Methods for performing DAF data filtering and padding

[00002] This application claims provisional priority to U.S. Provisional Application Ser. No. 60/077,860, filed Mar. 13, 1998.

[00003] This invention was supported in whole or in part, by grant No. 1-5-51749 from the National Science Foundation and grant number N-00014-K-0613 from the Department of the Navy, Office of Naval Research. The Government has certain rights in the invention.

[00004] 1. Field of the Invention

[00005] The present invention relates to methods using distributed approximating functionals (DAF), DAF-wavelets and DAF-neural networks for filtering, denoising, processing, restoring, enhancing, padding, or other signal processing procedures directed to images, signals, 1D, 2D, 3D . . . nD spectra, X-ray spectra, CAT scans, MRI scans, NMR, and other applications that require data processing at or near the theoretical limit of resolutions.

[00006] More particularly, the present invention relates to the use of infinitely smooth DAFs in combination with other signal processing techniques to provide methods and apparatuses utilizing such methods that can enhance image, spectral, or other signal data and decrease the time need to acquire images, spectra or other signals.

[00007] 2. Description of the Related Art

[00008] Many techniques currently exist for processing images, refining spectra, analyzing data or the like. Many of these techniques are well-known and used extensively. However, these techniques generally suffer from one or more limitation on their ability to enhance signal or image and construct or restore missing or lost data, especially if the user desires the error inherit in signal acquisition and the error introduced by the processing technique to be as small as possible, i.e., as close as possible to Heisenberg's uncertainty principle.

[00009] Thus, there is a need in the art for improved techniques for processing acquired data whether in the form on a image, a spectra, a multidimensional spectra or the like so that the error due to processing can be minimized which can increase resolution and decrease acquisition times.

[00010] The present invention provides a method implemented on a digital processing device or stored in a memory readable by a digital processing unit which uses distributed approximating functionals (DAFs) to enhance and improve signal, image and multidimensional data constructs processing and to decrease acquisition time of real world spectrometric techniques that operate on square wave type frequency domains which require a large acquisition time to capture the signal from noise. Shorten the acquisition time, the more noise and less resolution and definition the spectra will have. The limit for simultaneously acquiring frequency information in time is given by a variant of Heisenberg's uncertainty principle, i.e., ΔωΔt≦1. The methods of the present provides methods for image and signal processing where the accuracy and precision of the final signal and image closely approaches the uncertainty principles maximum accuracy and precision. The methods can be made to approach uncertainty principle accuracy via increased computational cost, but the real power of the methods of this invention is to yield improved measurements at a given Δω and Δt product.

[00011] The present invention also provides methods for improving X-ray and magnetic imaging techniques, especially mammogram images using the DAF and DAF processing techniques set forth herein.

[00012] The present invention also provides a mammogram imaging system of conventional design, the X-ray data derived thereform is then enhanced via DAF processing in an associated digital processing unit.

[00128] The inventors have found that a signals, images and multidimensional imaging data can be processed at or near the uncertainty principle limits with DAFs and various adaptation thereof which are described in the various section of this disclosure.

[00129] Experimental data sets encountered in science and engineering contain noise due to the influence of internal interferences and/or external environmental conditions. Sometimes the noise must be identified and removed in order to see the true signal clearly, to analyze it, or to make further use of it.

[00130] Signal processing techniques are now widely applied not only in various fields of engineering but also in physics, chemistry, biology, and medicine. Example problems of interest include filter diagonalization, solvers for eigenvalues and eigenstates [1-3], solution of ordinary and partial differential equations [4-5], pattern analysis, characterization, and denoising [6-7], and potential energy surface fitting [8]. One of the most important topics in signal processing is filter design. Frequency-selective filters are a particularly important class of linear, time-invariant (LTI) analyzers [9]. For a given experimental data set, however, some frequency selective finite impulse response (FIR) filters require a knowledge of the signal in both the unknown “past” and “future” domains. This is a tremendously challenging situation when one attempts to analyze the true signal values near the boundaries of the known data set. Direct application of this kind of filter to the signal leads to aliasing; i.e., the introduction of additional, nonphysical frequencies to the true signal, a problem called “aliasing” [9]. Additionally, in the implementation of the fast Fourier transform (FFT) algorithm [9], it is desirable to have the number of data values or samples to be a power of 2. However, this condition is often not satisfied for a given set of experimental measurements, so one must either delete data points or augment the data by simulating in some fashion, the unknown data.

[00131] Determining the true signal by extending the domain of experimental data is extremely difficult without additional information concerning the signal, such as a knowledge of the boundary conditions. It is an even tougher task, using the typical, interpolation approach, when the signal contains noise. Such interpolation formulae necessarily return data values that are exact on the grids; but they suffer a loss of accuracy off the grid points or in the unknown domain, since they reproduce the noisy signal data exactly, without discriminating between true signal and the noise. In this disclosure, an algorithm is introduced that makes use of the well-tempered property of “distributed approximating functionals” (DAFs) [10-13]. The basic idea is to introduce a pseudo-signal by adding gaps at the ends of the known data, and assuming the augmented signal to be periodic. The unknown gap data are determined by solving linear algebraic equations that extremize a cost function. This procedure thus imposes a periodic boundary condition on the extended signal. Once periodic boundary conditions are enforced, the pseudo-signal is known everywhere and can be used for a variety of numerical applications. The detailed values in the gap are usually not of particular interest. The advantage of the algorithm is that the extended signal adds virtually no aliasing to the true signal, which is an important problem in signal processing. Two of the main sources of aliasing are too small a sampling frequency and truncation of the signal duration. Another source of error is contamination of the true signal by numerical or experimental noise. Here we are concerned only with how to avoid the truncation induced and noisy aliasing of the true signal.

[00132] Distributed approximating functions (DAFs) were recently introduced [10-11] as a means of approximating continuous functions from values known only on a discrete sample of points, and of obtaining approximate linear transformations of the function (for example, its derivatives to various orders). One interesting feature of a class of commonly-used DAFs is the so-called well-tempered property [13]; it is the key to the use of DAFs as the basis of a periodic extension algorithm. DAFs differ from the most commonly used approaches in that there are no special points in the DAF approximation; i.e., the DAF representation of a function yields approximately the same level of accuracy for the function both on and off the grid points. However, we remark that the approximation to the derivatives is not, in general quite as accurate as the DAF approximation to the function itself because the derivatives of L²-functions contain an increased contribution from high frequencies. By contrast, most other approaches yield exact results for the function on the grid points, but often at the expense of the quality of the results for the function elsewhere [13]. DAFs also provide a well-tempered approximation to various linear transformations of the function. DAF representations of derivatives to various orders will yield approximately similar orders of accuracy as long as the resulting derivatives remain in the DAF class. The DAF approximation to a function and a finite number of derivatives can be made to be of machine accuracy with a proper choice of the DAF parameters. DAFs are easy to apply because they yield integral operators for derivatives. These important features of DAFs have made them successful computational tools for solving various linear and nonlinear partial differential equations (PDEs) [14-17], for pattern recognition and analysis [6], and for potential energy surface fitting [8]. The well-tempered DAFs also are low-pass filters. In this disclosure, the usefulness of DAFs as low pass filters is also studied when they are applied to a periodically extended noisy signal. For the present purpose, we assume that the weak noise is mostly in the high frequency region and the true signal is bandwidth limited in frequency space, and is larger than the noise in this same frequency region. To determine when the noise is eliminated, we introduce a signature to identify the optimum DAF parameters. This concept is based on computing the root-mean-square of the smoothed data for given DAF parameters. By examining its behavior as a function of the DAF parameters, it is possible to obtain the overall frequency distribution of the original noisy signal. This signature helps us to periodically extend and filter noise in our test examples.

[00133] The first example is a simple, noisy periodic signal, for which the DAF periodic extension is a special case of extrapolation. The second is a nonperiodic noisy signal. After performing the periodic extension and filtering, it is seen that the resulting signal is closely recreates the true signal.

[00134] DAFs can be viewed as “approximate identity kernels” used to approximate a continuous function in terms of discrete sampling on a grid [10-13]. One class of DAFs that has been particularly useful is known as the well-tempered DAFs, which provide an approximation to a function having the same order of accuracy both on and off the grid points. A particularly useful member of this class of DAFs is constructed using Hermite polynomials, and prior to discretization, is given by $\begin{array}{cc}{\delta }_M(x-x^{\prime }\sigma )=\frac{1}{\sigma }\exp -\left(\frac{{\left(x-x^{\prime }\right)}^2}{2{\sigma }^2}\right)\sum _{n=0}^{M/2}{\left(-\frac{1}{4}\right)}^n\frac{1}{\sqrt{2\pi }n!}{H}_{2n}\left(\frac{\left(x-x^{\prime }\right)}{\sqrt{2}\sigma }\right),& \left(1\right)\end{array}$
where σ, M are the DAF parameters and H_2n is the usual (even) Hermite polynomial. The Hermite DAF is dominated by its Gaussian envelope, exp(−(x−x′)²/2σ²), which effectively determines the extent of the function. The continuous, analytic approximation to a function ƒ(x) generated by the Hermite DAF is $\begin{array}{cc}f\left(x\right)\approx f_{\mathrm{DAF}}\left(x\right)={\int }_{-\infty }^{\infty }{\delta }_M(x-x^{\prime }\sigma )f\left(x^{\prime }\right)d{x}^{\prime }.& \left(2\right)\end{array}$

[00136] Given a discrete set of functional values on a grid, the DAF approximation to the function at any point x (on or off the grid) can be obtained by $\begin{array}{cc}{f}_{\mathrm{DAF}}\left(x\right)=\Delta \sum _j{\delta }_M\left(x-x_j|\sigma \right)f\left(x_j\right).& \left(3\right)\end{array}$
where Δ is the uniform grid spacing (non-uniform and even random sampling can also be used by an appropriate extension of the theory). The summation is over all grid points (but only those close to x effectively contribute). Similarly, for a two-dimensional function ƒ(x,y), one can write $\begin{array}{cc}{f}_{\mathrm{DAF}}\left(x,y\right)={\Delta }_x{\Delta }_y\sum _{j_1,j_2}{\delta }_{M_1}(x-x_{j_1}{\sigma }_1){\delta }_{M_2}(y-y_{j_2}{\sigma }_2)f\left(x_{j_1},y_{j_2}\right),& \left(4\right)\end{array}$
using a simple direct product. In FIG. 1, we plot Hermite DAFs obtained with two different sets of parameters, in (a) coordinate space, and (b) frequency space. The solid line (σ=3.05, M=88) is more interpolative compared to the DAF given by the dashed line (σ=4, M=12). The latter is more smoothing when applied to those functions whose Fourier composition lies mostly under the σ=3.05, M=88 DAF window. This results from the σ=4, M=12 DAF window being narrower in Fourier space than that of the DAF with σ=3.05, M=88. The discretized Hermite DAF is highly banded in coordinate space due to the presence of the Gaussian envelope, which means that only a relatively small number of values are needed on both sides of the point x in Equation (3), as can be clearly seen from FIG. 1(a). This is in contrast to the sinc function $\frac{\sin \left(\omega x\right)}{\pi x}.$
From FIG. 1(b), we see that the Hermite DAF is also effectively bandwidth-limited in frequency space. With a proper choice of parameters, the Hermite DAF can produce an arbitrarily good filter (see the dashed line in FIGS. 1(a-b). Once the boundary condition is fixed for a data set, Equation (3) or (4) can then be used to eliminate the high frequency noise of that data set. As long as the frequency distribution of the noise lies outside the DAF plateau (FIG. 1(b)), the Hermite DAF will eliminate the noise regardless of its magnitude.

