.S. Provisional Patent Application Ser. No. 62/057,103 entitled “A System and Method for Generating Feature Sets Using Tomographic Spectroscopy, for Analyte Detection,” filed on Sep. 29, 2014; and to U.S. Provisional Patent Application Ser. No. 62/057,496 entitled “Non-Invasive Glucose Monitoring,” filed on Sep. 30, 2014, the contents of each of which are incorporated herein by reference in their entirety.

[0002] In general, this disclosure relates to a system for detecting and/or quantifying the presence of very small amounts of material, the property or concentration of a material, or changes in the amount or properties of the material, or an event or anomaly of interest and, in one example, specifically to a system that performs measurement of biochemical analytes using diffuse reflectance tomographic spectroscopy in conjunction with collision computing.

[0003] The measurement of substances at extremely low concentrations in complex samples has absorbed the efforts of chemists for centuries. Modern spectroscopic techniques, which allow measurement of substances at parts-per-million (ppm) or even parts-per-billion (ppb) concentrations, have revolutionized this field and allowed the detection and measurement of naturally-occurring materials such as hormones, contaminating substances such as mercury, and pollutants such as atmospheric sulfur dioxide at levels far below those achieved using earlier methods of analysis. The direct measurement of many important substances in human tissue, however, has not been as successfully accomplished, and many other measurements where the substance to be measured is at a low concentration cannot be made accurately in a high noise and/or high clutter environment. One particularly challenging problem concerns the management of diabetes.

[0004] Diabetes is a condition in which the body's natural control of blood sugar (glucose) has been lost. Insulin is a hormone that is secreted by the pancreas that works with the body to process blood sugar. Typically, diabetes is caused by some problem with the body's ability to create or use insulin. Diabetes occurs in different medical conditions: type 1 diabetes (previously known as “juvenile diabetes”), type 2 (“adult onset”) diabetes, and gestational diabetes (a complication of pregnancy). In type 1 diabetes, the patient's pancreas is no longer able to produce insulin at normal rates, while in type 2 and gestational diabetes, a patient's cells are not able to properly utilize insulin. Some patients with type 2 or gestational diabetes can treat their conditions using diet, exercise, or a variety of pharmaceutical preparations. All patients with type 1 diabetes (and many with type 2, especially of longer duration) typically must control the disease with injections or infusions of insulin.

[0005] A body's normal creation and processing of insulin generally varies over the course of the day depending on a variety of factors, including when and what a person eats, whether that person is exercising, and the time of day. This variation in normal insulin production usually serves to maintain safe glucose levels in the body. If diabetes is left untreated, the complications that can arise can be extremely serious. In the absence of insulin, glucose in the blood can reach dangerously high levels. Extended periods of high blood glucose levels (“hyperglycemia”) can lead to a condition known as ketoacidosis, which, if untreated, can be fatal. Chronic poor control of glucose levels can also cause serious long-term complications, which include eye damage (resulting in blindness), kidney damage, cardiovascular disease, loss of feeling in the extremities, and slow healing of wounds. Frequently, diabetes may require amputations of toes, feet or legs. If blood glucose levels are allowed to drop below a threshold value (generally 50-60 milligrams per deciliter), the person can be in acute danger from this “hypoglycemia,” which can cause confusion, difficulty speaking, unconsciousness, and coma.

[0006] To allow better information and control, blood glucose measurement systems that can be used by individual users have been developed. These blood glucose measurement systems typically require the use of an electronic meter and disposable test strips. A test strip is inserted into the meter, and the user pricks himself or herself (usually on a finger) with a lancet to draw a small amount of blood which is applied to the test strip. The blood glucose meter, using one of a variety of analysis techniques, determines the amount of glucose in the small blood sample drawn from the user. However, because of the pain involved in lancing a body part to draw blood, the need to dispose of materials contaminated with blood, and the visibility and potential embarrassment of testing using conventional blood glucose monitoring systems, many investigators have tried to develop technologies that allow measurement of blood glucose without drawing blood or causing discomfort. These technologies have been termed “noninvasive” blood glucose measurement systems, or simply “noninvasive glucose.”

[0007] For example, U.S. Pat. No. 4,655,225, issued to Dahne et al., with a filing date of Apr. 18, 1985, appears to describe a relatively simple approach to measuring glucose in tissue. This patent states at column 1, lines 8 through 18: “This determination is carried out by measuring the optical near infrared absorption of glucose in regions of the spectrum where typical glucose absorption bands exist and computing the measured values with reference values obtained from regions of the spectrum where glucose has no or little absorption and where the errors due to background absorptions by the constituents of the surrounding tissues or blood containing the glucose are of reduced significance or can be quantitatively compensated.” When investigators eventually determined that no regions of the near-infrared NIR spectrum meeting these criteria could be found, they employed advanced techniques, augmented with sophisticated modifications to the instrumentation and complicated mathematical treatments. The NIR region used in these techniques is typically the region of the electromagnetic spectrum in the wavelength range of 700-2,500 nanometers

[0008] The most commonly-investigated technique for noninvasive glucose over the past twenty-five years or so has been near-infrared spectroscopy (NIRS), using light that is just beyond the visible portion of the spectrum, and generally considered to be a wavelength range of about 700 nanometers to about 2,500 nanometers. When near-infrared light is applied to tissue, generally the light is both scattered by cells and structures under the skin and is absorbed by substances in the tissue, including glucose, which can be termed an “analyte” (a substance whose concentration is being determined). The amount of absorbance due to glucose in this wavelength region, however, is extremely small and, coupled with (i) the low concentration of glucose in the fluids of tissue (about 50-500 milligrams per deciliter, equivalent to about 0.05% to 0.5%), (ii) the presence of many compounds with similar chemical structures and similar absorbance patterns in the near-infrared region, and (iii) the extremely high concentration of water in tissue, makes direct measurement of an analyte in this region of the spectrum very challenging, requiring the use of sophisticated spectroscopic imaging and computational techniques.

[0009] Many patents, such as U.S. Pat. No. 5,460,177, issued to Purdy, et al. in 1995, appears to describe approaches for using light from the near-infrared portion of the spectrum to provide “expressions” or spectra that can be examined to determine the concentration of glucose without drawing a sample of blood. Commonly-employed data reduction techniques in previous investigations generally include what are known as “multivariate” techniques, such as principal component analysis (“PCA”), partial least squares (“PLS”), support vector machine (“SVM”), and multiple linear regression (“MLR”). As a group, when these approaches are applied to the measurement of chemical substances, they are often referred to as “chemometrics.” These multivariate techniques may make it theoretically possible to create a correlation between measurable properties of a material like tissue, such as the amount of light absorbed or reflected as a function of wavelength (known as a “spectrum”), and the concentration of an analyte such as glucose. An example spectrum, which plots a variable related to light intensity (such as absorbance, transmittance, energy, etc.) as a function of the wavelength of light expressed in nanometers, is shown in FIG. 1.

[0010] In order to perform a multivariate analysis on a set of data, some techniques first build what is called a “model.” This may be done by first creating a number of like measurements (typically termed the “calibration set” in the context of multivariate analysis), which are spectra which contain the value of a parameter related to light intensity as a function of wavelength and which have known values for the concentration of the analyte. FIG. 2, for example, depicts an absorbance spectrum for one known concentration of glucose. The overall variance in the calibration data set may be separated into “factors” which typically represent decreasing amounts of variance. This model, once created using the multivariate technique, may represent the contribution of a measurement at each wavelength to the concentration of the analyte. The values of the model at each wavelength are often termed “final regression coefficients.” At some wavelengths, the value is positive, which may mean that the measurement at that wavelength contributes actual value to the calculation of the concentration, while at other wavelengths the value is negative, which may indicate an amount to be subtracted in the concentration calculation.

[0011] Once such a model, an example of which is shown in FIG. 3, has been created, a spectrum with an unknown analyte concentration can be analyzed using that model. With reference to FIG. 4, this can be done by multiplying the value of the new spectrum at each wavelength by the value of the final regression coefficient of the model at the same wavelength, and adding up all the results of those multiplications to give an estimated value of the analyte. The accuracy of this estimated concentration value usually depends on many factors—among them: the number of spectra used in the calibration set, the number of multivariate “factors” used to construct the model, the accuracy of the measurement process for the spectra, the similarity of the instrumentation used to generate the spectra, and the strength of the measured parameter. The measured parameter in the examples described above is the amount of absorbance of glucose molecules at each of the wavelengths, which is quite weak, and which is hidden under the strong absorbance of other components of tissue; the absorbance spectra of some tissue components are shown in FIG. 5, but these spectra are normalized—they are not based on a uniform scale.

[0012] Attempts at making measurements of glucose in tissue using these techniques have been made, but to the present time, these efforts have not succeeded in producing clinically accurate results. Several systems have used near-infrared spectroscopy and multivariate analysis techniques, but to date, no system for noninvasive measurement of glucose using near-infrared spectroscopy appears to have gained regulatory approval or appears to have been marketed in the U.S. In addition to multivariate and many other regression techniques, other practitioners have sought to extract a glucose signal from near-infrared tissue spectroscopy data using other methods of data reduction. Examples of these include: subtractive techniques where the spectra of interfering substances are sequentially removed from a tissue spectrum, analysis of a number of equations with several unknowns, neural networks, model trees, genetic algorithms, chaotic networks (and many related approaches that can be classified as fast learning algorithms), and an optical method known as Kromoscopy (Appl. Opt. 2000 Sep. 1: 39(25):4715-20). Many other approaches to measuring glucose in tissue have also been attempted, and a book has been written on the subject: The Pursuit of Noninvasive Glucose: “Hunting the Deceitful Turkey” by John L. Smith, 4^th Edition (Copyright 2015), the disclosure of which is incorporated herein by reference in its entirety. Like the multivariate techniques described above, these techniques have not succeeded in producing clinically accurate results for noninvasive measurement of glucose. A new inventive approach for the direct, noninvasive measurement of glucose in tissue, as well as other analytes of interest, is therefore required.

[0013] The term “collision” has been used in different fields and contexts. For example, in computer networking and telecommunications and during the execution of various algorithms processing data, data-packet collisions generally imply that two distinct pieces of data have the same hash value, checksum, fingerprint, or cryptographic digest. The hashing collisions typically allocate the same memory location to different data values. In the computational problem of determining the intersection of two or more computer animated objects, as encountered in simulations and/or video games, linear algebra and computational geometry methods (e.g., the axis-aligned bounding box method for an n-body collision described in Lin, Ming C (1993). “Efficient Collision Detection for Animation and Robotics (thesis)”. University of California, Berkeley), collision analysis techniques may be used to determine whether two animated objects would or have collided and the time of impact, and for post-collision trajectory estimation. Computational atomic physics, which introduced a class of stochastic algorithms, such as those used in game theory, molecular dynamics, social simulations, and econometrics, is inspired by techniques used in particle-field simulations. These techniques generally involve collisions of atomic and subatomic particles (such as those described in Sigurgeirsson et al. (2001), “Algorithms for Particle-Field Simulations with Collisions”, Journal of Computational Physics 172, 766-807) and algorithms used for calculating post-collision electron and positron scattering and excitation in atoms and ions, such as in non-perturbative close coupling approach.

[0014] Various methods inspired by the classical Boltzmann collision operator used in statistical physics or fluid simulations for describing the interaction between colliding particles in rarefied gas include Bobylev-Rjasanow's Integral Transform Method, Pareschi-Russo's Spectral Method, and Mouhot-Pareschi's method. These computational techniques typically exploit the fundamental properties of the Boltzmann binary collision operator (e.g., the Bhatnagar-Gross-Krook (BGK) operator described in P. L. Bhatnagar, E. P. Gross, M. Krook (1954). “A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems,” Physical Review 94 (3): 511-525) that are related to conservation of mass, momentum, and energy, to infer properties of the colliding entities. While the binary collision operator and its approximation have been used in computer simulations and modeling to infer properties of colliding entities, to date these techniques generally do not take into consideration the environment of the entities, such as noise, clutter from confounders, and ultra-weak magnitudes of signals associated with entities of interest.

[0015] Computational collision techniques, such as those described in Collision-Based Computing, Andrew Adamatzky, (Ed.), ISBN 978-1-4471-0129-1, generally appear to describe computer implementations of various collisions described above, such collisions between particles or collisions between physical entities. With reference to FIGS. 6-8B, collisions between two traveling waveforms can occur on a space-time grid or a grid in another Cartesian coordinate system. In general, the two traveling waveforms move toward each other along a straight line connecting their centers of mass, with each point of one discretized wavefront (102 in FIG. 6) engaging in a series of collision steps with a corresponding point of the other discretized waveform (104 in FIG. 6), with the result of each step being determined by a set of rules established at the line of collision L_C (100 in FIG. 6). Those rules determine the mathematical result of collision interactions and the shape of the resulting symbolic waveforms. Following completion of these partial steps in a collision between two waveforms, two resulting waveforms (106, 108 in FIGS. 7A-7F) are typically produced. The two resulting waveforms: (i) may be completely unchanged (except for a phase shift or delay) relative to the colliding waveforms, as shown in FIG. 7F), (ii) may be deformed (as shown in FIG. 8A), or (iv) either or both waveforms may be essentially completely destroyed (as shown in FIG. 8B). The nature of change in each waveform as a result of the collision (which can also be described as the degree of “elasticity” of the collision) generally depends on the composition of the waveforms, the relative energy of each waveform, and the rules established at the collision line L_C. A “soliton” is an example of a waveform that does do not change its properties other than a delay or phase shift during the collision process.

[0016] In various embodiments, the systems and methods described herein facilitate detection and/or measurement of the presence and concentration of one or more chemical constituents of interest (generally called analytes) in a specified environment or a medium to be analyzed (e.g., turbid layers of human tissue with inhomogeneities that scatter light in random directions) in the presence of one or more confounders. This is achieved in part by employing one or more nonlinear, non-invertible computational collisions between two entities—waveforms derived from sample observations, where the samples are represented in an observation domain (also called a data-collection domain), and a purposefully constructed waveform that is typically not based on the data-collection domain—yielding waveforms that can be processed to detect analyte presence and/or measure analyte concentration in the uncharacterized samples. Various embodiments described herein provide for precise and accurate detection and/or measurement of an analyte of interest in a medium to be analyzed. Various embodiments also facilitate detection and/or measurement of events in different types of environments.

[0017] Specifically, in certain embodiments a new computational collision technique is described whereby the presence and/or concentration of clinical and chemical analytes can be determined from sensor data in the presence of high noise, background interference, and interference from other substances with similar properties as that of the analyte(s). This computational collision process can be extended to other types of data where the information sought is considered or treated as the equivalent of an analyte. Thus, detection and/or measurement of an analyte, as described herein, may include the identification, characterization, and/or measurement of any material, property, magnitude, condition, anomaly, or an event of interest. The domain from which the source data for the uncharacterized samples may originate is generally referred to as the data-collection domain.

[0018] Although collisions actually occur only within a computer, with the interaction of what are termed “frequency components,” of two waveforms in the frequency domain, the collision process can be symbolically depicted as the collision between two discretized waveforms in a synthetic domain termed a “collision time domain.” For completeness and clarity, that process is described below in detail and illustrated in the referenced figures.

[0019] When the waveforms to be collided are properly constructed and are made co-dependent (as described below), energy changes (e.g., changes in spectral energy due to absorption of NIR radiation by the analyte molecules in a medium in certain embodiments) as derived from data corresponding to particular materials, conditions, or events, can be transferred from one waveform to the other, and amplified in successive collisions. In various embodiments, collision computing in which two co-dependent waveforms collide, allows for drawing inferences about the underlying properties of waveforms participating in the collision, based on the energy transfer from one waveform to the other, and the energy properties of the surviving waveform. In various embodiments, collision computing facilitates an estimation of a Net Analyte Signal (NAS) from spectral or other data sources. Through a projection process, the NAS can be used to determine the presence and/or the concentration of an analyte in a medium to be analyzed.

[0020] Thus, in various embodiments, methods and systems described herein provide for detecting the presence of and for estimating concentration of analytes from sensor data, by transforming the incoming sensor data to a waveform (Ψ_CF), i.e., a mathematical representation of the sensor data in the computer memory; colliding it with another waveform in the computer memory, referred to as the Zyoton (Ψ_Z), which is unrelated to and independent of the data domain of the sensor; and assessing the properties of the spectral energy of a modified Zyoton (Ψ_Z′) generated after the collision. Depending on the nature of input data, characteristics of the sensor used for observing the analyte, complexity and concentration of confounders, and expected concentration of the analyte, one or more iterations of computational collisions may be performed. After each collision operation between the two waveforms, the power spectral density of the collision-modified Zyoton Ψ_Z′ is estimated. Waveform collisions may be implemented in the analog or digital computing domain and in a time or frequency domain, i.e., the colliding waveforms may be expressed in the time domain or in the frequency domain. Analyte presence and concentration can be determined based on net gain or loss in the spectral energy after a selected number of collision iterations. Observability of consistent patterns in spectral energy gain or loss after each collision between the two waveforms is a prerequisite in various embodiments to concluding the integrity and consistency of the collision process. The processing unit in which the collisions are implemented is denoted as the collision computer.

[0021] Throughout this specification, data features which have been modified or conditioned by modulation (called conditioned features), as well as the Zyotons, are described as waveforms. When these waveforms collide, the collision operation takes place specifically at their wavefronts and, in that context, these waveforms may be described as wavefronts. These two terms are generally used separately to emphasize the special aspects of the collision computing process. Additionally, components of the original data that interfere with the determination of the analyte or obscure its signal are referred to as confounders and, taken as a group, are represented by the term “clutter.” The collision computing process generally functions to improve the signal to clutter ratio by decreasing the impact of the confounders present in the original data.

[0022] Unlike a collision between physical particles, which can be perfectly or imperfectly elastic or inelastic depending on how the kinetic energy and momentum are conserved and whether the two particles continue to exist as two entities, and unlike a collision between two waveforms that are not specifically constructed to be co-dependent (as described below), the waveform collisions between two co-dependent waveforms described herein are interferometric, where the two distinct but co-dependent waveforms combine post-collision and the interaction is dissipative, that is, the spectral energy of the combined system is less than the pre-collision energies of two waveforms. Also, as a result of this collision, the two waveforms combine to a single waveform, called the modified Zyoton. The purposeful dissipation during collisions is causal in character and serves as a probe to characterize analytes of interest because, as described below, generally the energy associated with the analyte of interest is transferred into the modified Zyoton while the energy associated with noise and clutter may be discarded. Precisely measuring a change in the energy of the modified Zyoton relative to the energy of the original Zyoton, in spectral regions (e.g., wavelength ranges) where the energy changes are most visible, allows us to quantify concentration of analytes with accuracy and reliability. As a design principle, the two waveforms and collision operators are designed based on analysis of sensor data to intentionally maximize changes in spectral energy in the spectral regions associated with the analyte, in each collision iteration. Also, the two waveforms are designed such that the Zyoton waveform typically has higher initial spectral energy than the other waveform with which it collides.

[0023] One important aspect of various methods and systems described herein is the discovery that very small portions of data (also called spectral data features) contain analyte-specific frequency components that persist, even in the presence of overwhelming amounts of clutter, and that these frequency components can be amplified and separated from that clutter. As used herein the term “frequency components” generally refers to spatial frequencies, i.e., those components of a traveling waveform that are periodic across position in space. These frequency components (or components) can be determined by a Fourier transform of a time-domain representation of the waveform.

[0024] Collision computing described herein facilitates the amplification and separation of data. To this end, a stable waveform called a Zyoton, which generally does not change its shape and morphology when it propagates through a propagation medium, is computationally collided with another waveform called a conditioned feature derived from a spectrum to be analyzed. The nature of the Zyoton and how it changes through the collision computing process are explained in detail below. A conditioned feature is obtained by modulating a pre-selected, fixed carrier waveform referred to as a carrier kernel having particular spectral properties, with the feature data to be analyzed. In particular, the spectral properties of the Zyoton and the carrier kernel are selected such that the Zyoton and the conditioned feature are co-dependent and can extract and amplify, in substance, the energy associated with the analyte but not with the noise and/or clutter. Thus, post-collision, the modified Zyoton waveform may represent, substantially, the energy loss in the radiation energy incident upon the medium due to the absorption of such energy by the analyte of interest, if present in the medium, or the presence of one or more confounders in the medium, or both, as well as, optionally, the radiant energy lost to dissipative processes such as scattering, specifically in the spectral region chosen for the feature.

[0025] Optionally, collision computing can be applied to analyze and characterize analyte presence or concentration from the emission spectra of a medium, which includes the spectrum of frequencies of electromagnetic radiation emitted due to an atom or molecule making a transition from a high energy state to a lower energy state. The energy of the emitted photon is equal to the energy difference between the two states. The energy states of the transitions can lead to emissions over a range of frequencies yielding an emission spectrum. By observing the frequencies and amplitudes of wavelengths in an emission spectrum, e.g., as spectral energy or changes therein at different wavelength ranges, elemental or molecular composition of the sample or concentration of an analyte in complex samples can be determined.

[0026] Example sources of emitted light are described in the table below:

[0027]

TABLE 1
Name	Source	Example
Chemiluminescence	Chemical reactions	Glow sticks
Triboluminescence	Friction energy	Light emission seen when
		pulling friction tape off the
		roll in the dark.
Bioluminescence	Biological	Light emission seen from
	processes	fireflies and some jellyfish
Thermoluminescence	Heat energy	Used for archeological
		dating
Electroluminescence	Electric voltage	Source of light seen in
		LEDs
Fluorescence	Light energy	Immediate re-emission after
		absorption of light
Phosphoresence	Light energy	Delayed re-emission after
		absorption of light
Blackbody emission	All materials above	Red-hot metal;
	0° K	the human body at 300° K

[0028] Collision computing treatment of spectra acquired in absorption spectroscopy or emission spectroscopy can be performed similarly as described above, to the extent of extracting spectral features from different wavelength regions, conditioning them using carrier kernel waveforms as described below, colliding them with Zyotons to estimate the absorbed or emitted energy change, and then transforming that result to conclude the presence of and/or to estimate the amount of the analyte. In part due to the shape and morphology preservation properties of the Zyoton, the collision produces a resulting waveform that can represent a change in energy substantially related only to the energy loss represented by the feature used in the collision, and not due to the collision operation itself. Moreover, the collision operator and one or more parameters thereof are selected such that the energy change represented by the feature is amplified when that change is represented in the resulting waveform. The resulting waveform (the modified Zyoton) can be collided iteratively with the same or different Zyotons, in order to further amplify the energy loss represented by the feature, without substantially introducing any noise or distortion.

[0029] Beyond the difficult problem of noninvasive glucose measurement described above, which is generally considered intractable using various known techniques, there are many other measurements of high value that cannot be performed affordably or reliably using various known analytical techniques and sensors, typically due to any of: lack of direct, unique markers or sensor signatures; an ultra-weak detected signal; overwhelming amount of interference or clutter that may obscure the signal of interest (where clutter can be described as other materials or sources of noise which may interfere with the signal in the data-collection domain. For example, in measurement of glucose in tissue, other materials in tissue that absorb in the same spectral region as the analyte, e.g., glucose, or the noise generated by the scattering of light by tissue, are generally considered to be clutter. Materials which absorb radiation in the same spectral region as the analyte are often referred to as confounders); rapidly changing background or tissue medium properties; inadequate or narrow measurement time window; or a combination of two or more of these exacerbating factors.

[0030] Nonlimiting examples of these measurement problems in various domains suitable for analysis with collision computing include:

• • [0031] a. Concentration measurements of biologically important molecules, either in-vivo or in-vitro, that have a cross-product of concentration and measurable property (absorbance, emission, fluorescence, magnetic cross-section, etc.) that is below the measurement limit of various known measurement systems;
  • [0032] b. Measurement of environmental toxins at concentrations below an existing detection threshold;
  • [0033] c. Accurate heart-rate and metabolic state tracking during exercise and variable motion using photoplethysmograph (PPG) devices;
  • [0034] d. In-vivo, early detection and continuous monitoring near-skin and deep tissue inflammation or infection e.g., detection and monitoring post-surgery, post stem-cell therapy, etc.;
  • [0035] e. Direct detection of circulating tumor cells using nanoparticle tagging;
  • [0036] f. Isolation and real-time detection of sophisticated cyberthreats including cyber-theft and injection of dormant, cyber viruses that can dramatically compromise privacy, information security, and infrastructure security;
  • [0037] g. Real-time steganography detection in video, audio, and digital imagery;
  • [0038] h. Remote identification of subsurface mineral, oil, gas, or water resources from the ground, aircraft or spacecraft, including water-table depletion mapping; and
  • [0039] i. The location, mapping, and characterizing of underground activities in hidden facilities.

[0040] Accordingly, in one aspect, a method is provided for determining concentration of glucose in a tissue sample. The method may include obtaining energy absorbed by glucose from energy directed to the tissue sample, as a result of a collision between two co-dependent waveforms, and projecting the energy absorbed by glucose to a glucose concentration.

[0041] In various embodiments, the method may include: receiving a feature corresponding to a specified wavelength range of a spectral signal, where the spectral signal generally represents at least one of: (i) absorption of energy, within the specified wavelength range, and (ii) loss of energy within the specified wavelength range due to scattering. Glucose in the tissue sample, and/or one or more confounders in the tissue may absorb the energy that is represented by the received feature. The method may also include generating a conditioned feature by modulating a carrier kernel using the received feature.

[0042] The step of receiving the feature may include directing near-infrared radiation to the tissue sample, collecting radiation from the tissue sample, generating an absorption spectrum from the collected radiation, and selecting a region of the absorption spectrum bounded by the wavelength range as the feature. Collecting the radiation may include receiving diffusely reflected radiation from the tissue sample and/or receiving radiation transmitted through the tissue sample. In some instances, one of the two co-dependent waveforms comprises an original Zyoton constructed to amplify energy absorbed by the feature, and the carrier kernel is constructed such that the original Zyoton and the conditioned feature obtained by modulating the carrier kernel using the feature are co-dependent. The step of colliding the original Zyoton with the conditioned feature may include, in a first iteration, colliding the original Zyoton with the conditioned feature to obtain a modified Zyoton, and renormalizing the modified Zyoton to obtain a renormalized Zyoton.

[0043] In various embodiments, the method also includes: in each of −1 additional iterations, where >1, colliding the original Zyoton with a renormalized Zyoton from an immediately preceding iteration, to obtain a remodified Zyoton, and renormalizing the remodified Zyoton to obtain a renormalized modified Zyoton. The method may further include computing energy gain of the renormalized modified Zyoton obtained after iterations relative to energy of the original Zyoton, to determine the energy absorbed by glucose. In some cases, the number of iterations is selected such that accuracy of the glucose concentration measurement, as determined by Absolute Relative Deviation (ARD) relative to a glucose concentration measurement obtained using a reference method, is less than 15%.

[0044] In other cases, the number of iterations is selected such that accuracy of a plurality of glucose concentration measurements, as determined by Mean Absolute Relative Deviation (MARD) relative to a plurality of glucose concentration measurements obtained using a reference method, is less than 15%. The conditioned feature may be derived from a glucose feature, where the glucose feature represents at least absorption of energy by glucose within a first specified wavelength range.

[0045] In various embodiments, the method includes: repeating the steps (a) through (c) with respect to another conditioned feature derived from a non-glucose feature paired with the glucose feature, forming a first feature pair. The non-glucose feature may represent absorption of energy, within a second specified wavelength range, by at least one confounder in the tissue sample. Alternatively or in addition, the non-glucose feature may represent loss of energy due to scattering within the second specified wavelength range. The method may further include: computing energy gain of the renormalized modified Zyoton corresponding to the non-glucose feature relative to energy of the original Zyoton, and computing net renormalized spectral energy gain (NRSEG) for the first feature pair based on energy gains corresponding to the glucose feature and the non-glucose feature of the first feature pair.

[0046] The method may also include normalizing the NRSEG for the first feature pair according to a weight designated to the first feature pair, to obtain a normalized NRSEG. In some instances, receiving the glucose feature and the non-glucose feature may include directing, via a source, near infra-red radiation to the tissue sample, and collecting, via a detector, radiation from the tissue sample. In various embodiments, the projecting step may include: (i) computing a number of NRSEG values for the first feature pair, where each NRSEG value corresponds to a respective illumination state of the near infra-red radiation from an illumination sequence. In general, different illumination states can target light paths of different lengths and/or at different depths through the tissue sample.

[0047] The method may further include (ii) computing a normalized absorption gradient (NAG) based on the several NRSEG values; and (iii) selecting from a number of individual projector curves one particular individual projector such that the computed NAG is within lower bound and upper bound NAG values associated with that individual projector curve. The method may also include (iv) determining glucose concentration using the selected individual projector curve and a representative energy absorption value based on a plurality of energy absorption values.

[0048] In certain embodiments, the method further includes repeating step (i) described above for each additional feature pair in a set of feature pairs that includes the first feature pair, and computing a number of NRSEG values for each additional feature pair in the set. Each NRSEG value of the various NRSEG values associated with a particular feature may correspond to a respective illumination state of the near infra-red radiation from an illumination sequence. The computing the NAG may include: for each feature pair in the set designated as acceptable: (A) computing an absorption gradient (AG), and (B) weighting the AG according to a weight associated with the corresponding acceptable feature pair. Computing the NAG may also include averaging the weighted AGs. The representative energy absorption value can be computed by: selecting a particular illumination state; for each feature pair in the set designated as acceptable, weighting an NRSEG value corresponding to the particular illumination state according to a weight associated with the corresponding acceptable feature pair; and averaging the weighted NRSEG values. The representative energy absorption value may then be set to the average of the weighted NRSEG values. In general, a set includes at least one member.

[0049] In various embodiments, the method includes, prior to computing the NAG, for the several NRSEG values corresponding to at least one feature pair in the set, computing a mean NRSEG value, and excluding an NRSEG value that is different from the mean NRSEG value by a specified threshold. The method may also include determining monotonicity of the plurality of NRSEG values corresponding to each feature pair in the set, and designating each feature pair having monotonicity as acceptable.

[0050] In some cases, determining the glucose concentration using the selected individual projector curve includes interpolation, which can include adding to a product of the representative energy absorption value and a slope of the selected individual projector curve, an intercept of the selected individual projector curve. In some cases, determining the normalized absorption gradient includes determining a slope of a regression of the plurality of NRSEG values, typically but not necessarily normalized, with respect to the illumination states, the illumination states being ordered such that successive illumination states represent monotonically changing distance between the source and at least one detector. In some cases, projecting the energy absorbed by glucose to glucose concentration includes mapping the computed NRSEG value to glucose concentration using a single projector curve. The method may also include designating the determined glucose concentration to one of a number of concentration bands. In certain instances, obtaining energy absorbed by glucose as a result of collision includes a single collision iteration. Monotonically changing distance can be monotonically increasing distance or monotonically decreasing distance.

[0051] In another aspect, a system is provided for determining concentration of glucose in a tissue sample. The system may include a first processor and a first memory in electrical communication with the first processor, the first memory including instructions which, when executed by a processing unit including at least one of the first processor and a second processor, and in electronic communication with a memory module comprising at least one of the first memory and a second memory, program the processing unit to: obtain energy absorbed by glucose from energy directed to the tissue sample as a result of collision between two co-dependent waveforms, and project the energy absorbed by glucose to a glucose concentration. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0052] In another aspect, an article of manufacture is provided that includes a non-transitory storage medium having stored therein instructions which, when executed by a processing unit program the processing unit, which is in electronic communication with a memory module, to determine concentration of glucose in a tissue sample. The instructions may program the processing unit to: obtain energy absorbed by glucose from energy directed to the tissue sample as a result of collision between two co-dependent waveforms, and project the energy absorbed by glucose to a glucose concentration. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0053] In another aspect, a method is provided for measuring a property associated with an information signal. The method may include colliding using a collision grid a first conditioned feature waveform that at least partially represents a property of the information signal with an original Zyoton, which includes a waveform that without a collision travels substantially unperturbed at a selected velocity in a substantially constant propagation medium at least over a length of a space-time grid, to obtain a first modified Zyoton, the first conditioned feature waveform and the original Zyoton being constructed to: (i) be co-dependent, and (ii) transfer the property of the information signal to the first modified Zyoton through the collision.

[0054] In various embodiments, the space-time grid includes at least one spatial dimension and a time dimension. The information signal may include one of a signal obtained from a sensor and a signal generated by data analysis. In some instances, the method further includes deriving the first conditioned feature waveform by modulating a carrier kernel using a first feature derived from the information signal. In some cases, the method includes generating the first feature by: selecting from the information signal one of: (i) a single contiguous range of wavelengths, and (ii) a combination of a plurality of discontiguous ranges of wavelengths.

[0055] In other cases, the method includes generating the first feature by: transforming the information signal into a spectral signal; and selecting from the spectral signal one of: (a) a single contiguous range of wavelengths, and (b) a combination of a plurality of discontiguous ranges of wavelengths. In some instances, modulating the carrier kernel includes frequency modulation of the carrier kernel using the first feature. In other instances, modulating the carrier kernel includes: generating an intermediate signal by modulating the first feature using a modulation signal; and modulating the carrier kernel with the intermediate signal. In other instances, modulating the carrier kernel includes interpolating the first feature to match a length of the interpolated first feature in time domain to a length of the carrier kernel in the time domain. In certain embodiments, constructing the original Zyoton and the first conditioned feature waveform to be co-dependent includes determining whether an absolute difference between a scaled velocity of the original Zyoton and a velocity of the first conditioned feature does not exceed a threshold κ_DV2.

[0056] In various embodiments, the method further includes adjusting at least one of: (i) the velocity of the original Zyoton, and (ii) the velocity of the first conditioned feature waveform, such that the absolute difference between the scaled velocity of the original Zyoton and the velocity of the first conditioned feature is within the threshold κ_DV2. In some instances, the original Zyoton includes an analyte-information-representing first group of frequency components and a non-analyte-information-representing group of frequency components, the first conditioned feature including corresponding groups of frequency components, the method further including: adjusting using a scaling vector at least one of: (i) a frequency domain amplitude of at least one component of the analyte-information-representing group of the original Zyoton, and (ii) a frequency domain amplitude of at least one component of the corresponding group of the first conditioned feature waveform, such that the absolute difference between the scaled velocity of the original Zyoton and the velocity of the first conditioned feature is within the threshold κ_DV2.

[0057] The method may also include renormalizing the first modified Zyoton to obtain a first renormalized Zyoton. In certain cases, the property of the information signal includes energy absorbed by at least one of an analyte and one or more confounders, the first renormalized Zyoton includes a set of analyte-information-representing components and a set of non-analyte-information-representing components, and the set of analyte-information-representing components of the first renormalized Zyoton represents the energy absorbed by at least one of the analyte and the confounder. Not all members of the set of analyte-information-representing components may represent analyte information, however. Similarly, one or more members of the set of non-analyte-information-representing components may represent analyte information.

[0058] The method may further include colliding using the collision grid the first renormalized Zyoton with the original Zyoton to obtain a second modified Zyoton, and renormalizing the second modified Zyoton to obtain a second renormalized Zyoton. The method may also include iterating the colliding and renormalizing steps (−2) times, >2, each iteration being based on a renormalized Zyoton from an immediately preceding iteration, and producing a new renormalized Zyoton, until a final renormalized Zyoton is produced after (−2) iterations. In certain cases, the at least partially represented property of the information signal includes energy absorbed by at least one of an analyte and a confounder in a wavelength range associated with a first feature corresponding to the first conditioned feature waveform, the final renormalized Zyoton includes a set of analyte-information-representing components and a set of non-analyte-information-representing components, and the set of analyte-information-representing components of the final renormalized Zyoton represents the energy absorbed by at least one of an analyte and a confounder in the wavelength range associated with the first feature.

[0059] In various embodiments, the method may further include: colliding using the collision grid a second conditioned feature waveform that at least partially represents the property of the information signal with the original Zyoton, to obtain a new first modified Zyoton, the second conditioned feature waveform and the original Zyoton being constructed to: (i) be co-dependent, and (ii) transfer the property of the information signal to the new first modified Zyoton through the collision; renormalizing the new first modified Zyoton to obtain a new first renormalized Zyoton; and iterating the colliding and renormalizing steps (−2) times, >2, each iteration being based on a new renormalized Zyoton from an immediately preceding iteration, and producing another new renormalized Zyoton, until a new final renormalized Zyoton is produced after (−2) iterations.

[0060] In such embodiments, the new final renormalized Zyoton includes a set of analyte-information-representing components and a set of non-analyte-information-representing components, and the set of analyte-information-representing components of the new final renormalized Zyoton represents the energy absorbed by at least one of an analyte and a confounder in a wavelength range associated with a second feature corresponding to the second conditioned feature waveform. The method may also include determining based on at least in part at least one of a specified signal-to-clutter ratio (SCR), a specified signal-to-noise ratio (SNR), a target precision, a target sensitivity, a target accuracy, a target stability, and an expected dynamic range of an analyte in a medium to be analyzed.

[0061] Renormalization can include downscaling, using a scaling vector, frequency domain amplitudes of a set of frequency components in an analyte-information-representing band of the first modified Zyoton to obtain the first renormalized Zyoton. The scaling vector can be a plurality of preset values, and can be determined according to frequency domain amplitudes of a set of frequency components in an analyte-information-representing band of the original Zyoton. In certain embodiments, renormalization includes truncating the first modified Zyoton by removing at least one frequency component in a non-analyte-information-representing band of the first modified Zyoton to obtain the first renormalized Zyoton, and distributing energy of the at least one removed frequency component among a set of remaining frequency components in at least one of a non-analyte-information-representing band and a transition band of the first renormalized Zyoton.

[0062] In various embodiments, the method further includes determining, prior to renormalizing, whether an absolute difference between a velocity of the original Zyoton and a velocity of the first modified Zyoton does not exceed a threshold κ_DV3. In some instances, the method includes determining, prior to renormalizing, whether divergence of the first modified Zyoton does not exceed a threshold τ. In some instances, the method includes determining, prior to renormalizing, whether energy of the first modified Zyoton is greater than energy of the original Zyoton. In some cases, the method includes determining, after renormalizing, whether an absolute difference between a scaled velocity of the original Zyoton and a velocity of the first renormalized Zyoton does not exceed a threshold κ_DV1.

[0063] The property of the information signal may include at least one of: (i) a spectral energy absorbed by at least one of an analyte and one or more confounders, (ii) another property of at least one of the analyte and the confounder, and (iii) an information measure. The information signal may include a sensor signal. In some cases, the sensor signal may include one of: an intensity spectrum signal and an absorbance spectrum signal. In some cases, the sensor signal may include an emission spectrum, where the emission can be fluorescence, phosphorescence, black-body emission, etc. In some cases, the sensor signal includes at least one of a reflectance signal, and a transmission signal. In some cases, the sensor signal includes electromagnetic radiation received from a medium to be analyzed. The electromagnetic radiation may include near infra-red radiation. The collision grid may include a frequency-domain grid, and colliding may involve a first bracketed conditional interaction, in frequency domain, between a first component of the original Zyoton and a first bracket of components of the first conditional feature waveform.