[00140] The approximate linear transformations of a continuous function can also be generated using the Hermite DAF. One particular example is derivatives of a function to various orders, given by $\begin{array}{cc}{f}^{\left(l\right)}\left(x\right)\approx f_{\mathrm{DAF}}^{\left(l\right)}={\int }_{-\infty }^{\infty }{\delta }_M^{\left(l\right)}\left(x-x^{\prime }|\sigma \right)f\left(x^{\prime }\right)d{x}^{\prime },& \left(5\right)\end{array}$
where δ_M⁽¹⁾ (x−x′|σ) is the lth derivative of δ_M (x−x′|σ), with respect to x, and is given explicitly by $\begin{array}{cc}=\frac{2^{-l/2}}{{\sigma }^{l+1}}\exp \left(\frac{-{\left(x-x^{\prime }\right)}^2}{2{\sigma }^2}\right)\sum _{n=0}^{M/2}{\left(-\frac{1}{4}\right)}^n{\left(-1\right)}^l\frac{1}{\sqrt{2\pi }n!}{H}_{2n+l}\left(\frac{x-x^{\prime }}{\sqrt{2}\sigma }\right)& \left(6\right)\end{array}$
When uniformly discretized by quadrature, Equation (5) gives $\begin{array}{cc}{f}^{\left(l\right)}\left(x\right)\approx f_{\mathrm{DAF}}^{\left(l\right)}\left(x\right)=\Delta \sum _j{\delta }_M^{\left(l\right)}\left(x-x_j|\sigma \right)f\left(x_j\right)& \left(7\right)\end{array}$
Expressions (5) and (7) are extremely useful in solving linear and nonlinear partial differential equations (PDEs) [14-17] because the differential operation has been expressed as an integration. With a judicious choice of the DAF parameters, it is possible to provide arbitrarily high accuracy for estimating the derivatives.

[00144] Case I. Filling a Gap

[00145] Suppose we have a set of uniformly spaced grid points on the infinite line, and a continuous function, ƒ(x), known on all grid points except for the set {x₁, . . . , x_K}. Assuming that ƒ(x) is in the DAF-class, we can estimate the unknown values by minimizing the cost function, $\begin{array}{cc}C\equiv \sum _{p=-\infty }^{\infty }{W_p\left(f\left(x_p\right)-f_{\mathrm{DAF}}\left(x_p\right)\right)}^2& \left(8\right)\end{array}$
where, W_p is a weight assigned to the point x_p, and in this disclosure it is chosen to be 1 on a finite grid and 0 elsewhere; ƒ_DAF(x_p) is the DAF approximation to the function at the point x_p. From Equations (3) and (8), we have $\begin{array}{cc}C\equiv \sum _{p=-\infty }^{\infty }{W_p\left(f\left(x_p\right)-\sum _{t=p-w}^{p+w}{\delta }_M\left(x_p-x_t|\sigma \right)f\left(x_t\right)\right)}^2& \left(9\right)\end{array}$
where w is the half DAF bandwidth. We minimize this cost function with respect to the unknown values, {ƒ(x_J), . . . , ƒ(x_K)}, according to $\begin{array}{cc}\frac{\partial C}{\partial f\left(x_l\right)}=0,J\le l\le K,& \left(10\right)\end{array}$
to generate the set of linear algebraic equations, $\begin{array}{cc}\sum _{p=-\infty }^{\infty }2W_p\left(f\left(x_p\right)-\sum _{t=p-w}^{p+w}{\delta }_M(x_p-x_l|\sigma \hspace{1em})f\left(x_t\right)\right)({\delta }_{\mathrm{pl}}-{\delta }_M\left(x_p-x_l|\sigma \right)\hspace{1em})=0,J\le l\le K,& \left(11\right)\end{array}$
where the unknowns are ƒ(x_p) and ƒ(x_t) for p=l or t=l. The symbol δ_pl is the kronecker delta. Solving these equations yields the predicted values of ƒ(x) on the grid points in the gap.

[00150] Case II. Extrapolation

[00151] A more challenging situation occurs when ƒ(x_l), l>J are all unknown. In this case, for points x_p beyond x_K, we specify a functional form for the unknown grid values, including some embedded variational parameters. It is simplest to choose linear variational parameters, e.g., $\begin{array}{cc}f\left(x\right)\approx \sum _{\mu =1}^L{\alpha }_{\mu }{\varphi }_{\mu }\left(x\right)& \left(12\right)\end{array}$
but this is not essential, and nonlinear parameters can also be embedded in the φ_μ(x). The choice of functions, φ_μ(x), can be guided by any intuition or information about the physical behavior of the signal, but even this is not necessary. This introduces additional variations of the cost function with respect to the additional parameters, so we impose $\frac{\partial C}{\partial {\alpha }_{\beta }}=0,1\le \beta \le L,$
and therefore obtain additional equations. We must also specify the choice of the W_p when one introduces both a gap and a “tail function”. There is enormous freedom in how this is done, and e.g., one can choose which points are included and which are not. In the present study, we shall take W_p=1 for 1≦p≦K (i.e., all known data points, plus all gap-points), and W_p≡1 for all other points (including tail-function points). Again, we emphasize that other choices are possible and are under study.

[00154] For case I, our procedure leads to Equation (11) and for case II, to the equations, $\begin{array}{cc}\sum _{p=1-w}^{\min \left(l+w,K\right)}2W_p\left(f\left(x_p\right)-\sum _{t=p-w}^{p+w}{\delta }_M(x_p-x_l|\sigma \hspace{1em})f\left(x_l\right)\right)\times \left({\delta }_{\mathrm{pl}}-{\delta }_M(x_p-x_l\sigma \right))=0,J\le l\le K,& \left(13\right)\\ \sum _{p=K+1-w}^K-2W_p\left(f\left(x_p\right)-\sum _{t=p-w}^{p+w}{\delta }_M(x_p-x_l|\sigma \hspace{1em})f\left(x_l\right)\right)\times \sum _{l=K+1}^{p+w}{\delta }_M\left(x_p-x_l\right){\varphi }_{\beta }\left(x_l\right)=0,1\le \beta \le L& \left(14\right)\end{array}$
These linear algebraic equations can be solved by any of a variety of standard algorithms [18]. Note that it is the well-tempered property of the DAFs that underlies the above algorithms. For standard interpolation algorithms, the value on each grid point is exact and does not depend on the values at other grid points, which means that the cost function is always zero irrespective of functional values.

[00156] The suitability of using Hermite DAFs to pad two isolated data sets has been tested for fitting one dimensional potential energy surfaces [8]. To explore further the algorithm in the case where only one data set is known, we show in FIG. 2 the extrapolation results for the arbitrarily chosen DAF-class function, $\begin{array}{cc}f\left(x\right)=4+2\sum _{j=1}^4\cos \left(\mathrm{jx}\right)+5\exp \left(-{\left(x-1\right)}^2\right)& \left(15\right)\end{array}$
Using values of the function on a discrete grid with uniform spacing, Δ≈0.024, on the domain shown in FIG. 2 (solid line), we attempt to determine the function at 100 uniformly distributed grid points in the range [−1.2, 1.2]. The tail function used is ƒ(x)≡1, multiplied by a linear variational parameter. From FIG. 2, it is seen that the predicted results are in almost total agreement with the actual function, for all three DAF parameters employed, for the points between −1.2≦x≦0.2. Larger errors occur for those x values which are further away from the known data boundary. The source of error simply is that one is forcing the function to join smoothly with a constant tail function, even though the constant is variationally determined. Had one employed the correct form for the tail function, with a multiplicative variational factor, the result would be visually indistinguishable for all three DAF parameters and the tail-variational constant would turn out to be essentially unity. It must be noted that, although we have discussed the algorithm in the context of one dimension, extending it to two or more dimensions is straightforward. One way to do this is with a direct product form, as given in Equation (4). However, such a direct two dimensional calculation is a time and memory consuming procedure because of the large number of simultaneous algebraic equations that must be solved. One alternative is to consider the two dimensional patterns as a grid constructed of many one dimensional grids, and then extrapolate each line separately. We expect this procedure may be less accurate than the direct two dimensional extrapolation algorithm because it only considers the influence from one direction and neglects cross correlation. However, for many problems it produces satisfactory results and it is a very economical procedure. Additionally, cross correlation can be introduced by DAF-fitting the complete (known plus predicted) data set using the appropriate 2D DAF.

[00158] The well-tempered property makes the DAFs powerful computational tools for extrapolation of noisy data, since DAFs are low-pass filters and therefore remove high frequency noise. In the next section, we will explore the use of the algorithm presented here for periodically extending a finite, discrete segment of data which may contain noise.

[00159] As described in the introduction to this disclosure, sometimes it is necessary to know the boundary conditions for a data set in order to apply noncausal, zero-phase FIR filters without inducing significant aliasing. Certain other numerical analyses require that the signal contain a number of samples equal to an integer power of 2. However, it is most often the case that the boundary conditions for the experimental data are unknown and the length of the data stream is fixed experimentally and typically not subject to adjustment.

[00160] A pseudo-signal is introduced outside the domain of the nonperiodic experimental signal in order to force the signal to satisfy periodic boundary conditions, and/or to have the appropriate number of samples. The required algorithm is similar to that for filling a gap, as discussed above. One can treat the period-length as a discrete variational parameter but we don't pursue this here. For a given set of experimental data {ƒ₁, ƒ₂, . . . , ƒ_J−1)}, we shall force it to be periodic, with period K, so that K−J+1 values {ƒ_J, ƒ_J+1, . . . , ƒ_K} must be determined. Since the extended signal is periodic, the values {ƒ_K+1, ƒ_K+2, . . . , ƒ_K+J−1} are also, by fiat, known to be equal to {ƒ₁, ƒ₂, . . . , ƒ_J−1}. Once the gap is filled in, the resulting signal can, of course, be infinitely extended as may be required for various numerical applications.

[00161] The pseudo-signal is only used to extend the data periodically retaining essentially the same frequency distributions. The utility of the present periodic extension algorithm is that it provides an artificial boundary condition for the signal without significant aliasing. The resulting signal can be used with any filter requiring information concerning the future and past behavior of the signal. In this disclosure, we also employ a Hermite DAF to filter out the higher frequency noise of the periodic, extended noisy signal. For doing this, we assume that the true signal is bandwidth limited and that the noise is mostly concentrated in the high frequency region.

[00162] As shown in the test example extrapolation in section III, there are infinitely many ways to smoothly connect two isolated signals using DAFs with different choices of the parameters. We require a procedure to determine the optimum DAF parameters in a “blind” manner. Fourier analysis is one way to proceed, but due to the structure of the Hermite DAF, we prefer to optimize the parameters while working in physical space rather than Fourier space. To accomplish this, we introduce a generalized signature for both noisy data extension and filtering, which we define to be $\begin{array}{cc}{S}_M\left(\sigma /\Delta \right)=\sqrt{\frac{\sum _n{\left(f_{\mathrm{DAF}}\left(x_n\right)-\stackrel{\_}{f_{\mathrm{DAF}}}\right)}^2}{N}}& \left(16\right)\end{array}$
where, M and σ are Hermite DAF parameters, and {overscore (ƒDAF)} is the arithmetic average of the ƒ_DAF(x_n) (the σ/Δ→∞ of ƒ_DAF(x_n)). The signature essentially measures the smoothness of the DAF filtered result.

[00164] A typical plot of S_M(σ/Δ) is shown in FIG. 4(b). We first note that it is monotonically decreasing. This is to be expected since increasing σ/Δ results in a smoother, more highly averaged signal. The second major feature of interest is the occurrence of a broad plateau. In this region most of the noise has been removed from the DAF approximation; however, the dominant portion of the true signal is still concentrated under the DAF frequency window. As a consequence the DAF approximation to the function is very stable in this region. As σ/Δ increases beyond the plateau the width of the DAF window in frequency no longer captures the true signal and as a result, the true DAF signal begins to be severely averaged. In the extreme, only the zero frequency remains and ƒ_DAF(x_n)={overscore (ƒDAF)} and hence S_M(σ/Δ)→0. As we discuss below, one can usefully correlate the transition behavior with the best DAF-approximation. The first extremely rapid decrease is due to the fact the DAF is interpolating and not well tempered. It is the region beyond the initial rapid decrease that is important (i.e., σ/Δ≧1.5). To understand the behavior in this region, we write S_M(σ/Δ) in the form $\begin{array}{cc}{S}_M\left(\sigma /\Delta \right)=\sqrt{\frac{\begin{array}{c}\sum _n(\hspace{1em}{\left(\Delta f\left(x_n\right)\right)}_{\mathrm{DAF}}^2+\\ {2{\left(\Delta f\left(x_n\right)\right)}_{\mathrm{DAF}}(\hspace{1em}\hspace{1em}\hspace{1em}{f}_{\mathrm{DAF}}^{\left(p\right)}\left(x_n\right)-\stackrel{\_}{f_{\mathrm{DAF}}}\hspace{1em})+(\hspace{1em}{f}_{\mathrm{DAF}}^{\left(p\right)}\left(x_n\right)-\stackrel{\_}{f_{\mathrm{DAF}}}\hspace{1em})}^2\end{array}}{N}}& \left(17\right)\end{array}$
where ƒ_DAF^(p)(x_n) is the DAF approximation using a σ/Δ in the middle of the plateau and (Δƒ(x_n))_DAF=ƒ_DAF(x_n)−ƒ_DAF^(p)(x_n). The cross term averages to zero because the DAF approximation is interpolating and hence the (Δƒ(x_n))_DAF fluctuate rapidly reflecting the presence of noise. Thus $\begin{array}{cc}{S}_M\left(\sigma /\Delta \right)=\sqrt{\frac{{\sum _n{\left(\Delta f\left(x_n\right)\right)}_{\mathrm{DAF}}^2+(\hspace{1em}{f}_{\mathrm{DAF}}^{\left(p\right)}\left(x_n\right)-\stackrel{\_}{f_{\mathrm{DAF}}}\hspace{1em})}^2}{N}}& \left(18\right)\end{array}$
which decreases rapidly since Σ_n (Δƒ(x_n))² _DAF is positive and rapidly decreasing as the high frequency noise is eliminated from the signal. The transition into the plateau reflects a change from an interpolative to a well tempered behavior. Although the algorithm presented in this section only refers to periodic extensions, we stress that this is only one possibility out of many.

[00167] Two numerical examples are presented in this section to show the usefulness of our algorithm.

[00168] Case I

[00169] The first one is the extrapolation of ƒ(x)=sin(5πx/128), to which noise has been added. The Hermite DAF parameters are M=6 for padding/extension and M=1 for smoothing in our numerical examples. The weight W_p, was taken as discussed above. The values at 220 evenly spaced grid points are input over the range [0,219], with 50% random noise added (ƒ=ƒ×[1+ random(−0.5,0.5)]. The continuation of the solid curve from points x₂₂₀ to x₂₅₆ shows the function without noise. We shall predict the remaining 36 points (excluding x₂₅₆ because the function there must equal the function at x₀) by the periodic extension algorithm presented in this disclosure. Because the original continuous function without noise is truly periodic, with period 256, this extension corresponds to filling the gap using noisy input data. The L_∞ error and the signature for periodic extension are plotted with respect to σ/Δ in FIGS. 4(a) and 4(b) respectively. From FIG. 4(b), we see that at σ/Δ≈10.5, the transition from the plateau to smoothing of the true signal occurs. As is evident from FIG. 4(a), the minimum extension error also occurs at σ/Δ around 10.5. In FIG. 3, we see the comparison of the true function (solid line) from the 220th to the 256th grid point, along with the periodic extension result (“+′” symbols). It is clearly seen that they agree very well.

[00170] We next use the padded, extended signal for low pass filtering. We plot in FIGS. 5(a) and 5(b) the L_∞ error and the signature of the filtered result for one complete period as a function of σ/Δ using M=12 rather than M=6 in the DAF. This is done for convenience for reasons not germane to the subject. The result is that the σ/Δ range for which the DAF is well-tempered changes and the transition from denoising to signal modifying smoothing occurs at σ/Δ=17 (FIG. 5(b)). However, we see from FIG. 5(a) that the L_∞ minimum error also occurs at about the same σ/Δ, showing the robustness of the approach with respect to the choice of DAF parameters. In FIG. 5(c), we show the resulting smoothed, denoised sine function compared to the original true signal. These results illustrate the use of the DAF procedure in accurately extracting a band-limited function using noisy input data. Because of the relatively broad nature of the L_∞ error near the minimum, one does not need a highly precise value of σ/Δ.

[00171] Case II

[00172] We now consider a more challenging situation. It often happens in experiments that the boundary condition of the experimental signal is not periodic, and is unknown, in general. However, the signal is approximately band-limited (i.e., in the DAF class).

[00173] To test the algorithm for this case, we use the function given in Equation (15) as an example. FIG. 6(a) shows the function, with 20% random noise in the range (−7,10) (solid line). These noisy values are assumed known at only 220 grid points in this range. Also plotted in FIG. 6(a) are the true values of the function (dashed line) on the 36 points to be predicted. In our calculations, these are treated, of course, as unknown and are shown here only for reference. It is clearly seen that the original function is not periodic on the range of 256 grid points. We force the noisy function to be periodic by padding the values of the function on these last 36 points, using only the known, noisy 220 values to periodically surround the gap.

[00174] As mentioned in section IV, for nonperiodic signals, the periodic extension is simply a scheme to provide an artificial boundary condition in a way that does not significantly corrupt the frequency distribution of the underlying true signal in the sampled region. The periodic padding signature is shown in FIG. 7. Its behavior is similar to that of the truly periodic signal shown in FIG. 4(b). The second rapid decrease begins at about σ/Δ=5.2 and the periodic padding result for this DAF parameter is plotted in FIG. 6(b) along with the original noisy signal. Compared with the original noisy signal, it is seen that the signal in the extended domain is now smoothed. In order to see explicitly the periodic property of the extended signal and the degree to which aliasing is avoided, we filter the noise out of the first 220 points using an appropriate Hermite DAF. The L_∞ error and the signature of the DAF-smoothed results are plotted in FIGS. 8(a) and 8(b) respectively. Again they correlate with each other very well. Both the minimum error and the starting point of the second rapid decrease occur at about σ/Δ=9.0, which further confirms our analysis of the behavior of the signature. In FIG. 9, we present the smoothed signal (solid line) along with true signal (dashed line), without any noise added. It is seen that in general, they agree with each other very well in the original input signal domain.

[00175] Application of the pseudo-signal in the extended domain clearly effectively avoids the troublesome aliasing problem. The minor errors observed occur in part because of the fact that the random noise contains not only high frequency components but also some lower frequency components. However, the Hermite DAF is used only as a low pass filter here. Therefore, any low frequency noise components are left untouched in the resulting filtered signal. Another factor which may affect the accuracy of filtering is that we only use M=12. According to previous analysis of the DAF theory, the higher the M value, the greater the accuracy [12], but at the expense of increasing the DAF bandwidth (σ/Δ increases). However, as M is increased, combined with the appropriate σ/Δ, the DAF-window is better able to simulate an ideal band-pass filter (while still being infinitely smooth and with exponential decay in physical and Fourier space.) Here we have chosen to employ M=12 because our purpose is simply to illustrate the use of our algorithm, and an extreme accuracy algorithm incorporating this principles is directly achievable.

[00176] This paper presents a DAF-padding procedure for periodically extending a discrete segment of a signal (which is nonperiodic). The resulting periodic signal can be used in many other numerical applications which require periodic boundary conditions and/or a given number of signal samples in one period. The power of the present algorithm is that it essentially avoids the introduction of aliasing into the true signal. It is the well-tempered property of the DAFs that makes them robust computational tools for such applications. Application of an appropriate well-tempered DAF to the periodically extended signal shows that they are also excellent low pass filters. Two examples are presented to demonstrate the use of our algorithm. The first one is a truncated noisy periodic function. In this case, the extension is equivalent to an extrapolation.

[00177] Our second example shows how one can perform a periodic extension of a nonperiodic, noisy finite-length signal. Both examples demonstrate that the algorithm works very well under the assumption that the true signal is continuous and smooth. In order to determine the best DAF parameters, we introduce a quantity called the signature. It works very well both for extensions and low pass filtering. By examining the behavior of the signature with respect to σ/Δ, we can determine the overall frequency distribution of the original noisy signal working solely in physical space, rather than having to transform to Fourier space.

[none]

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[00196] The theory of interpolating wavelets based on a subdivision scheme has attracted much attention recently [1, 9, 12, 13, 17, 22, 27, 29, 40, 42, 45, 46, 47, 48, 49, 54, 55, 56, 65 and 66]. Because the digital sampling space is exactly homomorphic to the multi scale spaces generated by interpolating wavelets, the wavelet coefficients can be obtained from linear combinations of discrete samples rather than from traditional inner product integrals. This parallel computational scheme significantly decreases the computational complexity and leads to an accurate wavelet decomposition, without any pre-conditioning or post-conditioning processes. Mathematically, various interpolating wavelets can be formulated in a biorthogonal setting.