[0064] In various embodiments, the first bracketed conditional interaction may include, for each component of the first conditional feature waveform in the first bracket: testing if frequencies of the component of the first conditioned feature waveform and the first component of the original Zyoton satisfy a first epsilon test; if the first epsilon test succeeds, generating a component of the first modified Zyoton, and setting an amplitude of the generated component of the first modified Zyoton as a product of amplitudes of the component of the first conditioned feature waveform and the first component of the original Zyoton; if the frequencies of the component of the first conditioned feature waveform and the first component of the original Zyoton satisfy a second epsilon test, setting a frequency of the generated component of the first modified Zyoton to a frequency of the first component of the original Zyoton; and otherwise setting the frequency of the generated component of the first modified Zyoton to a sum of frequencies of the component of the first conditioned feature waveform and the first component of the original Zyoton.

[0065] The first epsilon test may be selected based on to which group of components of the original Zyoton the first component of the original Zyoton belongs. In some cases, the first epsilon test is the second epsilon test. In certain instances, colliding may involve a second bracketed conditional interaction, in frequency domain, between a second component of the original Zyoton and a second bracket of components of the first conditional feature waveform. In such instances, the first bracketed conditional interaction may produce a first component of the modified Zyoton and the second bracketed conditional interaction may produce a second component of the modified Zyoton, and the method may further include: determining if the first and second modified Zyoton components are to be merged by testing if frequencies of the first and second modified Zyoton components satisfy a third epsilon test. The third epsilon test can be selected based on to which group of components of the modified Zyoton a larger one of the first and second components of the modified Zyoton belongs, the larger component being determined by comparing amplitudes of the first and second components of the modified Zyoton.

[0066] In various embodiments, the method may further include merging the first and second modified Zyoton components into the first modified Zyoton component by resetting an amplitude of the first modified Zyoton component to a sum of amplitudes of the first and second modified Zyoton components, and removing the second modified Zyoton component. In some such embodiments, merging the first and second modified Zyoton components into the first modified Zyoton component may further include, if a frequency of the first modified Zyoton component is smaller than a frequency of the second modified Zyoton component, setting a frequency of the first modified Zyoton component to the frequency of the second modified Zyoton component.

[0067] In other such embodiments, merging the first and second modified Zyoton components into the first modified Zyoton component may further include, if, prior to merging, the amplitude of the first modified Zyoton component is smaller than the amplitude of the second modified Zyoton component, setting a frequency of the first modified Zyoton component to the frequency of the second modified Zyoton component. The method may further include selecting a length of the first bracket of components of the first conditional feature waveform based at least in part on at least one of a specified signal-to-clutter ratio (SCR), a specified signal-to-noise ratio (SNR), a target precision, a target sensitivity, a target accuracy, a system stability, and an expected dynamic range of an analyte in a medium to be analyzed. The method may also include, prior to the first bracketed conditional interaction, at least one of: (i) shifting components of the original Zyoton according to a delay-shift parameter, (ii) scaling an amplitude of the first component of the original Zyoton according to at least one of (a) a precision of a computing device, (b) a dynamic range of a computing device, (c) a resolution of a sensor used to obtain a feature used to generate the first conditioned feature, (d) a dynamic range of the sensor, and (e) SNR of the sensor, and (iii) applying a phase rotation to the first component of the original Zyoton.

[0068] In various embodiments, the original Zyoton includes a plurality of components selected from a kernel of components of a base Zyoton generated from a family of waveforms, a number of components in the plurality being based on at least in part at least one of a specified signal-to-clutter ratio (SCR), a specified signal-to-noise ratio (SNR), a target precision, a target sensitivity, a target accuracy, a target stability, and an expected dynamic range of an analyte in a medium to be analyzed. The collision grid may include a frequency-domain grid, and a length of the original Zyoton may include a sum of: (i) a number (k) of analyte-information-representing components, (ii) a number (m) of transition components, and (iii) a number (j) of non-analyte-information-representing components of the original Zyoton.

[0069] In some embodiments, the original Zyoton is synthesized from at least one of a plurality of waveform families comprising: solitons; autosolitons; similaritons; custom solitons generated using a sine-Gordon-based equation; self-compressing similaritons; vortex-solitons; multi-color solitons; parabolic-similaritons; Ricci solitons; wavelets; curvelets; ridgelets; bions; elliptic waves comprising at least one of Jacobi elliptic functions and Weierstrass elliptic functions; and nonautonomous similinear wave equation.

[0070] In other embodiments, the original Zyoton is synthesized from at least one of a plurality of waveform generators comprising: meromorphic functions; Gamma functions; Riemann Zeta functions; regular instantons; Frobenius manifolds; harmonic oscillators; Hermite polynomials; polynomial sequences; asymptotic Hankel functions; Bessel functions; fractals; Neumann functions (spherical); poweroid coupled with sinusoidal functions; spatial random fields; cyclostationary series; random number generators; spherical harmonics; chaotic attractors; exponential attractors; multipoint Krylov-subspace projectors; Lyapunov functions; inertial manifolds of Navier-Stokes equation; evolution equation for polynomial nonlinear reaction-diffusion equation; evolution equation for Kuramoto-Savashinsky equation; evolution equation of exponential attractors; Fourier series; and Ramanuj an theta functions.

[0071] In some cases, the original Zyoton is synthesized from a random number generator having a replicability and at least one of: a periodicity of at least 1000, a correlation between a pair of generated values of at most 90%, and a spectral bandwidth of at least 100 Hz. In some cases, the original Zyoton is synthesized from a polynomial sequence, a Fourier transform of the polynomial sequence having a spectral bandwidth of at least 100 Hz. In certain instances, the original Zyoton is synthesized from a random number generator comprising at least one of: a linear congruential generator, a multiplicative congruential generator, an additive congruential generator, and a Fibonacci generator. In certain instances, the original Zyoton is synthesized from a polynomial sequence comprising at least one of an Abel polynomial sequence and a Bell polynomial sequence. In some embodiments, the original Zyoton is synthesized, at least in part, from at least one of: (i) a waveform family and (ii) a waveform generator, and the method further includes at least one of transforming and reducing a function to at least one of the waveform family and the waveform generator. In some cases, the original Zyoton is synthesized by at least one of: an addition, a multiplication, and a subtraction of at least two waveforms derived from at least two waveform families.

[0072] In another aspect, a system is provided for measuring a property associated with an information signal. The system may include a first processor and a first memory in electrical communication with the first processor, the first memory including instructions which, when executed by a processing unit including at least one of the first processor and a second processor, and in electronic communication with a memory module comprising at least one of the first memory and a second memory, program the processing unit to: collide using a collision grid a first conditioned feature waveform that at least partially represents a property of the information signal with an original Zyoton, which includes a waveform that without a collision travels substantially unperturbed at a selected velocity in a substantially constant propagation medium at least over a length of a space-time grid, to obtain a first modified Zyoton, the first conditioned feature waveform and the original Zyoton being constructed to: (i) be co-dependent, and (ii) transfer the property of the information signal to the first modified Zyoton through the collision. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0073] In another aspect, an article of manufacture is provided that includes a non-transitory storage medium having stored therein instructions which, when executed by a processing unit program the processing unit, which is in electronic communication with a memory module, to measure a property associated with an information signal. The instructions may program the processing unit to: collide using a collision grid a first conditioned feature waveform that at least partially represents a property of the information signal with an original Zyoton, which includes a waveform that without a collision travels substantially unperturbed at a selected velocity in a substantially constant propagation medium at least over a length of a space-time grid, to obtain a first modified Zyoton, the first conditioned feature waveform and the original Zyoton being constructed to: (i) be co-dependent, and (ii) transfer the property of the information signal to the first modified Zyoton through the collision. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0074] In another aspect, a method is provided for enabling extraction of a property of an element of a specified environment. The method may include selecting using at least one of: (i) at least one waveform family, and (ii) at least one waveform generator, a set of frequency components for an analyte-information-representing band and a set of frequency components for a non-analyte-information-representing band; and synthesizing a Zyoton by combining the set of frequency components selected for the analyte-information-representing band and the set of frequency components selected for the non-analyte-information representing band, where: at least one frequency component in the set of frequency components for the analyte-information-representing band corresponds to a property to be extracted; at least one of the frequency components for the non-analyte-information representing band does not correspond significantly to the property to be extracted; and the selected set of frequency components of the waveform family allow the Zyoton to propagate through a substantially constant propagation medium over a length of a collision grid without substantial change in morphology thereof. Not all members of the band of analyte-information-representing components may represent analyte information, however. Similarly, one or more members of the band of non-analyte-information-representing components may represent analyte information.

[0075] In some embodiments, after a collision of the Zyoton with a waveform constructed to perturb the Zyoton within specified limits, the selected set of frequency components allow: generation of a modified Zyoton via the collision; and subsequent propagation of the modified Zyoton through the substantially constant propagation medium over the length of the collision grid without substantial change in morphology of the modified Zyoton. In some cases, the at least one waveform family may include at least one of: optical solitons, autosolitons, similaritons, custom solitons generated using a sine-Gordon-based equation, self-compressing similaritons, vortex-solitons, multi-color solitons, parabolic-Similaritons, and Ricci solitons.

[0076] In some cases, the at least one waveform family includes at least one of: solitons; wavelets; curvelets; ridgelets; bions; elliptic waves comprising at least one of Jacobi elliptic functions and Weierstrass elliptic functions; and nonautonomous similinear wave equations. In some cases, the at least one waveform generator includes at least one of: meromorphic functions; Gamma functions; Riemann Zeta functions; regular instantons; Frobenius manifolds; harmonic oscillators; Hermite polynomials; polynomial sequences; asymptotic Hankel functions; Bessel functions; fractals; Neumann functions (spherical); poweroid coupled with sinusoidal functions; spatial random fields; cyclostationary series; random number generators; spherical harmonics; chaotic attractors; exponential attractors; multipoint Krylov-sub space projectors; Lyapunov functions; inertial manifolds of Navier-Stokes equation; evolution equation for polynomial nonlinear reaction-diffusion equation; evolution equation for Kuramoto-Savashinsky equation; evolution equation of exponential attractors; Fourier series; and Ramanuj an theta functions.

[0077] In certain instances, the specified environment includes a medium to be analyzed; and the property to be extracted includes at least one of: presence of an analyte, absence of the analyte, a quantity of an analyte, and a rate of change of the quantity of the analyte, in the medium to be analyzed. In various embodiments, the method may further include selecting from the at least one waveform family a set of frequency components for a transition band, and synthesizing the Zyoton by combining the set of frequency components for the analyte-information-representing band, the set of frequency components for the transition band, and the set of frequency components for the non-analyte-information-representing band.

[0078] In another aspect, a system is provided for enabling extraction of a property of an element of a specified environment. The system may include a first processor and a first memory in electrical communication with the first processor, the first memory including instructions which, when executed by a processing unit including at least one of the first processor and a second processor, and in electronic communication with a memory module comprising at least one of the first memory and a second memory, program the processing unit to: select using at least one of: (i) at least one waveform family, and (ii) at least one waveform generator, a set of frequency components for an analyte-information-representing band and a set of frequency components for a non-analyte-information representing band; and synthesize a Zyoton by combining the set of frequency components selected for the analyte-information-representing band and the set of frequency components selected for the non-analyte-information-representing band.

[0079] At least one frequency component in the set of frequency components for the analyte-information-representing band corresponds to a property to be extracted; at least one of the frequency components for the non-analyte-information-representing band does not correspond significantly to the property to be extracted; and the selected set of frequency components of the waveform family allow the Zyoton to propagate through a substantially constant propagation medium over a length of a collision grid without substantial change in morphology thereof. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0080] In another aspect, an article of manufacture is provided that includes a non-transitory storage medium having stored therein instructions which, when executed by a processing unit program the processing unit, which is in electronic communication with a memory module, to enable extraction of a property of an element of a specified environment. The instructions may program the processing unit to: select using at least one of: (i) at least one waveform family, and (ii) at least one waveform generator, a set of frequency components for an analyte-information-representing band and a set of frequency components for a non-analyte-information representing band; and synthesize a Zyoton by combining the set of frequency components selected for the analyte-information-representing band and the set of frequency components selected for the non-analyte-information-representing band.

[0081] At least one frequency component in the set of frequency components for the analyte-information-representing band corresponds to a property to be extracted; at least one of the frequency components for the non-analyte-information-representing band does not correspond significantly to the property to be extracted; and the selected set of frequency components of the waveform family allow the Zyoton to propagate through a substantially constant propagation medium over a length of a collision grid without substantial change in morphology thereof. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0082] In another aspect, another method is provided for enabling extraction of a property of an element of a specified environment. The method may include receiving a representative spectral signal from a data-collection domain, the representative spectral signal indicating a property of the element of the specified environment; generating a Zyoton from at least one waveform family, the at least one waveform family being independent of the data-collection domain; and generating a co-dependent carrier kernel for conditioning at least one feature corresponding to the representative spectral signal. In some instances, the element of the specified environment includes an analyte, and the property of the element of the specified environment includes concentration of the analyte. In some cases, the specified environment may include at least one confounder.

[0083] In some cases, the specified environment may include a portion of skin tissue, and the analyte may include glucose. In certain embodiments, the property of the element of the specified environment that is represented by the representative spectral signal includes energy loss via at least one of: (i) absorption by the analyte, (ii) absorption by the at least one confounder, (iii) noise generated by variation in absorption by the at least one confounder, and (iv) noise generated by scattering. The at least one confounder may include a dominant confounder. The representative signal may represent information about an analyte and/or one or more confounders.

[0084] In various embodiments, the representative signal includes a first model signal, and generating the Zyoton may include: obtaining a transformed signal from the first model signal representing energy loss via absorption by the analyte; adding a noise signal to the transformed signal to obtain a second signal; determining frequency components of the second signal; using the frequency components of the second signal, determining a number of frequency components required to separate a high analyte absorption region of the transformed signal from a low analyte absorption region of the transformed signal; selecting the at least one waveform family according to a morphology of one of the first model signal, the transformed signal, and the second signal; obtaining a base Zyoton based on the at least one selected waveform family; and selecting the determined required number of frequency components of the base Zyoton as frequency components of the Zyoton.

[0085] In some instances, the noise signal corresponds to an amount of noise that is a multiple of at least one of: (i) a noise associated with the first model signal, and (ii) a noise associated with the transformed signal; and the determined required number of frequency components represents a number of frequency components (k) required to separate the high analyte absorption region of the transformed signal from the low analyte absorption region of the transformed signal in presence of the specified amount of noise. In some cases, obtaining the transformed signal may include generating a second derivative of one of: the first model signal and an interpolated first model signal.

[0086] In other cases, obtaining the transformed signal may include generating a first composite signal by combining with the first model signal a second model signal representing energy loss via absorption by at least one non-dominant confounder, where the determined required number of frequency components represents a number of frequency components (m) required to separate the high analyte absorption region of the transformed signal from the low analyte absorption region of the transformed signal in presence of the at least one non-dominant confounder.

[0087] In still other cases, obtaining the transformed signal may include generating a second composite signal by combining with the first model signal a third model signal representing energy loss via variations in absorption by a dominant confounder; where the determined required number of frequency components represents a number of frequency components (j) required to separate the high analyte absorption region of the transformed signal from the low analyte absorption region of the transformed signal in presence of the dominant confounder and the specified amount of noise.

[0088] The Zyoton may include k analyte-information-representing components, k≧1, and the method may further include: setting amplitudes of the k analyte-information-representing components within a first range of amplitudes according to an amplitude profile. The first model signal may correspond to a pure-component analyte signal, and the transformed signal can be first, second, or higher-order derivative thereof.

[0089] In various embodiments, the method may further include generating the amplitude profile by: generating a plurality of wavelength-region pairs, each pair having a high absorption representing wavelength region corresponding to a respective wavelength range of the representative spectral signal and a low absorption representing wavelength region corresponding to a different respective wavelength range of the representative spectral signal; for each pair in the plurality of wavelength-region pairs, computing a ratio of a magnitude of a region of the transformed signal corresponding to the high absorption representing wavelength region and a magnitude of a region of the transformed signal corresponding to the low absorption representing wavelength region; determining a range of the magnitude ratios; and setting the amplitude profile as a function bounded by the range of magnitude ratios, the function being one of linear, quadratic, and exponential.

[0090] In some embodiments, the Zyoton includes k analyte-information-representing components, k≧1, and j non-analyte-information representing components, j≧1, the k analyte-information-representing components having amplitudes within a first range, and the method may further includes: randomly setting amplitudes of the j non-analyte-information representing components within a second range of amplitudes, any amplitude in the second range being less than 1/100 of any amplitude in the first range. In some instances, the Zyoton further includes m transition components, m≧1, and the method further includes: randomly setting amplitudes of the m transition components within a third range of amplitudes, any amplitude in the third range being at least two times any amplitude in the second range.

[0091] Generating the co-dependent carrier kernel may include: for each of the k analyte-information-representing major components of the Zyoton, including in the carrier kernel as a corresponding analyte-information-representing major component a frequency component having a frequency within a specified threshold σ of frequency of the corresponding analyte-information-representing k component of the Zyoton; and including in the carrier kernel as at least one non-major component, at least one of: at least one harmonic of at least one of an analyte-information-representing component, a transition component, and non-analyte-information-representing component included in the carrier kernel; and for at least one non-major component of the Zyoton, including in the carrier kernel a frequency component having a frequency within the specified threshold σ of frequency of the corresponding non-major component of the Zyoton.

[0092] In some instances, one of: the non-major component of the Zyoton is a transition component, and the corresponding non-major component of the carrier kernel is a transition component; and the non-major component of the Zyoton is a non-analyte-information-representing j component, and the corresponding non-major component of the carrier kernel is a non-analyte-information-representing j component. In certain embodiments, the method may further include setting amplitudes of the information-representing, transition, and non-information representing frequency components of the carrier kernel according to the amplitudes of the respective analyte-information-representing, transition, and non-analyte-information-representing Zyoton components scaled by a pre-set scaling factor.

[0093] The method may also include: selecting an amplitude scaling coefficient such that: (i) a difference between a scaled velocity of the Zyoton, scaled by the amplitude scaling factor, and a velocity of the carrier kernel does not exceed a specified velocity difference threshold κ^SYN; and setting amplitudes of the analyte-information-representing, transition, and non-analyte-information-representing frequency components of the carrier kernel according to the amplitudes of the respective analyte-information-representing, transition, and non-analyte-information-representing Zyoton components scaled by the selected scaling factor.

[0094] In another aspect, another system is provided for enabling extraction of a property of an element of a specified environment. The system may include a first processor and a first memory in electrical communication with the first processor, the first memory including instructions which, when executed by a processing unit including at least one of the first processor and a second processor, and in electronic communication with a memory module comprising at least one of the first memory and a second memory, program the processing unit to: receive a representative spectral signal from a data-collection domain, the representative spectral signal indicating a property of the element of the specified environment; generate a Zyoton from at least one waveform family, the at least one waveform family being independent of the data-collection domain; and generate a co-dependent carrier kernel for conditioning at least one feature corresponding to the representative spectral signal. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0095] In another aspect, another article of manufacture is provided that includes a non-transitory storage medium having stored therein instructions which, when executed by a processing unit program the processing unit, which is in electronic communication with a memory module, to enable extraction of a property of an element of a specified environment. The instructions may program the processing unit to: receive a representative spectral signal from a data-collection domain, the representative spectral signal indicating a property of the element of the specified environment; generate a Zyoton from at least one waveform family, the at least one waveform family being independent of the data-collection domain; and generate a co-dependent carrier kernel for conditioning at least one feature corresponding to the representative spectral signal. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0096] In another aspect, a method is provided for quantitating an analyte. The method may include: receiving from an uncharacterized sample an energy change value to be mapped, the energy change value corresponding to an uncharacterized sample; mapping the energy change value to be mapped to a quantity of an analyte in the uncharacterized sample, via an individual projector curve associating: (i) energy change values obtained from a synthetic reference system to analyte quantities of the reference system, and (ii) energy change values obtained from a non-synthetic reference system to the analyte quantities of the reference system.

[0097] The method may further include: computing a representative absorption gradient (AG) using a first set of energy change values corresponding to a first feature pair, each energy change value in the first set corresponding to a respective path of radiation through a medium to be analyzed; selecting the individual projector curve from a plurality of projector curves using the representative gradient value associated with the first feature pair; and determining a quantity of an analyte using the selected individual projector curve, the energy change value to be mapped being a representative energy change value associated with the first feature pair.

[0098] The energy change value to be mapped can be a net renormalized spectral energy gain, which may be optionally normalized. In some cases, the representative gradient value may include a normalized AG (NAG), and the method further includes: weighting the representative gradient associated with the first feature pair by a weight associated with the first feature pair, to obtain the NAG. Computing the representative energy change value can include weighting one energy change value from the first set of energy change values by a weight associated with the first feature pair. In certain instances, a plurality of feature pairs may include the first feature pair, and the method may further include: generating a set of acceptable feature pairs from the plurality of feature pairs; for each feature pair in the set of acceptable feature pairs computing a respective absorption gradient (AG) using a respective set of energy change values corresponding to the feature pair; computing the representative AG value as an average of the respective AGs associated with each of the acceptable feature pairs. The energy value change can be a net, renormalized spectral energy gain corresponding to a feature or a feature pair, which may be optionally normalized.

[0099] In various embodiments, generating the set of acceptable feature pairs may include testing for each feature pair in the plurality of feature pairs monotonicity of the corresponding set of energy change values across a plurality of illumination states. Computing the AG for each acceptable feature pair may include: determining a slope of a regression of the corresponding set of energy change values with respect to the illumination states, the illumination states being ordered such that successive illumination states represent monotonically changing distance between the source and at least one detector.

[0100] Prior to computing the average, the method may further include normalizing each of the AGs corresponding to the acceptable feature pairs using a weight associated with the corresponding acceptable feature pair. In some cases, computing the representative energy change value includes: for each acceptable feature pair, selecting one energy change value from the respective set of energy change values; weighting for each acceptable feature pair, the selected energy change value by a weight associated with the acceptable feature pair; and setting an average of the weighted energy change values as the representative energy change value. In some embodiments, the uncharacterized sample includes a portion of tissue; and the analyte includes glucose. In some embodiments, the energy change includes one of energy gain and energy loss.

[0101] In another aspect, a system is provided for quantitating an analyte. The system may include a first processor and a first memory in electrical communication with the first processor, the first memory including instructions which, when executed by a processing unit including at least one of the first processor and a second processor, and in electronic communication with a memory module comprising at least one of the first memory and a second memory, program the processing unit to: receive an energy change value to be mapped, the energy change value corresponding to an uncharacterized sample; and map the energy change value to be mapped to a quantity of an analyte in the uncharacterized sample, via an individual projector curve associating: (i) energy change values obtained from a synthetic reference system to analyte quantities of the reference system, and (ii) energy change values obtained from a non-synthetic reference system to the analyte quantities of the reference system. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0102] In another aspect, an article of manufacture is provided that includes a non-transitory storage medium having stored therein instructions which, when executed by a processing unit program the processing unit, which is in electronic communication with a memory module, to quantitate an analyte. The instructions may program the processing unit to: receive an energy change value to be mapped, the energy change value corresponding to an uncharacterized sample; and map the energy change value to be mapped to a quantity of an analyte in the uncharacterized sample, via an individual projector curve associating: (i) energy change values obtained from a synthetic reference system to analyte quantities of the reference system, and (ii) energy change values obtained from a non-synthetic reference system to the analyte quantities of the reference system. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0103] In another aspect, a method is provided for calibrating a measurement system for non-invasive analyte measurement. The method may include obtaining from a selected plurality of synthetic medium samples, each having a reference analyte concentration, a plurality of energy-value sets from the measurement system, each energy value in a particular energy-value set corresponding to a respective reference analyte concentration from the selected plurality of synthetic medium samples; generating a composite projector curve using the plurality of energy value sets and the plurality of reference analyte concentrations; and partitioning the composite projector curve into a set of non-overlapping individual projector curves according to a set of slopes of the composite projector curve, each individual projector curve being identified by a lower bound analyte concentration and an upper bound analyte concentration. In some instances, the plurality of energy-value sets includes a first energy-value set, each energy value in the first energy-value set corresponding to a first reference analyte concentration, and the method may further include: excluding from the first energy-value set an energy value that is different from a mean of the energy values by a specified threshold.

[0104] In various embodiments, the method may further include obtaining from representative subjects a plurality of groups of energy value vectors from the measurement system, each energy-value-vector group corresponding to a respective analyte concentration determined using a reference invasive measurement system, and each energy-value vector in a particular group corresponding to a respective feature pair; partitioning the plurality of groups of energy-value vectors into a plurality of projection sets, such that the analyte concentration corresponding to any energy-value-vector group in each projection set is within lower and upper bound analyte concentrations corresponding to a single respective individual projector curve from the set of individual projector curves; computing for each energy-value-vector group in each projection set, an average of normalized absorption gradients (NAGs), each NAG corresponding to an energy-value vector that corresponds to an acceptable feature pair; for each projector curve: designating as a lower bound NAG a minimum of averaged NAGs of all energy-value-vector groups of the corresponding projection set; and designating as an upper bound NAG a maximum of averaged NAGs of all energy-value-vector groups of the corresponding projection set. An energy-value vector may include a plurality of energy values, each energy value corresponding to an illumination state in an illumination sequence. In some instances, the partitioning step may include rejecting from a group of energy-value vectors any non-monotonic energy-value vector.

[0105] In another aspect, a system is provided for calibrating a measurement system for non-invasive analyte measurement. The system may include a first processor and a first memory in electrical communication with the first processor, the first memory including instructions which, when executed by a processing unit including at least one of the first processor and a second processor, and in electronic communication with a memory module comprising at least one of the first memory and a second memory, program the processing unit to: obtain from a selected plurality of synthetic medium samples, each having a reference analyte concentration, a plurality of energy-value sets from the measurement system, each energy value in a particular energy-value set corresponding to a respective reference analyte concentration from the selected plurality of synthetic medium samples.

[0106] The instructions may program the processing unit to generate a composite projector curve using the plurality of energy value sets and the plurality of reference analyte concentrations; and partition the composite projector curve into a set of non-overlapping individual projector curves according to a set of slopes of the composite projector curve, each individual projector curve being identified by a lower bound analyte concentration and an upper bound analyte concentration. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0107] In another aspect, an article of manufacture is provided that includes a non-transitory storage medium having stored therein instructions which, when executed by a processing unit program the processing unit, which is in electronic communication with a memory module, to calibrate a measurement system for non-invasive analyte measurement. The instructions may program the processing unit to: obtain from a selected plurality of synthetic medium samples, each having a reference analyte concentration, a plurality of energy-value sets from the measurement system, each energy value in a particular energy-value set corresponding to a respective reference analyte concentration from the selected plurality of synthetic medium samples.

[0108] The instruction may program the processing unit to generate a composite projector curve using the plurality of energy value sets and the plurality of reference analyte concentrations; and partition the composite projector curve into a set of non-overlapping individual projector curves according to a set of slopes of the composite projector curve, each individual projector curve being identified by a lower bound analyte concentration and an upper bound analyte concentration. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0109] In another aspect, a method is provided for computing analyte concentration in a medium. The method may include: selecting a member of a projector curve set according to a normalized absorption gradient (NAG) value, such that the NAG value is within a range of minimum and maximum NAG values associated with that member; and interpolating analyte concentration using a slope and an intercept of the selected member. The method may further include: receiving for a feature pair, a respective net renormalized spectral energy gain (NRSEG) value for each one of a plurality of illumination states; and designating the feature pair as acceptable feature pair, if the respective NRSEG values are monotonic across the illumination states, and rejecting the feature pair, otherwise.

[0110] The method may also include: receiving for a first acceptable feature pair, a respective net renormalized spectral energy gain (NRSEG) value for each one of a plurality of illumination states; computing a first absorption gradient (AG), typically but not necessarily normalized, across the illumination states using the NRSEG values associated with the first acceptable feature pair; and computing the NAG by applying to the first AG a first weight corresponding to the first acceptable feature pair. The method may also include: receiving for a second acceptable feature pair, a respective net renormalized spectral energy gain (NRSEG) value for each one of the plurality of illumination states; computing a second absorption gradient (AG) across the illumination states using the NRSEG values associated with the second acceptable feature pair; and computing the NAG by applying to the second AG a second weight corresponding to the second acceptable feature pair and averaging the first and second weighted AGs.

[0111] In another aspect, a system is provided for computing analyte concentration in a medium. The system may include a first processor and a first memory in electrical communication with the first processor, the first memory including instructions which, when executed by a processing unit including at least one of the first processor and a second processor, and in electronic communication with a memory module comprising at least one of the first memory and a second memory, program the processing unit to: select a member of a projector curve set according to a normalized absorption gradient (NAG) value, such that the NAG value is within a range of minimum and maximum NAG values associated with that member, and interpolate analyte concentration using a slope and an intercept of the selected member. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0112] In another aspect, an article of manufacture is provided that includes a non-transitory storage medium having stored therein instructions which, when executed by a processing unit program the processing unit, which is in electronic communication with a memory module, to compute analyte concentration in a medium. The instructions may program the processing unit to: select a member of a projector curve set according to a normalized absorption gradient (NAG) value, such that the NAG value is within a range of minimum and maximum NAG values associated with that member, and interpolate analyte concentration using a slope and an intercept of the selected member. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0113] In another aspect, a method is provided for generating features for measurement of heart beats. The method may include receiving a signal based on one of reflected and transmitted radiation from a tissue during a global time interval starting at a global start time and ending at a global end time; designating a first portion of the signal starting at the global start time and ending at a first end time less than the global end time as a first feature; designating a second portion of the signal starting at the global start time and ending at a second end time greater than the first end time as a second feature; and identifying a start of a heart-beat cycle within the first feature.

[0114] The method may further include determining that an expected end of the heart-beat cycle is absent within the first portion of the signal, designated as the first feature; and designating the first feature as a non-analyte feature. The method may also include determining that the expected end of the heat-beat cycle is present within the second portion of the signal, designated as the second feature; and designating the second feature as an analyte feature. In certain embodiments, the method may include determining a heart rate by: colliding a first conditioned feature waveform derived from the first feature with a co-dependent Zyoton to obtain a first energy change; colliding a second conditioned feature waveform derived from the second feature with a co-dependent Zyoton to obtain a second energy change; computing an end of the heart-beat cycle using the first and second energy changes; and computing the heart-rate using the start of the heart-beat cycle and the end of the heart-beat cycle.

[0115] In another aspect, a system is provided for generating features for measurement of heart beats. The system may include a first processor and a first memory in electrical communication with the first processor, the first memory including instructions which, when executed by a processing unit including at least one of the first processor and a second processor, and in electronic communication with a memory module comprising at least one of the first memory and a second memory, program the processing unit to: receive a signal based on one of reflected and transmitted radiation from a tissue during a global time interval starting at a global start time and ending at a global end time; designate a first portion of the signal starting at the global start time and ending at a first end time less than the global end time as a first feature; designate a second portion of the signal starting at the global start time and ending at a second end time greater than the first end time as a second feature; and identify a start of a heart-beat cycle within the first feature. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0116] In another aspect, an article of manufacture is provided that includes a non-transitory storage medium having stored therein instructions which, when executed by a processing unit program the processing unit, which is in electronic communication with a memory module, to generate features for measurement of heart beats. The instructions may program the processing unit to: receive a signal based on one of reflected and transmitted radiation from a tissue during a global time interval starting at a global start time and ending at a global end time; designate a first portion of the signal starting at the global start time and ending at a first end time less than the global end time as a first feature; designate a second portion of the signal starting at the global start time and ending at a second end time greater than the first end time as a second feature; and identify a start of a heart-beat cycle within the first feature. In various embodiments, the instructions can program the processing unit to perform one or more of the method steps described above.

[0281] A collision computer and a collision computing process according to various embodiments of the invention can detect and/or quantitate an analyte of interest within a medium, a property of a material, changes in the amount of the analyte and/or properties of the material, and/or an event or anomaly of interest. To this end, some embodiments of a collision computer receive intensity spectra representing the analyte, material, event, and/or anomaly and also the environment thereof, e.g., a medium in which the analyte may be present. In some embodiments, radiation (e.g., near-infra red (NIR) radiation) is directed to the environment/medium, and the radiation reflected from or passing through the environment/medium is detected by a detector, and is provided as one or more intensity spectra. In other instances, a stimulus (e.g., an electromagnetic signal, an acoustic signal, etc.) is directed to an environment and one or more signals generated in response to the stimulus are obtained from one or more detectors. The frequency of the electromagnetic signal may range from 300 EHz (where 1 exahertz or EHz=10¹⁸ Hz) down to 0.03 Hz or alternatively, with a wavelength ranging from 1 pm (picometer) to 10 Gm (gigameters). The detected signals are transformed to one or more intensity spectra and are presented to various embodiments of a collision computer.

[0282] An intensity spectrum presented to a collision computer represents, at least in part, an overall change in the incident radiation/stimulus that is caused by the medium/environment. In general, the change is caused by three factors. The first factor is the analyte and other materials that are present in the medium/environment, including the properties of the analyte or other materials, changes in concentration/properties of the analyte or other materials, an event of interest, and/or an anomaly. The second factor is one or more confounders, i.e., other materials that are present in the medium/environment and can cause a change in the incident radiation or stimulus in a manner similar to the manner in which the analyte causes a change. The third factor is the absorption, dispersion, and/or scattering of the incident radiation or stimulus by the medium/environment. Typically, the intensity spectra also include a significant noise component. The noise may be introduced by sensor errors or variability, analog-to-digital conversion (ADC) of the sensed signal, propagation of the radiation through the medium/environment along different paths, etc.

[0283] Both scattering and absorption remove energy from electromagnetic radiation, including light, traversing a turbid medium. Inhomogeneities in a specified environment scatter radiated light, to cause diffuse reflection from the medium by multiple reflections with structures in or particles of the environment. Structures in the medium act as scattering centers, and the overall arrangement of their shape, size, and composition, as well as time-varying fluctuations, including statistical thermal fluctuations, influence the attenuation due to scattering. This type of scattering, where the particles are of approximately the same size as the wavelength of light, is known as Mie scattering. Optionally, confounder molecules may also absorb energy in the same band where the analyte absorbs energy. In various embodiments, a difference between the energy of the incident radiation or stimulus and the energy of the intensity spectra received by a collision computer represents the overall change in the incident radiation or stimulus. The energy of the incident radiation (and the absorbance spectrum) can be computed as the spectral energy of the incident radiation signal (and the absorbance spectrum).

[0284] As described below with reference to FIGS. 9-11, the difference between the energies of the radiation incident upon the medium to be analyzed and the radiation received therefrom corresponds to one or more of the three factors, i.e., the energy difference may include the energy absorbed by the analyte or other materials, the energy absorbed by one or more confounders, and/or the energy absorbed or scattered by (also called lost within) the medium/environment. The energy difference may also be affected by one or more sources of noise described above. In various embodiments, a collision computer is constructed and operated to determine the energy difference due to only the first factor, i.e., absorption by the analyte, distinguishing such change in energy from the noise and from the changes in energy caused by absorption by one or more confounders and/or due to loss in the medium/environment. A relationship between the measured energy change and concentration of the analyte, property thereof, parameters of an event or anomaly, etc., can then be used to quantitate the analyte/material, properties thereof, events/anomalies, etc.

[0285] As an example of scattering, Rayleigh scattering, defined as dominantly-elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the radiation, generally does not involve absorption. Rayleigh scattering can result from the electric polarizability of the particles and may not change the state of material. Thus, it is considered a conservative process. The scattered particles may be individual atoms or molecules, and it occurs when light travels through transparent gaseous, solid, or liquid materials.

[0286] As described below, the colliding waveforms used in various embodiments of collision computing are spatio-temporal waveforms having a time dimension and at least one spatial dimension. A spatio-temporal representation of the waveform is also called a time-domain representation. These waveforms can also be represented in a frequency domain and may have one or more frequency components. The frequencies are spatial frequencies. Spectral energy density is one measurable observable of the waveform at various stages of processing—pre-collision, post-collision, and after the renormalization operation that is generally required in embodiments employing several collision iterations. Also, the spectral energy of waveforms can be characterized using one of more subsets of the frequency components.

[0287] In various embodiments of collision computing, a stable waveform called a Zyoton, which may change its shape and morphology when collided with another waveform only within limits as specified by certain parameters, is computationally collided with another waveform, called a conditioned feature, derived from the spectral data to be analyzed. The term spectral data generally refers to any data acquired from a spectroscopic measurement or a spectral sensor. A spectral signal can be directly obtained from a spectral sensor, or time-varying data can be transformed into a spectral signal by applying a Fourier transform or other mathematical transform to the data collected over some time window. Such a spectral signal may be referred to as a transformed signal. A conditioned feature is obtained by modulating a pre-selected, carrier waveform referred to as a “carrier kernel” with data (e.g., a feature).

[0288] A carrier kernel generally includes one or more (e.g., 2, 4, 6, 10, 15, 1000 or more) fundamental frequencies and/or their respective harmonics, i.e., integer multiples of the fundamental frequencies. The number of harmonics can range from 1 up to 4, 6, 10, 20, 32, 50, or more, and the number of included harmonics of different fundamental frequencies can be different. To illustrate, an exemplary carrier kernel includes three fundamental frequencies f₀, f₁, f₂. Ten harmonics of f₀, denoted h_0,j for j=0 . . . 9; four harmonics of f₁, denoted h_1,j for j=0 . . . 3; and 15 harmonics of f₂, denoted h_2,j for j=0 . . . 14, are also included in this exemplary carrier kernel. Moreover, the harmonics that are included need not start with the first harmonic, i.e., two times the frequency of the corresponding fundamental frequency, or the first overtone, and/or need not be consecutive. For example, only the odd harmonics of one fundamental frequency and only the even harmonics of another fundamental frequency may be included. In general, the harmonics can be selected according to any specified sequence of harmonic indices, such as a geometric sequence, Fibonacci sequence, etc. The one or more fundamental frequencies and their respective harmonics that are included in a carrier kernel are called frequency components of the carrier kernel.

[0289] The waveform resulting from the collision, called a modified Zyoton, represents, substantially, the energy loss in the radiation energy incident upon the medium due to the absorption of such energy by the presence of the analyte of interest in the medium, or the presence of one or more confounders in the medium, or both, specifically in the spectral region chosen for the feature. In part due to the shape and morphology preservation properties of the Zyoton, the collision produces a modified Zyoton that can represent a change in energy substantially related only to the absorption energy loss represented by the feature used in the collision, as extracted by the collision operation. Moreover, the collision operator and one or more parameters thereof are selected such that the energy change represented by the feature is amplified when that change is represented in resulting waveforms generated by several iterations of the collision. Therefore, the modified Zyoton can be collided iteratively, following renormalization, with the same or different Zyotons, in order to amplify the energy loss represented by the feature without substantially introducing any noise or distortion.