[00197] Following Donoho's interpolating wavelet theory [12], Harten has described a kind of piecewise biorthogonal wavelet construction method [17]. Swelden independently develops this method as the well-known “lifting scheme” [56], which can be regarded as a special case of the Neville filters considered in [27]. The lifting scheme enables one to construct custom-designed biorthogonal wavelet transforms by just assuming a single low-pass filter (a smooth operation) without iterations. Theoretically, the interpolating wavelet theory is closely related to the finite element technique in the numerical solution of partial differential equations, the subdivision scheme for interpolation and approximation, multi-grid generation and surface fitting techniques.

[00198] A new class of generalized symmetric interpolating wavelets (GSIW) are described, which are generated from a generalized, window-modulated interpolating shell. Taking advantage of various interpolating shells, such as Lagrange polynomials and the Sinc function, etc., bell-shaped, smooth window modulation leads to wavelets with arbitrary smoothness in both time and frequency. Our method leads to a powerful and easily implemented series of interpolating wavelet. Generally, this novel designing technique can be extended to generate other non-interpolating multiresolution analyses as well (such as the Hermite shell). Unlike the biorthogonal solution discussed in [6], we do not attempt to solve a system of algebraic equations explicitly. We first choose an updating filter, and then solve the approximation problem, which is a rth-order accurate reconstruction of the discretization. Typically, the approximating functional is a piecewise polynomial. If we use the same reconstruction technique at all the points and at all levels of the dyadic sequence of uniform grids, the prediction will have a Toplitz-like structure.

[00199] These ideas are closely related to the distributed approximating functionals (DAFs) used successfully in computational chemistry and physics [20, 21, 22, 65, 66, 67], for obtaining accurate, smooth analytical fits of potential-energy surfaces in both quantum and classical dynamics calculations, as well as for the calculation of the state-to-state reaction probabilities for three-dimension (3-D) reactions. DAFs provide a numerical method for representing functions known only on a discrete grid of points. The underlying function or signal (image, communication, system, or human response to some probe, etc.) can be a digital time sequence (i.e., finite in length and 1-dimensional), a time and spatially varying digital sequence (including 2-D images that can vary with time, 3-D digital signals resulting from seismic measurements), etc. The general structure of the DAF representation of the function, ƒ_DAF(x,t), where x can be a vector (i.e., not just a single variable), is
ƒ_DAF(x, t_p)=Σ_jφ(x−x_j)/|σ/Δ)ƒ(x_j,t_p)
where φ(x−x_j)|σ/Δ) is the “discrete DAF”, ƒ(x_j,t_p) is the digital value of the “signal” at time t_p, and M and σ/Δ will be specified in more detail below. They are adjustable DAF parameters, and for non-interpolative DAF, they enable one to vary the behavior of the above equation all the way from an interpolation limit, where
ƒ_DAF(x_j,t_p)≡ƒ(x_j,t_p)
(i.e., the DAF simply reproduces the input data on the grid to as high accuracy as desired) to the well-tempered limit, where
ƒ_DAF(x_j,t_p)≠ƒ(x_j,t_p)
for function ƒ(x,t_p)∈L²(R). Thus the well-tempered DAF does not exactly reproduce the input data. This price is paid so that instead, a well-tempered DAF approximation makes the same order error off the grid as it does on the grid (i.e., there are no special points). We have recently shown that DAFs (both interpolating and non-interpolating) can be regarded as a set of scaling functionals that can used to generate extremely robust wavelets and their associated biorthogonal complements, leading to a full multiresolution analysis [22, 46, 47, 48, 49, 54, 55, 66, 67]. DAF-wavelets can therefore serve as an alternative basis for improved performance in signal and image processing.

[00206] The DAF wavelet approach can be applied directly to treat bounded domains. As shown below, the wavelet transform is adaptively adjusted around the boundaries of finite-length signals by conveniently shifting the modulated window. Thus the biorthogonal wavelets in the interval are obtained by using a one-sided stencil near the boundaries. Lagrange interpolation polynomials and band-limited Sinc functionals in Paley-Wiener space are two commonly used interpolating shells for signal approximation and smoothing, etc. Because of their importance in numerical analysis, we use these two kinds of interpolating shells to introduce our discussion. Other modulated windows, such as the square, triangle, B-spline and Gaussian are under study with regard to the time-frequency characteristics of generalized interpolating wavelets. By carefully designing the interpolating Lagrange and Sinc functionals, we can obtain smooth interpolating scaling functions with an arbitrary order of regularity.

[00207] The basic characteristics of interpolating wavelets of order D discussed in reference [12] require that, the primary scaling function, φ, satisfies the following conditions.

• • [00208] (1) Interpolation: $\begin{array}{cc}\varphi \left(k\right)=\{\begin{array}{cc}1,& k=0\\ 0,& k\ne 0\end{array}k\in Z& \left(19\right)\end{array}$
    where Z denotes the set of all integers.
  • [00210] (2) Self-Induced Two-Scale Relation: φ can be represented as a linear combination of dilates and translates of itself, with a weight given by the value of φ at k/2. $\begin{array}{cc}\varphi \left(x\right)=\sum _k\varphi \left(k\text{/}2\right)\varphi \left(2x-k\right)& \left(20\right)\end{array}$

[00211] This relation is only approximately satisfied for some interpolating wavelets discussed in the later sections. However, the approximation can be made arbitrarily accurate.

• • [00212] (3) Polynomial Span: For an integer D≧0, the collection of formal sums symbol ΣC_kφ(x−k) contains all polynomials of degree D.
  • [00213] (4) Regularity: For a real V>0, φ is Hölder continuous of order V.
  • [00214] (5) Localization: φ and all its derivatives through order └V┘ decay rapidly.
     |φ^(r)(x)|≦A_s(1+|x|)^−s, x∈R, s>0, 0≦r≦└V┘  (21)
    where └V┘ represents the maximum integer that does not exceed V.

[00217] In contrast to most commonly used wavelet transforms, the interpolating wavelet transform possesses the following characteristics:

[00218] 1. The wavelet transform coefficients are generated by the linear combination of signal samplings,
S_j,k=2^−j/2ƒ(2^−jk), W_j,k=2^−j/2[ƒ(2^−j(k+½))−(P_jƒ)(2^−j(k+½))]  (22)
instead of the convolution of the commonly used discrete wavelet transform, such as
W_j,k=∫_R ψ_j,k(x)ƒ(2^−jk)dx  (23)
where the scaling function, φ_j,k(x)=2^j/2φ(2^jx−k), and wavelet function, ψ_j,k(x)=2^j/2 ψ(2^jx−k), P_jƒ as the interpolant 2^−j/2Σƒ(2^−jk)φ_j,k(x).

[00223] 2. A parallel-computing mode can be easily implemented. The calculation and compression of coefficients does not depend on the results of other coefficients. For the halfband filter with length N, the calculation of each of the wavelet coefficients, W_j,k, does not exceed N+2 multiply/adds.

[00224] 3. For a D-th order differentiable function, the wavelet coefficients decay rapidly.

[00225] 4. In a mini-max sense, threshold masking and quantization are nearly optimal approximations for a wide variety of regularity algorithms.

[00226] Theoretically, interpolating wavelets are closely related to the following functions:

[heading-00227] Band-limited Shannon wavelets

[00228] The π band-limited function, φ(x)=sin(πx)/(πx)∈C^∞ in Paley-Wiener space, generates the interpolating functions. Every band-limited function ƒ∈L²(R) can be reconstructed using the equation $\begin{array}{cc}f\left(x\right)=\sum _{k}f\left(k\right)\frac{\sin \pi \left(x-k\right)}{\pi \left(x-k\right)}& \left(24\right)\end{array}$
where the related wavelet function—Sinclet is defined as (see FIG. 10) $\begin{array}{cc}\psi \left(x\right)=\frac{\sin \pi \left(2x-1\right)-\sin \pi \left(x-1/2\right)}{\pi \left(x-1/2\right)}& \left(25\right)\end{array}$
Interpolating fundamental splines

[00231] The fundamental polynomial spline of degree D, η^D(x), where D is an odd integer, has been shown by Schoenberg (1972), to be an interpolating wavelet (see FIG. 11). It is smooth with order R=D−1, and its derivatives through order D−1 decay exponentially [59]. Thus,
η^D(x)=Σ_kα^D(k)β^D(x−k)  (26)
where β^D(x) is the B-spline of order D defined as $\begin{array}{cc}{\beta }^D\left(x\right)=\sum _{j=0}^{D+1}\frac{{\left(-1\right)}^j}{D!}(\begin{array}{c}D+1\\ j\end{array}){\left(x+\frac{D+1}{2}-j\right)}^{D}U\left(x+\frac{D+1}{2}-j\right)& \left(27\right)\end{array}$
Here U is the step function $\begin{array}{cc}U\left(x\right)=\{\begin{array}{cc}0,& x<0\\ 1,& x\ge 0\end{array}& \left(28\right)\end{array}$
and {α^D(k)} is a sequence that satisfies the infinite summation condition
Σ_kα^D(k)β^D(n−k)=δ(n)  (29)
Deslauriers-Dubuc functional

[00238] Let D be an odd integer, D>0. There exist functions F_D such that if F_D has already been defined at all binary rationals with denominator 2^j, it can be extended by polynomial interpolation, to all binary rationals with denominator 2^j+1, i.e. all points halfway between previously defined points [9, 13]. Specifically, to define the function at (k+½)/2^j when it is already defined at all {k2^−j}, fit a polynomial π_j,k to the data (k′/2^j, F_D(k′/2^j) for k′∈{2^−j[k−(D−1)/2], . . . , 2^−j[k+(D+1)/2]}. This polynomial is unique $\begin{array}{cc}{F}_D\left(\frac{k+1/2}{2^j}\right)\equiv {\pi }_{j,k}\left(\frac{k+1/2}{2^j}\right)& \left(30\right)\end{array}$

[00239] This subdivision scheme defines a function which is uniformly continuous at the rationals and has a unique continuous extension; F_D is a compactly supported interval polynomial and is regular; It is the auto-correlation function of the Daubechies wavelet of order D+1. It is at least as smooth as the corresponding Daubechies wavelets (roughly twice as smooth).

[heading-00240] Auto-correlation shell of orthonormal wavelets

[00241] If {hacek over (φ)} is an orthonormal scaling function, its auto-correlation φ={hacek over (φ)}*{hacek over (φ)} (−·) is an interpolating wavelet (FIG. 12) [40]. Its smoothness, localization and the two-scale relation are inherited from {hacek over (φ)}. The auto-correlations of Haar, Lamarie-Battle, Meyer, and Daubechies wavelets lead to, respectively, the interpolating Schauder, interpolating spline, C^∞ interpolating, and Deslauriers-Dubuc wavelets.