[0290] A modulated carrier kernel, i.e., a conditioned feature, and an original Zyoton are both traveling waveforms that can be represented in the computer memory. The propagation of these waveforms in space and time, and their collisions, are simulated by a programmed processor and/or by one or more hardware modules such as an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a systolic array, digital signal processor (DSP), etc. The result of the collision, i.e., a modified Zyoton, and the renormalized modified Zyoton derived from the modified Zyoton are also traveling waveforms, each of which can be represented in the computer memory. In various embodiments, the propagation of the renormalized modified Zyoton in space and collision thereof with the original Zyoton or another Zyoton are also simulated using a programmed processor and/or one or more hardware modules.

[0291] Typically, the medium being observed contains one or more analytes of interest that need to be characterized qualitatively or quantitatively. It may also have other substances present that are not of interest but spectroscopically absorb, emit, fluoresce, or otherwise interfere with the analyte signal in the data region chosen, in the same bands as the analyte, and may thereby act as confounders to the measurement of analytes of interest. The collision computing paradigm described here can be applied for detecting and characterizing analytes in concentrations from as high as moles/l or higher to concentrations as low as picomoles/l or even lower. Individually or collectively, confounders may be present with many orders of magnitude greater concentration compared to analytes of interest. For example, the concentration of glucose, a biochemical blood analyte of interest, can be overshadowed by the concentrations of many other substances present in blood or interstitial fluid.

[0292] Zyotons are conditioned feature-complement collision waveforms that are constructed to function as energy amplification mechanisms. Zyotons may be constructed using a variety of waveform families and generator function families suitable for detecting an analyte or compound of interest. Useful waveform families include, but are not limited to: solitons; autosolitons; similaritons; custom solitons generated using a sine-Gordon-based equation; self-compressing similaritons; vortex-solitons; multi-color solitons; parabolic-similaritons; Ricci solitons; wavelets; curvelets; ridgelets; bions; elliptic waves including either or both of Jacobi elliptic functions and Weierstrass elliptic functions; and nonautonomous similinear wave equations.

[0293] Useful Zyoton generator function families include, but are not limited to: meromorphic functions; Gamma functions; Riemann Zeta functions; regular instantons; Frobenius manifolds; harmonic oscillators; Hermite polynomials; polynomial sequences; asymptotic Hankel functions; Bessel functions; fractals; Neumann functions (spherical); poweroid coupled with sinusoidal functions; spatial random fields; cyclostationary series; random number generators; spherical harmonics; chaotic attractors; exponential attractors; multipoint Krylov-subspace projectors; Lyapunov functions; inertial manifolds of Navier-Stokes equation; evolution equation for polynomial nonlinear reaction-diffusion equation; evolution equation for Kuramoto-Savashinsky equation; evolution equation of exponential attractors; Fourier series; and Ramanuj an theta functions.

[0294] Random number generators (RNG) are also useful as generator functions to produce numerical sequences used to synthesize Zyotons. Desirable general properties of RNG used to develop Zyotons include replicability (i.e., generated number sequences would be replicable if the seeds used to generate the random numbers were known and fixed), and the longest practical cycle length or periodicity (i.e., the longest possible time before numbers in the generated sequence start to repeat). Other attributes such as independence (i.e., correlation between numbers generated by the RNG), uniformity, and computational speed are optional for the purpose of developing Zyotons. Classes of RNG that are used to develop Zyotons, in which r_n is the random number generated, include linear congruential generators, (where r_n=(ar_n−1+c) mod m, n=1, 2, . . . ; where m>0 is an integer coefficient used in the modulus operation, and a period up to 2³¹), multiplicative congruential generators (where r_n=(ar_n−1) mod m, m>0, n=1, 2, . . . and a period up to 2³¹−2), and additive congruential generators or Fibonacci RNG generators (where r_n=(r_n−1+r_n−k) mod m, n=1, 2, . . . ; m>0; k≧2 and with a period up to m^k).

[0295] RNG based on linear, multiplicative, or additive congruential generators may be used as Zyoton generator functions even though they can produce numbers that are not random (also known as pseudorandom), and can be relatively more easily controlled (compared to other methods) to exhibit desired spectral properties in the frequency domain, and thus can be used as a first step to generate numerical sequences used for the synthesis of one or more Zyotons.

[0296] Regardless of the generator function used, the generated numbers are collected, fixed (or “frozen”), and then Fourier transformed to frequency distributions with desired properties and amplitude profiles that are useful as Zyotons. RNG are particularly useful for developing collections of related Zyotons (with some common underlying properties in the frequency domain) but where each subsequent collision uses a different Zyoton from such a collection of Zyotons in multiple-iteration collision sequences.

[0297] The polynomial sequences described above typically are a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, . . . , in which each index is equal to the degree of the corresponding polynomial. Finite and infinite length numerical sequences can be encoded as polynomial sequences and can be Fourier transformed, and are differentiable and integrable and can be represented as continuous geometrical curves and manifolds. Zyotons generated using polynomial sequences can encode arbitrary numerical sequences of interest. Specifically, arbitrarily desired frequency component patterns can be encoded using polynomial sequences. Specific examples of polynomial sequences that can be used as Zyoton generators include Abel polynomials (where n-th term of the polynomial is of the form p_n(x)=x(x−an)ⁿ⁻¹), and Bell polynomials given by:

[0298] $\begin{array}{cc}{B}_{n,k}\left(x_1,x_2,\dots ,x_{n-k+1}\right)=\sum \frac{n!}{j_1!j_2!\dots j_{n-h+1}!}{\left(\frac{x_1}{1!}\right)}^{j1}{\left(\frac{x_2}{2!}\right)}^{j2}\dots {\left(\frac{x_{n-k+1}}{\left(n-k+1\right)!}\right)}^{jn-k+3},& \left(1\right)\end{array}$
where the sum is taken over all sequences j₁, j₂, j₃, . . . , j_n−k+1 of non-negative integers such that:
j₁+j₂+ . . . +j_n−k+1=k and j₁+2j₂+3j₃+ . . . +(n−k+1)j_n−k+1  (2)
As an example, Abel and Bell polynomials can be used to encode combinatorial sets of frequencies (such as the k, m, and j frequency component groups referred to below), and then Fourier transformed to derive Zyoton waveforms which can be used in collisions.

[0299] Zyotons can also be created by combining other individual Zyotons, e.g., by multiplying, adding, or phase-shifting (also called subtracting), two or more individual Zyotons. Generation of new Zyotons by dividing or subtracting one Zyoton waveform from another Zyoton waveform can be achieved by using multiplication or addition operation respectively. Zyotons so combined to form a new Zyoton can be from the same or different families of Zyotons. In addition, various waveform functions and numerical sequences generated by a sequence generator can be transformed or reduced to any of the functions listed above. Such other waveforms and/or numerical sequences, which can be transformed or reduced into a waveform family/numerical sequence that is suitable for Zyoton synthesis, may be used to synthesize a Zyoton to be used in the collision process.

[0300] Although waveforms derived from all the above families can propagate in a constant medium without substantial change to their morphology, Zyotons derived from the families of optical solitons, autosolitons, similaritons, custom solitons, self-compressing similaritons, vortex-solitons, multi-color solitons, parabolic-similaritons and Ricci solitons generally retain their morphology and exhibit post-collision propagation properties substantially similar to their pre-collision propagation properties.

[0301] The above-described pattern-forming systems may be used as generator functions to derive a Zyoton waveform based on three factors: (i) their tendency to produce spatially confined states on the same time-domain spatial (i.e., morphological) and spectral scale as the entire set of anticipated feature waveforms from a spectroscopic sensor, as characterized by the spectral bandwidth, spectral envelope, peak energy, and amplitude distribution; (ii) non-stationarity of confounders, i.e., are they constant or time-varying in concentration during the measurement process resulting in changes in the signal-to-clutter increase required for an accurate measurement; and (iii) cyclostationarity of feature data itself, i.e., degree to which the statistical properties of the feature change over time, as described below, including the morphology of the amplitudes of the feature in the time domain, distribution of spatial frequencies of the feature and distribution of relative amplitudes of frequencies in the frequency domain, and any phase rotation of the waveform in the frequency domain.

[0302] A purpose of modulation of a carrier kernel by a feature, which may be preceded by an optional precursor frequency modulation, is to achieve a desired level of cyclostationarity or poly-cyclostationarity in the conditioned feature waveform, and to force the conditioned feature waveform to be represented using a combination of one or more periodic (e.g., sigmoidal) functions. Poly-cyclostationarity can increase the probability of inducing and detecting morphological changes post-collision in specific frequency components. Carrier kernel waveforms with higher frequencies than those of the feature (or the optionally modulated feature) are generally used in the feature modulation step, to condition features by inducing poly-cyclostationary properties.

[0303] In summary, Zyotons can be represented as localized traveling-wave packets, which exist as propagating entities through space and/or time. In general, the space and/or time propagation of Zyoton waveforms is scale and shift invariant. As such, transformations may be used to vary space and/or time scales to match the precision and accuracy desired in the data-collection domain, and/or the spectral resolution of sensors producing the feature data. For features with low cyclo-stationarity, complex or more nuanced generator functions may be required, such as Lyapunov functions or meromorphic functions.

[0304] In one exemplary embodiment that can be used for non-invasive glucose analyte detection and measurement, Zyotons are derived using a base family of mathematical functions denoted as solitons. This choice of solitons as source family generator for Zyoton derivation, in general, is based on the above three properties for selection of waveforms, but is strongly influenced by the stationary nature of confounders in the data during the course of measurement (generally over a time period of a few milliseconds to a few seconds). Solitons are defined and represented as a permanent localized disturbance in a linear or non-linear wave. In physical systems, such as propagation of light waves in an optical fiber, solitons can result from the offsetting of nonlinear and dispersive effects in the propagation medium. Mathematically, solitons are a solution to weakly nonlinear dispersive partial differential equations describing physical systems, such as optical energy propagation in a telecommunication fiber cable. A detailed treatment and background to soliton dynamics is provided in Ablowitz, Mark J., Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons (Cambridge Texts in Applied Mathematics) (2011), and in Dauxois, T. and Peyrard, M., Physics of Solitons, ISBN-13: 973-0521854214.

[0305] Properties that make solitons attractive as a base family for designing collision waveforms or Zyotons include first, localization of the center of mass or confinement of peak energy to a region and resistance to weak external perturbations. Since collision computing involves deconstructed data or spectral features (i.e., fragments of spectra), local energy absorption or losses, and locality and stability of the soliton waveform are generally important. Second, permanency or morphological stability is important—solitons retain their shape. The accurate determination of spectral energy changes in the post-collision waveform, due to energy transfer from the colliding waveforms to the post-collision collided entity (i.e., the modified Zyoton), is important to analyte concentration estimation.

[0306] Power spectral density (PSD) of a signal waveform can be defined as the power contributed to the wave by a frequency, per unit frequency in watts/hertz (W/Hz). The PSD is the normalized limit of the energy spectral density (ESD) for a windowed signal x_N(t) and a measure of distribution of signal power. Thus, spectral energy corresponds to the total power within a frequency or wavelength interval. Alternately, spectral energy can also be computed for a set of discrete frequency components by using the power spectral density of the selected frequency components. This definition of power spectral density generalizes the estimation of total spectral energy S_xx(ω) for a discretized and digitized signal waveform (e.g., feature data), represented as a finite time-series X_n with 1≦n≦N, such as a signal sampled at discrete times x_n=x(nΔt) for a total measurement period T=NΔt. The spectral energy is given by:

[0307] $\begin{array}{cc}{S}_{\mathrm{xx}}\left(\omega \right)=\frac{{\left(\Delta t\right)}^2}{T}{\sum _{n=1}^N{x}_n{e}^{-i\omega n}}^2& \left(3\right)\end{array}$
where ω=2πf, and f denotes the frequency of the signal.

[0308] Given that mathematically one can compute the spectral energy density and the spectral energy for a signal waveform, waveform propagation and waveform collisions can be represented as energy entities and energy transfer operations, respectively. As Zyoton collisions are inherently non-linear events, energy representation provides a mathematical framework (or an “energy algebra”) for describing and inducing complex behaviors. Collision computing as described herein facilitates detection and/or quantitation of analytes and events using energy algebra. Morphological stability and properties of entire Zyotons, or selected frequency components with high retained fidelity, allows comparison and characterization both before and after collisions for large numbers of features extracted from large numbers of samples, collected under similar conditions, using the same or similar hardware platform and sensors.

[0309] Solitons are described in the art as “waves that act like particles.” An individual soliton can travel through a propagation medium over a certain distance without distortion of the soliton's morphology and without dissipation of the soliton's energy. If one soliton collides with another, typically the morphology of both solitons is preserved after the collision, and only a phase shift occurs in the respective propagation of one or both solitons as they continue to propagate through the propagation medium. Phase shift in this context generally means that there is an offset in arrival time and phase for either one or both of the colliding solitons relative to the case when no collision occurs, and a phase shift does not introduce any new waveform frequency components that would alter energies of the two solitons. As such, no significant (e.g., more than 0.01%) change generally occurs in the respective energies of the two solitons after a collision between them. In contrast, the morphological envelope of a soliton generally changes during collision with a non-soliton travelling wave. Typically, if a soliton collides with a non-soliton waveform of similar energy, the soliton is destroyed. Thus, in the first interaction (i.e., between two solitons), no measurable energy transfer occurs between the two solitons and, in the second interaction (i.e., between a soliton and a non-soliton waveform), although energy transfer occurs, it cannot be measured because the soliton is destroyed during the collision.

[0310] The selection of a soliton family to construct a Zyoton, and its time-domain and frequency-domain parameterization, is based on the consideration of at least one of several specific attributes, which include: the signal-to-noise ratio (SNR) of the measurement system, the anticipated signal-to-clutter ratio (SCR) or signal-to-clutter noise ratio (SCNR), the degree of desired SCR increase, the desired analyte quantitation accuracy and precision, resolution, specificity, dynamic range, sensitivity, and concentration range over which linearity is desired. The SCNR (signal to clutter noise power ratio or signal to coherent noise power ratio) is generally defined as the ratio of signal power to the power of the sum of clutter plus background non-coherent noise power. Clutter can be due to signal from confounders or from coherent noise. Coherent noise is a type of smooth pseudorandom noise which has been usually defined in the art as having three properties: (i) passing in the same input value will always return the same output value; (ii) a small change in the input value will produce a small change in the output value; and (iii) a large change in the input value will produce a random change in the output value. The sources of such coherent noise include: variability due to free space optics (i.e., variability in the radiated and/or collected light traveling through air as it is subjected to variable amounts of water vapor), detector electronic noise, light drift and variability, variable absorption through a fiber optic probe, etc.

[0311] As described above, a conditioned feature is particularly constructed to represent the energy absorbed by an analyte or by one or more confounders, and accurate estimation of the absorbed energies can lead to an accurate estimation of the analyte concentration. To facilitate accurate estimation of the absorbed energy represented in a conditioned feature in various embodiments, particularly designed Zyotons are employed. Specifically, a Zyoton is modified after a collision with a conditioned feature but the modification is particularly controlled such that the original Zyoton is transformed into a modified Zyoton that continues to propagate thorough the propagation medium and is not destroyed at least over a certain propagation distance. Therefore, the energy transferred to the original Zyoton during the collision can be measured from the modified Zyoton. As described below, the Zyoton and the conditioned feature are synthesized in a co-dependent manner, such that the energy transferred to the Zyoton over one or more collision iterations is proportional to the energy absorbed by an analyte and/or one or more confounders, as represented by the conditioned feature. The permanency or morphological stability of a Zyoton is important, in general, to properly estimate spectral energy changes therein, which are related to the analyte presence or concentration at the feature level. As described below, the energy changes at the feature level may be aggregated to estimate the net energy change due to the presence and quantity of an analyte.

[0312] Collisions between conditioned features and Zyotons lead to minor waveform distortions of the Zyoton that can be used to perform post-collision energy computations, because energy transfer is possible between a conditioned feature and a Zyoton. Therefore, the feature, derived from incoming or acquired data, generally acts as an energy modulator to the Zyoton. Propagation over grids with a sufficient number of points and over sufficient travel distances can filter out post-collision transients and dispersion in high frequency, lower-amplitude peaks. Typically, multiple collision iterations between a conditioned feature and a Zyoton or between a modified Zyoton and a Zyoton are involved in collision-based energy absorption estimation. So each post-collision result is a modified or energy-amplified Zyoton. Specifically, as described above, as a result of a collision between an original Zyoton and a conditioned feature, a modified Zyoton with increased energy is obtained. The amount of energy amplification during a collision is related to the properties of the conditioned feature, e.g., energy absorption at NIR wavelengths due to analyte concentration in the medium irradiated with NIR. The increase in energy is generally related to the energy absorbed by an analyte and/or one or more confounders, as represented by the conditioned feature. The absorbed energy can also be referred to as energy loss or energy lost as a result of the presence of the analyte and/or one or more confounders. As used herein, the absorbed energy or energy loss, however, relates only to analytes and/or confounder(s) and not to energy otherwise dispersed or lost in the environment or medium. It should be understood that a medium to be analyzed in which an analyte and one or more confounders may be present may be different than a propagation medium in which Zyoton(s) and conditioned feature(s) propagate.

[0313] In some embodiments, the modified Zyoton is renormalized as described below, and the renormalized Zyoton is collided with the original Zyoton. The result of this subsequent collision is another modified Zyoton, typically with more energy than the original Zyoton and the previously generated modified Zyoton and, again, the increase in the energy is related to the energy absorbed by an analyte and/or one or more confounders. In general, successive collisions between the original Zyoton and the modified, renormalized Zyotons can result in recurring energy increases in the modified Zyotons that relate to the energy absorbed by an analyte and/or one or more confounders, with increasing accuracy and observability. Thus, after a specified number of collision iterations, the energy gain in a modified Zyoton relative to the original Zyoton can be equivalent to or a multiple of the energy loss or the absorbed energy represented by a feature and/or a conditioned feature. Through one or more (e.g., tens, hundreds, thousands, tens of thousands, or more) collision iterations, it is important that underlying properties of the original and successively modified Zyotons are maintained, i.e., the morphological characteristics thereof change in a controlled manner, to enable accurate post-collision energy computations. Normal, non-solition waves generally flatten out, leading to loss of or change in character of dominant frequency components, thereby rendering post-collision energy computations ineffectual.

[0314] Assume that the total energy of the original Zyoton is E and that the energy of a conditioned feature is e. Further, assume that e includes the energy Δe, where Δe actually represents a cumulative loss of energy due to absorbance by an analyte and/or one or more confounders in the medium. Prior to the first collision, e is set to be around 0.001 times E and the gain in any individual collision iteration i, denoted δe_i, is less than 10⁻⁶ E. In the table below, δe₁ represents the gain in the post-collision spectral energy of the modified Zyoton due to transfer from the conditioned feature after the first collision, prior to the renormalization step. Similarly, δe_i represents the gain in the post-collision spectral energy of the modified Zyoton due to transfer from the modified Zyoton after the i^th collision, prior to the renormalization step. This gain, δe_i, is related to the NIR absorption by the analyte, within the wavelength boundary defining the feature. As described below, the collision computing process is designed so that the estimated per collision gain of δe_i is proportional to the absorption of energy by the concentration of analyte in the medium. This proportional relationship of E to δe_i (between a Zyoton and a renormalized Zyoton after the first collision iteration) is maintained throughout subsequent collisions, as shown below.

[0315]

TABLE 2
		Energy of
		Renormalized	Post-
		Zyoton (Energy	Collision
	Energy of	of Conditioned	Energy of
Iteration/	Original	Feature for	Modified
Collision	Zyoton	Iteration 1)	Zyoton	Comment
1	E	e	E + δe1	δe1 > 0
2	E	δe1	E + δe2	δe2 ≧ δe1
3	E	δe2	E + δe3	δe3 ≧ δe2
	E		E +	≧   ≈
				Δe

[0316] The number of iterations/collisions is selected such that after iterations/collisions, the increase in the energy of the original Zyoton, given by δeis approximately equal to Δe, or is a multiple thereof, i.e., the net loss of energy due to absorbance by an analyte and/or one or more confounders. Also, to preserve the propagation integrity of the Zyoton, the frequency components of the carrier kernel are scaled and selected to establish the spectral energy of the conditioned feature waveform to be around 1/100^th, 1/1,000, or 1/100,000^th of that of the Zyoton for the selected wavelength interval, so the feature serves only as a weak perturbation to the Zyoton.

[0317] The process of conditioning data features and the generation of Zyotons and their parameterization is driven by the overall analysis objectives—often detection of energy absorption by the analyte—and requires any conditioned features (in some cases just a single feature) and their Zyotons to become co-dependent. The feature conditioning and the Zyoton selection are both dependent on the expected signal-to-clutter ratio (SCR) improvement required, as described above. This co-dependency is captured by the frequency bandwidth and frequency components of both the carrier kernel used in feature conditioning and by the selection of Zyotons. The frequency bandwidth of the carrier kernel is generally set to be between

[0318] $\frac{1}{{\left(\mathrm{SCR}\right)}^2}\mathrm{and}\frac{1}{{\left(\mathrm{SCR}\right)}^3},$
so a measurement with a SCR of 0.01 would require a carrier kernel bandwidth of 10⁴ to 10⁶ Hz. A carrier kernel bandwidth of 100 KHz can be used in the analysis of spectroscopic data for a non-invasive glucose monitoring application, for example.

[0319] The total set of power-sorted frequency components for the Zyoton is defined as H=k+m+j where k>0, m≧0, and j>0, and the k and j frequency components do not overlap. In general, the k components are related to absorption by the analyte and are called analyte-information representing components, the j components are generally related to the energy lost due to scattering and are called non-analyte information representing components, and the m components serve as a transition zone to the j components, and are called transition components. In general, non-analyte information can be noise, clutter, etc., that this itself can be considered to be information, e.g., about noise, clutter, etc. Therefore, analyte-information-representing generally means representing a property, such as the presence or absence of or a quantity of an analyte and/or one or more confounders, energy absorbed by the analyte and/or one or more confounders, etc.

[0320] In various embodiments, the amplitudes of the analyte-representing k frequency components are higher relative to the amplitudes of the m and j components and, as such, in these embodiments, the k components are also called high-energy components. The non-analyte information representing j frequency components may have low amplitudes relative to the amplitudes of the k and m components, and may be called low-energy components. The transition components, i.e., the m frequency components, may have amplitudes greater than the amplitudes of the j components but lower than the amplitudes of the k components, and may be called medium-energy components. In some embodiments, the Zyotons and the corresponding carrier kernels are synthesized such that the analyte-information representing k components of the Zyoton and the co-dependent conditioned feature(s) are low-energy components, and the non-analyte information representing j components of the Zyoton and the co-dependent conditioned feature(s) are high-energy components.

[0321] Typically, the high-energy components of a Zyoton and the conditioned feature(s) have spectral energy that is several orders of magnitude greater than the spectral energy of the low-energy components, and the medium-energy components have spectral energy that is a multiple of the spectral energy of the low-energy components but that is not more than an order of magnitude greater than the spectral energy of the low-energy components. In some embodiments, the frequency components of the Zyoton (and also of the conditioned feature) can have amplitudes that can be represented by a continuum, where these are not broken up into specified k, m, and j subsets, each corresponding to a particular energy band. Instead, in these embodiments, certain frequency components can be selected which correspond to energy absorbed by an analyte, while other frequency components can be selected which correspond to absorption by confounders or sources of random noise, such as scattering or absorption variations of a confounder. Those frequency components corresponding to energy absorbed by an analyte may be amplified during the collision computing process, while those corresponding to noise may not be amplified.

[0322] The choice of the number k of the frequency components used in the spectral energy computations is driven in part by the dynamic range of spectral energy change desired across the dynamic range of concentration of the analyte of interest, and the desired precision. Example values of k range from 4 through 256 when a carrier kernel with a bandwidth of 100 KHz is used. The value of m ranges, for example, between 0 and a multiple of k, such as (4×k). Example values used in non-invasive glucose measurement are k=6 and m=8.

[0323] As described above, Zyotons may be selected from generator functions and/or waveform families that can produce waveforms that retain their properties, e.g., solitons that are stable over the desired frequency bandwidth. As such, and as further described below, the Zyoton and the conditioned feature waveforms, collectively referred to as the collision entities, are co-dependent. The energy scaling relationship between the Zyoton and the conditioned feature imposes the co-dependency condition between the two waveforms; and the scaling step as part of renormalization between the Zyoton and the modified Zyoton (prior to the second and any subsequent collisions) can re-establish the co-dependency condition between those two. Thus, by the algebraic transitive closure property, the Zyoton, the conditioned feature, the modified Zyoton, and the renormalized Zyoton are co-dependent.

[0324] As an example, one-dimensional Zyotons existing as a function of time t, Z_N(xt), can be constructed as:
Z_N(xt)=2∂_xTrB_x(1+B)⁻¹  (4)

[0325] where B is an N×N matrix with elements

[0326] $B_{\mathrm{mn}}=\frac{1}{p_m+p_n}{C}_m{C}_n,$

[0327] with variables C_m, C_n defined as

[0328] $C_m=\exp \left\{s_m-\frac{1}{2}\ln {\mathrm{ia}}_m\right\},C_n=\exp \left\{s_n-\frac{1}{2}\ln {\mathrm{ia}}_n\right\},$
respectively with m, n≦N;

[0329] $s_m=p_{m}x-p_m^{3}t+{\delta }_m,{\mathrm{ia}}_m=\prod _{n\left(\ne m\right)}^{}\frac{p_m-p_n}{p_m+p_n}\frac{1}{2p_m},$

[0330] and δ_m is a real-number constant,

[0331] p_m is a positive constant such that p₁>p₂> . . . >p_N.

[0332] The selection of p_m and δ_n, in the equations above is not based on the bandwidth of the Zyoton waveform but, instead, is based on the desired amplitude ratio of b sideband peaks on the left and right of the primary amplitude peak as shown in FIGS. 12A-12C. Note that b sideband peaks for the Zyoton waveform are defined as the portion of Zyoton wave amplitudes that are either above or below the frequency component corresponding to the peak amplitude of Z_N(xt). In the above equation for Z_N(xt), Tr is an algebraic operator that denotes the trace of an n-by-n square matrix and is defined to be the sum of the elements on the main diagonal of the matrix (the diagonal from the upper left to the lower right) of A, i.e.,

[0333] $\mathrm{tr}\left(A\right)=a_{11}+a_{22}+\dots +a_{\mathrm{nn}}=\sum _{i=1}^n{a}_{\mathrm{ii}}$
where a_nn denotes the entry on the n-th row and n-th column of A. This is an example of a Zyoton kernel that can be used for glucose analysis.

[0334] With reference to FIG. 13, a typical process of detecting the presence of an analyte in a medium and/or quantifying an amount of the analyte in the medium includes acquiring one or more intensity spectra in step 2. Using spectra and the corresponding reference concentration data (also called a reference calibration dataset) that is obtained in step 14, at least in part, from known amounts or concentrations of analyte, the presence of the analyte is detected and/or the quantity thereof is estimated in step 16 via a projection process that uses calibration tables, sets, and/or curves included in the reference calibration dataset. The presence of the analyte and/or the estimated quantity thereof is determined and reported in steps 18, 20.

[0335] The method described here utilizes one or more nonlinear, non-invertible computational collisions between two entities—a feature derived and extracted from sample observations, and a purposefully constructed waveform, where the collision yields waveforms that can be processed to determine analyte presence and to estimate analyte concentrations in the uncharacterized samples. This method of detecting the presence and estimating concentration of analytes from sensor data is performed by transforming the incoming sensor data to a wavefront called conditioned feature (Ψ_CF) (steps 4, 8); selecting another waveform, referred to as the Zyoton (Ψ_Z) (step 6); colliding the conditioned feature with the Zyoton to obtain a modified Zyoton (Ψ_Z′) and/or renormalized Zyoton (Ψ_Z′) (step 10), which may include additional collision iterations; and assessing the properties of the spectral energy distribution of the modified and/or renormalized Zyoton (also called a coupled waveform complex) (step 12). The step 12 generally includes determining the total spectral energy transferred to the Zyoton from the analyte-information representing components of the conditioned feature waveform after the collision. The wavelengths used in the synthesis of the Zyoton(s), and that may be present in the co-dependent conditioned feature and the modified and renormalized Zyoton waveforms are not typically the wavelengths associated with the incoming sensor data.

[0336] The conditioned-feature waveforms are “domain-based”, in that they are derived from features, which have themselves been derived from signals, spectral data, or sensor data in the data-collection domain. The Zyotons, however, are “non-domain-based” or are independent of the data-collection domain in that the waveform kernels they are synthesized from may have no direct relationship to the sensor data acquired in the data collection domain. For example, data collected can be of the form NIR radiation, heat radiation, visible light, etc., while a Zyoton can be generated from various waveform families and/or generator functions that waveforms and functions unrelated to these data domains.

[0337] Zyotons represent models of physical waveforms encountered in domains different from the data-collection domain (e.g., solitons in optical communication or cryptography). Non-domain generator functions may be optionally used to construct Zyotons where the generator functions are either closed-form analytical functions (such as Gamma functions, Riemann Zeta functions, Neumann Functions) or are represented as formal power series (e.g., Lambert series, Dirichlet series) in one indeterminate variable, whose coefficients encode information about a sequence of numbers that is indexed by the natural numbers. Zyotons, such as those derived from spherical harmonics, can be optionally derived from generator functions represented as formal power series in more than one indeterminate variable, to encode information about arrays of numbers indexed by several natural numbers. While these waveform families and the generator functions can be mathematical entities (generally functions and sequences), the particular frequencies of the Zyotons are selected, however, according to the properties of the analyte, the properties of one or more confounders, the properties of the medium/environment to be analyzed, and/or the properties of the measurement system.

[0338] The spectral energy distribution used in the step 12 for the detection and/or quantitation of the analyte can be described as the distribution of flux density versus frequency or spatial wavelength, or the energy as a function of the spatial wavelength. Depending on the nature of the input data, the characteristics of the sensor used for observing the analyte, the complexity and concentration of confounders, and the expected concentration of the analyte, one or more computational collisions may be implemented by the method described herein. After each collision iteration, the power spectral energy distribution of the collision-modified Zyoton (the resulting waveform: Ψ_Z′) is computed and compared to the power spectral density of the original Zyoton waveform to determine that the codependency condition (described below) is still met. The waveform collisions may be implemented either in the analog or digital computing domain, and in a time domain or in a frequency domain. Analyte presence and concentration are determined based on the net gain (which can also be represented as an energy loss in some computer implementations) in the spectral energy computed from spectral energy distribution of the modified and/or renormalized Zyoton after a selected number of collision iterations.

[0339] The computation of the spectral energy changes requires observability of energy states of the pre- and post-collision waveforms. As introduced in classical control theory, in general, “observability” is a measure of how well internal states of a system can be inferred by knowledge of its external, measurable outputs or observables. In some embodiments described herein, the frequency-domain amplitudes of the pre-collision original Zyoton and, post-collision, the frequency-domain amplitudes of the modified Zyoton and renormalized Zyoton waveforms are treated as system or “collision observables.” The collision process is implemented such that collision observables can be computed and compared when features are extracted from different measurements of the sample with different material concentrations, per the precision requirements of the system.

[0340] If a Zyoton does not possess the desirable properties described above, the collision observables, harvested from a single collision or from several collision iterations would offer no inferential utility i.e., the ability to determine the presence of and to estimate the concentration of materials in unknown, uncharacterized samples. Instead the computed spectral energy values may appear as random artifacts or noise given the very low signal-to-clutter ratios in the measured samples. The ability to observe consistent patterns in the spectral energy gained (or lost) after each collision iteration between two colliding waveforms is a prerequisite to concluding the integrity and consistency of the collision process. As described below, this is achieved, in part, by maintaining co-dependency between the colliding waveforms. The typical conditions for Zyoton synthesis (e.g., spectral energy 10³ or 10⁴ times higher than that of the conditioned feature) may set bounds for the changes in observed spectral energy after each collision.

[0341] The processing unit where the collisions are implemented is denoted as the collision computer. Three key elements of collision computing are: (i) the collision process on data, including the design of the collision waveforms, the collision operator, and number of collision iterations required to achieve the desired detection performance; (ii) the preprocessing of the acquired spectroscopic data, including deconstruction into spectral fragments that are a subset of the spectrum and their conditioning for use in the collision process; and (iii) the post-collision estimation of the change in spectral energy of the system as a mechanism to estimate the total energy change due to absorption by the analyte and its use to estimate analyte concentration.

[0342] An example of source data includes an intensity spectrum acquired by directing radiation to a medium and by detecting or collecting radiation reflected by the medium, or by directing radiation through a medium and detecting the transmitted radiation. In step 4, the spectral data may be normalized to eliminate amplitude variations, and one or more features (selected wavelength ranges of a spectrum) may be extracted from the optionally normalized spectral data. Different Zyotons may be required to collide with different features. The spectral energy of the Zyoton should be matched to both the spectral energy and the amplitude of the feature as described herein.

[0343] One or more Zyotons, with which the conditioned features are computationally collided to enable detection and/or measurement of the analyte, are generated or obtained in step 6. The Zyotons are generated according to one or more of the requirements for precision, accuracy, limit of detection, limit of quantization of desired analysis results, and characteristics of the spectral data acquisition system including: signal-to-noise ratio (SNR), system stability, spectral resolution, spectral bandwidth, illumination process and spatial sequence (if used), expected signal-to-clutter ratio, (SCR) due to the presence of confounders in the medium to be analyzed, and the required signal-to-clutter increase for an accurate measurement. As Zyotons are derived from inherently stable waveforms, the chromatic dispersion of the Zyoton waveform, the peak power and pulse energy (defined as the product of the power of the waveform peak and its width or the area under the peak) in its first k peak-power sorted frequency components (where k is typically <6), collectively control the limit of detection and quantitation that can be achieved in one or more collisions. In various embodiments, a reference to the “k frequency components” generally describes the first k peak-power-sorted frequency components of a waveform such as a Zyoton, a carrier kernel, and a conditioned feature.

[0344] In step 8, the features extracted in step 4 are conditioned, so that Zyotons and the conditioned features become co-dependent. Collectively, we refer below to the constraints on co-dependency imposed by the variables κ_DV1, κ_DV2, κ_DV3, and κ^SYN as the “Kappa Test.” Depending on whether the testing and/or scaling is performed in the time domain or in the frequency domain, a domain-specific value of the applicable Kappa variable is used for various tests of co-dependency in various embodiments.

[0345] In a waveform obtained by combining two or more sinusoids, such as in the construction of a Zyoton, construction of conditioned feature, etc., phase velocity is the velocity of a particular phase of the waveform and can be expressed as a ratio of the wave's angular frequency over wavenumber, i.e.,

[0346] $v_p=\frac{\omega }{k}.$
For or Zyotons constructed using soliton waveforms, the dispersion velocity is same as their phase velocity. In contrast, group velocity is the phase velocity of an envelope representing the combination of the two or more sinusoids, and can be expressed as

[0347] $v_g=\frac{d\omega }{dk}.$
The dispersion velocities of a Zyoton and a conditioned feature are generally proportional to the time domain amplitudes of the sideband peaks thereof.

[0348] Assume that during the collision process, a conditioned feature waveform CF_i, corresponding to a feature F is propagating with dispersion velocity ω₁, and a Zyoton Z_i is propagating with dispersion velocity ω₂. Further assume that the collision of the two waveforms produced a modified Zyoton Z_i′, propagating with dispersion velocity ω₃. The difference between the pre- and post-collision dispersion velocities of the Zyotons, i.e., the magnitude of (ω₂-ω₃) should be less than a preset Kappa parameter κ_DV3, if the conditioned feature waveform CF_i and the original Zyoton Z_i are co-dependent. If a Kappa test is performed in the time domain, the velocity can be obtained as either the L1 norm or the L2 norm of the time-domain amplitudes of a selected number of sideband peaks of the two Zyoton waveforms Z_i and Z′_i.

[0349] Optionally, a Kappa test can be performed pre-collision, by comparing the velocities of the conditioned feature and the original Zyoton, i.e., ω₁ and ω₂, respectively. As a conditioned feature is generally constructed to have only a fraction of the energy of the original Zyoton, ω₁ is typically only a fraction of ω₂. As such, in comparing ω₁ with ω₂, either one or both of the velocities are scaled, and it is tested whether the magnitude of a difference between the velocities, after scaling one or both, is less than a Kappa parameter κ_DV2.

[0350] As described below in various embodiments, the modified Zyoton Z′_i is renormalized to produce a renormalized Zyoton Z″_i, having dispersion velocity ω₄ that is comparable to the dispersion velocity ω₁ of the conditioned feature CF_i. The energy of the renormalized Zyoton is also expected to be comparable to that of the conditioned feature. Therefore, in some embodiments, Kappa test can be performed by comparing the velocities of the renormalized Zyoton and the original Zyoton, i.e., ω₄ and ω₂, respectively. Here again, as the renormalized Zyoton is expected to have only a fraction of the energy of the original Zyoton, ω₄ is typically only a fraction of ω₂. As such, in comparing ω₄ with ω₂, either one or both of the velocities are scaled, and it is tested whether the magnitude of a difference between the velocities, after scaling one or both, is less than a Kappa parameter κ_DV1. In some embodiments, κ_DV1 can be the same as or approximately equal to κ_DV2. In general, the difference in the velocities to be compared is an absolute difference, i.e., a magnitude of the difference regardless of whether one velocity is greater than or less than the other is compared with an applicable Kappa parameter.

[0351] As described above, co-dependency condition can be tested pre-collision, i.e., by testing whether |α_T*(ω₂)−(ω₁)| is less than a Kappa designated κ_DV2, where α_T is the scaling factor applied during testing to the velocity of the original Zyoton. As described above, the velocities can be compared by scaling ω₁ instead of or in addition to scaling ω₂. In order to satisfy this test, a nominal scaling coefficient α_C is applied to achieve a pre-selected ratio (α_Z) of spectral energy of Zyoton to the spectral energy of a conditioned feature. The ratio α_Z can be, e.g., 100; 500; 1,000; 2,000; 3,500, etc. In some embodiments, the spectral energies in determining the ratio are computed using only a limited number (e.g., 2, 6, 8, 15, etc.) of frequency components of the Zyoton and the conditioned feature. The nominal scaling coefficient α_C is applied to the original Zyoton or to the conditioned feature, or to both waveforms such that the pre-selected ratio (α_Z) of spectral energies of the Zyoton and the conditioned feature are achieved.

[0352] The term nominal scaling coefficient is used because different features from the same or different spectra from the same or different media to be analyzed are likely to result in somewhat different scaling coefficients to establish the ratio α_Z of spectral energy of the Zyoton to the spectral energy of the conditioned feature prior to the first collision. Generally, the following condition is imposed on the ratios of the nominal scaling coefficient:

[0353] $\frac{\left(\alpha C-\alpha \mathrm{Fi\_MAX}\right)}{\alpha C}<0.1,0.2,\mathrm{or}0.5,\mathrm{and}$ $\frac{\left(\alpha C-\alpha \mathrm{Fi\_MIN}\right)}{\alpha C}<0.1,0.2,\mathrm{or}0.5,$
where α_{Fi_MIN}, α_{Fi_MAX} respectively denote the maximum and minimum scaling coefficients encountered to achieve a desired spectral energy ratio α_Z (e.g., 1000) with an α_C (e.g., 0.001). The parameters α_{Fi_MIN} and α_{Fi_MAX} are any members of a set of all spectral energy scaling coefficients {α₁, α₂, α_∝} required in a collision computing sequence used to analyze spectral features from a spectroscopically analyzed sample in the frequency domain. It is to be understood that these coefficients {α₁, α₂, α_∝} and α_Z may be computed and applied in the time domain or in the frequency domain, and are distinct from the computing-coefficient α described below in the discussion of the collision operator.

[0354] Another important co-dependency condition is that the width of the post-collision Zyoton waveform divergence be less than a preset threshold τ. A width of a Zyoton waveform can be described as a time lag between any two phases of a Zyoton waveform, such as time lags t1, t2, and t3 depicted in FIG. 14A. After collision, one or more of these widths can change, as in t1′, t2′, or t3′ shown after a collision in FIG. 14B, but the second co-dependency condition requires that each of these differences, i.e., Δt1=t1′−t1, Δt2=t2′−t2, and Δt3=t3′−t3, be less than τ.

[0355] The selection of values for κ and τ are related to the collision grid, bandwidth of the Zyoton, and peak energy of the Zyoton. As the values of κ and τ bound the post-collision spectral energy change that can occur to preserve Zyoton properties, they also bound the original spectral energy of the colliding entity, i.e., the conditioned feature waveform for the first collision by imposing a design constraint on the carrier kernel spectral energy. The number and specifics of frequency components and their amplitudes of the carrier kernel are thus constrained by κ and τ. An increase in the κ and τ values may allow for an increase in the amplitude(s) of the first k frequency components, as well as the number k of analyte-information representing components that are used in the pre-collision and post-collision computations described below. Once κ and τ are established, the properties of the optimized carrier kernel used for conditioning the features can be locked. Exemplary values for κ_DV2 and τ for noninvasive glucose measurement are 0.6×10⁻⁶ and 1.4×10⁻⁶, respectively, as determined, for example, by an L2 norm. The parameters κ and τ are generally represented as unit-less numbers.

[0356] These conditions generally need to be met for every conditioned feature-Zyoton pairing. If these two co-dependency conditions are not achieved, the collisions may not have computational utility or the collision operator may become unstable. In various embodiments, the variables κ_DV1, κ_DV2, κ_DV3, and κ^SYN are used as tests of collision integrity before the first collision iteration as a limit of the difference in dispersion velocity (or spectral energy) between the Zyoton and the conditioned waveform. Thereafter, before and/or after one or more subsequent collision iterations, these variables can be used as tests of collision integrity between the Zyoton and a modified Zyoton or between a Zyoton and a renormalized Zyoton. Optionally, these variables can also be used to test a limit of the difference in dispersion velocity (or spectral energy) between the Zyoton and the carrier kernel, when they are initially synthesized, in order to verify their codependency. Upon completion of the conditioning process, each conditioned feature can be represented in the computer memory as a parameterized waveform. One or more Zyotons and one or more conditioned features are computationally collided in step 10 (FIG. 13).

[0357] Typically, the absorption energy loss due to the analyte is much less than that due to one or more confounders, and the total energy losses due to the analyte and confounders are significantly smaller than the loss due to scattering and/or dispersion of the radiation in the medium. As such, the signal-to-clutter ratio (SCR) of a signal corresponding to the energy loss due to the analyte of interest to the overall energy loss due to confounders and scattering is often very low—e.g., as small as 10⁻⁴ or 10⁻⁶. By tuning the Zyotons, the collision operator, one or more parameters thereof, and the modulation of the feature to a required SCR increase, the energy loss represented by the feature can be amplified, without introducing noise or distortion, in one or more collisions, and can then be measured.

[0358] One key objective of the Zyoton tuning process is to configure the Zyoton such that energy loss estimated from a Zyoton collided with a specific conditioned feature, after the one or more prescribed number of collision iterations, is strictly monotonic, that is, trending in the same direction throughout the entire interval, as the concentration of the analyte in the sample over the analyte concentration range of interest, when the feature concentrations and computed energy loss are represented as ordered pairs. It is to be understood that the term monotonic, while used here in the sense of a constantly increasing trend of a data set in which the analyte concentrations are constantly increasing, can also be used to describe the constantly decreasing trend of a data set in which the analyte concentrations are constantly decreasing. The collision process can be thus described as a monotonic transformation of an acquired spectroscopic dataset. A calibration table and/or curve can be used to determine concentration of the analyte in the medium according to the amplified energy loss or change, as represented by the final result of one or more collision iterations yielding a monotonic response.

[0359] In various embodiments, several collision iterations are used to expand the dynamic range of the estimated spectral energy changes so that greater precision can be obtained during the post-collision projection process for predicting analyte concentration. Specifically, in various embodiments, the collision process is iterative in which the first collision iteration in a collision sequence is between a Zyoton and a conditioned feature. Subsequent collision iterations may be between the renormalized output of the preceding collision (i.e., a renormalized Zyoton) and the same or a different Zyoton. Whenever more than one collision iteration is used (i.e., the number of iterations >1), the colliding entities are usually a renormalized Zyoton and the original Zyoton or a new Zyoton.

[0360] Different Zyotons may be used to collide with different conditioned features. Different features extracted from varying regions of the original source data can have vastly varying differences in their measurement scale, sometimes varying over several orders of magnitude. With such variations, in some embodiments, the co-dependency condition cannot be achieved using the same Zyoton for all selected spectral features. Thus, different Zyotons are designed and tested to ensure that each conditioned feature and Zyoton pair satisfies the codependency condition described above.

[0361] The analyte in the medium to be analyzed generally induces a change in the measured parameters of the medium (such as absorption or emission of energy) relative to those parameters if no analyte were present in the medium. This change is represented in the acquired spectral data of the medium and, hence, in one or more extracted features. The Zyotons, the conditioning of the features, and the collision operator are designed such that one or more collisions in step 10 (FIG. 13) generally induce a measurable change in one or more properties (e.g., spectral energy reduction due to incident light absorption by the analyte in the measured spectral bandwidth) of the waveform resulting from the collisions. In step 12 (FIG. 13), after a preset number of collisions between a conditioned feature and the corresponding Zyoton have been completed, such change in the resulting waveform is computed to determine the net spectral energy gain or loss for each collided feature. In some embodiments, the process is configured to be represented as an energy gain, but represents a loss in spectral energy due to absorption of energy by the analyte.

[0362] Following a collision process involving one or more collision iterations, the changes in the properties of the post-collision waveform (the modified Zyoton) before and after the collisions may be analyzed as a mechanism to infer properties of the feature. Impacted properties may include changes in propagation velocity, peak energy, dispersion velocity, and changes in the spectral envelope. These changes can be quantified through changes in the spectral energy of the modified Zyoton generated after each collision relative to the spectral energy of the original Zyoton. In the first iteration, at least a portion of the energy of the feature that is represented by the conditioned feature is transferred to the modified Zyoton and then to the renormalized Zyoton. In subsequent collision iterations, the energy of the feature is transitively transferred from a renormalized Zyoton generated in a previous iteration to a renormalized Zyoton generated in the current iteration.

[0363] In effect, the collision computing process examines how the energy from the incoming, uncharacterized feature interacts with the properties of the stable Zyoton waveform to produce a post-collision waveform with the analyte information content of the original feature. Depending on how the conditioned feature waveform itself has been influenced by the underlying analyte concentration, and by confounders and the media, its interaction with the Zyoton can be different. As such, a Zyoton can be described as a nonlinear amplifier system and a feature as a perturbation. The repeated collision process is thus a protocol for characterizing analyte properties and estimating concentration in the uncharacterized samples.

[0364] The absorbed spectral energy, represented as a net gain in the energy of a modified Zyoton relative to the energy of the original Zyoton, is transformed into an analyte concentration using a projection operator. Projection is a generalized process of representing the average estimated post-collision spectral energy absorption in terms of net, renormalized spectral energy changes corresponding to an uncharacterized sample, and associating these net energy changes with those corresponding to known, analyte levels (represented in a “reference system”), as described below in detail. The reference system may or may not have any physical relationship with the samples being analyzed. For example, in non-invasive glucose measurement, as described below, the reference system may be a collection of tissue simulating dispersive liquid or gel samples (also referred to as “phantoms” or “tissue phantoms”), and containing specified glucose concentrations. The projection operator can relate the observed spectral energy absorbed in spectra from uncharacterized human subjects to known concentrations of glucose in an Intralipid fat emulsion or gel-based tissue phantom system. To this end, calibration curves can be used to relate the two disparate systems. As a generalized mechanism, projection can be accomplished by mapping or transforming the computed post-collision spectral energy absorption to an absolute reference system of known concentrations. Projection may optionally involve transforming the estimated spectral energy change (gain or loss) through one or more intermediate systems through the use of calibration curves prior to projecting the spectral energy change onto the reference system.

[0365] With reference to FIG. 15, in various embodiments, collision computing is performed for selected features, sequentially or in parallel. Features may be conditioned and introduced into the collision computer as one-dimensional vector data objects including one or more numerical vector elements. Prior to introduction into the collision computer at step 4, the complete spectrum (or other data set) may be preprocessed and standardized. The features are transformed at step 6 into a spectral waveform using a form of frequency modulation called the conditioning process.

[0366] An optional preprocessing performed on feature data involves numerical scaling of the feature data amplitude vector with respect to another reference data vector. The reference data vector can be derived from an in-line, in-measurement property used to compensate for variability in the measurement. This may be done in systems where there is high measurement-to-measurement variability during the acquisition of feature data (such as when the acquisition involves the detection of NIR light diffusely reflected from tissue). During preprocessing, the raw sensor data can be transformed into a data representation that is appropriate for the follow-on conditioning and collision computing processes. As an example, the acquired raw sensor data could be in the form of discrete counts, current, voltage or some form of flux intensity, with values ranging from a minimum to a maximum as defined by the sensor's dynamic range. The sensor or data acquisition platform may be optionally equipped with a hardware reference channel that can simultaneously collect raw reference data, also in the form of discrete counts, current, voltage or some form of flux intensity under a fixed condition (such as dark current or when there is no sample exposed to the sensor). The preprocessing step can combine sensor data and its reference counterpart to create a data object (“absorbance” or “pseudo-absorbance,” as described below) that can be used in subsequent computation. One example is the preprocessing of diffusely reflected spectral intensity amplitudes by combining them with reference amplitudes to transform the intensity data into absorbance or transmittance representations in spectroscopic analysis.

[0367] The standardization step can involve computations that are implemented to further transform the preprocessed data to remove the effects of sensor to sensor variability, variability due to data formats and variability due to hardware changes or degradation. Standardization generally helps to ensure syntactic integrity and can enhance semantic equivalence of the data, irrespective of differences in the hardware that was used to collect it. Both preprocessing and modulation are described in detail below.

[0368] An optional frequency modulation may be performed as part of the sensor data conditioning process at step 6. This frequency modulation step, referred to as an optional “precursor frequency modulation,” has a different purpose compared to preprocessing and standardization. In dealing with sensor measurements to observe materials where interference from clutter is high (SCR<<1), frequency modulation may be used as a mechanism to expand the observability state space, i.e., observable states from which the properties of the material (e.g., an analyte) can be classified, quantified, or inferred. By coupling the sensor data features or preprocessed, standardized sensor data features with carriers of different frequencies in a precursor modulation, the state space over which the data are analyzed can be transformed so as to increase the probability of detection of the amount, event, material or property of material of interest. Unlike the conventional frequency modulation in signal processing where a carrier wave is used to encode the signal or information by varying the instantaneous frequency of the wave, the precursor modulation can increase the observability space of events and materials.

[0369] The conditioned feature is then introduced into the collision computer at step 4, along with a complementary entity—the collision waveform or Zyoton, generated at step 8 using the criteria described above. The synthesis of the Zyoton and the carrier kernel used to generate the conditioned feature is also described below. Within the collision computer, the conditioned feature waveform and Zyoton waveforms are represented as waveforms propagating toward each other on a collision grid. During the collision at step 4, the conditioned feature wavefront collides and couples with the Zyoton wavefront, thereby changing the properties of the Zyoton and producing a resulting waveform called a modified Zyoton. The Zyoton and the feature waveforms are fed into the collision computer that performs bracketed interactions between the two propagating waveforms and, in particular, the colliding wavefronts, producing the new post-collision waveform i.e., the modified Zyoton waveform. Unlike conventional modulation techniques, which typically involve multiplication and/or convolution operations, the bracketed interaction operator involves conditional operations and is non-invertible.

[0370] The amplitudes of the Zyoton and/or the conditioned feature may be optionally modified prior to the collision thereof, e.g., to ensure the co-dependency conditions described above, and the Zyoton and/or conditioned feature components may be optionally shifted and aligned prior to the collision thereof. Phase modulation may be applied to the modified Zyoton. The collision operation is controlled, as described below, such that a measureable change occurs in a property (e.g., spectral energy) of the modified Zyoton relative to the corresponding property of the original Zyoton, and the change is attributable to the corresponding property of the feature, as represented by the conditioned feature. To this end, various collision operators such as shifting, scaling, phase rotation, number of iterations, etc., can be determined at step 12 based on the properties of the medium to be analyzed, the analyte, the sensor, and the computing system.

[0371] In various embodiments, the conditioned feature waveform may be unused after the first collision at step 10. A second collision may occur, at step 4, between the post-collision modified waveform following renormalization thereof (called the renormalized Zyoton), and the original Zyoton or a different Zyoton. The different Zyoton, in general, is selected such that it is co-dependent, as described above, with the renormalized Zyoton. Similarly, additional collisions can be performed between the waveform resulting from the previous collision, after renormalization thereof, and the original Zyoton or a different Zyoton.

[0372] In generalized collisions, one of three additional collision modalities can be implemented: (i) successive collisions between the original conditioned feature and the post-collision modified and renormalized Zyoton waveform from the preceding collision; (ii) successive collisions between the original conditioned feature with a new Zyoton waveform; and (iii) successive collisions between the post-collision modified and renormalized waveform from the previous collision and a new Zyoton waveform. Whenever successive collisions are described, a renormalization process described below is generally included as an integral part of the iterative collision process. The renormalization, in general, includes energy scaling and/or removal of one or more frequency components, and redistribution of the energy of the removed component(s). When a modified Zyoton is renormalized and collided with the original Zyoton in a subsequent iteration, the energy scaling that is applied can be different from the energy scaling applied when the modified Zyoton is renormalized and collided with the original conditioned feature.

[0373] The measurable changes in post-collision waveform properties relative to the corresponding property of the original Zyoton, e.g., the net gain in spectral energy of the coupled waveform (i.e., the modified Zyoton and/or the renormalized Zyoton), as computed using the difference between the spectral energy of the Zyoton waveform and post-collision modified and renormalized Zyoton waveform following one or more collisions, can be used to estimate the presence and concentration of an analyte in a medium.

[0374] An example illustrating the utility of this technique is the non-invasive estimation of glucose (an analyte of interest) in human skin tissue, using optical diffuse reflectance spectroscopy, whereby the concentration of glucose can be determined based on the measurement of selective spectral absorption of energy when skin is illuminated with near-infrared radiation. The analyte concentration can be determined by computing the spectral energy distribution of the post-collision Zyoton waveforms after all collision iterations are completed. The spectral energy gain between the energies of the modified Zyoton obtained after the last iteration and the original Zyoton, as represented by the spectral energy of the last renormalized Zyoton, is related to the feature properties as represented by the conditioned feature waveform properties. The collision process is used to estimate how much of the loss of energy of the incident NIR in the original feature waveform is due to the presence and concentration of the analyte. In a properly selected and designed Zyoton, carrier kernel, and collision operation, an ultra-small net energy loss due to absorption by the analyte can be detected and amplified on a dynamic range that extends to as much as fourteen orders of magnitude, using Zyotons with frequency bandwidths in the MHz region.

[0375] One embodiment, used for a non-invasive analyte measurement, employs 20,000 repeated collisions to estimate the net energy gain due to absorption at the feature level and to overcome the high level of interference from confounders. In some embodiments, the collision operator may be modified at step 16, if required, and then the result is re-introduced into the collision computer at step 4, for the next collision at step 10. In some embodiments, the collision operator is not modified between one or more or all of the subsequent collision iterations. The process is repeated until the required number of collisions are completed, as determined at step 14, and the results may be output (reported) and/or may flow at step 20 to the next step for estimating the net energy loss due to absorption.

[0376] With reference to FIG. 16, the spectral energy of a post-collision result (e.g., a Zyoton perturbed by a conditioned feature) is computed at step 6 to estimate the change due to the presence and concentration of material exhibited in the spatial, temporal, or spectral window defined by the feature. For systems with several collision iterations, the output after each collision is generally renormalized in steps 8, 10, and 12 before a subsequent collision can occur.

[0377] This renormalization is performed for two reasons: (i) removal of the energy of the original Zyoton, and (ii) redistribution of energy lost by a truncation operation. All collisions following the first collision may be between a Zyoton and a renormalized, modified Zyoton resulting from the preceding collision interaction, so the original spectral energy of the Zyoton needs to be removed from the modified Zyoton prior to the next collision by properly scaling the frequency domain amplitudes of the analyte-information representing (e.g., the first k) frequency components.

[0378] In statistical field theory and quantum field theory, the pileup of contributions from a large number of measurement scales involved in a measurement problem can lead to intractability of measurements. A renormalization of the measurement scales can address the intractability associated with a large number of measurement dimensions in problem solving, as described in K. G. Wilson, “The renormalization group: critical phenomena and the Kondo problem,” Reviews in Modern Physics, 47, 4, 773 (1975). The collision computing process does not generally involve analysis of direct measurements of an analyte at different measurement scales and, as such, unlike the renormalization employed in statistical fields theory and quantum fields theory, the renormalization described herein is directed to readjustment of the observable, e.g., the spectral energy, across successive iterations.

[0379] In general, collision computing also transforms signal detection and quantitation to a spatio-temporal (space-time) framework, which allows waveform interactions which may not be fully accounted for or modeled, and may include approximations used for numerical stability in computation. The term framework is used here as the space-time scale for implementing collisions and may not be related to the space-time dimension associated with acquiring the data. The framework can be a computational framework, or it may be related to the apparatus for implementing a collision computer such as a digital computer, graphic processing unit processor, or optical computer, or any such examples implemented in firmware.

[0380] The result of every collision interaction is baselined observable energy scalar i.e., the spectral energy gain of the modified Zyoton relative to the spectral energy of the original Zyoton as estimated from a fixed subset of frequency components. The spectral energy of the original Zyoton selected from a database 1, and used in the collision, is determined at step 2. That spectral energy is used in step 8 to determine a normalization coefficient, so that the energy of the original Zyoton can be removed from the modified Zyoton at step 10.

[0381] As described above, the second objective of the renormalization is truncation of the modified Zyoton, so that the renormalized Zyoton can be effectively collided again with the original Zyoton, or with another Zyoton. New frequency components can be created in the transition and non-analyte information representing bands during a collision iteration. The introduction of these components may cause a change in dispersion velocity and/or divergence, so that the co-dependency condition may not be satisfied for the next collision iteration. As such, one or more of these components may be removed during truncation of the modified Zyoton. In general, subportion(s) of the non-analyte information representing and/or transition portions (these terms are described below) of the modified Zyoton are removed, and the energy thereof is redistributed among the surviving subportions. This energy redistribution can be achieved via amplitude balancing at step 12. If renormalization is implemented in the frequency domain, the amplitude balancing can compensate for the impact of any removal of one or more frequency components of any entity participating in a collision.

[0382] The truncation generally results in a loss of spectral energy of the modified Zyoton. The energies of the removed frequency components of the modified Zyoton are computed and redistributed across the surviving frequency components of the modified Zyoton. This is implemented using an amplitude balancing operation at step 8. Specifically, in the frequency domain, the amplitude re-balancing (or re-distribution) operation entails distribution of the removed spectral energy contribution from the removed frequency components over the remaining m and j frequency components of the modified Zyoton. Thus, the frequency domain amplitudes of the surviving m and j components are adjusted at step 8 during the renormalization process.

[0383] Alternatively or in addition, renormalization may further include an additional truncation of the modified Zyoton to select the length thereof. To this end, relatively smaller-amplitude frequency components of the modified Zyoton may be removed to ensure that the frequency-domain length of the renormalized Zyoton waveform matches the frequency-domain length of the original Zyoton and/or a new Zyoton with which the renormalized Zyoton is to be collided. This truncation may also be followed by an additional amplitude balancing operation in step 12, in which the amplitudes of all the remaining m and j frequency components of the truncated modified Zyoton are adjusted to compensate for the energy of the frequency components removed in the additional or alternative truncation process. This renormalized post-collision Zyoton waveform may then be reused in the next collision iteration as described with reference to FIG. 15. Renormalization can be implemented in the time domain by modifying the time-domain velocity and/or dispersion velocity of the modified Zyoton.

[0384] In general, the renormalization of the post-collision waveform of the modified Zyoton allows removal therefrom of the energy contained in the original Zyoton after each collision iteration so that energy changes can be successfully accumulated in a multi-collision protocol for quantifying analyte presence and concentration. In some embodiments, two renormalization parameters are derived using spectra generated from a model of waveform propagation over a collision grid. The first renormalization parameter includes a set of numbers associated with the respective amplitudes of every frequency-domain component of the waveform, matched to the length of the truncated (e.g. down-sampled) result after collision. The first renormalization parameter modifies the frequency domain amplitudes of the surviving frequency components, such that the spectral energy of all frequency components of the modified Zyoton, ignoring the first k analyte-information representing components, is the same as the spectral energy of all but the first k frequency components of the conditioned feature used in the collision. This can be tested by comparing the spectral energy of all but the first k components of the renormalized Zyoton with the spectral energy of all but the first k components of the conditioned feature.

[0385] The renormalization process also computes a second parameter, i.e., a renormalizing scaling factor, which changes the frequency domain amplitudes of the first k components to provide the next waveform for the next collision. The renormalizing scaling factor is designed such that, the frequency domain amplitudes of the first k frequency components of the renormalized modified Zyoton are within 0.1%, 1%, 5%, 10%, 20%, 30%, or 50%, 100%, etc., of the frequency domain amplitudes of the first k frequency components of the conditioned feature. As a result, the energy of the renormalized modified Zyoton becomes substantially similar to that of the conditioned feature just prior to collision and is within 0.1%, 0.4%, 1%, 2%, 5%, or 10%, etc. of the energy of the conditioned feature. This can also ensure that the Kappa test of co-dependency condition is satisfied for the next collision iteration. In some embodiments, this computation is done up to the fifteenth decimal digit for high precision and to accommodate SCR below 0.0001. In some embodiments that employ only a single iteration of a collision operation, if renormalization is performed, the frequency domain amplitudes of the first k frequency components of the renormalized modified Zyoton are adjusted such that the resulting energy of these k components can be up to two or three orders of magnitude of the energy of the corresponding frequency components of the conditioned feature to expand the dynamic range of energy change. Applications where only one collision iteration is used in conjunction with renormalization include classification of analyte levels into 3, 5, 7, 12 etc., bands or regions.

[0386] The renormalization process allows accumulation of energy changes generally only due to absorption of energy by the analyte in each feature. The purpose is to transform all the energy changes to a relative scale and dynamic range which can then be projected onto analyte concentrations using a dataset that provides concentration-specific references. FIG. 17 shows the morphological envelope of an exemplary Zyoton and conditioned feature used in collision computing. FIG. 18 shows the Zyoton and the conditioned feature waveform on a “spatio-temporal” grid and the Zyoton waveform before a collision. FIG. 19 illustrates the progression of a collision between the conditioned feature and the Zyoton wavefronts during a collision. FIG. 20 shows the modified Zyoton after the collision. FIG. 21 shows the modified and renormalized Zyoton after the collision and preceding the next collision in a multi-iteration collision protocol. FIG. 22 shows the modified and renormalized Zyoton after a finite number of collisions in a multi-iteration collision protocol. FIG. 23 shows a typical energy gain trajectory due to the net spectral energy change resulting from analyte-specific absorption evolving over several collision iterations in a collision computer for two features drawn from samples with two different material concentrations (for example, tissue glucose). The two curves show the amplification resulting from collision computing.

[0387] With reference to FIGS. 24-27, the collision and renormalization processes are illustrated schematically in the time domain. FIG. 24 illustrates a collision between a Zyoton Z_i 1 schematically represented by a bell-shaped envelope and a conditioned feature γ(FWi) 8. In the time domain, a Zyoton typically includes several peaks such as those shown in FIG. 12A, which, together, can be represented schematically by an envelope such as that shown in FIG. 24. Each Zyoton peak itself can be a schematic envelope, i.e., each peak can be a group of one or more peaks, as shown in FIG. 12B. The conditioned feature may initially have length ′ in the time domain, while the zyoton has length in the time domain. The length of the conditioned feature 6 is adjusted, e.g., by interpolation, down-sampling, and/or truncation, to match the length of the Zyoton to allow the collision therebetween. The result of the first collision is shown in FIG. 25 as a modified Zyoton envelope 14, Ω(γ(FW_i),Z_i)t₁. The energy of the modified Zyoton is E₁, the energy of the original Zyoton is E₀, and the energy gain (E₁−E₀) or loss, where (E₁−E₀) is less than zero, is proportional to ΔE, i.e., is the energy of the analyte-information representing portion of the conditioned feature FWi. FIG. 25 shows that the modified Zyoton with energy E₁ still has length . In some instances, however, the length of the modified Zyoton can be different from , e.g., the length can be ″.

[0388] In FIG. 26, the modified Zyoton Ω(γ(FW_i),Z_i)t₁ is renormalized to a renormalized Zyoton N(Ω₁), 20. The renormalization includes removal of the energy of the original Zyoton and redistribution of the portion of the modified Zyoton removed by truncation. The energy of the analyte-information representing portion of the renormalized Zyoton is ΔE₁. The renormalized Zyoton N(Ω₁), 20 is collided again with the original Zyoton Z_i, to yield the new modified Zyoton shown in FIG. 27 as Ω(N(Ω_i),Zi)t₂, 22, having energy E₂. The energy gain of the new modified Zyoton relative to the original Zyoton, i.e. (E₂−E₀) is proportional to the energy ΔE₁ of the analyte-information representing portion of the renormalized Zyoton N(Ω₁), which is proportional to the energy ΔE of the analyte-information representing portion of the conditioned feature.

[0389] In each subsequent collision iteration, the energy gain of the modified Zyoton generated in that iteration relative to the energy of the original Zyoton, represented by (E_n−E₀) for the n-th collision iteration, is proportional to the energy of the analyte-information representing portion of the renormalized Zyoton generated in the previous collision iteration, i.e., ΔE_i, which itself is transitively proportional to the energy ΔE of the analyte-information representing portion of the conditioned feature.

[0390] In FIG. 9, E_I represents the spectral energy of the radiation incident upon a medium, in a wavelength band [λ¹₁, λ¹₂] For simplicity, E_I is assumed to be constant across all wavelength bands and, as such, the spectral energy of the incident radiation (e.g., in the near infra-red region of the electromagnetic spectrum) in another wavelength band [λ²₁, λ²₂] is also E_I. In the medium (e.g., a portion of skin), a portion of the incident radiation is absorbed by one or more confounders. That portion can be less than 1%, 2%, 5%, 10%, etc. A portion of the incident radiation, e.g., less than 0.1%, 0.2%, 0.6%, 1%, 1.5%, 3%, 7%, etc., may be absorbed by an analyte of interest, and is denoted by E_A. In some instances (e.g., in tissue measurement), an additional portion of the radiation may be scattered within the tissue and lost, and is considered to be the lost portion E_L. A portion of the incident radiation is not absorbed by the analyte, any confounders, or otherwise lost (e.g., due to scattering) present in the medium, and is reflected from (or passes through) the medium. The reflected radiation can be detected by a detector.

[0391] The analyte of interest may absorb radiation at some wavelengths and may not absorb the radiation, at least at a significant level, at some other wavelengths. For example, in FIG. 9 it is assumed that the analyte absorbs radiation in the wavelength band [λ¹₁, λ¹₂], designated as Analyte Feature F1, and that the analyte does not substantially absorb radiation in the wavelength band [λ²₁, λ²₂], designated as Non-Analyte Feature F2. For the Analyte Feature F1, the energy absorbed by the analyte is E_A1, the energy absorbed by one or more confounders is E_C1, energy lost (i.e., absorbed by elements of the medium other than the confounders and the analyte) is E_L1, and energy reflected by the medium and detected by the detector (denoted E_O1) is E_R1. For the Non-Analyte Feature F2, the energy absorbed by the analyte is E_A2 (which could be a negligible amount), the energy absorbed by one or more confounders is E_C2, energy lost is E_L2, and the energy reflected by the medium and detected by the detector (denoted E_O2) is E_R2. Therefore:
ΔE₁=E_I−E_O1=E_L1+E_C1+E_A1;
ΔE₂=E_I−E_O2=E_L2+E_C2+E_A2=E_L2+E_C2, because E_A2≃0; and
ΔE₁−ΔE₂=(E_L1−E_L2)+(E_C1−E_C2)+E_A1
In the equations above, E_I is known, and E_O values can be obtained by direct measurement from the detector. E_L, E_C, and E_A, are unknowns for each feature.

[0392] The spectral energy of feature F1 is represented by ΔE₁=(E_L1+E_C1+E_A1). The spectral energy of a conditioned feature is also represented by ΔE₁=+E_C1+E_A1). The Zyoton and the collision operator are designed such that after the first collision between the Zyoton and conditioned feature, the energy of the modified Zyoton (E_Z¹) changes relative to the energy of the original Zyoton (E_Z⁰), but the change corresponds, in substance, only to E_C1 or E_A1, and not as much to E_L1. In this way, E_C1 and/or E_A1 can be extracted, even though E_L1 is unknown.

[0393] In both ΔE₁ and ΔE₂, typically, E_L>>E_C>E_A. As such, if ΔE₁ and ΔE₂ are amplified using a conventional nonlinear amplifier, in the difference between amplified ΔE₁ and amplified ΔE₂, the difference between E_L would dominate. Collision computing can selectively amplify E_C1 and/or E_A1 and not E_L, so that the difference between ΔE₁ and ΔE₂ can be used to determine the energy absorbed by the analyte, which in turn can be used to determine accurately the analyte concentration.

[0394] Referring to FIG. 10A, in one embodiment, radiation of energy E₁ is directed to a medium and the radiation reflected by the medium is detected at a detector. An absorbance spectrum “O,” representing E_O across a range of wavelengths, is derived from the detected radiation. Waveforms L, C¹, C², and A represent, respectively, the absorbance spectra corresponding to the radiation lost (representing E_L), the radiation absorbed by a confounder C¹ (representing E_C¹), the radiation absorbed by a confounder C² (representing E_C²), and the radiation absorbed by the analyte A (representing E_A). In general, the properties of the medium, which may determine the radiation that is lost, the concentrations of the confounders C¹ and C², and the concentration of the analyte are not known. As such, the absorbance spectra L, C¹, C², and A are also not known. These absorbance spectra, however, are the components of the absorbance spectrum O, which can be derived from the radiation detected at the detector. In FIG. 10A, spectra corresponding to two confounders are shown for illustration. In general, a medium may include no confounders, only one confounder, or more than two (e.g., 3, 5, 8, etc.) confounders.

[0395] The observed spectrum O is divided into several (e.g., 2, 4, 5, 10, 15, 18, 24, 32, 50, etc.) regions called features. With reference to FIG. 10B, a feature F₁ includes a region O₁ of the observed absorbance spectrum O. Feature 1 also includes regions L₁, C¹₁, C²₁, and A₁ of the absorbance spectra L, C¹, C², and A, respectively. Here again, the spectrum O₁ includes the spectra L₁, C¹₁, C²₁, and A₁, but these individual spectra are not known. The spectrum L₁ represents E_L1; spectrum C¹₁ represents E_C11; spectrum C²₁ represents E_C21; and spectrum A₁ represents E_A1. FIG. 9 also depicts the energy absorbed by one or more confounders as E_C1. In general, for N confounders, the absorption is additive and can be expressed as E_C1=Σ_i=1^N E_Ci1, where N≧0. This additive absorption is non-invertible and components E_Ci1 generally cannot be accurately recovered using a linear transformation, spectral deconvolution, or other passive signal processing technique.

[0396] Although the actual spectra L₁, C¹₁, C²₁, and A₁ are not known, from spectroscopic analysis of the analyte alone (in the solid or liquid phase or in solution), also referred to as a “pure component spectrum,” certain parameters of features from these spectra, called spectral parameters (such as the amplitude envelope or spatial frequency distribution obtained by computing a Fourier transform of a feature vector), can be determined. Pure component spectra represent the absorbance-wavelength relationship for the analyte of interest and/or for one or more confounders.

[0397] A carrier kernel, with at least an order of magnitude higher spatial frequency compared to the frequencies associated with Fourier transformed feature F1, is modulated using the waveform O₁ that is called a feature waveform and that represents the feature F₁. As an example, a system with an expected concentration range of about three orders of magnitude could require a five-to-six order of magnitude range in post-collision spectral energies, and a carrier kernel with a 1 MHz to 100 MHz bandwidth would generally be suitable. The various spectral parameters are used to select one or more parameters of the carrier kernel and/or one or more parameters of the modulation. The desired target dynamic range of post-collision spectral energies and precision desired for the measurement determine the fundamental frequencies and number of frequency components of the carrier kernel, collectively called modulation parameters. The modulation of the carrier kernel by the feature waveform O₁ yields a modulated waveform called a conditioned feature.

[0398] In general, a conditioned feature includes several components, each of which is associated with a different spatial frequency. The modulation parameters are selected such that different frequency component combinations of the conditioned feature represent different spectral energies associated with the feature. For example, with reference to FIG. 10C, in the conditioned feature F₁, some frequency components represent E_C21, i.e., the energy absorbed by the confounder C² at the wavelengths defining the feature F₁. Similarly, some frequency components represent E_C11; some frequency components represent E_A1; and some frequency components represent E_L1.

[0399] With reference to FIG. 10D, in some instances, the frequency components representing the energy absorbed by the analyte and the energy components representing the energy absorbed by one or more confounders (e.g., confounders 1 through N1 of a total of N confounders, where N1≦N), overlap. Some other frequency components, such as E_C^Ni+1, represent the energy absorbed by the confounders (N1+1) through N, and yet other frequency components represent the lost energy. With reference to FIG. 10E, the frequency components (some of which are fundamental frequencies and some of which are harmonics of those fundamental frequencies) representing the lost energy may overlap with the frequency components representing the energy absorbed by one or more confounders, e.g., confounders N2 through N of a total of N confounders, where N2≧1. The term frequency components is used here to describe the sinusoidal spatial frequency components that make up a waveform. Such waveforms result from “adding” together the series of sine wave frequencies that would result from a Fourier transform of the waveform. Thus, whatever its shape, a complex waveform can be split up into its individual spatial frequency components.

[0400] The representation of a particular energy (e.g., E_C11, E_A1, a combination thereof, or E_L1), is a function of: (i) the number of frequency components representing that particular energy in the modulated waveform (i.e., the conditioned feature); (ii) the frequencies of those frequency components; and (iii) the amplitudes of those frequency components. In general, for the j-th feature (denoted F_j), the relationship between the frequency components and their respective contribution to an energy such as E_Cij, E_Aj, E_Lj, where i denotes the i-th confounder, is not known. If it were known, a simple Fourier analysis of the conditioned feature would suffice to reveal the various energies of interest including E_Aj.

[0401] The parameters of a Zyoton with which the conditioned feature is collided and the parameters of the collision operator are also selected using the spectral parameters, i.e., the parameters derived from pure component analysis based on published or experimentally measured pure component spectra. If published spectra are used, knowledge about the noise of the measurement system and the sources of the noise can be used. The frequency bandwidth and amplitude envelope of the b Zyoton sideband peaks shown in FIG. 12C depend on the morphological profile (e.g., the frequency spectrum of the second derivative of the waveform) of the pure component absorption spectrum waveform features. FIGS. 28A and 28B show the pure component absorbance spectra for the analyte and one confounder of interest: glucose and collagen, respectively. FIGS. 29A and 29B show the first derivative of pure component absorbance spectra for the analyte and one confounder of interest: glucose and collagen respectively. FIGS. 30A and 30B show the second derivative of pure component absorbance spectra for the analyte and one confounder of interest: glucose and collagen respectively, computed from the result of FIGS. 29A and 29B.

[0402] As can be seen from FIGS. 30A and 30B, the two second derivative graphs for the two compounds have different characteristics. The frequency components and bandwidth of the pure component spectra and/or the first, second, or higher-order derivatives thereof, obtained by taking a Fourier transform of the selected waveform and by determining the range of the frequency components in the Fourier transform, can be used in determining the Zyoton frequencies and carrier kernel frequencies for conditioning the feature data, as described below.

[0403] In the example of glucose, the second derivative of a spectrum such as FIG. 30A, after optional interpolation by an amount of about 2¹⁰ to 2¹⁶, is added to random noise with a root mean square (rms) amplitude of the random noise set to be 1000 times the rms amplitude of the second derivative, to yield a combined second derivative waveform. In one embodiment, the random white noise used is a colored grey noise with a C-weighting and correlation time of 10⁻⁶ seconds, but other weightings and/or correlation times could also be used. In the preceding condition, the rms amplitude of the second derivative of the analyte is calculated by taking the square root of the sum of the squares of the instantaneous amplitude values of the second derivative (amplitude as a function of wavelength) over a window size that is twice the length of a feature in the time domain.

[0404] The frequency bandwidth of the resulting combined waveform (noise+the second derivative of the interpolated data vector) and the relationship among the frequency component amplitudes over the wavelength region corresponding to the features are used to establish the bandwidth and frequency components of the carrier kernel, as described below. Thus, the Zyoton and carrier kernel frequencies are related to the frequency components of the analyte pure component spectrum or the first, second, or higher-order derivatives thereof, with added noise.