[heading-00242] Lagrange half-band filters

[00243] Ansari, Guillemot, and Kaiser [1] used Lagrange symmetric halfband FIR filters to design the orthonormal wavelets that express the relation between the Lagrange interpolators and Daubechies wavelets [7]. Their filter corresponds to the Deslauriers-Dubuc wavelet of order D=7 (2M−1), M=4. The transfer function of the halfband symmetric filter h is given by
H(z)=½+zT(z²)  (31)
where T is the trigonometric polynomial. Except for h(0)=½, at every even integer lattice point h(2n)=0, n≠0, n∈Z. The transfer function of the symmetric FIR filter h(n)=h(−n), has the form $\begin{array}{cc}H\left(z\right)=1/2+\sum _{n=1}^{M}h\left(2n-1\right)\left(z^{1-2n}+z^{2n-1}\right)& \left(32\right)\end{array}$

[00246] The concept of an interpolating wavelet decomposition is similar to “algorithm a trous”, the connection having been found by Shensa [42]. The self-induced scaling and interpolation conditions are the most important characteristics of interpolating wavelets. From the following equation
ƒ(x)=Σ_nƒ(n)φ(x−n)  (33)
and Equation (19), the approximation to the signal is exact on the discrete sampling points, which does not hold in general for commonly used non-interpolating wavelets.

[00249] Interpolating wavelets with either a Lagrange polynomial shell or Sinc functional shell are discussed in detail. We call these kinds of window modulated wavelets generalized interpolating wavelets, because they are more convenient to construct, processing and extend to higher dimensional spaces.

[heading-00250] Generalized Lagrange Wavelets

[00251] Three kinds of interpolating Lagrange wavelets, Halfband Lagrange wavelets, B-spline Lagrange wavelets and Gaussian-Lagrange DAF wavelets, are studied here as examples of the generalized interpolating wavelets.

[00252] Halfband Lagrange wavelets can be regarded as extensions of the Dubuc interpolating functionals [9, 13], the auto-correlation shell wavelet analysis [40], and halfband filters [1]. B-spline Lagrange Wavelets are generated by a B-spline-windowed Lagrange functional which increases the smoothness and localization properties of the simple Lagrange scaling function and its related wavelets. Lagrange Distributed Approximating Functionals (LDAF)-Gaussian modulated Lagrange polynomials, have been successfully applied for numerically solving various linear and nonlinear partial differential equations. Typical examples include DAF-simulations of 3-dimensional reactive quantum scattering and the solution of a 2-dimensional Navier-Stokes equation with non-periodic boundary conditions. In terms of a wavelet analysis, DAFs can be regarded as particular scaling functions (wavelet-DAFs) and the associated DAF-wavelets can be generated in a number of ways [20, 21, 22, 65, 66, 67].

[heading-00253] Halfband Lagrange Wavelets

[00254] A special case of halfband filters can be obtained by choosing the filter coefficients according to the Lagrange interpolation formula. The filter coefficients are given by $\begin{array}{cc}h\left(2n-1\right)=\frac{{\left(-1\right)}^{n+M-1}\prod _{m=1}^{2M}\left(M+1/2-m\right)}{\left(M-n\right)!\left(M+n-1\right)!\left(2n-1\right)}& \left(34\right)\end{array}$
These filters have the property of maximal flatness in Fourier space, possessing a balance between the degree of flatness at zero frequency and the flatness at the Nyquist frequency (half sampling).

[00256] These half-band filters can be utilized to generate the interpolating wavelet decomposition, which can be regarded as a class of the auto-correlated shell of orthogonal wavelets, such as the Daubechies wavelets [7]. The interpolating wavelet transform can also be extended to higher order cases using different Lagrange polynomials, as [40] $\begin{array}{cc}{P}_{2n-1}\left(x\right)=\prod _{m=-M+1,m\ne n}^M\frac{x-\left(2m-1\right)}{\left(2n-1\right)-\left(2m-1\right)}& \left(35\right)\end{array}$
The predictive interpolation can be expressed as $\begin{array}{cc}\Gamma S_j\left(i\right)=\sum _{n=1}^M{P}_{2n-1}\left(0\right)\left[S_j\left(i+2n-1\right)+S_j\left(i-2n+1\right)\right],i=2k+1& \left(36\right)\end{array}$
where Γ is a projection and S_j is the jth layer low-pass coefficients. This projection relation is equivalent to the subband filter response of
h(2n−1)=P_2n−1(0)  (37)
The above-mentioned interpolating wavelets can be regarded as the extension of the fundamental Deslauriers-Dubuc interactive sub-division scheme, which results when M=2. The order of the Lagrange polynomial is D=2M−1=3 (FIG. 15(a)).

[00261] It is easy to show that an increase of the Lagrange polynomial order D, will introduce higher regularity for the interpolating functionals (FIG. 16(a)). When D→+∞, the interpolating functional tends to a band-limited Sinc function and its domain of definition is on the real line. The subband filters generated by Lagrange interpolating functionals satisfy the properties:

[00262] (1) Interpolation: h(ω)+h(ω+π)=1

[00263] (2) Symmetry: h(ω)=h(−ω)

[00264] (3) Vanishing Moments: ∫_Rx^pφ(x)dx=δ_p

• • [00265] Donoho outlines a basic subband extension to obtain a perfect reconstruction. He defines the wavelet function as
    ψ(x)=φ(2x−1)  (38)
    The biorthogonal subband filters can be expressed as
     {tilde over (h)}(ω)=1, g(ω)=e^−iω, {tilde over (g)}(ω)=e^−iω{overscore (h(ω+π))}  (39)
    However, the Donoho interpolating wavelets have some drawbacks. Because the low-pass coefficients are generated by a sampling operation only, as the decomposition layer increases, the correlation between low-pass coefficients become weaker. The interpolating (prediction) error (high-pass coefficients) strongly increases, which is deleterious to the efficient representation of the signal. Further, it can not be used to generate a Riesz basis for L²(R) space.

[00270] Swelden has provided an efficient and robust scheme [56] for constructing biorthogonal wavelet filters. His approach can be utilized to generate high-order interpolating Lagrange wavelets with higher regularity. As FIG. 13 shows, P₀ is the interpolating prediction process, and the P₁ filter is called the updating filter, used to smooth the down-sampling low-pass coefficients. If we choose P₀ to be the same as P₁, then the new interpolating subband filters can be depicted as $\begin{array}{cc}\{\begin{array}{c}{h}_1\left(\omega \right)=h\left(\omega \right)\\ {\tilde{h}}_1\left(\omega \right)=1+\tilde{g}\left(\omega \right)\stackrel{\_}{P\left(2\omega \right)}\\ g_1\left(\omega \right)=e^{-i\omega }-h\left(\omega \right)P\left(2\omega \right)\\ {\tilde{g}}_1\left(\omega \right)=\tilde{g}\left(\omega \right)\end{array}& \left(40\right)\end{array}$
The newly developed filters h₁, g₁, {tilde over (h)}, and {tilde over (g)} also generate the biorthogonal dual pair for a perfect reconstruction. Examples of biorthogonal lifting wavelets with regularity D=3 are shown in FIG. 14. FIG. 15 gives the corresponding Fourier responses of the equivalent subband decomposition filters.
B-Spline Lagrange Wavelets

[00273] Lagrange polynomials are natural interpolating expressions for functional approximations. Utilizing a different expression for the Lagrange polynomials, we can construct other forms of useful interpolating wavelets as follows. We define a class of symmetric Lagrange interpolating functional shells as $\begin{array}{cc}{P}_M\left(x\right)=\prod _{i=-M,i\ne 0}^M\frac{x-i}{-i}& \left(41\right)\end{array}$
It is easy to verify that this Lagrange shell also satisfies the interpolating condition on discrete, integer points, $\begin{array}{cc}{P}_M\left(k\right)=\{\begin{array}{cc}1,& k=0\\ 0,& \mathrm{otherwise}\end{array}& \left(42\right)\end{array}$
However, simply defining the filter response as
h(k)=P(k/2)/2, k=−M, M  (43)
leads to non-stable interpolating wavelets, as shown in FIG. 16.

[00278] Including a smooth window, which vanishes at the zeros of the Lagrange polynomial, will lead to more regular interpolating wavelets and equivalent subband filters (as shown in FIGS. 16 and 17). We select a well-defined B-spline function as the weight window. Then the scaling function (mother wavelet) can be defined as an interpolating B-Spline Lagrange functional (BSLF) $\hspace{1em}\begin{array}{cc}{\varphi }_M\left(x\right)=\frac{{\beta }^N\left(x/\eta \right)}{{\beta }^N\left(0\right)}{P}_M\left(x\right)=\frac{{\beta }^N\left(x/\eta \right)}{{\beta }^N\left(0\right)}\prod _{i=-M,i\ne 0}^M\frac{x-i}{-i}& \left(44\right)\end{array}$
where N is the B-spline order and, η is the scaling factor to control the window width. To ensure coincidence of the zeroes of the B-spline and the Lagrange polynomial, we set
2M=η×(N+1)  (45)
To ensure the interpolation condition, the B-spline envelope degree M must be odd number. It is easy to show that if B-spline order is N=4k+1, η can be any odd integer (2k+1); if N is an even integer, then η can only be 2. When N=4k−1, we can not construct an interpolating shell using the definition above. From the interpolation and self-induced scaling properties of the interpolating wavelets, it is easy to verify that
h(k)=φ_M(k/2)/2, k=−2M+1, 2M−1   (46)
Gaussian-Lagrange DAF Wavelets

[00284] We can also select a class of distributed approximation functional—Gaussian-Lagrange DAFs (GLDAF) as our basic scaling function to construct interpolating wavelets as: $\hspace{1em}\begin{array}{cc}{\varphi }_M\left(x\right)=W_{\sigma }\left(x\right)P_M\left(x\right)=W_{\sigma }\left(x\right)\prod _{i=-M,i\ne 0}^M\frac{x-i}{-i}& \left(47\right)\end{array}$
where W_σ(x) is a window function. It is chosen to be a Gaussian,
W_σ(x)=e^−x2/2σ2   (48)
because it satisfies the minimum frame bound condition in quantum physics. Here σ is a window width parameter, and P_M(x) is the Lagrange interpolation kernel. The DAF scaling function has been successfully introduced as an efficient and powerful grid method for quantum dynamical propagations [40]. Using Swelden's lifting scheme [32], a wavelet basis is generated. The Gaussian window in our DAF-wavelets efficiently smoothes out the Gibbs oscillations, which plague most conventional wavelet bases. The following equation shows the connection between the B-spline window function and the Gaussian window [34]: $\begin{array}{cc}{\beta }^N\left(x\right)\cong \sqrt{\frac{6}{\pi \left(N+1\right)}}\exp \left(\frac{-6x^2}{N+1}\right)& \left(49\right)\end{array}$
for large N. As in FIG. 21, if we choose the window width
σ=η√{square root over ((N+1)/12)}  (50)
the Gaussian-Lagrange wavelets generated by the lifting scheme will be similar to the B-spline Lagrange wavelets. Usually, the Gaussian-Lagrange DAF displays a slightly better smoothness and more rapid decay than the B-spline Lagrange wavelets. If we select more sophisticated window shapes, such as those popular in engineering (Bartlett, Hanning, Hamming, Blackman, Chebychev, and Bessel windows), the Lagrange wavelets can be generalized further. We shall call these extensions Bell-windowed Lagrange wavelets.
Generalized Sinc Wavelets