[0405] Specifically, as Zyotons are derived from inherently stable waveforms, the chromatic dispersion of the Zyoton waveform, the peak power, and the pulse energy in its first k-peak-power sorted frequency components (where k may be ≦6), may collectively control the limit of detection and quantitation that can be achieved in one or more collisions. Therefore, in some embodiments, Zyoton waveforms are generated that are soliton based and show analyte-specific absorption effects more strongly in changes in the spectral energy of the first k frequency components and scattering effects more strongly in changes in the spectral energy of the j frequency components. As mentioned previously, the total set of frequency components for the Zyoton is defined as H=k+m+j where k>0, m≧0 and j>0, and the k and j frequency components, i.e., the analyte-information representing and non-analyte information representing components do not overlap.

[0406] Once the number of frequency components that can yield an estimate for analyte-specific absorption with the desired performance parameters (e.g., accuracy, precision, dynamic range) is established (e.g., 6 in one embodiment for non-invasive glucose measurement), the parameters of the collision grid, e.g., the bracket length in terms of numbers of frequency components of both the conditioned feature or renormalized Zyoton, as described below, grid spacing for propagation of the Zyoton waveform, a window length, etc., are computed. In various embodiments, the window length specifies a total number of points (in the time domain) or a total number of components (in the frequency domain) of the original Zyoton, conditioned feature, and/or renormalized Zyoton that are used in the collision operation. Examples of these parameters for a noninvasive glucose measurement are a window length of 2048 and a grid spacing of 10⁻⁶ seconds. The grid spacing can be additionally based on characteristic of the computing system, e.g., available numerical range and precision, available processing capacity and time, etc.

[0407] The window length for the collisions is generally set such that at least those frequency components of a conditioned feature (and a renormalized Zyoton) that are used in the computation of spectral energy change due to analyte-specific collision interactions can be propagated. Changes in the spectral energy of acquired features due to scattering losses may be intentionally filtered out from the post-collision spectral energy computations through the use and choice of the window length that is smaller than the length of all of the data object containing the frequency components of the Zyoton waveform.

[0408] Limiting the collision window length can thus reject incorporation of scattering-related spectral energy changes by filtering out the non-analyte information representing spatial frequency components (e.g., L>2048, where L<H=k+m+j) present in the conditioned feature which generally represent the frequencies related to scattering events. The length of a resulting modified Zyoton vector after a collision is typically greater than L (in the frequency domain) and greater than (in the time domain, as described with reference to FIGS. 24-27), and may contain additional frequency components as a result of the conditional bracketed interactions described below. These additional components may be removed by truncation during renormalization. Each collision interaction in one collision iteration, as described below, may correspond to a point on the collision grid and can describe at least in part the propagation of the two colliding waveforms. In the time domain or in the frequency domain, the length of the collision grid can take on values such as 10, 100, 1000, or up to 1,000,000 grid points or more.

[0409] Thus, the original Zyoton and the collision operator are synthesized such that the first k frequency components of the modified Zyoton resulting from the collision between the original Zyoton and the conditioned feature can represent a feature energy component to be quantitated. For example, referring to FIG. 9, in one instance the first k frequency components may represent E_A1, i.e., the energy absorbed by the analyte at the wavelengths associated with Feature F1. In another instance, the first k frequency components may represent, E_C11, i.e., the energy absorbed by the confounder C¹ at the wavelengths associated with the Feature F1. The first k frequency components may also represent the total energy absorbed by two or more, or all confounders, e.g., E_C1. In some instances, the first k frequency components of the conditioned feature may represent essentially the total energy absorbed by the analyte and by one or more confounders, e.g., (E_A1+E_C1). In general, with a properly designed Zyoton and collision operator, the first k frequency components do not represent the lost energy E_L1.

[0410] In a collision computer, in the first iteration, the original Zyoton and the conditioned feature, each of which represents a traveling waveform in the computer memory, and are synthesized as described below, are computationally collided. This collision is “nearly elastic.” In a perfectly elastic collision between a Zyoton and another waveform, the dispersion velocity of the Zyoton after the collision would be the same as the dispersion velocity of the Zyoton before the collision. This would typically occur only when a Zyoton collides with another soliton waveform.

[0411] In a generally inelastic collision, the post-collision dispersion velocity of a Zyoton colliding with another waveform would be significantly different (e.g., greater than 1%, 2%, 5%, 10%, 20%, etc.) than the dispersion velocity of the original Zyoton, and the Zyoton may be destroyed during the collision. In the nearly elastic collision described herein, the colliding waveforms, i.e., a Zyoton, and a conditioned feature or a renormalized Zyoton colliding with the Zyoton, are constructed such that during the collision the conditioned feature or the renormalized Zyoton perturb the Zyoton only within specified limits. These limits are generally established by the limits on a permissible change in dispersion velocity, as specified by one or more Kappa parameters and/or by a permissible change in waveform divergence, as specified by the divergence parameter τ. In general, the difference between the velocity of the original Zyoton and the velocity of the modified Zyoton resulting from the collision is limited to a specified threshold numerical limit, which can be tested using a suitable Kappa test as described above. In various embodiments, to ensure a near-elastic collision, the collision operator is constructed such that the divergence of the post-collision waveform, i.e., the modified Zyoton, is not greater than a pre-set divergence parameter τ, as described in detail below.

[0412] As such, in various embodiments, the difference between the pre- and the post-collision dispersion velocities, i.e., a scaled velocity of the original Zyoton and the velocity of the renormalized Zyoton do not exceed a specified numeric threshold, denoted κ_DV1. A similar constraint can also be applied to the difference between the scaled dispersion velocity of the original Zyoton and the velocity of a conditioned feature, where the threshold is designated κ_DV2. For noninvasive glucose measurement, where the spectral energy of the Zyoton waveform before scaling is about three orders of magnitude higher than the spectral energy of the conditioned feature waveform, an example of the threshold κ_DV2 is 1×10⁻⁶.

[0413] The frequency domain amplitudes of the k, m, and j frequency components of an original Zyoton are denoted A_k, A_m, and A_j, respectively. The frequency domain amplitudes of the corresponding k, m, and j frequency components of a co-dependent conditioned feature are denoted as a_k, a_m, and a_j, respectively. Once A_k are chosen, the corresponding a_k can be determined using a scaling coefficient α. The scaling coefficient α may be selected such that the two co-dependency conditions are satisfied. Specifically, the difference between the scaled velocity of the original Zyoton and the velocity of the conditioned feature do not exceed a specified threshold κ_DV2, and the divergence of the post-collision modified Zyoton do not exceed a specified threshold τ. The velocity condition can be tested by scaling the velocity of the conditioned feature instead of or in addition to scaling the velocity of the original Zyoton. A suitable Kappa threshold may be used accordingly.

[0414] In various embodiments, if the κ_DV2 test fails, the frequency domain amplitudes of the frequency components of the conditioned feature are adjusted by applying a scaling coefficient to the corresponding frequency domain amplitudes of the frequency components of the original Zyoton. Thus

[0415] $a_k=\frac{1}{\alpha }{A}_k;a_m=\frac{1}{\alpha }{A}_m;\mathrm{and}{a}_j=\frac{1}{\alpha }{A}_j,$
where A_k, A_m, and A_j are vectors.

[0416] The time-domain amplitudes of the Z peaks of the original Zyoton are denoted as and the time-domain amplitudes of the CF peaks of the conditioned feature are denoted as . The respective velocities of the original Zyotons and the conditioned feature are functions of and , respectively, denoted as ν() and ν(), respectively. Once a conditioned feature is generated, the velocity-based co-dependency condition can be tested in the time domain as: |α_C*ν()−ν(|≦κ_DV2, where α_C*ν() is the scaled velocity of the original Zyoton. In the time domain, the velocity of a waveform associated with the collision process (e.g., a Zyoton, a conditioned feature, a modified Zyoton, and/or a renormalized Zyoton) can be computed as L1 or L2 norm of the amplitudes of a selected number of sideband envelopes, such as those of a Zyoton that are shown in FIGS. 12A-12C. In the frequency domain, the velocity of a waveform can be represented by L1 or L2 norm of the frequency domain amplitudes of the first k, i.e., the analyte-information representing components of the waveform.

[0417] As discussed above, α_C is the nominal scaling coefficient applied to achieve a desired ratio (α_Z) of spectral energy of the Zyoton to the spectral energy of a conditioned feature computed using a fixed number (e.g., 6 in one embodiment) of k frequency components, by adjusting the amplitude of the frequency components of the Zyoton so that the spectral energy difference between the scaled Zyoton and the conditioned feature is less than κ_DV2. As noted above, depending on whether the testing and/or scaling is performed in the time domain or in the frequency domain, a domain-specific value of the applicable Kappa variable is used in various embodiments.

[0418] The term nominal scaling coefficient is used because different features from the same or different spectra, from the same or different samples, are likely to result in slightly different scaling coefficients to maintain a fixed relationship, α_Z, between the spectral energy of Zyoton and spectral energy of the conditioned feature prior to the first collision. The following condition is imposed on the nominal ratio: |(α_C−α_{Fi_MAX})/α_C|≦e.g., 0.1, 0.2, 0.5, or 1.0, and |(α_C−α_{Fi_MIN})/α_C|≦e.g., 0.1, 0.2, 0.5, or 1.0, where α_{Fi_MIN}, α_{Fi_MAX} respectively denote the maximum and minimum scaling coefficients required to achieve the desired ratio of the spectral energy of the conditioned feature to the spectral energy of the Zyoton, computed using a preset number of frequency components, for all conditioned features to be collided with the same Zyoton. In some embodiments, |(α_C−α_{Fi_MAX})/α_C|≦0.1 and (α_C−α_{Fi_MIN})/α_C|≦0.1.

[0419] It is to be understood that in this disclosure, the terms “strongly absorbing” and “weakly absorbing” spectral regions (or similar terms, such as “high absorbing” spectral regions and “low absorbing” spectral regions) as used in connection with the measurement of an analyte, refer only generally to absorption within the pure-component spectrum of a substance which may be either an analyte or a confounder. Thus, for a particular substance, the spectral regions which are strongly absorbing generally have a higher level of absorbance than the spectral regions that are weakly absorbing, but those terms do not imply any sort of strict mathematical relationship between the absorption values of the two regions, nor do they imply any relationship among the absorption values of two different substances. In some cases, a region identified as strongly absorbing in one context could be considered weakly absorbing in a different context and vice versa. For example, in a general broad region of low analyte absorption, some sub-regions may have higher absorption than some other sub-regions. Similarly, high analyte absorption can be significantly low, in absolute terms, relative to a low confounder absorption.

[0420] Similarly, the terms “analyte regions” or “analyte features” and “non-analyte regions” or “non-analyte features” as applied to spectral regions or features, refer only generally to variations in absorbance due to the analyte. Generally, the absorbance of an analyte in an analyte feature is greater than in non-analyte features, but in some cases, the absorbance of the analyte in a non-analyte feature or spectral region may be actually higher than the absorbance in an analyte feature or region. This situation can occur because the designations “analyte feature” and “non-analyte feature” may take into consideration the absorbance of one or more confounders, or even a dominant confounder. As noted above in describing “strongly absorbing” and “weakly absorbing” spectral regions, the terms “analyte regions” or “analyte features” (e.g., GL features, for glucose measurement, PPG features for PPG measurement, etc.) and “non-analyte regions” or “non-analyte features” (e.g., NO-GL features, NO-PPG features, etc.), as applied to spectral regions or features, do not imply any sort of strict mathematical relationship between the absorption values of the analyte in those regions or features. Rather, they serve to describe the manner in which the regions or features are used, as further described below.

[0421] The table below describes energies at various stages for a feature in a spectral region where the analyte, if present, is known to absorb strongly, using nominal example values to illustrate the process.

[0422]

TABLE 3
	Example
Description	Value	Symbol
Energy of original Zyoton	1000	E
Energy absorbed by an	1	Δe
analyte, as represented by
one feature
The goal is to obtain net energy gain of 1 after   collisions
Energy of conditioned	10	e; represents Δe, i.e., 1
feature
Energy of modified Zyoton	1000.10	{E, e} → E1; E1 depends on
after the first collision		e which is the energy of the
		conditioned feature.
		represents the
		collision operation.
		As the collision is inelastic,
		E1 < E + e
Energy of the renormalized	0.15	e1 =   (E1) ≈ (E1 − E) where
modified Zyoton after the		e1 is obtained by removing,
first collision		approximately, the energy (E) of
		the original Zyoton from the
		modified Zyoton.   represents
		an embodiment of the
		renormalization operation
		described herein.
		As the collision is inelastic E1 <
		E + e, e1 < e
Energy of modified Zyoton	1000.12	{E, e1} → E2, where   is
after the second collision		the collision operator
Energy of the renormalized	0.18	e2 =   (E2) ≈ (E2 − E), where
modified Zyoton after the		is the renormalization
second collision		operator
Energy of modified Zyoton	1000.67	{E, ek−1} → Ek
after k collisions
Energy of the renormalized	0.81	ek =   (Ek) ≈ (Ek − E)
modified Zyoton after k
collisions
. . .	. . .	. . .

[0423] The table below describes energies at various stages for a collision involving a feature in a spectral region where the analyte, if present, is known to absorb weakly. This feature could be from a spectral region where confounders show high absorption, using nominal example values to illustrate the process.

[0424]

TABLE 4
Description	Example Value	Symbol
Energy of original Zyoton	1000	E
Energy absorbed by feature	0.001	Δe
The goal is to obtain no net energy gain or a gain of ≦0.001 after   collisions
Energy of conditioned	10	e; represents Δe, i.e., 1
feature
Energy of modified Zyoton	1000.0004	{E, e} → E1; E1 depends on e
after the first collision		which is the energy of the
		conditioned feature.
		As the collision is inelastic,
		E1 < E + e
Energy of the renormalized	0.00006	e1 =   (E1) ≈ (E1 − E) where
modified Zyoton after the		e1 is obtained by removing,
first collision		approximately, the energy (E) of
		the original Zyoton from the
		modified Zyoton.
		As the collision is inelastic E1 <
		E + e, e1 < e
Energy of modified Zyoton	1000.0007	{E, e1} → E2
after the second collision
Energy of the renormalized	0.0009	e2 =   (E2) ≈ (E2 − E)
modified Zyoton after the
second collision
Energy of modified Zyoton	1000.0009	{E, ek−1} → Ek
after k collisions
Energy of the renormalized	0.002	ek =   (Ek) ≈ (Ek − E)
modified Zyoton after k
collisions
. . .	. . .	. . .
Energy of modified Zyoton	1000.004 or 1000.000081	{E,   } →
after   collisions	(could be greater or less than
	0.00001)
Energy of the renormalized	0.004 or 0.000032 (could be	=   (   ) ≈ (   − E)
modified Zyoton after	greater or less than 0.00001)	≈ Δe
collisions

[0425] In the same example, the “net renormalized spectral energy gain,” after collisions, for a feature pair (with one feature covering the analyte absorption spectral region and a second feature covering a spectral region where the analyte is known not to absorb strongly, but with strongly absorbing confounders), can be computed as Δe=1.060−0.004=1.056. Using calibration tables, this result value, i.e., 1.056, can be used to obtain the concentration of the analyte present in the sample as used in this example.

[0426] In the table and discussion below, upper case letters represent Zyoton amplitudes; lower case letters represent conditioned feature amplitudes. Plain symbols without any prime represent the original Zyoton; symbols with a single prime represent a modified Zyoton; and symbols with double prime represent a renormalized modified Zyoton. Symbols representing the k-th iteration, representing a collision between the original Zyoton and the renormalized Zyoton generated after the (n−1)-th collision, include superscript “(n).” For the sake of convenience, the superscript (1), indicating the modified and renormalized Zyotons generated after the first collision, is omitted. Ordinary, non-stylized letters represent frequency domain amplitudes; stylized letters represent time domain amplitudes. Thus:

[0427]

TABLE 5
Symbol	Meaning
Ak, Am, and Aj	Frequency domain amplitudes of the k, m, and
	j frequency components of the original Zyoton,
	respectively. In general, k, m, and j are greater
	than one and, as such, Ak, Am, and Aj are row
	or column vectors of k, m, and j elements,
	respectively.
ak, am, and aj	Frequency domain amplitudes of the k, m, and
	j frequency components of the conditioned
	feature, respectively
A′k, A′m′, and A′j′	Frequency domain amplitudes of the k, m, and
	j′ frequency components of the modified
	Zyoton after the first collision/iteration,
	respectively. Compared to the original Zyoton,
	the modified Zyoton may have different m and
	j frequency components, denoted m′ and j′
	frequency components
A″k, A″m, and A″j	Frequency domain amplitudes of the k, m, and
More specifically,	j frequency components of the renormalized
A″k = A″k(1); A″m = A″m(1); and A″j = A″j(1)	Zyoton after the first iteration or collision,
	respectively. Due to renormalization, the
	renormalized and original Zyotons have the
	same k, m, and j frequency components in
	terms of frequencies, though their respective
	frequency domain amplitudes are different.
	Typically, A″k is less than Ak by a factor of
	1,000.
A″k(2); A″m(2); and A″j(2)	Frequency domain amplitudes of the k, m, and
	j frequency components of the renormalized
	Zyoton after the second iteration or collision,
	respectively.
= [   , . . . ,   ]	Time domain amplitudes of lz peaks of the
	original Zyoton containing l peaks. Dispersion
	velocity of the original Zyoton is proportional
	to the time domain amplitude of one or more b
	sideband peaks (as exemplified in FIG. 12C)
	excluding the strongest peak of the Zyoton
	waveform, denoted   . For example
	bz = lz − 1.
= [   , . . . ,   ]	Time domain amplitudes of lCF peaks of the
	conditioned feature. Dispersion velocity of the
	conditioned feature is proportional to the time
	domain amplitude of one or more b sideband
	peaks (as exemplified in FIG. 12C excluding
	the strongest peak of the conditioned feature
	waveform, denoted   . For example,
	bCF = lz − 1.
	In the general case, it is assumed that prior to a
	collision, even though the number of space-
	time points in the Zyoton and the conditioned
	feature are the same, the number of peaks in
	the Zyoton (lz) can be different than the
	number of peaks in the conditioned feature
	(lCF).

[0428] In various embodiments, the nearly elastic collision described herein satisfies two conditions: (i) that the collision must not be perfectly elastic, and (ii) that the collision must not be generally inelastic. According to the first condition, the collision is not perfectly elastic so that the original Zyoton and the post-collision, modified Zyoton are not identical. Specifically, with reference to FIG. 11, the collective energy of the first k frequency components of the modified Zyoton, denoted by scalar SE_A′_k is greater than the collective energy of the first k frequency components of the original Zyoton, denoted by scalar SE_A_k. The spectral energy SE_A_k can be computed as the L1 or L2 norm of the amplitude vector A_k, and the spectral energy SE_A_k can be computed as the L1 or L2 norm of the amplitude vector A′_k. Similarly, the spectral energy of the conditioned feature, denoted SE_a_k, can be computed as the L1 or L2 norm of the amplitude vector a_k.

[0429] In general, the dispersion velocity of a waveform (also called the velocity) is proportional to the spectral energy thereof. Therefore, the velocity of a Zyoton is proportional to the L1 or L2 norm of the frequency-domain amplitudes of all k, m, and j components of the Zyoton, represented by the combined amplitude vector [A_k, A_m, A_j]. In many embodiments, the spectral energy of the k components (the analyte-information representing components) is significantly greater than the spectral energy of the m transition components and the j non-analyte information representing components. As such, in some embodiments, the velocity of a waveform associated with a collision is proportional to the spectral energy of the analyte-information representing k components of that waveform.

[0430] Thus, in some embodiments the velocity of the original Zyoton, denoted ν(Z), is represented by SE_A_k. The velocity of the conditioned feature, denoted ν(CF), is represented by SE_a_k. Therefore, in various embodiments, the second requirement of near elasticity, i.e., the collision must not be inelastic, can be met by synthesizing the Zyoton and the carrier kernel and, optionally, by scaling the Zyoton, such that the scaled velocity of the Zyoton is not different from the velocity of the conditioned feature by κ_DV2. This test can be expressed as: |α_T*SE_A_k−SE_a_k|≦κ_DV2. Instead of scaling the velocity of the Zyoton, or in addition to such scaling, the velocity of the conditioned feature can be scaled during this test.

[0431] In the time domain, the dispersion velocity of a waveform is proportional to the L1 or L2 norm of the sideband peaks excluding the highest peak by amplitude. The velocity can be approximated by a selected number of prominent sideband peaks instead of considering all peaks. Therefore, in some embodiments the velocity of the original Zyoton, ν(Z), is represented by the L1 or L2 norm of the sideband amplitudes [, . . . , ], where the amplitude of the highest peak of the Zyoton waveform is excluded. There is no requirement for one-to-one or direct correspondence between the first k frequency components of a Zyoton and the l_Z peaks of the time-domain Zyoton waveform. Similarly, the velocity of the conditioned feature, ν(CF), can be represented by the L1 or L2 norm of the sideband amplitudes [, . . . , ], where the amplitude ₁ of the highest peak of the conditioned feature waveform is excluded. As such, in some embodiments, the velocity test to ensure the second requirement of inelasticity can be expressed as: |α_T*∥[, . . . , ]∥−∥[, . . . , ]∥|≦κ_DV2. The double bars represent the L1 or L2 norm of the amplitudes. The scaling factor α_T and the Kappa parameter κ_DV2 can take on different values based on whether the test is performed in the frequency domain or in the time domain.

[0432] After the first collision iteration, a modified Zyoton is generated. The co-dependency of the original Zyoton and the modified Zyoton can be tested in the frequency domain, to ensure that the collision was near elastic, by testing the condition: |SE_A_k−SE_A′_k|≦κ_DV3. In the time domain, velocity test can be performed as: |∥[, . . . ]∥−∥[, . . . , ,]∥|≦κ_DV3. The Kappa parameter κ_DV3 can take on different values based on whether the test is performed in the frequency domain or in the time domain. In addition, the divergence test can be performed in the time domain by determining the widths of selected b peak envelopes of the original Zyoton, denoted [δ₁, δ₂, . . . , δ_b] and the widths of the corresponding peak envelopes of the modified Zyoton, denoted [δ′₁, δ′₂, . . . , δ′_b]. FIG. 19 depicts δ₁, δ₂, and δ₃ as t1, t2, and t3, and δ₁′, δ₂′, and δ₃′ as t1′, t2′, and t3′. For each peak-envelope pair, a difference can be computed. In some embodiments, the divergence condition is satisfied if |δ_p−δ′_p|≦τ, for each p=1, . . . , b. In some embodiments, an L1 or L2 norm of the differences (δ_p−δ′_p) may then be computed. If the L1 or L2 norm is less than the specified threshold τ, the divergence condition is determined to be satisfied.

[0433] After the modified Zyoton is renormalized, the co-dependency of the original Zyoton and the renormalized Zyoton can be tested in the frequency domain, to ensure that the next collision iteration would be near elastic, by testing the condition |α_T*SE_A_k−SE_A″_k|≦κ_DV3. Instead of scaling the velocity of the Zyoton, or in addition to such scaling, the velocity of the renormalized Zyoton can be scaled during this test. In some embodiments, the scaling coefficient α_T used to scale the velocity of the original Zyoton, represented in the frequency domain by the spectral energy of the analyte-information representing k components of the original Zyoton, may be different than that used when the co-dependency between the original Zyoton and the conditioned feature was tested, to account for the difference between the energies of the conditioned feature and the renormalized Zyoton. In some embodiments, the difference between the values of κ_DV2 and κ_DV3 can account for this energy difference. In some embodiments, the values of κ_DV2 and κ_DV3 can be the same. In the time domain, the velocity test between the original Zyoton and the renormalized Zyoton can be performed as: |α_T*∥[, . . . , ]∥−[, . . . , ]∥|≦κ_DV3. Here again, the scaling factor α_T and the Kappa parameter κ_DV3 can take on different values based on whether the test is performed in the frequency domain or in the time domain.

[0434] After the first collision between the original Zyoton and the conditioned feature, in some embodiments, the subsequent collisions occur between a renormalized Zyoton from a preceding iteration (typically the previous iteration), and the original Zyoton or a new Zyoton. For the i-th collision iteration, the velocity of the original Zyoton, if used in that iteration, is ν(Z). The velocity of the other colliding waveform is the renormalized Zyoton from the previous collision iteration, and the velocity thereof is denoted ν(Z″_(i−1)). The velocity of the modified Zyoton generated in the i-th iteration is denoted ν(Z′i) and the velocity of the renormalized Zyoton generated in the i-th iteration is denoted ν(Z″i). As described above, these velocities can be computed in the frequency domain or in the time domain.

[0435] Optionally, prior to the i-th collision iteration, the velocity test |α_T*ν(Z)−ν(Z″_(i−1))|≦κ_DV1 is performed. Optionally, after the i-th collision iteration, but before the corresponding renormalization, the velocity test |ν(Z)−ν(Z′_(i))|≦κ_DV3 is performed. Optionally, the divergence test using the envelope peaks of the original Zyoton Z and the modified Zyoton Z′_i is performed. Optionally, after the renormalization corresponding to the i-th collision iteration, the velocity test |α_T*ν(Z)−ν(Z″_i)|≦κ_DV1 is performed. One or more tests may be performed in each collision iteration and in different collision iterations, different tests may be performed. The collective objective of these tests, however, is that each collision iteration is near elastic as described above, and that the colliding waveforms are co-dependent.

[0436] To illustrate a collision of the two traveling waveforms in the frequency domain, the k, m, and j frequency components of both the Zyoton and the conditioned feature are depicted (as shown in FIG. 11) as ordered from left to right. A Zyoton or a conditioned feature can be represented as frequency components increasing from left to right or could be represented as frequency components increasing from right to left. The ordering of the frequency components is based on the power spectral densities (PSDs) of the respective frequency components. In some embodiments, the PSD of the k components>the PSD of the m components>the PSD of the j components. In some embodiments, the PSD of the k components<the PSD of the m components<the PSD of the j components.

[0437] The conditioned feature and the original Zyoton are synthesized such that the increase in the collective energy of the first k frequency components of the original Zyoton due to a collision is proportional to the collective energy represented by the first k frequency components of the conditioned feature. As described above, the first k frequency components of the conditioned feature may capture E_A1, E_C1, (E_A1+E_C1), a fraction of E_C1, e.g., E_C[n]=Σ_i=1ⁿ E_Ci1, where there are a total of N confounders and n<N, or (E_A1+E_C[n]). As such, after the first collision, (∥A′_k∥−∥A_k∥) can be proportional to any one of E_A1, E_C1, (E_A1+E_C1), E_C[n], and (E_A1+E_C[n]). These feature energy quantities, generally referred to as E_M1, represent the concentration of a material (e.g., the analyte, a combination of the analyte and at least some confounders, a single confounder, or a combination of at least some confounders) in the medium. Therefore, a change in the spectral energy of a post-collision result (e.g., a Zyoton perturbed by a conditioned feature) relative to the spectral energy of the original Zyoton can be used to estimate the concentration of material exhibited in the spatial, temporal or spectral window defined by the feature.

[0438] According to the second condition, the collision is not generally inelastic, i.e., at a minimum, the collision operation does not introduce into the modified Zyoton any new frequency components within the first k frequency components that were not present in the original Zyoton. Introduction of such frequency components can represent energies that are not related to the energies absorbed by the analyte and any confounders. Therefore, if any such frequency components are introduced, the difference (∥A′_k∥−∥A_k∥) may not accurately represent the concentration of material exhibited in the spatial, temporal, or spectral window defined by the feature. A Zyoton design rule requires that the conditioned feature and the Zyoton not introduce new frequency components in the set of the first k Zyoton frequency components. In some embodiments, this is ensured by scaling the Zyoton waveform energy to be approximately three or more orders of magnitude higher than the spectral energy of the conditioned feature with which it is collided.

[0439] In an embodiment for glucose measurement, where: (i) the absorbance level due to the analyte may be less than 0.01% of the total absorbance, and (ii) there can be up to a three orders of magnitude dynamic range of the analyte (i.e., glucose) concentration range, in the first collision, the threshold value κ_DV2 used to ensure that the Zyoton energy is at least three orders of magnitude greater than the energy of the conditioned feature can be 1×10⁻⁶. Recall, the Kappa threshold is also used in the velocity test described above, to ensure the co-dependency of the Zyoton and the conditioned feature to be collided therewith. In general, the threshold κ can be decreased as the collision iteration count is increased. Thus, as the number of collision iterations increases, the amount of dispersion permitted in each iteration can become smaller.

[0440] While the difference (∥A′_k∥−∥A_k∥) after the first collision can represent E_M1 (i.e., any one of E_A1, E_C1, (E_A1+E_C1), E_C[n], and (E_A1+E_C[n])), the accuracy of such representation can be improved, especially when the signal-to-clutter ratio (SCR) is low, using several collision iterations. In various embodiments, the elasticity of a single collision, as measured by κ_DV1, which corresponds to a collision between the original Zyoton and a renormalized Zyoton, is generally inversely proportional to the SCR. Low SCR generally implies that the value of κ_DV1 for insuring compliance with co-dependency condition is small, e.g., a value of 1×10⁻⁶ used for a SCR of 10⁻³ or a value of 1×10⁻⁸ for a SCR of 10⁻⁴ and, as such, the change in the velocity of the modified Zyoton, and the corresponding renormalized Zyoton, relative to the velocity of the original Zyoton is small. For an order of magnitude reduction in SCR, the threshold κ_DV1 may be generally reduced by two orders of magnitude. As the permitted change in the dispersion velocity after one collision is small, the observed change in energy after one collision, i.e., (∥A′_k∥−∥A_k∥), is also small. Therefore a single collision iteration may not be sufficient to produce a discernible change in the spectral energy due to analyte absorption estimated using the first k frequency components. The energy change after a single collision may be only visible if k were large or if the window length approaches the total number of frequency components of the conditioned feature. The latter case, however, can represent change in energy due to not only analyte and/or confounder absorption, but also due to energy loss or absorption of energy in the medium. Therefore, several collision iterations may be required to induce a change in the spectral energy computed using the first k frequency components to discern and characterize changes due to absorption by the analyte and/or one or more confounders.

[0441] In one embodiment, the subsequent collisions involve the modified Zyoton and the original Zyoton. Therefore, before the next collision, the modified Zyoton is renormalized to remove therefrom the spectral energy of the original Zyoton so that the gain in the energy of the original Zyoton caused in the next iteration is proportional substantially only to K_m, as represented by the modified Zyoton produced in the subsequent iteration. As used herein, substantially means the contribution from noise and/or clutter is no more than 0.01%, 0.2%, 0.5%, 1%, 10%, etc. As explained above, in applications with extremely low SCR (e.g., SCR<10⁻⁴) the (∥A′_k∥−∥A_k∥) may be nearly zero post renormalization. This can lead to numerical instabilities in the computation, such as divide-by-zero instabilities. To prevent such instabilities, book-keeping is employed, e.g., by ensuring that after each collision, or at least after a selected number of successive iterations (e.g., 2, 3, 5, 10, 25, 50, etc.), the total change between the energies of the first k frequency components of the modified Zyoton and the energy of the first k frequency components of the original Zyoton monotonically increases. This change may be tracked as a change in the ratio of energies of the k frequency components of the modified Zyoton and original Zyoton from a starting value of unity (i.e., 1) or 100%.

[0442] In addition, post-collision renormalization can compensate for the impact of any pre-collision up-scaling of any entity participating in a collision (e.g., the Zyoton, the conditioned feature, and/or the renormalized Zyoton from the previous iteration) or the down-scaling of any of these entities by the factor α, as described below. For example, the down-scaling of the modified Zyoton waveform, to obtain a renormalized Zyoton, removes some of the inherent waveform energy.

[0443] As described above, this renormalization computation typically targets two objectives in the same computational step. Specifically, in the first objective of renormalization, it is necessary to remove the spectral energy of the original Zyoton. In addition, it is often necessary to compensate for the energy of frequency components that were removed in any post-collision down-sampling step, also called truncation. Without this compensation, numerical artifacts and energy inaccuracies may be introduced into follow-on collision iterations. The first aspect of the renormalization is generally directed to the removal of the energy of the k components of original Zyoton from the modified Zyoton, often producing a renormalized Zyoton in which the amplitudes of the first k frequency components are similar to the respective amplitudes of the first k frequency components of the conditioned feature.

[0444] The second aspect is generally directed to preserving the energy contributions of any frequency components in the modified Zyoton that were removed post-collision for computational efficiency, and/or to ensure that the subsequent collision iterations would be near-elastic, as described above. In particular, any new frequency components that are transition (i.e., m) components and/or non-analyte-information representing (i.e., j) components, can cause a subsequent iteration to be inelastic. As such, one or more of these new frequency components are removed, but the energy thereof is redistributed among the surviving m and/or j components. Alternatively or in addition, one or more transition components and/or one or more non-analyte-information representing components that were present in the original Zyoton and/or the conditioned feature may be removed, and the energy thereof may be redistributed among the surviving m and/or j components.

[0445] Therefore, referring again to FIG. 11, the amplitudes of all of the k, m, and j components of the modified Zyoton, denoted [A′]=A′[1, . . . , k+m+j], are scaled using a renormalized scaling vector α, computed based on the amplitudes of the k components of the original Zyoton and the corresponding k components of the modified Zyoton. In some embodiments, the scaling vector α is based on a ratio S₁ of the spectral energy of the k components of the original Zyoton and the spectral energy of the corresponding k components of the modified Zyoton, expressed as:

[0446] $S_1=\frac{A_k}{A_k^{\prime }}.$
In some embodiments, this can cause the respective amplitudes of various or all components of the renormalized modified Zyoton, denoted [A″]=A″[1, . . . , k+m′+j′] to be within 10%, 20%, or 50%, etc., respectively, of the amplitudes of the corresponding frequency components of the conditioned feature, denoted [a]=a[1, . . . , k+m+j]. Thus:
[A″]=S₁[A′]A[1, . . . ,k+m+j′]≈a[1, . . . ,k+m+j]  (9)
In some embodiments, the a preset amount of energy, e.g., E* is removed. To this end, a pre-set scaling vector S₁=[S₁¹, S₁², . . . , S₁^k, . . . , S₁^k±m±j] may be used to scale the amplitudes of the components of the modified Zyoton [A′]. The energy level E* can be selected according to the energy of the original Zyoton or independently thereof.

[0447] The cardinality of the transition components of the modified Zyoton, denoted m′, and the cardinality of the non-analyte-information representing components of the modified Zyoton, denoted j′, indicate that the modified Zyoton may have some new transition and/or non-analyte-information representing components. Any of these transition and/or non-analyte-information representing components, including new and/or previously existing ones, may be removed during truncation. The amplitudes of some or all of the surviving transition and/or non-analyte-information representing components of the modified Zyoton may then be adjusted further using another scaling factor S₂, such that the spectral energy of the surviving m and j components of the renormalized Zyoton is approximately equal to the spectral energy of all of the transition and/or non-analyte-information representing components of the modified Zyoton after scaling thereof but its truncation. Thus:
[A″_m,j]=S₂[A′_m,j]_TR  (10)
where [A′_m,j]_TR represents a modified Zyoton as truncated. The scaling factor S₂ may be based on the ratio

[0448] $\frac{S_1\times A_{m^{\prime },j^{\prime }}^{\prime }}{A_{m,j}}.$

[0449] A subsequent collision between the original Zyoton and the renormalized modified Zyoton yields a second modified Zyoton. The spectral energy of the first k frequency components of the second modified Zyoton is denoted ∥A′⁽²⁾_k∥, and the difference (∥A′⁽²⁾_k∥−∥A_k∥) can represent E_M1⁽²⁾, which is a refined estimate of E_M1, i.e., a refined estimate of any one of E_A1, E_C1, (E_A1+E_C1), E_C[n], and (E_A1+E_C[n]). After a selected number of iterations, , which can vary from one embodiment to another, as described below, E_M1⁽⁾ can provide an accurate estimate of the energy absorbed at the wavelengths associated with the feature F1 by the concentration of a material in the medium to be analyzed. As described above, in the general case, the material can be only the analyte, a single confounder, a combination of one or more confounders, or a combination of the analyte and one or more confounders.

[0450] The above described process can be repeated for another feature F2 to obtain . In some embodiments, the two features F1 and F2 are selected such that there is a difference between the absorption of at least one constituent of the material (i.e., the analyte and/or at least one confounder) in the respective wavelengths bands of the two features. Therefore, the difference between and or a function of these two variables can be used in determining the concentration of the constituent that absorbs differently in the two wavelength bands. In one embodiment for noninvasive glucose measurement, in which the analyte does not absorb substantially in the wavelength bands of the Feature F2 and if the total absorption of the confounders remains substantially unchanged between the wavelength bands of Features F1 and F2, and where represents (E_A1+E_C1) and represents E_C2, the difference (−) can represent E_A1, which can be used in determining the concentration of the analyte. These specific conditions generally may not be true, however. Therefore, particular functions or relations, as described below, based on the estimates of energies corresponding to one or more features and/or one or more feature pairs can be used to determine more accurately the concentration of the analyte.

[0451] With reference to FIG. 10B, a feature is generally related to the intensity of radiation reflected from or transmitted through a medium, or to the absorption of incident radiation within the medium to be analyzed, corresponding to a specified range of wavelengths [λ1, λ2], also called feature boundaries. Typically, a feature (e.g., Feature F1) does not have a smooth profile as shown in FIG. 10B. Examples of intensity and absorbance spectra are shown in FIGS. 31A and 31B. The features obtained from any of these spectra, typically have a saw-toothed, somewhat jagged profile, such as that shown in FIG. 32. Such a profile generally results from sampling errors, device and detector noise, ADC discretization artifacts, and/or path length effects of summing absorbance from many individual NIR photons. Noise in an intensity spectrum produced from radiation collected at a detector typically results from several sources. The quantization error of the analog-to-digital converter (ADC) is one source. Another source of noise is the variation in total travel distances of individual photons as they propagate through a medium to be analyzed (e.g., the skin tissue), and this variation in combination with scattering losses within the medium can introduce measurement noise in a received or detected intensity spectrum.

[0452] Moreover, the noise in an intensity or absorbance spectrum may not be uniform across the spectrum, and may vary according to the wavelength. As such, the noise in a feature within the wavelength range [λ1, λ2] can be non-uniform and may vary non-monotonically with the wavelength. The variations in noise across a spectrum can be determined by the difference between any single spectrum and a mean spectrum obtained from multiple spectra generated using the same incident radiation and medium. FIG. 31A provides an example of a diffuse reflectance intensity and FIG. 31B shows the associated absorbance from a NIR spectrum from human skin. An example of noise in a captured spectrum is shown in FIGS. 33 through 35.

[0453] FIG. 33 provides an example of a set of mean-centered absorbance spectra to show instrument related spectral variability as a function of NIR wavelength. FIG. 34 shows an example of wavelength-by-wavelength RMS noise, based upon a set of spectra. FIG. 35 depicts a difference between a single spectrum and the mean spectrum and provides an illustration of the variation in noise across wavelengths. FIG. 36 shows the scaled intensity and absorbance spectra associated with a single feature on the same plot. As shown in the legend for FIG. 36, different absorbance and intensities profiles have been normalized by multiplying with different factors. The ultra-weak absorbance of pure-component spectrum of glucose, in the feature region delineated by 1540 nm to 1554 nm, has been multiplied by a factor of 40,000 to plot at the same scale as the absorbance from confounders: fat (multiplied by factor of 8) and water (multiplied by factor of 2). These scaled absorbances for glucose and the confounders fat and water are overlaid on the same plot. The legend provides all the factors used for normalizing the spectra (in intensity or absorbance units) to plot them on the same graph.

[0454] The fluctuations in the feature absorbance profiles shown in FIG. 36 typically result from sampling errors, device and detector noise, analog-to-digital conversion (ADC) discretization artifacts, and path length effects of summing absorbance from the travel-path of many individual NIR photons launched in each illumination of the sample. A feature such as that shown in FIG. 36, thus may have both periodicity and frequency content.

[0455] The process of generating a conditioned feature begins with the synthesis of a Zyoton waveform and a carrier kernel waveform. The Zyoton is generated using a number of frequency components. The first k (e.g., six) frequency components are selected, e.g., for measuring the energy absorbed by an analyte (such as energy absorbed by glucose in tissue), according to pure component spectrum of the analyte and the characteristics of instrument noise, such as the RMS value in the amplitude domain, power spectral density in frequency domain, and/or periodicity.

[0456] In various embodiments, the selected frequency components are related to frequencies of radiation that are absorbed strongly by the analyte and those absorbed weakly or negligibly by the analyte. The analyte-specific frequency components may be selected following a Fourier transform of the selected spectral feature either before or after an optional interpolation, a sort of the frequency components by peak power, and a selection of the first k (e.g., 6) frequency components. A Fourier transform of a feature includes sinusoidal waveforms at different frequencies/frequency components that, when summed, represent the feature waveform.

[0457] The feature data extracted from the spectrum has low spatial frequency content, often due to a limited data vector length. The spatial frequency content may therefore be insufficient to characterize changes in the feature properties due to analyte concentration and interference from confounders. Coupling the feature with a complex carrier kernel with a higher spatial frequency content can provide additional degrees of freedom to characterize the underlying analyte. Pre-cursor modulation 10 in FIG. 37 of the feature can also increase the degrees of freedom by increasing the bandwidth of the feature waveform prior to modulating the carrier kernel. Different features, associated with different wavelength regions, with different underlying analyte concentrations generally modulate the carrier kernel differently.

[0458] A carrier kernel typically includes k high amplitude frequency components, where k≧1, and j low amplitude frequency components, where j>1, and may include m medium-amplitude components. While the specific values of the amplitudes of these frequency components can be selected according to the amplitudes of the frequency components of the Zyoton to be used in the collision process and/or other parameters of the collision process, the amplitudes of the k high amplitude (also called high energy) frequency components of the carrier kernel are typically two or three orders of magnitude greater than the amplitudes of the j frequency components of the carrier kernel. The high energy k frequency components are employed primarily for extraction of the energy absorbed by the analyte from a feature.

[0459] The carrier kernel may optionally include m frequency components (called medium energy components or transition zone components) having amplitudes that are generally two to three times the amplitudes of the low energy j components. The Zyoton used during the collision process also has high energy k components, low energy j components and, optionally, medium energy m components. Likewise, a conditioned feature generated by modulating the carrier kernel by a feature also has the high energy k components, low energy j components and, optionally, medium energy m components. During various frequency-domain implementations of the collision process, the respective Zyoton and conditioned feature components are amplitude sorted. Therefore, the discussion below usually refers to the high-energy components of a waveform (e.g., a Zyoton, a carrier kernel, a conditioned feature, a modified Zyoton, a renormalized Zyoton, etc.), as the first k frequency components or the first k components.

[0460] In various embodiments, the first k frequency components of a Zyoton are determined by examination of the second derivative of the pure component spectrum (e.g., glucose) in order to identify areas of the greatest rate of change of energy absorption within the features derived from the spectrum. For example, FIG. 28A depicts an absorbance spectrum of glucose, and FIG. 30A shows the second derivative of that absorbance spectrum. The ratio of the magnitude of the second derivative in the regions that correspond to high and low analyte absorption regions (as shown in FIG. 28A) may be used to establish the relation (e.g., relative ratio of magnitudes of first k frequency components of the Zyoton and/or the carrier kernel) between the first k frequency components, as described below with reference to FIG. 38.

[0461] The second derivative may also be used, as discussed below, as an indicator of how confounder interference will distort high and low absorbing regions in the frequency domain for the specific wavelength region from which features are selected. In the absence of noise, the second derivative can be used to provide a numerical classifier for identifying the areas of greatest change in either of the high and low absorbance wavelength regions of the pure component spectrum of the analyte. Use of the second-order derivative is generally described in Ferraty, F. and Vieu, P, “The Functional Nonparameteric Model and Application to Spectrometric data”, Computational Statistics, 17(4), 2002, which shows that by taking the second- and higher-order derivatives, one can focus on curvature rather actual values taken by functions to build more robust classifiers and improve prediction performance in the presence of noise and other uncertainties.

[0462] The number k, i.e., the cardinality of the set of high-energy components, can be determined by estimating the number of frequency components required to separate (as defined below) spectral regions of the second derivative corresponding to higher and lower absorbing regions of the pure component spectrum in the presence of at least three orders of magnitude of added RMS pseudo-random noise (PRN). Commonly known techniques for generating pseudo-random noise may be employed and addition is typically represented as wavelength-by-wavelength addition of the PRN value to the pure component absorbance amplitude value. In various embodiments discriminant classification is further used to develop a quadratic discriminant classifier using combinations of different numbers of frequency components, i.e., using 2, 3, 4, . . . k components.

[0463] A quadratic classifier can be used to separate two or more classes of objects or events by a quadric surface (where quadric is any D-dimensional hypersurface in (D+1)-dimensional space defined as the locus of zeros of a quadratic polynomial). In coordinates x₁, x₂, . . . , x_D+1, the general quadric is defined by the algebraic equation:

[0464] $\begin{array}{cc}\sum _{i,j=1}^{D+1}{x}_i{Q}_{\mathrm{ij}}{x}_j+\sum _{i=1}^{D+1}{P}_i{x}_i+R=0& \left(11\right)\end{array}$
where x=(x₁, x₂, . . . , x_D+1) is a row vector, x′ is the transpose of x (a column vector), Q is a (D+1)×(D+1) matrix and P is a (D+1)-dimensional row vector and R a scalar constant. The values Q, P and R are often taken to be over real numbers or complex numbers. The set of “vectors of observations” used in the discriminant analysis includes features drawn from the selected regions of the second derivative waveform combined with noise (PRN), each of which is associated with a higher or lower absorption region of the analyte pure component spectrum.

[0465] To generate the vectors of observation to be used in discriminant analysis: (i) the values of the second derivative of the pure component absorbance spectrum, to which PRN has been added, in the high and low absorbance regions as defined above, are extracted and copied into two new vectors HighAbs and LowAbs respectively; (ii) in some embodiments, the HighAbs and LowAbs vectors from the pure component absorbance spectrum for the analyte are then both up-sampled to the twice the up-sampled length used for up-sampling a spectral feature vector (e.g., 2*2048=4096) as used in collision computing. The resulting vectors are denoted as up-sampled-HighAbs and up-sampled-LowAbs vectors respectively; (iii) up-sampled HighAbs and up-sampled LowAbs vectors are then Fourier transformed into spatial frequencies. The results are denoted as F{Up-sampled HighAbs)} and F{Upsampled LowAbs} vectors.

[0466] Additional steps include: (iv) magnitude sort F{Upsampled HighAbs} and F{Upsampled LowAbs} on the basis of the amplitude of frequency components to yield F′{Upsampled HighAbs} and F′{Upsampled LowAbs} vectors; (v) construct observation test vectors, to be used in discriminant analysis test, by selecting subsets of F′{Upsampled HighAbs} and F′{Upsampled LowAbs} vectors with variable number of frequency components such as v=2, 3, 4, . . . k components, where each value v is a candidate k value; and (vi) optionally repeat this sequence of step (i)-step (v) 2, 5, 10, 50, 100, or 500 times, where different amounts of random noise are added to the pure component spectrum to generate different test sets of observation vectors of different numbers of frequency components.

[0467] The value of “k” may then be set to the number of components that can separate the high and low absorbance noisy second derivative regions of the analyte pure component spectra, in the case of a single test, or the maximum number of components that are required to separate these high and low absorbing regions in repeated tests. In various embodiments, the quadratic discriminant technique is used to build a classifier that can associate combinations of frequency components with the absorption regions associated with an analyte of interest. In general, high and low absorbing spectral regions are said to be separated when the result of the discriminant classifier, normalized to a value of 1, is less than 0.1, 0.2, or 0.5.

[0468] In one embodiment, the number of high energy components (k) of the original Zyoton (and of the carrier kernel, as well), where the k high energy components represent absorption by the analyte, is determined using: (a) the Fourier transform of a noisy signal obtained by adding a certain amount of noise to the second derivative of the analyte pure component spectrum, and (b) discrimination classification. With reference to FIG. 39, the pure component spectrum of the analyte of interest, denoted XA, is obtained in step 2. This spectrum is typically generated using the source of radiation and the detector (e.g., an optical probe, examples of which are described below) to be used in the non-invasive measurement system. The spectrum XA may be optionally interpolated in step 2. In step 4, a second derivative of the spectrum XA, denoted XA″ is obtained.

[0469] Pseudorandom noise N^A is added to the second derivative XA″, in step 6, to obtain a noisy signal S^A. The noise N^A can be gray noise with a correlation time of 10⁻⁶ seconds, and may have a root-mean-squared (rms) value that is a multiple of (e.g., 100, 1000, 1500, 5000, 10,000 times, etc.), the rms value of the noise introduced by the sensor, i.e., the noise of the spectrum signal XA. Alternatively, the rms value of NA can be a multiple of the RMS value of the first or higher-order derivative of XA. In step 8, Fourier transform of the noisy second derivative signal S^A is computed to identify the various frequency components therein. These components are sorted by amplitude in step 10, though in some embodiments, this step is optional.

[0470] Discriminant classification is performed in step 12 to determine the value of k, i.e., the cardinality of the set of high-energy components of the Zyoton and the carrier kernel. Specifically, k is set to be equal to the number of frequency components of the noisy signal that are needed to separate the high and low absorbance regions as defined above. Thus, the determination of k for the Zyoton (and for the carrier kernel, as well) is based, at least in part, on the SNR of the measurement system. In some embodiments, the value of k may be incremented by a suitable integer constant, e.g., 1, 2, 4, etc. The actual frequencies of the k Zyoton and the carrier kernel frequencies are generally determined by the selected Zyoton family/generator and need not be related to the frequencies of the components of the noisy second derivative signal. In summary, k is concluded when the discriminant classification technique using k-components can separate regions of the second derivative of the pure component spectrum corresponding to higher and lower absorbance regions of the pure component spectrum in the presence of noise.

[0471] The next m (e.g., 4, 6, etc.,) frequency components are designed as a transition zone to the lower amplitude j frequency components. The discriminant classification technique may be used to determine m, i.e., the number of frequency components in the transition zone of the Zyoton (and the carrier kernel, as well). With reference to FIG. 40, pure component spectra from one or more confounding materials, e.g., urea and collagen if glucose is the analyte of interest, are added in step 4 to the pure component spectrum of the analyte (XA) obtained in step 2, to produce a composite spectrum, denoted XC, in step 4. The second derivative of the composite spectrum XC, denoted XC″, is obtained in step 6.

[0472] In step 8, XC″ is further combined with random white noise N^C (e.g., colored grey noise with correlation time of 10⁻⁶ seconds) having an rms value equal to a multiple (e.g., 100, 1000, 2500, 10,000 times, etc.), the rms value of the signal-to-noise ratio of the second derivative of the composite spectrum, i.e., XC″, where all the absorbances of the confounders are normalized to the maximum absorbance of the analyte. In some embodiments, the rms value of N^C is a multiple of the rms value of XC, a first derivative thereof, or third and higher-order derivatives of XC. In some embodiments, the rms value of N^C is a multiple of the rms value of sensor noise, i.e., the rms value of the pure component spectrum of the analyte XA, or a first or higher-order derivative of XA. The number m of the frequency components required to separate, according to the discriminant classification described above, the spectral regions of the second derivative of the composite spectrum, with added noise, where the separated spectral regions correspond to higher and lower absorbance regions of the pure component spectrum of the analyte can be determined in steps 10-14. The discriminant classification in step 14 may include several trials of different candidate values of m. This provides the number of frequency components k+m. The discriminant analysis used to determine the number m of the medium energy components is similar to the discriminant analysis described above that is performed in the determination of the number k of the high-energy components. In finding m, however, the candidate values for the length of the observation vectors for discriminant analysis are set to be of length k+1, k+2, k+3, . . . , k+m frequency components.

[0473] For the determination of j, a dominant confounder may be used in the analysis. For example, if glucose is the analyte of interest, the spectrum of water (generally known to have much higher absorbance than glucose in tissue) can be used to create additional distortion. With reference to FIG. 41, the pure component spectrum of the analyte (i.e., XA), one or more confounders, and the water spectrum (all normalized to the absorbance of water, because the water spectrum dominates the spectrum of glucose) are combined to obtain a composite spectrum or signal, denoted XC⁺ in steps 2 and 4. In step 10, the second derivative of this composite spectrum, denoted XC⁺″ is mixed with random noise N⁺ having, e.g., the A-weighting and correlation time of 10⁻⁶ seconds. The noise N⁺ is constructed such that the rms value of N⁺ is a multiple of (e.g., 400, 800, 1000, 2000, 7500, 12,000, times, etc.), the rms value of the SNR of the second derivative of the composite spectrum XC⁺″.

[0474] In some embodiments, the rms value of N⁺ is a multiple of the rms value of XC⁺, a first derivative thereof, or third and higher-order derivatives of XC⁺. In some embodiments, the rms value of N⁺ is a multiple of the rms value of XC used to determine m, or the rms value of the pure component spectrum of the analyte XA, or a first or higher-order derivative of XC or XA. Fourier analysis followed by discriminant classification are applied in steps 10-14, as in the determination of the values of k and m, to determine the number j of the low-energy components required to separate the spectral regions of the second derivative with added noise corresponding to regions of higher and lower absorbance of the analyte pure component spectrum, as described above. Including the water spectrum generally increases the range of the frequency components beyond those corresponding to the random noise from scattering. Because of the magnitude of the absorbance of the dominant confounder (water), very small local variations in in its concentration typically manifest as noise in the absorbance that are added to the noise contribution from light scattered by the medium.

[0475] In some embodiments, instead of adding a dominant confounder, random white noise denoted N⁺⁺ (e.g., grey colored random noise with, e.g., A-weighting and correlation time of 10⁻⁶ seconds) with a magnitude that is a multiple (e.g., 20, 50, 100, 125, times etc.) the noise N⁺ that was added in step 8 is added in step 16 and, optionally, pure component spectra of one or more confounders are also added to the pure component spectrum of the analyte XA. Thus, the rms value of the noise N⁺⁺ can be 100,000 times the rms value of the SNR of the second derivative of the analyte pure component spectrum without confounders, i.e., the signal XA″. In some embodiments, the rms value of N⁺⁺ is a multiple of the rms value of XA, a first derivative thereof, or third and higher-order derivatives of XA. In some embodiments, the rms value of N⁺⁺ is a multiple of the rms value of XC⁺, or XC used to determine m, or a first or higher-order derivative of XC⁺ or XC. Instead of adding a dominant confounder spectrum, such as water spectrum, such noise may be added to increase the frequency range to include the scatter-related frequency components.

[0476] The discriminant analysis used to determine the number j of the low energy components is similar to the discriminant analysis described above that is performed in the determination of the number k of the high-energy components and the number m of the medium energy components. In finding j, however, the candidate values for the length of the observation vectors used for the discriminant analysis are set to be of length k+m+1, k+m+2, . . . , k+m+j frequency components.

[0477] After the values of the number of Zyoton and carrier kernel components in the high, medium, and low energy regions, i.e., k, m, and j, respectively, are determined, the actual components of the waveforms are selected. With reference to FIG. 38, the bandwidth and/or a morphological profile of the noisy signal S⁺ (generated in step 8 of FIG. 41) is computed in step 2 (of FIG. 38). A Zyoton waveform family and/or a generator is selected in step 2 according to the morphological properties of: (a) the pure component spectrum of the analyte, optionally combined with the pure component spectrum/spectra of one or more confounders, or (b) the first or higher-order derivative of the pure component or combined spectrum, optionally with added noise. The following examples illustrate the selection of a waveform family and/or generator.

[0478] If the spectral bandwidth of the signal used in step 2 is high, e.g., greater than 1 kHz, 100 kHz, 1 MHz, 100 MHz, etc., high spectral bandwidth Zyotons can be synthesized using waveform families such as solitons; vortex-solitons; wavelets; or multi-color soliton and/or, optionally, using waveform generator functions such as harmonic oscillators or chaotic attractors.

[0479] If the spectral envelope of the signal used in step 2 is symmetrical, Zyotons with symmetrical spectral envelop can be synthesized using waveform families such as solitons; wavelets; ridgelets; or multi-color solitons and/or, optionally, using generator functions such as cyclostationary series or Lyapunov functions. On the other hand, if the spectral envelope of the signal used in step 2 is monotonic, Zyotons with a monotonic spectral envelop can be synthesized using waveform families such as solitons; autosolitons; similaritons; wavelets; curvelets; ridgelets; bions; or elliptic waves and/or optionally, using generator functions such as Frobenius manifolds; harmonic oscillators; Hermite polynomials; polynomial sequences; asymptotic Hankel functions; or Neumann spherical functions. If the spectral envelope of the signal used in step 2 is random, Zyotons with a random spectral envelop can be synthesized using waveform families such as solitons; multi-color solitons; Ricci solitons; or nonautonomous similinear wave equations and/or, optionally, using generator functions such as random number generators.

[0480] If the peak spectral energy of the signal used in step 2 is greater than a selected threshold, Zyotons with high peak energy can be synthesized using waveform families such as solitons; autosolitons; self-compressing similaritons; vortex-solitons; multi-color solitons; and/or, optionally, using waveform generators such as Gamma functions; Riemann Zeta functions; polynomial sequences; spatial random fields; spherical harmonics; chaotic attractors; exponential attractors; or evolution equation of exponential attractors.

[0481] If the spectrum of the signal used in step 2 has linearly distributed amplitudes, Zyotons with a linear amplitude distribution can be synthesized using waveform families such as solitons; autosolitons; similaritons; wavelets; curvelets; ridgelets; and/or, optionally, using waveform generators such as Gamma functions; Riemann Zeta functions; polynomial sequences; or spherical harmonics. If the spectrum of the signal used in step 2 has non-linearly distributed amplitudes, however, e.g., exponentially distributed amplitudes, Zyotons with an exponential amplitude distribution can be synthesized using waveform families such solitons; vortex-solitons; multi-color solitons; and/or, optionally, using waveform generators such as Fractals; poweroid coupled with sinusoidal functions; chaotic attractors; exponential attractors; evolution equation for polynomial nonlinear reaction-diffusion equation; evolution equation for Kuramoto-Savashinsky equation; or evolution equation of exponential attractors.

[0482] In step 4, “k” high energy components are obtained from the selected Zyoton family/generator. This process is repeated in steps 6, 8, respectively, to obtain the m transition components and the j low energy components. Each component (e.g., the i-th component) can be represented in terms of its frequency Ω(i) and frequency-domain amplitude A(i).

[0483] In some embodiments, a base Zyoton obtained from a Zyoton family/generator may be represented in the time domain, as described above, as:
Z_N(xt)=2∂_xTrB_x(1+B)⁻¹  (12)
By applying the Fourier transform to this time-domain representation, the frequency components thereof can be determined. These frequency components of the base Zyoton, each represented as an amplitude-frequency pair <A*_i, Ω*_i>, sorted by the amplitude, can be expressed in terms of k, m, and j as shown in Table 6:

[0484]

TABLE 6
Amplitudes	A1*	A2*	. . .			. . .			. . .
Frequencies	Ω1*	Ω2*	. . .			. . .			. . .

[0485] The amplitude-sorted components of the base Zyoton can be grouped into high, medium, and low-energy components (i.e., k, m, and j components) according to the amplitudes thereof. The boundaries between the high-energy and the medium-energy components and between the medium-energy and low-energy components are not absolute, and can change, depending on the embodiment. Thus, the designations of one or more frequency components to a particular group can change.

[0486] Using the value of k, computed, e.g., as described with reference to FIG. 39, the first k components of the base Zyoton, i.e., the components {<A*₁, Ω*₁>, . . . <A*_k, Ω*_k>} may be selected to be included as the high-energy k components in the Zyoton to be synthesized. Using the value of m, computed, e.g., as described with reference to FIG. 40, the first m components of the base Zyoton from the medium-energy group thereof, i.e., the components {<, >, . . . <, >} may be selected to be included as the medium-energy m components in the Zyoton to be synthesized. Similarly, using the value of j, computed, e.g., as described with reference to FIG. 41, the first j components of the base Zyoton from the low-energy group thereof, i.e., the components {<₊₁, ₊₁>, . . . <_+j, _+j>} may be selected to be included as the medium-energy m components in the Zyoton to be synthesized. In general, >k; >m; and >j.

[0487] In some embodiments, the amplitudes A(i) of the Zyoton thus formed, require no further adjustment. In other embodiments, however, the amplitudes of these components, initially set to be A*_i according to the selected components of the base Zyoton, are adjusted in step 10. These adjustments may be performed according to the ratio of the average, median, minimum, or maximum magnitude of the second derivative in the regions that correspond to high and low analyte absorption wavelength regions. To this end, the pure component spectrum XA of the analyte of interest is partitioned into a number of wavelength regions, where each wavelength region corresponds to a particular range of wavelengths [λ₁, λ₂] of the spectrum XA. The wavelength regions are thus similar to features but, in general, features correspond to wavelength regions of the spectral signals obtained from a medium to be analyzed. The wavelength regions described herein are generally derived from pure component spectra.

[0488] Some of these wavelength regions, where the absorption of energy by the analyte of interest is high are designated as analyte wavelength regions. Some wavelength regions, where the absorption of energy by the analyte is low, are designated as non-analyte wavelength regions. The terms high absorption of energy and low absorption of energy should be understood in the context of analysis, as described above. One or more of the several analyte wavelength regions are selected to interact (via conditioning thereof) with the Zyoton and the carrier kernel to be synthesized. The number of the selected analyte wavelength regions is n. Each of these n selected analyte wavelength regions is denoted Fi, and is paired with a corresponding non-analyte wavelength region denoted NFi, forming n wavelength region pairs FP.

[0489] For each wavelength-region pair FPi, two regions of the second derivative of XA, i.e., XA″, are identified, where the first region of the second derivative corresponds to the wavelength boundaries of Fi, i.e., [λ_Ii, λ_2i], and the second region of the second derivative corresponds to the wavelength boundaries of NFi. A ratio Ri of the average, median, minimum, or maximum magnitude of the first region of the second derivate to the average, median, minimum, or maximum magnitude of the second region of the second derivative is then computed. These steps are performed for each of the n wavelength-region pairs, to obtain n ratios. The maximum and minimum of these ratios, denoted R^max and R^min are then computed.

[0490] The amplitudes of the k Zyoton components, i.e., the high-energy or analyte-information representing components, are then selected such that the pair-wise ratios thereof are distributed in the range [R^max, R^min]. In particular, if the amplitudes of the k Zyoton components are [A1, A2, . . . , Ak], A1 is designated a specified value, e.g., (1,000; 1,600; 4,000; 10,000; 50,000; 125,000, etc.). In some embodiments, the value of the amplitude A*₁ of the base Zyoton may be retained. A2 is then selected such that A1/A2=R^max. An intermediate ratio value R2 less than R^max but greater than R^min is then selected, and A3 is selected such as A2/A3=R2. Another intermediate ratio value R3 less than R2 but greater than R^min is then selected, and A4 is selected such as A3/A4=R3. The amplitudes of the k Zyoton components are thus selected such that, finally, the ratio A(k−1)/Ak=R^min. In some embodiments, the ratios are distributed in the ascending order, in the range [R^min, R^max]. In various embodiments, the distribution of ratios, i.e., the selection of the intermediate ratios R2, R3, etc., can linear, quadratic, or exponential, for example.

[0491] In various embodiments, the amplitudes of the j frequency components, i.e., the non-analyte-information representing components, may be selected randomly as a fraction (e.g., in a range from about 1/100 to 1/1000^th) of the amplitude of any of the k Zyoton components. In some embodiments, the amplitudes of the j components can be a percentage such as 2%, 5%, 10%, 50% of the amplitudes of the k components. The amplitudes of the m transition component can be a multiple (e.g., 2, 3, 5, 10, etc.) of the amplitude of any of the j Zyoton components. The (k+m+j) frequency components may then be sorted by amplitude in step 12. In the time domain, a Zyoton thus constructed is a traveling waveform that can be perturbed if collided with another waveform such as an unmodulated carrier kernel or a conditioned feature. The perturbation generally results in a change in the morphological profile of the Zyoton, e.g., a change in dispersion velocity, divergence, etc. Without a collision with another waveform, however, the Zyoton can propagate substantially unperturbed, i.e., without a substantial change in dispersion velocity, divergence, etc., at least over a preselected distance, such as the length of a collision grid (e.g., 2000 points; 10,000 points; 20,000 points; 100,000 points; or more) in the time domain. Substantial in this context can be a more than 0.001%, 0.005%, 0.02%, 0.1%, 1%, 5%, 10%, 20%, etc. change in a parameter of interest such as dispersion velocity over the distance of propagation.

[0492] The frequency components of the carrier kernel may be selected to correspond to those of the Zyotons in step 14. In general, the amplitudes of the k, m, and j components of the carrier kernel are selected such that the spectral energy of the carrier kernel is only a fraction (e.g., 1/100, 1/1,000, etc.) of the spectral energy of the Zyoton. In some embodiments, the amplitudes of the frequency components of a Zyoton are determined as described above, and then the amplitudes of the corresponding frequency components of the carrier kernel are set using a constant scaling factor such as 0.25, 0.2, 0.1, 0.08, 0.03, 0.01, etc.

[0493] In some embodiments, the scaling factor is selected such that the Zyoton and the unmodulated carrier kernel are co-dependent. This can be achieved by applying a Kappa test to these two waveforms. In particular, if a magnitude of a difference between a scaled velocity of the Zyoton and the velocity of the carrier kernel is not greater than the value of a Kappa parameter κ_SYN, used during synthesis, the Zyoton and the carrier kernel are likely co-dependent. The co-dependence typically cannot be ascertained during waveform synthesis because a conditioned feature obtained by modulating the synthesized carrier kernel with one or more features, and used in a collision operation, typically has a different energy and velocity than that of the carrier kernel.

[0494] In various embodiments the parameter κ^SYN is similar to the parameter κ_DV2 discussed above. In performing the above-described Kappa test, the velocity of the unmodulated carrier kernel may be scaled during the comparison, instead of or in addition to scaling the velocity of the Zyoton. The Kappa test using the parameter κ^SYN can be performed in the time domain or in the frequency domain. If this Kappa test fails, the frequency-domain amplitudes of the frequency components of the Zyoton, the carrier kernel, or both, may be scaled using a scaling coefficient α_SYN, which is similar to the normalized scaling coefficient α_C discussed above, such that the Kappa test would succeed using the adjusted Zyoton and/or the carrier kernel.

[0495] An example of some of the frequencies and amplitudes in a carrier kernel for the k, m, and j components is shown in Table 7 below.

[0496]

TABLE 7
	Frequency	Frequency Component Amplitude
Region	(KHz)	(dBm/Hz)
K	218.7	625.4878
	277.4	260.5532
	317.2	233.3075
	123.4	233.3075
	87.16	194.6868
	412.9	194.6868
M	497.3	38.5267
	511.1	38.5267
	732.42	35.9625
	614.9	35.9625
	507.2	33.6208
	843.1	33.6208
	328.6	25.5264
J	1832.4	6.3816
	7322.1	5.0936
	5376.2	5.0936
	11236.5	4.0355
	9032.5	4.0355
	18043.1	3.8207
	7344.5	3.8207
	11264.4	2.5359
	14090.2	2.5359
	9433.7	1.2619
	. . .	. . .

[0497] In the table above, the first six frequency components are the k components, the next seven components are the m components, and the rest are a subset of the j frequency components. To design a carrier kernel, a desired amplitude profile for the k, m, and j components is set. The example relationship among the amplitudes described here, assigned to their respective frequency components, was established for an embodiment of glucose measurement, such that all the frequency components interact and mix with the selected Zyoton. A function generator and/or a waveform synthesizer can generate the carrier kernel from the frequency components selected as described above. The carrier kernel is then modulated using the feature waveform to obtain a corresponding conditioned feature. In some embodiments, a time-domain representation of the carrier kernel is discretized into a selected number (e.g., 2048) of points.

[0498] A feature as extracted from an absorbance spectrum may be initially represented as a 32 point waveform. FIG. 32 shows an absorbance profile of a single feature, extracted from a reflectance intensity spectrum in the region between 6493 cm⁻¹ (1540 nm) and 6433 cm⁻¹ (1554 nm). The initial feature length here, as extracted from an absorbance spectrum, is 32 data points (60 cm⁻¹) for a resolution of the spectrum of 2 cm⁻¹ (including the two endpoints of the spectral range). In various embodiments, that feature waveform is up-sampled to the length of the carrier kernel (e.g., 2048) using e.g., spline interpolation to match the length of the two waveforms. FIG. 42 shows the feature absorbance of FIG. 32, up-sampled by interpolation to 2048 data points from the initial 32-data point spectral feature. This up-sampling may be done to match the length of the carrier kernel and Zyoton waveforms, and is followed by the optional precursor frequency modulation described above. Features can be obtained from reflectance/intensity signals and then such features can be transformed into absorbance-representing features. Alternatively, as described above, the reflectance/intensity signal can be transformed into an absorbance signal and the features can be obtained therefrom.

[0499] A Fourier transform is then applied to the up-sampled and optionally modulated feature (also called a feature vector) to obtain a Fourier transformed feature. FIG. 43 shows an example of the first 10 frequency components associated with the up-sampled feature shown in FIG. 42. A Fourier transform of the carrier kernel is also generated to obtain a Fourier transformed carrier kernel, which includes the k, m, and j frequency components, as described above. In some embodiments, the Fourier transformed feature and carrier kernel are convolved to obtain a Fourier transformed conditioned feature waveform. In some embodiments, the carrier kernel is frequency and/or phase modulated by the original or up-sampled feature, to obtain the conditioned feature. This modulation can be performed in the time (wavelength) domain or in the frequency domain. If the modulation is performed in the frequency domain, a time-domain representation of the conditioned feature waveform can be generated using the inverse Fourier transform of the frequency-domain representation of the conditioned feature, e.g., the result of a convolution of the Fourier transformed carrier kernel and the Fourier transformed up-sampled feature.

[0500] In various embodiments, the co-dependency between a Zyoton and a conditioned feature to be collided therewith is achieved by ensuring that: (i) an absolute difference between their respective dispersion velocities is less than a specified threshold κ_DV2, and (ii) divergence of the envelopes of a selected number of sidebands of the modified Zyoton created as a result of a collision is less than another specified threshold τ. As described above, the first condition can be tested prior to a collision between the original Zyoton and the conditioned feature, using the constraint variable κ_DV2. Additionally, or in the alternative, the first condition may be tested post collision, between the original Zyoton and a modified Zyoton using the constrain variable κ_DV3, or between the original Zyoton and a renormalized Zyoton, using the variable κ_DV1. It should be noted that while κ_DV1, κ_DV2, κ_DV3, and τ may optionally take on different values, they all serve to enforce the co-dependency conditions.

[0501] The first type of constraints, κ_DV1, κ_DV2, and κ_DV3, are related to the dispersion velocity, which can be represented in the time domain as the amplitudes of the selected sideband peaks of the time-domain representation of the waveforms considered, as discussed above. In the frequency domain, the dispersion velocity can be represented by the amplitudes of the first k frequency components or, as described above, by (∥A′_k∥−∥A_k∥). The second type of constraint, τ, is related to the divergence of the sideband envelopes which may be caused by the introduction of new m and/or j frequency components in the post-collision modified Zyoton which can result in a subtle change in the morphology (e.g., the overall shape) of the modified Zyoton.

[0502] In general, the dispersion velocity can be described as the velocity with which the narrow-band peak envelope of a Zyoton (e.g., the strongest peak of the underlying soliton kernel as shown in FIG. 12A) propagates in a medium over time, ignoring higher order chromatic dispersion and nonlinear effects such as those arising from the propagation of lower amplitude pulses (or higher frequency components) in the Zyoton. Chromatic dispersion of the Zyoton generally results from the introduction of new m and/or j frequency components and the nonlinear effects from changes in the amplitudes of the m and/or j frequency components. Energies associated with peaks at these new frequencies can distort the shape of the waveform. The computation of dispersion velocity of solitons, which can be used to analyze Zyotons, is discussed in Haus and Ippen, “Group velocity of solitons,” Optical Letters 26 (21), 1654 (2001), the entirety of which is incorporated herein by reference.

[0503] Changes in the post-collision dispersion velocity, i.e., the dispersion velocity (or the velocity) of a modified Zyoton can be constrained by the asymmetric energy relationship between the conditioned feature waveform and the Zyoton waveform, or between the renormalized Zyoton waveform from the previous iteration and the Zyoton, as described above, where the energy of the Zyoton may be three or more orders of magnitude more than that of the conditioned feature and/or the renormalized Zyoton. In various embodiments, the conditioned feature waveform and the renormalized Zyoton can be scaled (as described below in the collision equation) to a small fraction of the total spectral energy of the selected Zyoton, in order to achieve the desired energy balance.

[0504] As described above, the introduction of additional m and/or j frequency components can change the shape of the modified Zyoton relative to the shape of the original Zyoton. This shape divergence can be described as a change in the width of the post-collision Zyoton waveform envelope. FIGS. 14A-14B are time-domain representations of a Zyoton and a modified Zyoton resulting from a collision of the Zyoton and a conditioned feature. FIG. 14A shows that the time-domain length of the original Zyoton is and FIG. 14B shows the time-domain length of the modified Zyoton is ₁. The post-collision, pre-renormalization time-domain length of the modified Zyoton can be stated as ₁=+Δl₁, where Δl₁ is the width of divergence after the first collision. In order for the collision to be nearly elastic, Δl₁ is required to be less than a specified threshold τ. An example of a numerical value of τ is /100.

[0505] The renormalization can change the width of the modified Zyoton such that the width of the renormalized Zyoton is reset to . Such renormalization can be achieved via truncation and/or re-sampling in the time domain. If the modified Zyoton is represented in the frequency domain, re-sampling can be implemented by performing an inverse Fourier transform of the frequency components of the modified Zyoton, followed by truncation and/or down-sampling, to adjust the width of the modified Zyoton. Then a Fourier transform followed by an amplitude sort of the frequency components can establish the amplitudes that may be used to obtain the renormalized Zyoton. The co-dependency condition requires that the amplitude vectors described above and/or the corresponding frequency components be selected during the synthesis of the Zyoton and the carrier kernel such that conditions described above in terms of the thresholds κ_DV1, κ_DV2, κ_DV3 and τ are satisfied.

[0506] The process for ensuring co-dependency is related to the process of synthesizing a carrier kernel, as described above, and to the process of generating one or more conditioned features. FIGS. 44A and 44B respectively show the distribution and a detailed profile of base frequencies used in the synthesis of one carrier kernel. FIG. 44A shows the plot of amplitudes of base sinusoidal frequencies as a function of a frequency index, that can be used to generate the carrier kernel. FIG. 44B illustrates the distribution of the first 16 frequency components as an example. A frequency index is created by sorting the frequency components on the basis of the amplitudes of their Fourier components.

[0507] In one embodiment, pure-component spectra of the analyte of interest over the wavelengths of interest are deconstructed into spectral fragments (also referred to as wavelength regions or spectral regions of). The boundaries of these spectral fragments generally correspond to those of spectral features to be used to generate conditioned features using the carrier kernel. The pure-component-spectra-based spectral fragments can be used to determine the values of one or more Kappa parameters described above. To this end, interpolation and/or pre-cursor modulation are optionally applied to the spectral fragment. A carrier kernel is modulated using the spectral fragment to obtain a conditioned fragment. The conditioned fragment is collided with a Zyoton, and a change in dispersion velocity of the modified Zyoton relative to the dispersion velocity of the original Zyoton is computed.

[0508] Noise such as randomly generated gray noise with a correlation time of 10⁻⁶ seconds may then be added to the spectral fragment. The amount of noise can be a fraction of (e.g., 2%, 25%, 50%, etc.), equal to, or a multiple of (e.g., 2, 5, 10, 100, 1000, 10000 times, or more) the expected and/or measured noise of the non-invasive measurement system. The above-described steps may then be repeated using the noisy spectral fragments to obtain corresponding changes in the dispersion velocity of the modified Zyoton. The steps of adding noise, generating a conditioned fragment, colliding it with a Zyoton, and computing a change in the dispersion velocity may be repeated for different amounts of noise. The parameter κ_DV1 may then be set according to these observed changes. For example, κ_DV1 can be a minimum, average, median, or maximum of different observed changes in the dispersion velocity. Other Kappa parameters discussed above can be set relative to κ_DV1.

[0509] FIGS. 45A through 45D illustrate the distribution of amplitudes of carrier kernel frequencies and their frequency components in the time and frequency domains. The profile of the full carrier kernel and a zoom-in to the high-amplitude frequency components are shown. FIG. 45A shows the time domain representation of a carrier kernel waveform generated using the frequencies shown in FIG. 44A. FIG. 45B provides a distribution of carrier kernel waveform frequency components, specifically a distribution of the amplitudes of these components, as a function of spatial frequency. FIG. 45C shows a time-domain representation of the waveform shown in FIG. 45A, down-sampled to 2048 data points. The down-sampling is optional and a down-sampled waveform may include less than 2048 (e.g., 2000, 1024, 1000, 600, 512, 200, etc.) or more than 2048 (e.g., 3000, 5000, 10,000, 40,960, etc.) points. FIG. 45D shows the distribution of the spatial frequencies of the down-sampled carrier kernel waveform and amplitudes of the frequency components as a function of spatial frequencies.