[00292] As we have mentioned above, the π band-limited Sinc function,
φ(x)=sin(πx)/(πx)C^∞  (51)
in Paley-Wiener space, constructs an interpolating function. Every π band-limited function ƒ∈L²(R) can be reconstructed by the equation $\begin{array}{cc}f\left(x\right)=\sum _{k}f\left(k\right)\frac{\sin \pi \left(x-k\right)}{\pi \left(x-k\right)}& \left(52\right)\end{array}$
where the related wavelet function—Sinclet is defined as (see FIG. 10) $\begin{array}{cc}\psi \left(x\right)=\frac{\sin \pi \left(2x-1\right)-\sin \pi \left(x-1/2\right)}{\pi \left(x-1/2\right)}& \left(53\right)\end{array}$
The scaling Sinc function is the well-known ideal low-pass filter, which possesses the ideal square filter response as $\begin{array}{cc}H\left(\omega \right)=\{\begin{array}{cc}1,& \omega \le \pi /2\\ 0,& \pi /2<\omega \le \pi \end{array}& \left(54\right)\end{array}$
Its impulse response can be generated as
h[k]=∫_{(−π/2,π/2)}e^jkωdω/2π=sin(πk/2)/(πk)  (55)
The so-called half-band filter possess a non-zero impulse only at the odd integer sampler, h(2k+1), while at even integers, h[2k]=0 unless a k=0.

[00300] However, this ideal low-pass filter is never used in application. Since the digital filter is an IIR (infinite impulse response) solution, its use as a digital cutoff FIR (finite impulse response) will produce Gibbs phenomenon (overshot effect) in Fourier space, which degrades the frequency resolution (FIG. 20). The resulting compactly supported Sinc scaling and wavelet functions, as well as their biorthogonal dual scaling and wavelet functions, are shown in FIG. 21. We see that the regularity of the cutoff Sinc is obviously degraded with a fractal-like shape, which leads to poor time localization.

[heading-00301] B-Spline Sinc Wavelets

[00302] Because the ideal low-pass Sinc wavelet can not be implemented “ideally” by FIR (finite impulse response) filters, to eliminate the cutoff singularity, a windowed weighting technique is employed to adjust the time-frequency localization of the Sinc wavelet analysis. To begin, we define a symmetric Sinc interpolating functional shell as $\begin{array}{cc}P\left(x\right)=\frac{\sin \left(\pi x/2\right)}{\pi x}& \left(56\right)\end{array}$
Utilizing a smooth window, which vanishes gradually at the exact zeros of the Sinc functional, will lead to more regular interpolating wavelets and equivalent subband filters (as shown in FIGS. 22 and 23). For example, we illustrate using a well-defined B-spline function as the weight window. Then the scaling function (mother wavelet) can be defined as an interpolating B-spline Sinc functional (BSF) $\hspace{1em}\begin{array}{cc}{\varphi }_M\left(x\right)=\frac{{\beta }^N\left(x/\eta \right)}{{\beta }^N\left(0\right)}P\left(x\right)=\frac{{\beta }^N\left(x/\eta \right)}{{\beta }^N\left(0\right)}\frac{\sin \left(\pi x/2\right)}{\pi x}& \left(57\right)\end{array}$
where N is the B-spline order and, η is the scaling factor to control the window width. To ensure the coincidence of the zeroes of the B-spline and the Sinc shell, we set
2M+1=η×(N+1)/2   (58)
To maintain the interpolation condition, h(2k)=0, k≠0, it is easy to show that when the B-spline order N=4k+1, η may be any odd integer (2k+1). If N is an even integer, then η can only be 2. FIG. 24 shows a mother wavelet comparison for a B-spline Sinc and a Guassian Sinc for N=4 and η2. When N=4k−1, we can not construct interpolating shell using the definition above. The admissibility condition can be expressed as $\begin{array}{cc}\{\begin{array}{cc}\eta =2,& N=2i\\ \eta =2k+1,& N=4i+1\end{array}& \left(59\right)\end{array}$
From the interpolation relation $\begin{array}{cc}\varphi \left(k\right)=\{\begin{array}{cc}1,& k=0\\ 0,& k\ne 0\end{array},k\in Z& \left(60\right)\end{array}$
and the self-induced two-scale relation $\begin{array}{cc}\varphi \left(x\right)=\sum _k\varphi \left(k/2\right)\varphi \left(2x-k\right)& \left(61\right)\end{array}$
it is easy to show that
h(k)=φ_M(k/2)/2, k=−2M+1, 2M−1   (62)
Gaussian-Sinc DAF Wavelets

[00312] We can also select a class of distributed approximation functionals, i.e., the Gaussian-Sinc DAF (GSDAF) as our basic scaling function to construct interpolating scalings, $\hspace{1em}\begin{array}{cc}{\varphi }_M\left(x\right)=W_{\sigma }\left(x\right)P\left(x\right)=W_{\sigma }\left(x\right)\frac{\sin \left(\pi x/2\right)}{\pi x}& \left(64\right)\end{array}$
where W_σ(x) is a window function which is selected as a Gaussian,
W_σ(x)=e^−x2/2σ2  (64)
Because it satisfies the minimum frame bound condition in quantum physics, it significantly improves the time-frequency resolution of the Windowed-Sinc wavelet. Here σ is a window width parameter, and P(x) is the Sinc interpolation kernel. This DAF scaling function has been successfully used in an efficient and powerful grid method for quantum dynamical propagations [40]. Moreover, the Hermite DAF is known to be extremely accurate for solving the 2-D harmonic oscillator, for calculating the eigenfunctions and eigenvalues of the Schrodinger equation. The Gaussian window in our DAF-wavelets efficiently smoothes out the Gibbs oscillations, which plague most conventional wavelet bases. The following equation shows the connection between the B-spline and the Gaussian windows [34]: $\begin{array}{cc}{\beta }^N\left(x\right)\cong \sqrt{\frac{6}{\pi \left(N+1\right)}}\exp \left(\frac{-6x^2}{N+1}\right)& \left(65\right)\end{array}$
for large N. As in FIG. 15, if we choose the window width
σ=η√{square root over ((N+1)/12)}  (66)
the Gaussian Sinc wavelets generated by the lifting scheme will be similar to the B-spline Sinc wavelets. Usually, the Gaussian Sinc DAF displays a slightly better smoothness and rapid decay than the B-spline Lagrange wavelets. If we select more sophisticated window shapes, the Sinc wavelets can be generalized further. We call these extensions Bell-windowed Sinc wavelets. The available choices can be any of the popularly used DFT (discrete Fourier transform) windows, such as Bartlett, Hanning, Hamming, Blackman, Chebychev, and Besel windows.

[00319] The above mentioned generalized interpolating wavelet is defined on the domain C(R). Many engineering applications involve finite length signals, such as image and isolated speech segments. In general, we can define these signals on C[0,1]. One could set the signal equal to zero outside [0,1], but this introduces an artificial “jump” discontinuity at the boundaries, which is reflected in the wavelet coefficients. It will degrade the signal filtering and compression in multi scale space. Developing wavelets adapted to “life on an interval” is useful. Periodization and symmetric periodization are two commonly used methods to reduce the effect of edges. However, unless the finite length signal has a large flat area around the boundaries, these two methods cannot remove the discontinuous effects completely [4,6,11].

[00320] Dubuc utilized an iterative interpolating function, F_D on the finite interval to generate an interpolation on the set of dyadic rationals D_j. The interpolation in the neighborhood of the boundaries is treated using a boundary-adjusted functional, which leads to regularity of the same order as in the interval. This avoids the discontinuity that results from periodization or extending by zero. It is well known that this results in weaker edge effects, and that no extra wavelet coefficients (to deal with the boundary) have to be introduced, provided the filters used are symmetric.

[00321] We let K_j represent the number of coefficients at resolution layer j, where K_j=2^j. Let 2^j>2D+2, define the non-interacting decomposition. If we let j₀ hold the non-interaction case 2^j0>2D+2, then there exist functions φ_j,k^Interval, ψ_j,k^Interval, such that for all ƒ∈C[0,1], $\begin{array}{cc}f=\sum _{k=0}^{2^{j0}-1}{S}_{j_0}\left(k\right){\varphi }_{j_{0,k}}^{\mathrm{Interval}}+\sum _{j\ge j_0}\sum _{k=0}^{2^j-1}{W}_j\left(k\right){\psi }_{j,k}^{\mathrm{Interval}}& \left(67\right)\end{array}$
The φ_j,k^Interval, ψ_j,k^Interval, are called the interval interpolating scalings and wavelets, which satisfy the interpolation conditions $\begin{array}{cc}\{\begin{array}{cc}{\varphi }_{j,k}^{\mathrm{Interval}}\left(2^{-j}n\right)=2^{j/2}{\delta }_{k,n},& 0\le n<K_j\\ {\psi }_{j,k}^{\mathrm{Interval}}\left(2^{-j-1}n\right)=2^{j/2}{\delta }_{2k+1,n},& 0\le n<K_{j+1}\end{array}& \left(68\right)\end{array}$
The interval scaling is defined as $\begin{array}{cc}{\varphi }_{j,k}^{\mathrm{Interval}}=\{\begin{array}{c}{\varphi }_{j,k}^{\mathrm{Left}},0\le k\le D\\ {{\varphi }_{j,k}}_{\left[0,1\right]},D<k<2^j-D-1\\ {\varphi }_{j,k}^{\mathrm{Right}},2^j-D-1\le k<2^j\end{array}& \left(69\right)\end{array}$
where φ_j,k|_[0,1] is called the “inner-scaling” which is identical to the fundamental interpolating function, and φ_j,k^Right and φ_j,k^Left are the “left-boundary” and the “right-boundary” scalings, respectively. Both are as smooth as φ_j,k|_[0,1]. Interval wavelets are defined as $\begin{array}{cc}{\psi }_{j,k}^{\mathrm{Interval}}=\{\begin{array}{c}{\psi }_{j,k}^{\mathrm{Left}},0\le k<\lfloor D/2\rfloor \\ {{\psi }_{j,k}}_{\left[0,1\right]},\lfloor D/2\rfloor \le k<2^j-\lfloor D/2\rfloor \\ {\psi }_{j,k}^{\mathrm{Right}},2^j-\lfloor D/2\rfloor \le k<2^j\end{array}& \left(70\right)\end{array}$
ψ_j,k|_[0,1] is the inner-wavelet, and ψ_j,k^Left and ψ_j,k^Right are the left and right-boundary wavelets, respectively, which are of the same order regularity as the inner-wavelet [13].

[00326] The corresponding factors for the Deslauriers-Dubuc interpolating wavelets are M=2, and the order of the Lagrange polynomial is D=2M−1=3. The interpolating wavelet transform can be extended to high order cases by two kinds of Lagrange polynomials, where the inner-polynomials are defined as [14] $\begin{array}{cc}{P}_{2n-1}\left(x\right)=\prod _{m=-M+1,m\ne n}^M\frac{x-\left(2m-1\right)}{\left(2n-1\right)-\left(2m-1\right)}& \left(71\right)\end{array}$
This kind of polynomial introduces the interpolation in the intervals according to $\begin{array}{cc}{P}_{j}S\left(i\right)=\sum _{n=1}^M{P}_{2n-1}\left(0\right)\left[S_j\left(i+2n-1\right)+S_j\left(i-2n+1\right)\right]i=2k+1& \left(72\right)\end{array}$
and the boundary polynomials are $\begin{array}{cc}{L}_d\left(x\right)=\prod _{m=0,m\ne d}^{2M-1}\frac{x-m}{d-m},0\le d\le D& \left(73\right)\end{array}$
which introduce the adjusted interpolation on the two boundaries of the intervals. That is, $\begin{array}{cc}{P}_{j}S\left(i\right)=\sum _{d=0}^D{L}_d\left(i/2\right)S_j\left(i+2d-1\right),i=2k+1,0\le k\le \lfloor D-1\rfloor /2& \left(74\right)\end{array}$
The left boundary extrapolation outside the intervals is defined as $\begin{array}{cc}{P}_{j}S\left(-1\right)=\sum _{d=0}^D{L}_d\left(-1/2\right)S_j\left(2d\right),i=2k+1,0\le k\le \lfloor D-1\rfloor /2& \left(75\right)\end{array}$
and the right boundary extrapolation is similar to the above. The boundary adjusted interpolating scaling is shown in FIG. 25.

[00332] Although Dubuc shows the interpolation is almost twice differentiable, there still is a discontinuity in the derivative. In this disclosure, a DAF-wavelet based boundary adjusted algorithm is introduced. This technique can produce an arbitrary smooth derivative approximation, because of the infinitely differentiable character of the Gaussian envelope. The boundary-adjusted scaling functionals are generated as conveniently as possible just by window shifting and satisfy the following equation.
φ_m(x)=W(x−2m)P(x), m=−└(M−1)/2┘, └(M−1)/2┘  (76)
where φ_m(x) represents different boundary scalings, W(x) is the generalized window function and P(x) is the symmetric interpolating shell. When m>0, left boundary functionals are generated; when m<0, we obtain right boundary functionals. The casem=0 represents the inner scalings mentioned above.

[00335] One example for a Sinc-DAF wavelet is shown in FIG. 26. We choose the compactly-supported length of the scaling function to be the same as the halfband lagrange wavelet. It is easy to show that our newly developed boundary scaling is smoother than the commonly used Dubuc boundary interpolating functional. Thus it will generate a more stable boundary adjusted representation for finite-length wavelet transforms, as well as a better derivative approximation around the boundaries. FIG. 27 is the boundary filter response comparison between the halfband Lagrange wavelet and our DAF wavelet. It is easy to establish that our boundary response decreases the overshoot of the low-pass band filter, and so is more stable for boundary frequency analysis and approximation.

[heading-00336] Eigenvalue Solution of 2D Quantum Harmonic Oscillator

[00337] As discussed in Ref. [22], a standard eigenvalue problem of the Schrodinger equation is that of the 2D harmonic oscillator, $\begin{array}{cc}\left[-\frac{h^2}{2m}\sum _{i=1}^2\frac{{\partial }^2}{\partial x_i^2}+\frac{1}{2}\left(x_1^2+x_2^2\right)\right]{\Phi }_k\left(x_1,x_2\right)=E_k{\Phi }_k\left(x_1,x_2\right)& \left(77\right)\end{array}$
Here Φ_k and E_k are the kth eigenfunction and eigenvalue respectively. The eigenvalues are given exactly by
E_k1,k2=1+k₁+k₂, 0≦k<∞,0≦k₁≦k₂   (78)
with a degeneracy (k_d=k+1) in each energy level E_k=1+k. The 2D version of the wavelet DAF representation of the Hamiltonian operator was constructed and the first 21 eigenvalues and eigenfunctions obtained by subsequent numerical diagonalization of the discrete Sinc-DAF Hamiltonian. As shown in Table 1, all results are accurate to at least 10 significant figures for the first 16 eigenstates. It is evident that DAF-wavelets are powerful for solving eigenvalue problems.

[00002]

TABLE 1
Eigenvalues of the 2D harmonic oscillator
	k = kx + ky	kd	Exact Solution	Sinc-DAF Calculation
	0	1	1	0.99999999999835
	1	2	2	1.99999999999952
				1.99999999999965
	2	3	3	2.99999999999896
				2.99999999999838
				2.99999999999997
	3	4	4	3.99999999999943
				3.99999999999947
				3.99999999999986
				3.99999999999994
	4	5	5	4.99999999999907
				4.99999999999953
				4.99999999999989
				5.00000000000674
				5.00000000000813
	5	6	6	5.99999999999982
				6.00000000000018
				6.00000000000752
				6.00000000000801
				6.00000000011972
				6.00000000012005

Target Extraction

[00342] Military target extraction, including such applications as high-resolution radar (aerial photograph) imaging, radar echo, and remote sense detection, is a difficult subject. National Defense Agencies (e.g. the Navy, Army, and Air Force) have great interest in technical advances for reconnaissance, earlier warning and target recognition. Some of our DAF-wavelet schemes represent a significant breakthrough for these very difficult tasks. Compared with other methods, our algorithm possesses a very high resolution for target localization and high efficiency for clutter/noise suppression, as well as computational efficiency.

[00343] Detecting a military target in a low luminance environment is challenging work for image processing. To improve the target discrimination, the visibility of differences between a pair of images is important for modern image restoration and enhancement. We construct a method for detectability using a multiple channel enhancement technique. The images were captured on a color monitor at a viewing distance giving 95 pixels per degree of visual angle and an image size of 5.33*5.05 degrees. The mean luminance of the images was about 10cd/m² [39]. Using our newly developed visual enhancement techniques, visual targets can be extracted very accurately in a low-luminance environment for detection and warning. The technique combines the response of human vision system (HVS) with multiresolution enhancement and restoration methods. The simulation of tank-target detection in a low-luminance environment is shown in FIG. 28.

[heading-00344] Image Filtering

[00345] Image de-noising is a difficult problem for signal processing. Due to the complicated structure of image and background noise, an optimal filtering technique does not currently exist. Generally, the possible noise sources include photoelectric exchange, photo spots, image communication error, etc. Such noise causes the visual perception to generate speckles, blips, ripples, bumps, ringing and aliasing. The noise distortion not only affects the visual quality of images, but also degrades the efficiency of data compression and coding. De-noising and smoothing are extremely important for image processing.

[00346] We use a DAF-wavelet to generate a more efficient, human-vision-system-based image processing technique, which processes the advantages of 1) long range de-correlation for convenience of compression and filtering; 2) high perceptual sensitivity and robustness; 3) filtering that takes account of the human visual response. It therefore can enhance the most important visual information, such as edges, while suppressing the large scale of flat regions and background; 4) it can be carried out with real-time processing.

[00347] Biorthogonal interpolating wavelets and corresponding filters are constructed based on Gauss-Lagrange distributed approximating functionals (DAFs). The utility of these DAF wavelets and filters is tested for digital image de-noising in combination with a novel blind restoration technique. This takes account of the response of human vision system so as to remove the perceptual redundancy and obtain better visual performance in image processing. The test results for a color photo are shown in FIG. 29. It is evident that our Color Visual Group Normalization technique yields excellent contrast and edge-preservation and provides a natural color result for the restored image [48].

[heading-00348] Imaging Enhancement

[00349] Mammograms are complex in appearance and signs of early disease are often small and/or subtle. Digital mammogram image enhancement is particularly important for solving storage and logistics problems, and for the possible development of an automated-detection expert system. The DAF-wavelet based mammogram enhancement is implemented in the following manner. First we generate a perceptual lossless quantization matrix Q_j,m to adjust the original transform coefficients C_j,m(k). This treatment provides a simple human-vision-based threshold technique for the restoration of the most important perceptual information in an image. For grayscale image contrast stretching, we appropriately normalize the decomposition coefficients according to the length scale, L, of the display device [16] so that they fall in the interval of [0,1] of the device frame
NC_j,m(k)=Q_j,mC_j,m(k)/L.  (79)
We then use a nonlinear mapping to obtain the desired contrast stretching
{overscore (NC_j,m)}=γ_j,mX_j,m(NC_j,m)  (80)
where the constant γ_j,m and function X_j,m are appropriately chosen so that the desired portion of the grayscale gradient is stretched or compressed.

[00354] To test our new approach, low-contrast and low quality breast mammogram images are employed. A typical low quality front-view image is shown in FIG. 30(a). The original image is coded at 512×512 pixel size with 2 bytes/pixel and 12 bits of gray scale. We have applied our edge enhancement normalization and device-adapted visual group normalization. As shown in FIG. 30(b), and FIG. 30(c), there is a significant improvement in both the edge representation and image contrast. In particular, the domain and internal structure of high-density cancer tissues are more clearly displayed. FIG. 31(a) is an original 1024×1024 side-view breast image which has been digitized to a 200 micron pixel edge with 8 bits of gray scale. The enhanced image result is shown in FIG. 31(b). In this case we again obtain a significant improvement in image quality as described herein.

[00355] In summary, a new class of wavelets—generalized symmetric interpolating wavelets were described, which are generated by a window modulated interpolating shell. Due to the absence of a complicated factorization process, this kind of interpolating wavelet is easily implemented and possesses very good characteristics in both time (space) and spectral domains. The stable boundary adjustment can be generated by a window shifting operation only. It overcomes the overshoot of the boundary response introduced by other boundary processing, such as Dubuc Lagrange wavelet and Daubechies boundary filters. Many successful applications of DAF-wavelets have been reported to illustrate its practicality and its mathematical behavior.