[0510] In ordinary frequency modulation (FM), a signal modulates the frequency of a carrier wave where the modulated output does not have any nonlinear dependence on the original signal. In various embodiments described herein, the carrier kernel is frequency modulated by a feature. The carrier kernel is created with several selected frequencies, and the result of modulation by the feature creates a nonlinear relationship to the feature. A non-linear relationship is achieved because the carrier wave is a summation of several frequencies instead of a single one. Other types of modulation, e.g., amplitude modulation, phase modulation, etc., and/or combinations of two or more modulation techniques can be used to modulate a carrier kernel using a feature.

[0511] In general, a carrier kernel includes a set of distinct carrier frequencies F_C=[f_c1, f_c2, . . . , f_cNk], where N_K is the number of component carrier frequencies in the kernel. Optionally, there are several harmonics of each frequency f_c1 embedded in the carrier kernel. As stated above, the feature data extracted from a spectrum has low spatial frequency content, generally due to a limited data vector length of the feature. This spatial frequency content is often insufficient to characterize changes in the feature properties due to low analyte concentration and interference from confounders. Coupling the feature with a complex carrier kernel with a high spatial frequency content, after an optional precursor modulation, can provide additional degrees of freedom to characterize the underlying analyte. Different features, associated with different wavelength regions, with different underlying analyte concentrations and corresponding absorption of energy, modulate the carrier kernel differently. When the carrier kernel, modulated by the feature (and thus transformed to a conditioned feature) with an expanded range of spatial frequencies, is collided with the Zyoton (which also has an expanded range of corresponding spatial frequencies), the analyte absorption properties can be extracted and amplified. In some embodiments, the same carrier kernel is used for all features, regardless of the wavelength regions from which those features are derived. Different carrier kernels and/or different Zyotons can be optionally used for different features.

[0512] A set of carrier frequency component amplitudes C=[C₁, C₂, . . . , C_NK] is associated with F_C. As described above in the discussion of Zyoton and carrier kernel synthesis, the choice of the frequencies f_ci for i=1, . . . , N_K, and N_K relate to the relationship between absorbing and non-absorbing feature wavelength regions in the Fourier transform of the analyte's pure component spectrum, and/or noise characteristics of the feature, and by the anticipated confounder interference, as expressed by the SCR. Specifically, the selection of at least some of the carrier kernel frequency components was described above, relating them to the number of components of a frequency-component based discriminant classifier required to overcome the introduced distortion of the second derivative of the pure component spectrum for an analyte of interest in the presence of confounders and added random noise. All the frequency components and their amplitudes can be provided to a signal generator or synthesizer to generate the carrier kernel waveform.

[0513] Spectral features need to be appropriately prepared before modulating a carrier kernel by those features. FIGS. 46A through 46C show a feature prepared in the absorbance domain and the distribution of spatial frequencies and their amplitudes in sorted order. In particular, FIG. 46A shows the absorbance spectrum of a feature, which can obtained by transforming an intensity spectrum obtained from a sensor, as described below. FIG. 46B shows all frequency components of the feature in the amplitude-sorted order, and FIG. 46C shows the first ten frequency components of the feature in the amplitude-sorted order.

[0514] When a carrier kernel is modulated by a feature waveform, a Fourier transform of the resulting conditioned feature typically includes more than N_K frequency components. In general, the modulated carrier kernel, i.e., the conditioned feature includes (k+m_CF+j_CF) frequency components. The first k frequency components represent the energy absorbed by the analyte and/or one or more confounders. The amplitudes of these frequency components are denoted a_k, a_mCF and a_jCF, as depicted in FIG. 11. The Zyoton with which the conditioned feature is collided generally includes (k+m_Z+j_Z) frequency components, and the amplitudes of the frequency components of the Zyoton are denoted A_k, A_mZ, and A_jZ, as also depicted in FIG. 11. The respective frequency components of the Zyoton and the conditioned feature, however, need not have exactly the same frequencies.

[0515] FIGS. 47A through 47D illustrate the distribution of frequencies and the normalized amplitudes of the frequency components of a conditioned feature after modulation of the waveform by a feature. All of the frequency components of the conditioned feature waveform, sorted by amplitude, are shown in FIG. 47A. The analyte-information-representing k components of the conditioned feature are shown in FIG. 47B. FIG. 47C zooms in to the transition zone or mid-amplitude range m components of the conditioned feature, and FIG. 47D zooms in to non-analyte-information representing j components of the conditioned feature. FIGS. 48A and 48B show a time-domain representation of the conditioned feature having a frequency domain profile shown in FIGS. 47A through 47D. A zoom-in to the overall time-domain profile of the conditioned feature waveform from FIG. 48A is shown in FIG. 48B.

[0516] FIGS. 12A through 12C show the morphological profile and amplitude envelopes of an example one-dimensional Zyoton, derived from solitons, and corresponding to the Zyoton_D1 (shown in FIG. 49). FIG. 12A shows a time-domain representation of the Zyoton and FIG. 12B shows a zoomed-in representation of the Zyoton. FIG. 12C shows the primary peak in the time domain, and selected b sideband peaks on both sides of the primary peak. FIGS. 50A through 50C show time-domain representations of other Zyoton waveforms that may be used in collision computing. Specifically, these Zyotons waveforms correspond, respectively, to Zyoton D3, Zyoton D1, and Zyoton MM1 (shown in FIG. 49).

[0517] FIGS. 51A through 51C show the frequency domain representation of the different Zyoton waveforms, i.e., Zyoton_D1, Zyoton_D3, and Zyoton_MM1. These frequency-domain representations correspond, respectively, to the time-domain representations shown in FIGS. 50A through 50C. FIG. 51D shows a zoom-in to the frequency components of Zyoton_MM1. FIGS. 52A through 52C show the profile of amplitude-sorted frequency components for the Zyoton waveforms Zyoton_D1, Zyoton_D3, and Zyoton_MM1, respectively.

[0518] In various embodiments, the parameters δ and φ (as described below in the collision equation), are applied to the Zyoton and/or conditioned feature waveform vectors. In various embodiments, for computational efficiency, Zyotons are expressed as one-dimensional traveling waves in the x, y or z dimension over time. The Zyotons may be synthesized, however, using complex-valued waveform families (e.g., solitons) and/or generator functions that are plane-wave, i.e., two-dimensional constructs (having x and y dimensions over time), or even three-dimensional constructs (having x, y, and z dimensions over time). As the corresponding Zyotons may be represented as one-dimensional traveling waveforms, a compensating mechanism may be employed for this dimensional simplification between Zyotons and the travel models of the waveforms from which they are derived. As an example, a general form of a two-dimensional soliton plane wave is S(x, t)=Ae^iρe^i(kx-ωt), where A is a positive constant called the amplitude, ρε[0, 2π) (which symbolizes that ρ can vary from 0 to 2π) is called the initial phase, k and ω are two real-valued parameters called the wave number and the angular frequency, and

[0519] $\frac{k}{2\pi }$
is the number of waves per unit length, while

[0520] $\frac{\omega }{2\pi }$
is the number of waves per unit time. For one-dimensional Zyotons, the dispersion velocity along the propagation axis is expressed as

[0521] $\frac{\omega \left(k\right)}{k}.$

[0522] In various embodiments, the collision operation is notionally one-dimensional, but the complete collision operations may need to be planar or three dimensional. Therefore, a phase rotation operation can capture the phase distribution effects. In some embodiments, a total phase rotation space of 2 π is set for the entire collision process. The total 2 π phase rotation is then divided by the number of collision iterations, to compute the rotation on a per collision iteration basis. This can compensate for the random variability in the collision process that may result from restricting the collision operation to a single dimension.

[0523] The power spectral density profile of the unscaled Zyoton (Zyoton_D1) is shown in FIGS. 53A-53B. An amplitude-sorted frequency component profile of a conditioned feature to be collided with Zyoton_D1 as shown in FIG. 53C. FIGS. 54A through 54C respectively show amplitude time-domain profile, frequency distribution, and power spectral density profile of Zyoton_D1 waveform, prior to a collision. The scaling coefficient α_Z was used match the spectral energy ratio of the Zyoton waveform to the spectral energy of the conditioned feature as described above. FIGS. 55A and 55B show the amplitude-sorted frequency component profile of a modified Zyoton waveform obtaining by colliding the scaled Zyoton_D1 with the conditioned feature described with reference to FIG. 53C. A zoom-in to the amplitude-sorted frequency component profile of the modified Zyoton is shown in FIG. 55B.

[0524] With reference to FIG. 56, in various embodiments, one or more of certain specific analysis objectives, 2a, 2b, 2c, 2d, and 2e are used for the selection of waveform families and/or waveform generators to be used in Zyoton and/or carrier kernel synthesis, and/or in the synthesis of Zyoton and/or carrier kernel waveforms. One or more of these objectives may also be used in selecting parameters for conditioning the extracted features, and the parameters defining the process of collision computing. The analysis objectives include: analyte detection sensitivity, i.e., how small a change in analyte quantity is required to be detected, the output concentration precision, and the accuracy of measurement with respect to a reference standard for the analyte.

[0525] One or more sensor parameters, such as resolution, sampling interval, dynamic range, sensor bandwidth, and the expected clutter absorption (i.e., expected signal-to-clutter ratio (SCR) increase that is required), may also be used for Zyoton and carrier kernel synthesis and/or in the selection of collision parameters and configuration of the collision computer. The specific Zyoton waveform selection depends on the expected required signal-to-clutter increase and the overlap between the signal and clutter observables. Both magnitude and shape similarity between the analyte and the confounders overall may drive the selection of the waveform families and/or generator functions used in Zyoton synthesis, as described above.

[0526] One or more of the sensor resolution, drift, precision, and noise characteristics may be used to set the number of frequency components of the synthesized Zyoton waveform to be used in the spectral energy computations as well as the number of frequency components of the carrier kernel waveform used for feature conditioning. The relative scale of peak energy of the Zyoton waveform and its co-dependent conditioned feature waveform are used in the determination of the scaling coefficients and scaling vectors for the conditioning process and for the collision process. The number of collision iterations required is generally proportional to the expected required SCR increase. In some embodiments, the number of collision iterations may be increased by a partial or full order of magnitude for each order of magnitude increase in the expected required SCR.

[0527] The number of collisions (also referred to as collision count parameter) may be set depending on the sensor signal-to-noise characteristics. In some embodiments, the number of collisions is initially set to 100 when the signal to noise ratio (SNR) of the sensor is 1000:1. As an example, the SNR of a spectroscopic standard can be determined by the ratio of power of a selected band (e.g. a 1 nm band) at a wavelength of interest (e.g., 1650 nm) to the ratio of background power in the same band using an NIR Diffuse Reflectance Standard with known reflectance value. The collision count parameter may be altered during calibration of the collision computer to accommodate sampling variability. The number of collisions is generally increased by an order of magnitude if the SNR of the sensor or SCR decreases by an order of magnitude. For example, in an embodiment for non-invasive glucose measurement in tissue, with SCR of 10⁻⁴, a collision count of 20,000 was used with a sensor SNR of 10³. In a similar manner, the number of collisions can be decreased if the SNR and/or SCR were to increase. The minimum number of collision iterations required is one. In various embodiments, the expected required SCR increase is determined by analyzing expected overlap and energy absorption due to dominant confounders which have spectroscopic absorption peaks in the same region as the analyte in the spectral bandwidth where the spectrum is acquired.

[0528] As described above, the specific properties of the collision process that are selected to estimate accurately the presence, absence, concentration, and/or rate of change of concentration over a fixed time interval (e.g., in seconds or minutes) of an analyte include the selection (or synthesis) of the collision waveform or Zyoton 10, which depends, at least in part, on the expected required SCR increase. This includes the selection of a base family of usable waveforms and the setting of one or more Zyoton parameters during synthesis thereof. Selecting the collision parameters, 20, includes selecting the number of collision iterations to be performed to assess the net absorption of energy due to the analyte of interest, and setting the collision operator, i.e., the delay shift, scaling, and phase adjustment parameters. The interaction between the frequency components of a conditioned waveform and a co-dependent Zyoton waveform generally provides a robust estimate of energy changes due to selective energy absorption by the analyte. In some embodiments, the net energy change is an energy gain in successive collision iterations. The collision computer can be configured and the colliding waveforms can be constructed such that the energy change in successive collision iterations is an energy loss.

[0529] In general, the collision between a Zyoton and a conditioned feature or between a Zyoton and a renormalized Zyoton resulting from the previous collision iteration creates a finite-range frequency component perturbation. As scattering due to the medium is inherently random, but statistically bounded in time and space, in some embodiments, Zyoton waveforms are selected that are soliton-based and which show absorption effects more strongly in the analyte-information representing k frequency components, which in some embodiments are high-amplitude components. The scattering effects are generally represented more strongly in non-analyte information representing j frequency components, which in some embodiments are low-amplitude components. The total set of frequency components of the Zyoton is described as H=k+m+j where k>0, m≧0, and j>0, and k and j frequency components generally do not overlap.

[0530] Such a Zyoton waveform can be constructed, as described above, using the pure component spectra of analytes (such as glucose) and pure component spectra of one or more confounders, including dominant confounders, and by modeling the scattering effects, if any, of the medium to be analyzed. The wavelength dependent absorption and scattering properties of the medium may be modeled in order to synthesize the Zyotons and carrier kernels to be used in the collisions. This knowledge from bio-optical models and chemistry generally yields suitable waveform families and generator functions for synthesizing Zyoton waveforms, and can also guide the number of frequency components that are used in the estimation of analyte-specific absorption energy.

[0531] Once the number of frequency components that can yield a robust estimate for analyte-specific absorption is established for the analyte-information representing (i.e., k), transition (i.e., m), and non-analyte information representing (i.e., j) regions of the Zyoton and carrier kernel waveforms, (e.g., k=6, 7, 10, etc., in some embodiments for non-invasive glucose measurement), the parameters of the collision grid can be computed. In addition, for computational efficiency and to limit the computer memory requirements, windowed and/or bracketed interactions may be used to implement the waveform collisions.

[0532] In various embodiments, the window length is set such that interaction of frequency components that are used in the computation of spectral energy changes due to the analyte-specific spectral absorption can be propagated from one collision to the next. In some embodiments, the window length is set such that the frequency component changes resulting from scattering effects are substantially eliminated during the truncation operation built into the collision process. In some embodiments, variations in the scattering properties of the medium may contain analyte information and, as such, at least some frequency components corresponding to the scattering properties of the medium may be retained. The renormalization performs a truncation according to the window length. Optionally, the collision parameters can be altered, 30, during the collisions. Specifically, one or more properties of Zyotons may be modified and the constraints or boundary conditions for the collision operation, e.g., the Kappa parameters described above, may be modified, to ensure that each collision is a valid interaction.

[0533] The overall process of constructing a collision computer can be summarized as follows: (a) Frequency components of the second derivative of a signal generated from pure component spectra, or frequency components of the pure component spectra themselves or of the first or other higher order derivatives of the pure component spectra, with added random noise may be used to determine the numbers (k, m, and j) of frequency components of the carrier kernel; (b) The same analysis can be used to determine the numbers (k, m, and j) of frequency components of the Zyoton, as well; (c) The spectral properties (e.g., bandwidth, morphology, etc.) of the signals and/or frequency components in step (a) above can be used to select a suitable Zyoton family; (d) The frequency of the peak component of the Zyoton can be any frequency within the family that supports the bandwidth of the signals and/or frequency components; (e) Frequencies of other Zyoton components can be selected according to the numbers k, m, and j.

[0534] Moreover, (f) Frequencies of the frequency components of the carrier kernel in some embodiments must match respective frequencies

[0535] $\left(e.g.,\mathrm{within}a\mathrm{value}\sigma ,\mathrm{where}\sigma \mathrm{is}\mathrm{defined}\mathrm{as}\text{:}\frac{\mathrm{BW}}{2\left(k+m+j\right)}\right)$
of the corresponding Zyoton components, where BW is the bandwidth of the carrier kernel (in Hz) and k, m and j refer to the number of frequency components of the carrier kernel as described above; (g) The amplitudes of the frequency components of the Zyoton and the carrier kernel determine the relative spectral energies of the Zyoton and the carrier kernel. (h) The amplitudes of the respective time-domain waveforms corresponding to the frequency components determine the respective velocities of the Zyoton and the carrier kernel.

[0536] During the collision iterations, truncation of a modified Zyoton, i.e., the removal of one or more frequency components of the modified Zyoton during the post-collision renormalization may cause the renormalized Zyoton (which is typically the “feature proxy” for the next collision), to be clipped. The amplitudes of the surviving components of the renormalized Zyoton frequency components are generally rescaled to reintroduce the effect of this clipping. In various embodiments, renormalization performs the rescaling (also called amplitude balancing).

[0537] Some Zyotons, however, have severe nonlinearities and such rescaling is difficult to apply to a modified Zyoton. Therefore, in some embodiments, the rescaling is applied to the original Zyoton waveform instead, to offset the clipping of the modified Zyoton that was performed to obtain the renormalized Zyoton, which is then collided with the original Zyoton as rescaled, in the next collision iteration. This may allow for a consistent, repeatable correction to be applied in each collision iteration, so as to ensure the co-dependency of the colliding waveforms. Truncation of the modified Zyoton (and downscaling thereof to remove the energy of the original Zyoton), to obtain a renormalized Zyoton, and a modification of the original Zyoton instead of amplitude rebalancing of the renormalized Zyoton, can improve the computational efficacy of the collision process and/or may reduce latency in producing the result for real-time analysis of sensor data.

[0538] In some embodiments, e.g., for glucose detection, the Zyoton properties are not changed between collisions, i.e., the same Zyotons are used. The collision operator is also not altered. In some configurations of the collision computer, however, the collision operator may be modified after each successive collision for certain applications. The selection of the parameters as described above generally specifies the collision computer 50.

[0539] As a Zyoton represents a traveling wave, its instantaneous amplitude coefficients (i.e., the absolute value of the magnitude of amplitude peaks as a function of different frequencies in the envelope of the waveform at a particular time where the envelope function of wave has a smooth curve outlining its extremes, as shown in FIGS. 12A-12C serve as initial conditions for analyzing propagation behavior through a propagation medium over time. In various embodiments, the Zyotons are synthesized such that numerical differences in propagation properties before and after collision are small and propagation of the Zyoton remains substantially unchanged in terms of the dispersion velocity and divergence thereof.

[0540] To ensure that Zyoton properties are preserved, i.e., the Zyoton is perturbed but only within certain limits, the post-collision velocity of the modified Zyoton is compared to that of the original Zyoton. Unless the difference is within a given threshold κ_DV3, the original Zyoton and the modified Zyoton are not acceptable for additional collisions, and the collisions are not “nearly elastic,” as described above. In some embodiments, the modified Zyoton is renormalized and the velocities of the original and renormalized Zyotons can be compared to ensure that the difference therebetween does not exceed the threshold κ_DV1. In addition, it is also tested whether divergence of the modified Zyoton is not greater than the threshold τ. If the kappa and tau tests are both met, the collision is determined to be nearly elastic, and the perturbation of the Zyoton is determined to be within limits specified by the kappa and tau parameters. In some embodiments, the collision operator is constructed such that the divergence test need not be performed explicitly. In various embodiments, the velocity and divergence tests may be performed in the time domain and/or in the frequency domain and, accordingly, the various kappa and tau parameters may take on suitable values.

[0541] If the velocity difference and/or the divergence are outside the respective thresholds, the Zyoton may be sub-optimal as a collision entity. A new Zyoton may then be selected or synthesized, or the feature data may have a gross artifact and may be examined. One or more new features may be obtained from additional samples.

[0542] In some embodiments, in addition to selecting/synthesizing Zyotons and carrier kernels and configuring the collision computer, the extracted features are conditioned for collisions using the process shown in FIG. 37. In particular, in some embodiments, a selected feature 4 is modulated in step 10 using a signal such as 16.5 Hz, 30 Hz, 75 Hz, 125 Hz, 500 Hz, 1 kHz, 40 kHz, 100 kHz, 260 kHz, 1 MHz, or a signal having even higher frequency. This modulation in step 10, called precursor modulation, is optional. The precursor modulation can be amplitude, frequency, or phase modulation or a combination of any two or all three of these. Alternatively, or in addition, in some embodiments, the feature is interpolated at step 10, to match its time-domain length with that of the carrier kernel. The interpolation of the feature is also optional. The feature (which may be pre-cursor modulated and/or interpolated) is used to frequency modulate a carrier kernel at step 30, to turn that feature into a conditioned feature. To this end, frequency modulation is performed.

[0543] As described above, in some embodiments, the spectral properties of one or more spectral fragments obtained from one or more pure component spectra are used in the synthesis of the Zyoton and/or carrier kernel. In general, a spectral fragment is the portion of a spectrum between two wavelengths within the total wavelength range of the spectrum. The wavelength boundaries of these spectral fragments may correspond to the wavelength boundaries of one or more features to be conditioned using the carrier kernel. In some embodiments, the spectral properties (e.g., bandwidth, frequency distribution, etc.) of the spectral fragments are obtained at step 20 by applying a Fourier transform to a spectral fragment of the pure component spectra of the analyte of interest and/or one or more confounders typically present in the medium to be analyzed. Examples of analytes and confounders include glucose, urea, fat, collagen, water, if the medium to be analyzed is tissue or blood; gases such as radon, helium, etc., if the medium or environment to be analyzed is the atmosphere of the earth, another planet, etc.; or a chemical, if the medium to be analyzed is water in a reservoir, ocean, etc.

[0544] In step 22, one or more carrier kernels are synthesized using the spectral properties obtained in step 20 and/or additional properties such as the expected required SCR increase, and/or the SNR of the system to be used for data acquisition and measurement. In some embodiments, these various properties are used to synthesize a carrier kernel directly, while in some embodiments, these properties are used to design a Zyoton. A Zyoton may be obtained from a database 50, and the properties thereof (e.g., the frequency distribution, amplitude profile, and numbers of k, m, and j components) may be determined at step 55. As described above, a carrier kernel may then be synthesized according to the properties of the Zyoton such that the carrier kernel and the Zyoton are co-dependent.

[0545] Based on the selection of a Zyoton to be used in collision, a co-dependent carrier kernel is selected, e.g., in terms of the frequency components and amplitude profile thereof, in step 24. The selected carrier kernel is modulated in step 30 using the feature, which may be precursor modulated and/or interpolated, to obtain a conditioned feature. The conditioned feature may be resampled at step 40 to match its length with that of the Zyoton. The resampling may result in removal of one or more frequency components that were not included in the carrier kernel and/or the Zyoton, but were introduced by the modulation in step 30. The modulation in the step 30 may be performed in the time domain or in the frequency domain.

[0546] The conditioned feature may then be adjusted to ensure co-dependency thereof with the Zyoton. If the adjustment is performed in the time domain, the time-domain amplitudes of the envelope peaks of the conditioned feature are adjusted such that the absolute difference between the velocity of the conditioned feature and the scaled velocity of the Zyoton is not greater than a kappa parameter κ_DV2. If the adjustment is performed in the frequency domain, the frequency domain amplitudes of the k, m, and j components of the conditioned feature are adjusted such that the absolute difference between the spectral energy of the analyte-information representing k components of the conditioned feature and scaled spectral energy of the k components of the Zyoton is not greater than a frequency-domain value of kappa parameter κ_DV2. Alternatively or in addition, the velocity/spectral energy of the conditioned feature may be scaled and, accordingly, a suitable kappa parameter can be used to adjust the conditioned feature. The numeric scaling coefficient that establishes the spectral energy of the Zyoton with respect to the spectral energy of the conditioned feature, is denoted α_Z.

[0547] In one embodiment for non-invasive glucose measurement, the various parameters described above were selected such that: (i) the spectral energy of the conditioned feature was adjusted to match 1/100,000^th the spectral energy of the Zyoton with which the conditioned feature was to be collided. (ii) the conditioned feature waveform was propagated on a 10,000 point grid, and (iii) in this embodiment, precursor modulation using a modulation signal at a frequency of 16.5 Hz was used in step 10.

[0548] The optional precursor modulation in step 10 is different from modulation of the carrier kernel in step 30. In systems having low instrument resolution, low SNR, and/or low SCR, (i) the number of k frequency components selected is typically greater than six, in order to create a collision computing system that may achieve the desired accuracy, and (ii) the frequency of the signal used for the optional precursor modulation in step 10 is also generally greater than 16.5 Hz, generally to bring the frequency components of the feature to a range comparable to those of the Zyoton and the carrier kernel. For example, the modulating frequency used for a system where the SNR is reduced by an order of magnitude and resolution reduced by a factor of two is 100 KHz.

[0549] In various embodiments, the collision grid is not related to the wavelength or wavenumber scale associated with the acquired data from which the features are generated. Instead, it is typically related to the maximum of the cardinality of the total set of Zyoton frequency components used in collision computing. The extracted feature can be treated as a one-dimensional data object including a vector of numbers. Once conditioned, the feature can represent an object on a spatial grid. The collision process is generally implemented on a spatio-temporal collision grid having a time dimension and at least on spatial dimension, such as x, y, or z dimension, that captures waveform propagation over time and a distance. The spatio-temporal collision grid can thus represent traveling waveforms, i.e., the Zyoton, the conditioned feature, and the modified and renormalized Zyotons. The number of spatial dimensions of the collision grid is equal to the number of spatial dimensions of the waveforms used in the collisions. Thus, if the waveforms are planar, the collision grid has two spatial dimensions. If the waveforms have three spatial dimensions, the collision grid also has three spatial dimensions and the time dimension. The conditioned feature, the Zyoton, the modified Zyoton, and the renormalized Zyoton, are all travelling waveforms on the spatio-temporal collision grid. A single collision iteration may analyzed over e.g., a 1,000; 2,000, 5,000; 10,000; 20,000; 100,000 point collision grid.

[0550] The carrier kernel and the Zyoton properties are related and the two are co-dependent, because, among other reasons as described above, they both depend on the expected required SCR increase and/or the SNR of the measurement system. Thus, both waveforms are dependent on the expected SCR increase required to discern energy loss due to absorption at the feature level. For some media/environments to be analyzed, the expected required SCR increase can range from four to six orders of magnitude using Zyoton frequency bandwidths of hundreds of kHz to tens of MHz. In some embodiments, the carrier kernel may optionally be frequency matched to the Zyoton or may have a higher frequency bandwidth.

[0551] As described above, a conditioning process is applied to all of the features of the acquired spectra, which generally involves frequency modulation of a carrier kernel. Unlike a constant-frequency carrier signal typically used in typical amplitude modulation (AM) or frequency modulation (FM), as in modulation of radio-frequency (RF) signals, the carrier kernel modulated by the feature is a complex waveform itself, having several frequency components having different amplitudes. A carrier kernel has three specific properties: (i) it is frequency matched, at least to some degree, to the Zyoton; (ii) the Zyoton is designed so that its properties are unchanged within limits specified by dispersion velocity and divergence parameters when collided with the conditioned feature, in terms of morphology and peak frequency components. As the preservation of Zyoton properties is a key design constraint, the carrier kernel modulated by each feature is selected such that Zyoton propagation after collision is not significantly altered, as specified by the dispersion velocity and divergence parameters; and (iii) the carrier kernel is constructed to mitigate interference from random noise, sampling variability, and other measurement transients in the feature and to improves the dynamic range of measurements.

[0552] Modulation of a carrier kernel generally provides smoothing of random noise in the feature and can increase dynamic range of measurement at the same time. An example of a waveform family used to generate a carrier kernels, denoted G(t), is:

[0553] $\begin{array}{cc}G\left(t\right)=\frac{1}{2\mathrm{\pi i}}{\int }_{-i\infty }^{i\infty }[{E\left(s\right)}^{-1}{e}^{\mathrm{st}}ds\left(-\infty <t<\infty \right),\mathrm{where}E\left(s\right)=\prod _1^{\infty }\left(1-\frac{s}{a_k}\right)e^{s/b_k},& \left(12\right)\end{array}$
and a_k=b_k+ic_k (k=1, 2, . . . ) being a sequence of complex numbers such that
Σ_k=1^∞(1/b_k)²<∞,and
Σ_k=1^∞(c_k/b_k)²<∞.
The constant k in the above equation (which is different from the number of analyte-information representing components k in a carrier kernel and/or a Zyoton), is set to 2000 in some embodiments so that the conditioning process can eliminate sampling-related transients that may occur.

[0554] A conditioned feature denoted as a Feature Waveform (FWi) in the collision equation below, i.e., a carrier kernel modulated using a feature, must optionally match the time-domain length of its paired Zyoton (Zi). But the two, the Zyoton waveform and the conditioned feature waveform, have typically different lengths in the time domain. To match the morphology, including the waveform lengths, a conditioned feature may be down-sampled as described above in step 40 (FIG. 37), e.g., from 10,000 points to 2,000 points. Additionally, or in the alternative, the Zyoton can be down-sampled to match the lengths of the two waveforms. In some embodiments, the conditioned feature and/or the Zyoton are up-sampled so that the lengths of their time-domain representations match. Optional truncation of FWi is implemented to prune the collision spatio-temporal grid, e.g., for computational efficacy. The choice of collision grid parameters is related to the selected properties of the Zyoton. For example, the length of the collision grid must be greater than the length of a time-domain representation of the Zyoton.

[0555] As described above, the co-dependency condition requires the change in post-collision dispersion velocity to be less than a threshold κ_DV1. The collision grid can establish a reference scale for the variable κ_DV1. In general, the parameter κ_DV1 is inversely proportional to both the time and space dimensions of the collision grid. This has the general effect of numerically establishing κ_DV1 as a target value for each system.

[0556] Amplitude adjustment of the conditioned feature is typically performed as described above, generally to adjust the amplitudes of the frequency components of the conditioned features FWi to a fraction of the peak amplitudes of all Zyotons that may be used in the collision. Amplitude adjustment completes the feature conditioning process, and is implemented for all features e.g., those extracted from all acquired spectra in a multiple illumination sequence (MIS), as described below.

[0557] The collision operation generally entails six steps. The first step is initialization of the collision time scale. The next step is computation of the collision-computing parameters. The generalization of the collision operator is given by equation (13):

[0558] $\begin{array}{cc}{\Omega \left({\stackrel{\_}{\mathrm{FW}}}_i,\stackrel{\_}{Z_l}\right)}_{t_k}=\{\begin{array}{c}\varepsilon \left(\eta \left(\varpi \left(\stackrel{\_}{Z_l},\stackrel{\_}{\alpha },\stackrel{\_}{\delta },\varphi \right)\otimes \rho \left({\stackrel{\_}{\mathrm{FW}}}_l,\stackrel{\_}{\beta },\stackrel{\_}{\delta }\right)\right)\right)\\ \varepsilon \left(\eta \left(\varpi \left(\stackrel{\_}{Z_l},\stackrel{\_}{\alpha },\stackrel{\_}{\delta },\varphi \right)\otimes \rho \left(\Omega ,\stackrel{⋓}{\beta },\stackrel{\_}{\delta }\right)\right)\right)\end{array}❘\begin{array}{c}k=1\\ k>1\end{array}& \left(13\right)\end{array}$
where Ω(FW_i, Z_l)_t1 is the result of a collision interaction between the feature wavefront FWi and Zyoton Z_l after the first collision. Also, Ω(FW_i, Z_l)_tk is the result of collision interaction between the renormalized feature wavefront resulting from the (k−1)th collision and Zyoton Zi.

[0559] The term k denotes collision iteration count in the above equation, and covers the case that includes only a single collision, as well as multiple (or =k>1) collisions, where is the total number of collision iterations, and is a collision-computing parameter.

[0560] Variable t_k ranging from 1 to k denotes the index on the grid for collision, with t_k, representing the kth point on the grid. k typically ranges from 1 to 100,000 depending on the signal-to-clutter increase desired.

[0561] Ω( are the renormalized results of (k−1)th collision that is used in the kth collision. The collision iterations can be performed in the time domain or in the frequency domain.

[0562] In the time domain, the collision-computing parameter α denotes the scaling vector. The scaling depends on the desired dynamic range in the post-collision energy change desired. In one embodiment, the scaling vectors are chosen to amplify the change in detected energy over a dynamic range that extends over eight orders of magnitude. Such a dynamic range can be required, e.g., if the original feature were extracted from a spectrum acquired from a human subject with tissue glucose concentrations varying between 20 mg/dl and 600 mg/dl. In the mathematics of Zyoton wavefront propagation dynamics, the scaling vector relates to the speed at which the zyoton wavefront propagates in media, some of which may be anisotropic or time-varying. In physical terms, the scaling vector α, can control the speed of the Zyoton wavefront propagation, and thus controls the amplification applied to the spectral energy of the conditioned feature during the collision operation. Values of elements of a typically range between 10⁵ and 10⁻⁵, depending on the amplitude of the Zyoton waveform peaks in the time domain.

[0563] The collision-computing parameters of delay and phase rotation can mitigate noise. Some portion of the noise may be correlated with the overall measurement system, e.g., the radiation emission and detection subsystem. Some portion of the noise, e.g., noise resulting from properties of the medium to be analyzed, may be uncorrelated to the overall measurement system. δ denotes the delay shift vector used to align the elements of the Zyoton and the conditioned feature wavefronts. The delay δ and phase rotations φ may be optionally applied to the time-domain representation of the Zyoton or conditioned feature. Zyoton morphology is related to the delay shift parameter, in that specific zyotons may have substantial numbers of closely-related frequency components that require an appropriate selections of δ. This delay is related, in the glucose example, to the actual spectral wavelength resolution of the instrument used to acquire the spectrum from which the feature is extracted, with lower resolutions requiring larger delay values. This same δ may be used if all features have the same cardinality in terms of the number of spectral elements, e.g., 32 points in the noninvasive glucose embodiment. If the cardinality or feature length is different, then a different δ is used for each feature. This delay can also capture the expansion of the frequency bandwidth in the original feature by the optional precursor modulation by the conditioning operation. Physically, δ compensates for the delay in transferring energy from principal frequency components of the original feature and those of the Zyoton. The numerical elements of the delta vector are related functions of the length of the Zyoton waveform in the time domain, and to the amount of sensor noise. If a delay is not desired, all values of δ are set to null. Proper δ selections allow precise time-scale alignment of the feature waveform and Zyoton in the computer memory prior to collision.

[0564] Collision-computing parameter φ represents the phase rotation applied to the Zyoton in the time domain, with values below π radians and typically ranging between 0.1 radian and 0.0001 radian. If no phase rotation is to be applied, then φ is set to 2 π radians. In addition to compensating for the dimensional deficiency of the zyoton described above, phase rotation is used to eliminate the impact of spectrum-to-spectrum variations of random noise in the data, changes in scattering which could be due to changes in the medium, and changes in the absorption of confounders whose absorption overlaps with that of the analyte in the spectral region as of the feature.

[0565] Optionally, β and {hacek over (β)} are used to insure that the kappa test is met in the time domain. Collision-computing parameter β denotes the optional scaling vectors for the conditioned feature wavefront, and can be used if the Zyoton is not scaled in performing the Kappa test. If scaling is not desired, all values of β are set to unity. The β scaling generally impacts the amplitude of the time-domain peaks of the conditioned feature waveform, and can adjust the velocity of the conditioned feature waveform. Implicitly, this results in the scaling of the frequency domain amplitudes of the conditioned feature, leading to an adjustment of the energy of the conditioned feature waveform.

[0566] Collision-computing parameter {hacek over (β)} denotes the optional scaling vectors for the renormalization of the modified Zyoton following the collision. If scaling is not desired, all the values of {hacek over (β)} are set to unity. This {hacek over (β)} scaling of amplitudes of the modified Zyoton is used to adjust its velocity in the time domain, and results in the scaling of frequency domain amplitudes of the renormalized modified Zyoton, and thereby and adjusts the energy of the Zyoton. In various embodiments, this scaling is done so that kappa tests, e.g. κ_DV1 and/or κ_DV3 tests, are met in the next collision iteration. The scaling vectors β and {hacek over (β)} are not necessarily related; the former is used in testing the velocity condition of the two co-dependency conditions, and the latter is used to adjust the velocity during renormalization.

[0567] The parameter ω represents the fully conditioned Zyoton vector prior to a collision, wherein the conditioning operation includes of scaling using α, and delay-shift δ and phase rotation φ operations. The parameters α, δ and φ are interrelated and chosen such that the spectral energy change during a collision between, for example, a feature with strong analyte signal and one with weak analyte signal content derived from the same spectrum could be separated by six to eight orders of magnitude. Calibration set data, using known concentrations of the analyte of interest in a calibration medium, can be used to derive these parameters. Improper or inconsistent selection will yield limited utility in detection of analytes in the presence of confounders and biological or instrument noise. If no scaling, delay shift or phase rotation is applied, then ω is set to unity. Typically ω is set to unity for in-vitro analysis embodiments where the medium is much simpler than in the tissue glucose embodiment.

[0568] The parameter ρ represents the fully conditioned feature-derived wavefront FWi vector prior to a collision, wherein the conditioning operation includes optional scaling using β, and delay-shift δ operations. The coefficients β and shift δ are selected in the similar manner as for the Zyoton Wavefront. The collision operator, Zyoton design and selection, feature conditioning and collision operator parameters are codependent. The parameters β and δ are interrelated and chosen such that the spectral energy change during a collision between for example, a feature with a strong analyte signal and one with a weak analyte signal content derived from the same spectrum can be separated by six to eight orders of magnitude during a collision. Calibration set data, using known concentrations of the analyte of interest in calibration media are used to derive these parameters. If no scaling or delay shift is applied, then ρ is set to unity.

[0569] As discussed herein, scaling takes two forms in the collision process. Prior to the first collision, the spectral energy of a Zyoton waveform is initially designed set to a multiple of the spectral energy of an expected conditioned feature. After a collision, the modified Zyoton is renormalized, which entails down-scaling in the frequency domain. The design of the carrier kernel and zyotons is an iterative process, and the initial waveforms chosen are refined as indicated by performance of the collision-computing process. The scaling vectors need to be computed prior to and after each collision, and are feature dependent.

[0570] The collision-computing parameter η denotes the bracket operator to which the delay-shift δ is applied during collision. It represents the localization parameter for the collision. Different bracket lengths, e.g., of 100, 1000, or 2000 may be used. For example, a bracket length of 2000 frequency components is used for non-invasive measurement of the concentration of some analytes. If no bracket operator is applied then, η is set to unity and the collision reduces to a phase rotation with a shift delay operation.

[0571] The collision-computing parameter ε denotes the compression operator applied to the collision process. This is used to squeeze the result (i.e., down-sample) the results of the wavefront to a fixed length post-collision. If no compression is applied, then ε is set to unity. The generalized collision operator Ω(FWi, Zi) achieves a localized excitation of the Zyoton wavefront, propagating in a system with constant velocity and colliding with a feature-derived wavefront such that the feature-wavefront presents a modulating influence on the stable Zyoton. The energy transfer, i.e., loss during the collision, is related to the energy of the feature wavefront.

[0572] The collision grid is established prior to the collision. Optional scaling of the Zyoton by α is also performed prior to the first collision. Any optional scaling of the conditioned feature waveform by β is also conducted prior to collision. Alignment of the frequency components, achieved via delay and phase shift operations, is performed during collisions. The alignment could optionally be performed before or after the collisions in the frequency domain as well. The collision-computing parameters, including phase shift, delay, bracket, compression are used during the collision process.

[0573] Bracketing in the time domain can be achieved by truncating the time-domain representations of a Zyoton waveform, a conditioned feature, and/or a renormalized Zyoton according to the time-domain length of the bracket. However, bracketing in the frequency domain is preferred. Alternatively or in addition, bracketing in the time domain can restrict the zyoton dimensionality and/or or phase. The selected points of a time-domain waveform generally result in a corresponding bracket in the frequency domain. As such, in various embodiments, bracketing is implemented in the frequency domain by selecting a subset of fixed number of frequency components k, m, and j of the conditioned feature. The frequency domain bracket length specifies the total number of frequency components of a waveform that are selected. In various embodiments, the k components are not removed during the truncation that is performed during the implementation of a bracket. A key difference between truncation as part of renormalization and frequency domain bracketing is that in renormalization the energy of the removed components is redistributed, while in bracketing such redistribution is not done because frequency components (typically the j components) from the Zyoton and the conditioned feature or renormalized Zyoton are removed.

[0574] During the overall collision process, the truncation operation is optionally performed at three steps: In step (i) during the feature conditioning process when convolving the feature with the carrier kernel for the purpose of making a conditioned feature and the Zyoton the same length in the time domain, and optionally for computational efficacy as truncation reduces the size of the collision grid; in step (ii) during the collision operation, through the bracketing operation in the frequency domain to constrain the set of resulting frequency components to capture energy changes in the spectral feature due to absorption by the analyte and to delete frequency components that are only related to random scattering losses; and in step (iii) during the renormalization step, in the frequency domain, to remove frequency components that not are not related to energy absorption by the analyte.

[0575] In some embodiments, the collision process may be implemented in the frequency domain, where the symbol represents a bracketed interaction operator which effects the collision process described below. The bracketed interaction typically involves from one to several thousand frequency components of the Zyoton and the conditioned-feature waveforms. For computational efficacy, efficient use of storage, and to immunize the system against random noise, typically only selected frequency components are used in the post-collision energy change computations. The bracketed interactions are set such that all frequency components in the conditioned feature waveform and original Zyoton that contribute to the computation of energy gain are used.

[0576] More importantly, the truncation window (or the number of frequency components retained after collision) is typically chosen to be long enough to accommodate all the frequency components of original waveforms that are likely to contribute to the computation of the energy gain due to the analyte over the complete set of collisions in each collision iteration and over all collision iterations. For example, in one embodiment where the energy computation uses 6 to 20 k-frequency components, but involves 20,000 collisions, the number of retained frequency components was chosen to be 1,000, after several trials, to amplify energy absorbed by the analyte while eliminating the absorption of energy by confounders and the effects of scattering and random noise.

[0577] The table below describes different stages of the overall collision process for different types of collisions. (All operations are assumed to occur in the frequency domain):

[0578]

TABLE 8
				Number of
	Number of	Number of	Number of	Frequency
	Frequency	Frequency	Frequency	Components in
	Components in	Components in	Components in	the
	the Original	the Conditioned	the Modified	Renormalized
	Zyoton (Z)	Feature (CF)	Zyoton (Z′)	Zyoton (Z″)
At Synthesis,	(k + m + j) = 10,000	(k + m′ + j′) =	(Z′) does not exist	(Z″) does not
Prior to Any		11,000		exist
Collisions
After the	(k + m + j) = 2,000	(k + m + j) = 2,000	(Z′) does not exist	(Z″) does not
Bracketing	Energy of the	Energy of the		exist
Performed Prior	8,000 frequency	9,000 frequency
to the First	components that	components that
Collision	are removed is	are removed is
	not redistributed	not redistributed
	across the	across the
	remaining 2,000	remaining 2,000
	components	components
After the First	With respect to	CF does not exist	(k + m″ + j″) =	(Z″) does not
Collision	the completed		2,200	exist prior to
	collision			renormalization
	iteration, Z does
	not exist.
	For the next
	collision
	iteration,
	previously
	bracketed Z can
	be extracted from
	memory
	(k + m + j) = 2,000
After the First	Same as above.	CF does not exist	(Z′) does not exist	(k + m + j) =
Renormalization,	(k + m + j) = 2,000		after	2,000
Which Is Same			renormalization	Energy of the
As Prior to the				200 frequency
Next Collision				components
Iteration				that are
				removed is
				redistributed
				across the
				surviving (m + j)
				components

[0579] Collision computing provides a mechanism whereby a source waveform is collided, (through the pseudo-convolution process described above in a computer processor) with a Zyoton, a waveform constructed to function as an energy amplification mechanism. The medium properties may be constant or time-varying (due to diffusion, flow, etc.). Also, the medium may be isotropic or anisotropic, and of uniform composition or spatially varying composition.

[0580] As described earlier, a Zyoton is a traveling waveform, i.e., a Zyoton has a temporal dimension and at least one spatial dimension. A representative Zyoton Z includes k high energy components having amplitudes and frequencies as: (A₁, Ω_Z1), (A₂, Ω_Z2), . . . , (A_k, Ω_Zk), where k>0; optionally, m medium energy components having amplitudes and frequencies as: (A_k+1, Ω_Zk+1), (A_k+2, Ω_Zk+2), . . . , (A_k+m, Ω_Zk+m), if m>0; and j low energy components having amplitudes and frequencies (A_k+m+1, Ω_Zk+m+1), (A_k+m+2, Ω_Zk+m+2), . . . , (A_k+m+j, Ω_Zk+m+j), where j>0. A representative conditioned feature CF that is to be collided with the representative Zyoton Z, where CF is also a traveling waveform, generally has at least k+m′+j′ frequency components: (a₁, Ω_CF1), (a₂, Ω_CF2), . . . , (a_k, Ω_CFk), (a_k+1, Ω_CFk+1), (a_k+2, Ω_CFk+2), . . . (a_k+m′, Ω_CFk+m′), (a_k+m′+1, Ω_CFk+m′+1), (a_k+m′+2, Ω_CFk+m′+2), . . . , (a_k+m′+j, Ω_{CFk+m′+j′}), where m′+j′≧m+j.

[0581] FIG. 17 illustrates the collision between a representative zyoton Z and a representative conditioned feature CF on a synthetic “collision space-time” domain grid. FIG. 18 symbolically depicts the synthetic-domain collision, where the wavefronts of Z and CF collide at a synthetic line of collision on the synthetic space-time grid at time t₀. In the real time domain, the velocity of a Zyoton is several times (e.g., a few hundred, a few thousand, a few hundred thousand times, etc.) greater than the velocity of the conditioned feature. Therefore, in the real time-domain collision, the conditioned feature is virtually stationary relative to the Zyoton, and during collision, the Zyoton sweeps across the conditioned feature.

[0582] In various embodiments, the Zyoton-conditioned feature collision is computationally implemented in the frequency domain. To this end, a magnitude spectrum of the frequency components of a Zyoton can be obtained by applying a Fourier transform to a time-domain Zyoton waveform, and, similarly, a magnitude spectrum of the frequency components of a conditioned feature can obtained by applying a Fourier transform to a time-domain conditioned feature waveform. Zyotons synthesized from generator functions, polynomial sequences, or one or more waveform families enumerated above, are first generated as numerical sequences, where the elements of the numerical sequence are considered to be analogous to a time domain representation of the Zyoton.

[0583] In various embodiments, the frequency-domain collision of Z and CF is implemented using the respective sorted magnitude spectra thereof, i.e., magnitude spectra sorted by amplitudes. In a frequency-domain collision, the frequency components of the Zyoton interact (“collide”) with selected components of the conditioned feature in a carefully controlled process called bracketed collision interaction. A pictorial representation of the collision process is shown in FIG. 57 in a synthetic grid, i.e., regardless of the domain of implementation, where the components of the zyoton Z are shown as moving from left to right, and the components of the conditioned feature, CF move from right to left toward a line of collision or interaction at the center.

[0584] Each successive row, moving from top to bottom, represents one “time unit” of this synthetic grid, i.e., the in the top row, the two components Z1 and CF1 have not yet interacted; on the second row Z1 and CF one have interacted, indicated by “Z1CF1” in the center. The “Z1CF1” designation and similar designations “ZiCFi” in the subsequent rows in FIG. 57 does not imply a conventional convolution or a simple multiplication of the two elements. Rather this designation symbolically represents the collision interaction between elements Zi and CFi. In actuality, the collision interaction with each Zi involves more than one conditioned feature components, as controlled by the collision bracket length, as described below, but in the symbolic representation only CFi is shown for clarity. At the right of the second row is shown a new frequency component, Z′1, of the modified Zyoton. Here again, in actuality as described below, a number of new modified Zyoton components may be produced in each collision interaction, but only one is shown in this symbolic representation for clarity.

[0585] Each pair of components interacts successively as described below, producing new frequency components of the modified zyoton, Z′2, Z′3, etc., from the interaction. These new components include the energy transferred from the conditioned feature to the modified Zyoton during the collision process. FIGS. 46B-47C depict a sorted (in descending order) magnitude spectrum of a Zyoton. The Y axis shows the amplitudes and the X axis shows frequencies according to an index and not actual frequencies because in a sorted magnitude spectrum, the frequencies are typically not in a sorted order.

[0586] In the representations of Zyotons and conditioned-feature described above, without the loss of generality:
A₁>A₂> . . . >A_k>A_k+1>A_k+2> . . . >A_k+m>A_k+m+1>A_k+m+2> . . . >A_k+m+j; and
a₁>a₂> . . . >a_k>a_k+1>a_k+2> . . . >a_k+m′>a_k+m′+1>a_k+m′+2> . . . >a_k+m′+j′.
In general, for a component index ν, amplitude A_ν>A_ν+1 and amplitude a_ν>a_ν+1, but Ω_Zv can be greater than or less than Ω_Zv+1 and, similarly, Ω_CFv can be greater than or less than Ω_CFv+1.

[0587] The Zyoton and the conditioned feature to be collided therewith are constructed, however, such that for each of the k components: |Ω_Zνk−Ω_CFνk|≦ε_kΩ_Zνk, ν^k==1, . . . , k. In various embodiments, ε_k can range from 0.0001 up to 0.0005. Thus, if ε_k=0.005, for each k component, having a component index ν^k, the frequency of the conditioned feature, denoted Ω_CFνk, is within (i.e., neither greater than nor less than) 0.05% of the frequency of the Zyoton, denoted Ω_Zνk. Furthermore, for each of the m components: |Ω_Zνm−Ω_CFνm|<ε_mΩ_Zνm, ν^m=(k+1), . . . , (k+m), and for each of the j components: |Ω_Zνm−Ω_CFνm|<ε_mΩ_Zνm, ν^j=(k+m+1), . . . , (k+m+j). Collectively, we define the constraint imposed by ε_k, ε_m and ε_j as Epsilon Tests. In various embodiments, ε_m can range from 0.001 up to 0.025, and ε_j can range from 0.001 up to 0.1. Thus, if ε_m=0.025, for each m component, having a component index ν^m, the frequency of the conditioned feature, denoted Ω_CFνm, is within 2.5% of the frequency of the Zyoton, denoted Ω_Zνm. Similarly, if ε_j=0.1, for each j component, having a component index ν^m, the frequency of the conditioned feature, denoted Ω_CFνj, is within 10% of the frequency of the Zyoton, denoted Ω_Zνj.

[0588] In some situations, a conditioned feature CF may have extra m and/or j components. In one example, the Z has six k components (Z1-Z6), three m components (Z7-Z9), and five j components (Z10-Z14). The CF to be collided with Z has six k components (indexed CF1-CF6), but four m components (CF7-CF10), and six j components (indexed CF11-CF16). In such cases the frequency relationship |_ΩZνm−Ω_CFνm|<ε_mΩ_Zνm and |_ΩZνj−Ω_CFνj|<ε_jΩ_Zνj is imposed as before for the corresponding Z and CF components identified by the indices ν^m and ν^j. For some CF components, however, there is no corresponding Z component. In the foregoing example, CF10, an m component, does not correspond to Z10, which is a j component of the Zyoton. The Epsilon Tests are nevertheless satisfied in various embodiments for these unmatched components by ensuring that the frequency of an unmatched CF component of the “m” group is within ε_m of at least one m component of Z. Similarly, it is ensured in various embodiments that the frequency of an unmatched CF component in the “j” group is within ε_j of at least one j component of Z. For example, CF10 must be within ε_m of any one of Z7 through Z9, and each of CF15 and CF16 must be within ε_j of any one of Z10 through Z14. If this condition is not met, the kappa test(s), described elsewhere, may fail and a redesign of either the zyoton, the carrier kernel, or both, may be required.

[0589] The conditioned feature is obtained via a frequency modulation of the carrier kernel by the feature. The feature may have been optionally frequency modulated, using a pre-cursor modulation signal such as a sinusoidal signal having a frequency of 16.5 Hz, 60 Hz, 100 Hz, 1.2 kHZ, 100 kHz, 175 kHz, etc. Even if frequencies of all k (and m and/or j) components of the carrier kernel are initially selected to be identical to the corresponding Zyoton component frequencies, this modulation can create slight differences between the frequencies of the k (and m and/or j) components of the conditioned feature and those of the unmodulated carrier kernel. The amplitude/magnitude sorted Zyoton and conditioned feature, where the frequencies of the respective Zyoton and conditioned feature components are related according to ε_k, ε_m, or ε_j are denoted F(Z) and F(CF), respectively.

[0590] The frequency-domain collisions are implemented on a frequency domain collision grid, with collision coefficients including a collision grid length K≦(k+m+j) and a bracket length CB<(k+m+j). If the collision begins at the interaction step g₀ on the collision grid, at step g_(i) the frequency component F_Z(i) of the Zyoton F (Z) interacts with a corresponding frequency component F_CF(i) of the conditioned feature F (CF). Note that these interaction steps do not necessarily correspond to the real time associated with the traveling waveforms in the time domain. In a computer implementation, the time period between successive grid points, i.e., computation steps can be set to the reciprocal of the length of the Zyoton frequency vector used in the collision times the square of the length of the collision bracket, i.e.

[0591] $\mathrm{time}\mathrm{period}=\frac{1}{\left(k+m+j\right)*{\mathrm{CB}}^2}.$
“Time” here is the duration of computation/interaction steps involved in a collisions that includes all interactions between Zyoton and conditioned feature frequency components during a single iteration of the overall collision process. A single collision iteration can produce a maximum of (K−δ)*CB frequency components of the modified Zyoton, where δ represents a shift-delay. If no shift-delays are needed, δ is equal to zero. The frequency-domain collision grid can therefore be represented as a numerical vector of time points or indices {g₀, g₁ . . . g_(K-δ)*CB}. This number of interaction steps in a single collision iteration is different from the number of steps required in an element-by-element multiplication, and in a conventional convolution, as well, for the reasons described below.

[0592] The collision interactions can be carried out in configurations ranging from completely parallel implementation, where each Zyoton frequency component interacts simultaneously with each conditioned feature frequency component from the corresponding bracketed set of CF frequency components, to a completely sequential implementation, where each operation in each interaction occurs one at a time, in sequence. In various embodiments, the number of grid points of the collision grid in the frequency domain can vary depending on the computational configuration employed, but more grid points are generally required as the configuration becomes more sequential.

[0593] The collision-computing coefficient, δ≧0, is an index shift parameter (i.e., it shifts frequency components of the Zyoton) or a delay coefficient in frequency domain collisions. If δ takes values greater than 0, it can result in shifts of the Zyoton frequency components that interact with the frequency components of the conditioned feature.

[0594] With reference to FIGS. 38-41, in one example, N=k+m+j=10 and CB=4, and δ=0, and, as such, collisions can be performed for the Zyoton components Z1 through Z7. The collision grid length K=N−CB+1 is 7, for a partially sequential, partially parallel implementation. The minimum number of interaction steps, however, is (K)*CB where each frequency component interaction is performed in a sequential manner over all K Zyoton components and associated K brackets of CF components; or K where each CF component of the collision bracket interacts with the corresponding Zyoton component in parallel. In this example, the shift coefficient δ is zero. For computational efficacy, a collision grid can be set to be evenly spaced with a time interval of 1 picosecond, 1 nanosecond, 1 microsecond, 1 millisecond, or 1 second, up to 1 minute. In some embodiments, the time interval is set to be less than

[0595] $\frac{1}{\left(k+m+j\right)*{\mathrm{CB}}^2}$
seconds.

[0596] The result of a collision interaction between the Zyoton Z and the conditioned feature CF according to various embodiments, when converted back to the time domain, is a non-invertible combination of the Zyoton and the conditioned feature. In the frequency domain, this interaction can be described as bracketed conditional amplitude multiplications between each Z and a set of CF components, and conditional summations of the amplitudes of two or more components resulting from the bracketed multiplication. As described below, the conditions for the conditional amplitude multiplication and for the conditional amplitude summations are whether the frequencies of interacting components match.

[0597] For the conditional multiplication, the amplitudes of a pair of Z and CF components are multiplied if and only if the frequencies of those two components satisfy an applicable Epsilon test. The applicable Epsilon value (i.e., ε_k, ε_m, or ε_j) is determined according to the energy group (i.e., k, m, or j) to which the Z component of the interaction belongs. Similarly, for a conditional summation the amplitudes of two or more resulting components of the overall collision interaction are summed if and only if the frequencies thereof satisfy the Epsilon test applied to the Zyoton component with the largest magnitude. The conditional multiplications and summations generally makes the collision interaction non-invertible. While conventional modulations techniques are generally invertible, the overall collision process described herein, which may additionally include scaling, shifting, and/or phase rotation of frequency components, is not a conventional modulation, and is generally not invertible.

[0598] Post-collision, the modified Zyoton generally includes new frequency components that were not present in the original Zyoton and the conditioned feature. Interactions between the Zyoton and the conditioned feature described below may introduce new frequency components, but because of the amplitude mismatch between the frequency components of the Zyoton and those of the conditioned feature, the amplitudes of those new frequency components are typically smaller than the amplitudes of the k components of the Zyoton. The renormalization process is designed to remove some of these components via truncation.

[0599] The frequency-domain collision of the Zyoton and the conditioned feature may then be expressed in the form of a pair-wise (i.e., component by component) conditional multiplication of the (optionally shifted, scaled, and/or phase-rotated) components of the zyoton by all the components of the conditioned feature within a bracket. This can be expressed as:

[0600] $\begin{array}{cc}{F}^{\prime }\left(Z^{\prime }\right)={\coprod }_1^k\left({\coprod }_{t-\delta }^{t-\delta +\mathrm{CB}}\left({\stackrel{\_}{\alpha }}^f\left({\varphi }^F\left(Z_{\left(g_{\left(t-{\delta }^f\right)}\right)}\right)\right)\right)\otimes {\mathrm{CF}}_{\left(g_{\left(t\right)}\right)}\right)),t=0,1,\dots ,\left(k+l+m\right)-\mathrm{CB}& \left(14\right)\end{array}$
where F′ (Z′) is the unsorted magnitude spectrum of modified Zyoton frequency components produced by the collision, and may include additional frequency components. The operator ␣₁^CB( . . . ) represents the set of conditional co-products that includes the product terms resulting from the interactions between a single Zyoton component Z_g with all the conditioned feature components CF_g, CF_g+1, . . . , CF_(g+CB−1) over the bracket length CB, and the operator ␣₁^K(..) represents the conditional co-product over the interactions of all the components of the Zyoton Z_g1, Z_g2, . . . , Z_K, with all the conditioned feature components within the respective brackets [g₁, g_1+CB−1], [g₂, g_2+CB−1], . . . , [g_K, g_K+CB−1] that occur during all the steps of a single collision. The term co-product generally refers to one or more multiplications and sums of the products. Unlike the conventional co-product, multiply-add, and/or multiply-accumulate operations, however, these co-products are conditional, as described herein.

[0601] A bracketed conditional multiplication operator, , of Z_gp and [CF_gp, CF_gp+1, . . . , CF_(gp+CB−1)] can yield [Z′_gp, Z′_gp+1, . . . Z′_(gp+CB−1)] as follows:

[0602] Case (a) Zgp is a k component: By the construction of Z and CF, as described above, the frequencies of the pair Ω_Zgp and Ω_CFgp must satisfy the ε_k test. Therefore, the frequency of the result of an interaction between Z_gp and CF_gp, which would produce the modified Zyoton component Z′_gp, is designated the frequency Ω_Zgp; thus, Ω_Z′gp=Ω_Zgp. The amplitude of Z′_gp is computed as A′_gp=A_gp×a_gp.

[0603] For the remainder of the collision bracket, i.e., for each CF component CF_gp+l, where l=1 . . . CB−1, unless the frequencies Ω_Zgp and Ω_CFgp+l satisfy the ε_k test, no interaction between Z_gp and CF_gp+l occurs. If the ε_k test would succeed for a particular value of 1, by construction, Ω_Zgp+l would be within ε_k times Ω_Zgp. In this situation, the shift-delay operator, as described below, may be applied and, interaction of the collision bracket may not occur with either Z_gp and Z_gp+l. Thus, in general, for the k components, Z_gp interacts only with CF_gp in the collision bracket and not with CF_gp+l, l=1 . . . CB−1. Thus, no new k components are generated, in general.

[0604] Case (b) Zgp is an m component: By the construction of Z and CF, as described above, the frequencies of the pair Ω_Zgp and Ω_CFgp must satisfy the ε_m test. Therefore, Z_gp and CF_gp would interact, producing a modified Zyoton component Z′_gp. The amplitude of Z′_gp is computed as A′_gp=A_gp×a_gp. If the frequencies Ω_Zgp and Ω_CFgp satisfy the stricter ε_k test, the frequency of Z′_gp is designated the frequency Ω_Zgp; thus, Ω_Z′gp=Ω_Zgp. Otherwise, the frequency designated to of Z′_gp is the sum of Ω_Zgp and Ω_CFgp.

[0605] For the remainder of the collision bracket, i.e., for each CF component CF_gp+l, where l=1 . . . CB−1, unless the frequencies Ω_Zgp and Ω_CFgp+l satisfy the ε_m test, no interaction between Z_gp and CF_gp+l occurs. If the frequencies satisfy the ε_m test, a new modified Zyoton component Z′_gp+l is generated, and the amplitude thereof is computed as A′_gp=A_gp×a_gp. If Ω_Zgp and Ω_CFgp+l satisfy the ε_k test, the frequency of Z′_gp+1, i.e., Ω_Z′gp+l, is set to Ω_Zgp; otherwise Ω_Z′gp+l=Ω_Zgp+Ω_CFgp+l. If all CF components in the collision bracket satisfy the ε_m test, CB−1 new modified Zyoton components would be created. If no CF component in the collision bracket satisfies the ε_k test, however, all of the newly created modified Zyoton components would be designated frequencies according to Ω_Z′gp+l=Ω_Zgp+Ω_CFgp+l. Unless all of these frequencies strictly match (i.e., they satisfy the ε_k test) with frequencies of the original Zyoton and/or conditional feature, the modified Zyoton would have one or more new frequencies. Often, one or more but not all of the CF components of the bracket may interact with Ω_Zgp and one or more new frequencies may be generated.

[0606] Case (c) Zgp is a j component: By the construction of Z and CF, as described above, the frequencies of the pair Ω_Zgp and Ω_CFgp must satisfy the ε_j test. Therefore, Z_gp and CF_gp would interact, producing a modified Zyoton component Z′_gp. The amplitude of Z′_gp is computed as A′_gp=A_gp×a_gp. If the frequencies Ω_Zgp and Ω_CFgp satisfy the stricter ε_k test, the frequency of Z′_gp is designated the frequency Ω_Zgp; thus, Ω_Z′gp=Ω_Zgp. Otherwise, the frequency designated to of Z′_gp is the sum of Ω_Zgp and Ω_CFgp.

[0607] For the remainder of the collision bracket, i.e., for each CF component CF_gp+l, where l=1 . . . CB−1, unless the frequencies Ω_Zgp and Ω_CFgp+l satisfy the ε_j test, no interaction between Z_gp and CF_gp+l occurs. If the frequencies satisfy the ε_j test, a new modified Zyoton component Z′_gp+l is generated, and the amplitude thereof is computed as A′_gp=A_gp×a_gp. If Ω_Zgp and Ω_CFgp+l satisfy the stricter ε_k test, the frequency of Z′_gp+l, i.e., Ω_Z′gp+l, is set to Ω_Zgp; otherwise Ω_Z′gp+l=Ω_Zgp+Ω_CFgp+l. If all CF components in the collision bracket satisfy the ε_j test, CB−1 new modified Zyoton components would be created. If no CF component in the collision bracket satisfies the ε_k test, however, all of the newly created modified Zyoton components would be designated frequencies according to Ω_Z′gp+l=Ω_Zgp+Ω_CFgp+l. Unless all of these frequencies strictly match (i.e., they satisfy the ε_k test) with frequencies of the original Zyoton and/or conditional feature, the modified Zyoton would have one or more new frequencies. Often, one or more but not all of the CF components of the bracket may interact with Ω_Zgp and one or more new frequencies may be generated.

[0608] After the bracketed interactions for all Zyoton components are generated, the components of the modified Zyoton are amplitude sorted, and designated as k, m, j components according to the amplitudes thereof. Thereafter, all of these components are compared in a pairwise manner. For a component designated as a k component, if the frequency of that component matches with the frequency of any other component, i.e., the frequencies satisfy the ε_k test, those two components are merged. To this end, the amplitudes of these two components are summed the sum is set as the amplitude of one of the components in the matching pair, and the other component is removed. In some embodiments, the component that is retained is designated the greater of the frequencies of the components of the pair. In some embodiments, the lesser frequency may be designated to the retained component, while in some embodiments, the frequency of the retained component is not changed.

[0609] In the pairwise comparisons, for a component designated as an m component, the frequency comparison and conditional merging, i.e., conditional sum of the amplitudes, is performed as described above, except the ε_m test is applied instead of the ε_k test. In the pairwise comparisons, for a component designated as an j component, the frequency comparison and conditional merging, i.e., conditional sum of the amplitudes, is performed as described above for a k component, except the ε_j test is applied instead of the ε_k test. The retained components may be amplitude sorted, yielding a modified Zyoton.

[0610] To illustrate the collision interactions, with reference to FIGS. 58A-58D, for the Zyoton component Z1, the collision bracket includes the CF components: {CF1, CF2, CF3, CF4}. With reference to FIG. 58B, the only collision interaction that occurs for the component Z1, however, is Z1CF1, because CF1 is the only CF component in the bracket that satisfies the ε_k test. For the Zyoton component Z2, the collision bracket includes the CF components {CF2, CF3, CF4, CF5}. Only the Z2CF2 interaction occurs, because CF2 is the only component in Z2's bracket that satisfies the ε_k test with respect to Z2. Similarly, for the Zyoton component Z3, only the Z3CF3 interaction occurs. In each of these interactions, the amplitude of the resulting modified Zyoton components, denoted Z1′, Z2′, Z3′, respectively, are the respective products of the amplitudes of the corresponding Z and CF components. For example, the amplitude of Z1′ is the product of the amplitudes of Z1 and CF1. The frequency of Z1′ is set to be the frequency of Z1 because, by definition, the frequencies of Z1 and CF1 satisfy the ε_k test.

[0611] In FIG. 58B, this is illustrated by the computation of Z′1, Z′1_1, Z′1_2, and Z′1_3. As the bracket length is 4, the collision bracket corresponding to Z1 includes four CF components, namely {CF1, CF2, CF3, and CF4}. By construction, the frequencies of Z1 and CF1 satisfy the ε_k test and, as such, Z′1 is designated the frequency of Z1, 260 kHz. In some embodiments, the frequency of the Zyoton component is chosen as the frequency of the modified Zyoton component, if a k component of the Zyoton is involved in the collision interaction. Otherwise, if an m or j component of the Zyoton is involved, the frequency of either the Zyoton component or the conditioned feature component may be chosen as the frequency of the resulting modified Zyoton component. The amplitude of Z′1 is the product of the amplitudes of Z1 and CF1. Z1 conditionally collides with the other CF components in the bracket, but the frequencies of these CF components do not satisfy the ε_k test when compared to the frequency of Z1 and, as such, the results of these unperformed interactions are shown as shown as “--” entries. The interaction of Z1 and CF2 fails to meet the ε_k test and, hence, the corresponding entry Z′1_1 is designated “--”; similarly, Z′1_2 and Z′1_3 are also designated “--”. The frequency of Z′1 is the frequency of Z1, and no k components having new frequencies are created.

[0612] With reference to FIG. 58C, for the m component Z5, the collision bracket includes the CF components {CF5, CF6, CF7, CF8}. The components Z5 and CF5 interact because their frequencies satisfy the ε_m test. There is no interaction between Z5 and the components CF6 and CF7, because the pairs (Z5, CF6) and (Z5, CF7) do not satisfy the ε_m test. The components Z5 and CF8 interact, however, because they satisfy the ε_k test. Thus, the collision of Z5 with the CF components within Z5's collision bracket produces two modified Zyoton components, namely, Z5′ and Z5′_3. The amplitudes of these two modified Zyoton components are determined by the product of the amplitudes of Z5 and CF5 and the product of the amplitudes of Z5 and CF8, respectively.

[0613] The components Z5 and CF8 satisfy the stricter ε_k test and, as such, the frequency of Z5′ is set to be the frequency of Z5. While the components Z5 and CF8 satisfy the ε_m test, they do not satisfy the ε_k test and, as such, the frequency of Z5′_3 is set to be the sum of the frequencies of Z5 and CF8. As such, a modified Zyoton component having a frequency different from the frequencies of any of the Zyoton and conditioned feature components is created. The amplitudes of these new components are most likely to be within the m region of the modified Zyoton, but depending on the amplitudes of Z5, CF5, and CF8, one or both of these amplitudes may be in the j region of the modified Zyoton. In general, because the m-component amplitudes are much lower than the k-component amplitudes, the new component magnitudes typically do not reach the magnitude region of the k components.

[0614] With reference to FIG. 58D, for the j component Z7, the collision bracket includes the CF components {CF7, CF8, CF9, CF10}. Here again, by construction, the frequencies of Z7 and CF7 satisfy the ε_j test and, as such, an interaction between these two components is permitted. That interaction produces the modified Zyoton component Z′7. The components CF8 and CF9 do not satisfy the ε_k test with respect to Z7 and, as such, these two CF components do not interact with Z7. The component CF10 satisfies the ε_k test, however, and interacts with Z7, producing the modified Zyoton component Z′7_3.

[0615] As described above, the amplitude of Z′7 is the product of the amplitudes of Z7 and CF7, and the amplitude of Z′7_3 is the product of the amplitudes of Z7 and CF10. Neither CF7 nor CF10 satisfies the stricter ε_k test and, as such, the frequency of Z′7 is set to be the sum of the frequencies of Z7 and CF7 (i.e., 510 kHz), and the frequency of Z′7_3 is set to be the sum of the frequencies of Z7 and CF8 (i.e., 480 kHz). While 480 kHz is a new frequency, the frequency of Z′5 is 500 kHz. Z′5 is likely an m component of the modified Zyoton, and the components Z′5 and Z′7 satisfy the ε_m test. Therefore, these two components can be merged. As Z′5 is the larger component by amplitude, the amplitude of Z'S is reset as the sum of the initially computed amplitude of Z′5 (i.e., 1482) and the amplitude of Z′7 (i.e., 12.54). Thus, the new amplitude of Z′5 is set to be 1494.94. The merged component Z′7 is removed from the modified Zyoton. In this example, the frequency of the retained component Z′5 is not modified, and remains to be 500 kHz. In some embodiments, the larger of the frequencies of the two components may be designated to the retained components. In other embodiments, the smaller frequency may be designated, and in some embodiments the frequency of the removed component may be designated to the retained component.

[0616] In general, for each of K Zyoton components, up to CB modified Zyoton components may be generated initially, as described above, for a total of (K*CB) initial modified Zyoton components, where CB is the bracket length. Frequencies of these components are checked pairwise to determine pairs having matching frequencies, where the match can be determined according to the applicable Epsilon test. To determine the applicable Epsilon test, for a pair of components of the modified Zyoton that are under evaluation for a possible frequency match, the energy region (i.e., the k, m, or j region) to which the component having the greater amplitude likely belongs is determined. If the component likely belongs of the k region, the ε_k test is applied. If the component likely belongs to the m region, ε_m test is applied. Otherwise, the ε_j test is applied. The matching pairs are merged, similarly as Z′5 and Z′7 are merged, by summing the respective amplitudes thereof. Two or more initially generated modified Zyoton components may be conditionally merged together. In various embodiments, the final components of the modified Zyoton, obtained after conditional merging, are sorted by amplitudes. The conditional merging typically removes one or more modified Zyoton components. The renormalization process describe above may additionally remove one or more components of the modified Zyoton and may redistribute the energy of such additionally removed components.

[0617] As described above, these collision interactions, even during a single iteration of the overall collision process, typically results in new modified Zyoton frequency components having frequencies that are not present in the original Zyoton and/or the conditioned feature in the j and m regions.

[0618] In general, in the bracketed interaction, the Zyoton components are ordered by amplitudes thereof and, as such, gp through gp+K−1 correspond to the component indices of Z. One or more new frequencies may be associated with the modified Zyoton components identified by the indices Z′_gp1 through Z′_{gp_(CB−1)}, for all gp in the range from 1 through K. All of these modified Zyoton components may be sequentially indexed from 1 through K+U, where U is the number of new components of the modified Zyoton in the m and j regions. All of the K+U components of the modified Zyoton are not necessarily amplitude-sorted after the collision interactions. As such, the collision of Z and CF produces modified Zyoton components that are not necessarily magnitude sorted. In some embodiments, prior to renormalization, these components are sorted according to the amplitudes thereof. The frequencies of the k components of the Zyoton Z are preserved during the generation of the modified Zyoton, as described above, and these k components are not removed by truncation during renormalization.

[0619] In Expression (14) for the collision operation, optionally, one or more of the collision computing parameters: shift delay δ^f; scaling vector α^f; and phase shift φ^f; are applied to the Zyoton frequency components in the frequency domain. The superscript “f” indicates that the collision computing parameters are frequency-domain coefficients; and each Ω represents a frequency component undergoing interaction at a time point on the collision grid. Sorting the components of F′ (Z′) on a descending amplitude order yields the resulting magnitude spectrum of the modified Zyoton F′ (Z′).

[0620] As shown in the expression for F′(Z′), the collision computing parameter shift delay δ^f is optionally introduced as an integer shift delay in the index of the Zyoton components that interact with the components of conditioned feature or modified zyoton. Values of δ^f can range from 0 thru k−1 where k is the number of k-frequency components. As an example, shift delay δ^f may be used in Zyotons derived from waveforms such as parabolic-similaritons; wavelets; curvelets; ridgelets; or elliptic waves, any of which can yield multiple components which are very close in frequency to each other, or yield groups of components that are close in frequency to each other.

[0621] With reference to FIG. 59A, the three k components of the Zyoton are close in frequencies. The closeness for the delay shift operation need not be determined by the ε_k test and, instead, another suitable threshold such as frequency difference of no more than 0.2%, 0.5%, 1%, 2% etc., of the larger or smaller frequency can be used to determine the closeness. If such close frequency components are observed, the collision process may start by setting the shifting value equal to the number of close components minus one. Thus, in the example shown in FIG. 59A, δ=2. In one embodiment, the collision begins not with the component Z1, but with the component Z3. The bracket of the CF components to be collided with the component Z3, however, is not shifted. Instead, the collision bracket for Z3 includes the CF components {CF1, CF2, CF3, CF4}. Once the value of the shift-delay 6 is set, it is uniformly applied for all collision interactions. Thus, the collision bracket for the Zyoton component Z8 includes the CF components {CF6, CF7, CF8, CF9}.

[0622] In some embodiments, the first of the close Zyoton components may be used and the remaining components or not used in the collision process. Instead, the collision continues with the next Zyoton component in the ordered set that is not determined to be close in frequency. In some embodiments, the center component is selected and the remaining close components are not used in the collision process. Here again, the collision continues with the next Zyoton component in the ordered set that is not determined to be close in frequency. For each of these collisions, the corresponding bracket of the CF components is shifted by the delay-shift value δ. For example, FIG. 59B shows that the collision of Z2, the center component, is performed using the bracket {CF1, CF2, CF3, CF4} and the next collision operation involves the Zyoton component Z4, which is not close to Z1, Z2, and Z3, and the collision bracket for Z4 is {CF2, CF3, CF4, CF5}. The center component can be determined according to the frequencies of the close components or according to the amplitudes of the close components.

[0623] The phase shift φ^f shown in the expression for F′(Z′) can be introduced, either as a constant phase shift (which reduces to a scalar) in the frequency domain; or as a rotation vector where the amplitude of each frequency component of the Zyoton

[0624] $Z_{\left(g_{\left(t-{\delta }^f\right)}\right)}$
participating in the collision is multiplied by a variable phase shift coefficient

[0625] ${\varphi }_{g_{\left(t-{\delta }^f\right)}}^f={\varphi }_{g_{\left(t-{\delta }^f-1\right)}}^f+\left(t-{\delta }^f\right)\frac{2\pi }{360*\left(k+m+j\right)}$ $\mathrm{where}$ ${\varphi }_1^f=\frac{2\pi }{360*\left(k+m+j\right)}.$

[0626] For subsequent (second and later) collisions, the frequency components of the conditioned feature would be replaced by the frequency components of the renormalized modified zyoton, and the other collision computing coefficients would optionally remain unchanged. The expression can be written as:

[0627] $\begin{array}{cc}{F}^{\prime }\left(Z^{\mathrm{\prime \prime }}\right)={\coprod }_1^k\left({\coprod }_{t-\delta }^{t-\delta +\mathrm{CB}}\left({\stackrel{\_}{\alpha }}^F\left({\varphi }^F\left({\Omega }_{Z\left(g_{\left(t-{\delta }^F\right)}\right)}\right)\right)\otimes {\Omega }_{Z\prime \left(g_{\left(t\right)}\right)}\right)\right),t=0,1,\dots ,\left(k+l+m\right)-1& \left(15\right)\end{array}$
where Z″ is the new resulting modified zyoton, Z′ is the renormalized modified zyoton resulting from a previous collision, and other terms are as defined above.