[none]

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[00424] Local averaging is one of the simplest filtering techniques. It preserves the mean gray level while suppressing the variability in flat regions. However, simple linear local averaging is undesirable for image smoothing because it is incapable of preserving image edges. Linear local averaging filters are essentially low pass filters, which tend to blur the edges and fine structures in the original image. In order to preserve the edges while achieving some degree of smoothing, it is desirable to employ nonlinear filters. A number of nonlinear selective averaging methods [1-8] have been devised for this purpose. The basic idea for these methods is to select only a portion of the gray level values in the local window to use in a (weighted) average. In references [3-5], an alpha-trimmed mean filter (α-TMF), which uses a “median basket” to select a predetermined number of pixels above and below the median pixel to the sorted pixels of the moving window, was proposed.

[00425] The values in the basket are averaged to give the α-TMF filtering output. An asymmetric way to select the averaging pixels whose values are close to that of the median pixel was presented in references [4-6] and was named the modified trimmed mean filter (MTMF) in reference [4]. Recently, we developed a generalization of the α-TMF, which we denoted as GTMF [8].

[00426] It employs the same way to select the pixels from the window for inclusion in the median basket. However, the selected pixels and the center pixel in the window are weighted and averaged to give the filtering output. It has been shown [8] that the GTMF performs better than other well known filters for the removal of either impulse noise or additive noise.

[00427] A new nonlinear filtering technique is disclosed, called the “dual window selective averaging filter” (DWSAF), to remove additive noise (e.g., Gaussian noise). Assuming implicitly that the ideal image is piecewise flat, two normal concentric moving windows and a pixel container are employed to determine the values to be used in replacing the gray level value of the center pixel I_c. Three steps are employed in this filtering algorithm. First, the GTMF is implemented within the smaller window W_S to give an intermediate output G_c at the center pixel of the window. Second, only the gray level values close to G_c are selected from the larger window W_L and put to the container C. Third, if the number of pixels in C equals zero, the final DWSAF output D_c is just G_c, otherwise, the gray level values in C are averaged to give the final DWSAF output D_c.

[00428] In contrast to the α-TMF, the GTMF also includes the center pixel I_c in the averaging operation and its weight is usually larger than those of other pixels in the median basket, which is important for the removal of additive noise.

[00429] A threshold T based on the GTMF output G_c is used to select the pixels for the container. A pixel is selected for inclusion in the container if its value is in the range of [G_c−T, G_c+T]. When T=0, the DWSAF is equivalent to the GTMF within the smaller window, and when T=255 (for an 8bpp gray scale image), it becomes a simple moving average within the larger window.

[00430] The implementation of the generalized trimmed mean filter (GTMF) has been described in detail in [17]. The pixels {I₁, I₂, . . . , I_m−1, I_m, I_m+1, . . . I_n} in the local window associated with a pixel I_c, have been arranged in an ascending (or descending) order, with I_m being the median pixel. The key generalization to median filtering introduced in the alpha-trimmed mean filter (α-TMF) [18] is to design a median basket to combine a group of pixels whose gray level values are close to the median value of the window.

[00431] An averaging operation is then utilized to generate an adjusted replacement A_c for I_c. For example, a 3-entry median basket α-TMF is implemented according to $\begin{array}{cc}{A}_c=\frac{\left(I_{m-1}+I_m+I_{m+1}\right)}{3}& \left(81\right)\end{array}$

[00432] It is evident that a single-entry median basket α-TMF is equivalent to the median filter and a n-entry median basket α-TMF is equivalent to the simple moving average filter.

[00433] In general, the α-TMF outperforms the median filter. However, it is still not optimal when filtering either the additive noise corrupted images or highly impulse noise corrupted images. For example, when removing additive noise, the α-TMF does not take the I_c as a special pixel. As is well known, for an image corrupted by additive noise, I_c has the largest probability of being the closest to the true value among all the pixels in the window. Neglecting the influence of the center pixel is a mistake if one desires to filter additive noise. Another difficulty for the α-TMF, when it is used to remove impulse noise from highly corrupted images, is that the pixels selected for the median basket may also be corrupted by impulse noise. It is therefore unreasonable for all the pixels in the basket to have the same weight.

[00434] The GTMF uses a median basket to collect a group of pixels from the sorted pixel values of the window associated with I_c, in the same way as the α-TMF.

[00435] The values of the selected pixels and I_c are then weighted and averaged to give the GTMF output.

[00436] For a 3-entry median basket, the output of the GTMF is given by $\begin{array}{cc}{G}_c=\frac{\left(w_1{I}_{m-1}+w_2{I}_m+w_3{I}_{m+1}+w_c{I}_c\right)}{\left(w_1+w_2+w_3+w_c\right)}& \left(82\right)\end{array}$
where G_c is the GTMF output, w₁, w₂, w₃, and w_c are the weights. It is interesting to see that when w_c=0 and w₁=w₂=w₃≠0, the GTMF reduces to the α-TMF. When w₁=w₂=w₃=0, it becomes the standard median filter. Since the GTMF takes the center pixel I_c into account, we expect that it should also work better for removing additive noise.

[00438] In this section, we will discuss how to implement our dual window selective averaging filter (DWSAF), based on the GTMF. The standard selective averaging algorithm [2] is not optimal because it only computes the average (or weighted-average) of the gray level values close to that of the center pixel in a window. However, sometimes it is unreasonable to choose the gray level value of the center pixel as a criteria because the pixel may be highly corrupted. Using the standard selective averaging algorithm to treat such a pixel will result in it remaining highly corrupted.

[00439] The key idea of the DWSAF is to find an alternative criteria for the standard selective averaging algorithm to use. In reference [4], a modified trimmed mean filter (MTMF) was proposed to average only those pixels whose gray level values fall within the range [M_c−q, M_c+q], where M_c is the median value (M_c=I_m) and q is a preselected threshold.

[00440] Here, we propose a new filtering algorithm which employs two different sizes of concentric moving windows W_S and W_L, and a pixel container C. Three steps are involved in the filtering technique. First, the GTMF is employed in the smaller window W_S according to Equation (82) to give an intermediate output G_c, which is used as the criteria for the standard selective averaging algorithm.

[00441] Second, from the larger window W_L, only the pixels with gray level values close to G_c are selected and placed in a container C. In doing so, a switching threshold T is employed to determine if a pixel in W_L is close enough to G_c. We determine if a pixel I_k in W_L belongs to C as follows:
I_k∈C, if |I_k−G_c·|<T,  (83)
where T is the threshold.

[00444] Third, if the number of pixels N in the container C is greater than zero, the gray level values of the pixels in the container are averaged with the same weight for all pixels to give the final output of the DWSAF. If N=0, the final output equals G_c·. We summarize this as $\begin{array}{cc}{D}_c=\{\begin{array}{cc}{A}_c,& \mathrm{if}N\ne 0\\ G_c,& \mathrm{if}N=0\end{array}& \left(84\right)\end{array}$
where D_c is the output of the DWSAF to replace I_c and A_c is the average of the gray level values in the container. The reason for employing two windows is as follows. As in the case of median filter, the implementation of the GTMF using a larger window blurs the image more than using a smaller window. However, the output of the GTMF in a smaller window can be used as a criteria to improve the filtering result by use of the standard selective averaging algorithm in the larger window.

[00446] This reduces the image blurring because we only average those gray level values that are close to the GTMF output. For different choices of the threshold T, the DWSAF can be made to be either a GTMF with respect to the smaller window W_S (T=0) or a simple moving average within the larger window W_L (T=255 for an 8bpp gray-scale image).

[00447] The DWSAF algorithm can be employed iteratively, which generally improves its performance. If the weight w_c of the center pixel is too high, the output values at some pixels may remain close to their original input values after a number of iterations, and thus some isolated impulse-like noise may be present. Numerical experimentation shows that changing the threshold T to zero after a number of iterations greatly alleviates this problem.

[00448] Two numerical examples are presented here for testing our filtering algorithm. The first one is a standard one-dimensional blocked signal, which is shown in FIG. 32(a). We degrade it with Gaussian noise of mean square error MSE=1.00 and mean average error MAE=0.80 (see FIG. 33(b)). The sizes of the two moving windows we used are 7 and 19 for W_S and W_L respectively. The weights for the 3-entry basket which we used for the GTMF are w₁=w₃=0, w₂=3 and w_c=4. The switching threshold is chosen to be T=1.5. To obtain improved filtering results, the DWSAF algorithm is implemented recursively.

[00449] In FIG. 32(c), we show the filtered signal. It is seen from FIG. 32(c) that the filtered result is in excellent agreement with the original noise-free signal. The MSE and MAE of our filtered result are 6.19E-2 and 6.49E-3 respectively! The DWSAF is extremely efficient and robust for removing additive noise from blocked signals.

[00450] To further confirm the usefulness of the algorithm, we next consider a two-dimensional benchmark 8 bit, 512 \times 512, “Lena” image, which has been degraded with two different amounts of Gaussian noise. FIG. 33(a) shows the corrupted Lena image with peak signal-to-noise ratio PSNR=22.17 dB and FIG. 34(a) shows the corrupted image with PSNR=18.82 dB. Since the original noise-free image is not exactly piecewise flat, it is not desirable to produce a filtering performance as efficient as in the first example. However, we expect that the DWSAF will yield improved results.

[00451] The sizes of the two windows for both corrupted images are 3 for W_S and 5 for W_L. Three-entry median basket is used for both corrupted images, and the weights are w₁:w₂:w₃:w_c=1:1:1:10 for the image shown in FIG. 33(a) and w₁:w₂:w₃:w_c=1:1:1:50 for the image shown in FIG. 34(a). The initial switching threshold T is 50 for both corrupted images, and is changed to 0 after the first iteration for the lower noise image and after the second iteration for the higher noise image. FIGS. 33(b) and 34(b) show the filtered images produced by our numerical algorithm. Comparing them with the degraded images, it is clear that the algorithm can effectively remove noise and significantly preserve edges simultaneously. For a quantitative evaluation of the performance of the algorithm, the PSNR, mean square error (MSE) and mean absolute error (MAE) comparison for different filtering algorithms are listed in TABLE 2.

[00002]

TABLE 2
Comparative Filtering Results for Lena Image
Corrupted with Different Amount of Gaussian Noise
	Noisy Image I	Noisy Image II
Algorithm*	PSNR	MSE	MAE	PSNR	MSE	MAE
No Denoising	22.17 dB	394.37	15.97	18.82 dB	853.07	23.71
Median (3 × 3)	29.38 dB	75.09	6.26	27.60 dB	112.97	7.72
Median (5 × 5)	28.60 dB	89.66	6.53	27.39 dB	118.41	7.53
α-TMP (3 × 3)	29.84 dB	67.46	6.07	28.13 dB	100.10	7.04
MTMF (3 × 3)	29.91 dB	66.46	5.79	28.23 dB	97.74	7.39
New Approach	30.69 dB	55.50	5.34	28.96 dB	82.69	6.45
*All results are implemented recursively for optimal PSNR performance. The threshold used for the MTMF algorithm is optimized to q = 65 for image I and q = 100 for image II.

[00452] The PSNR is increased about 8.52 dB for the lower noise image and 10.14 dB for the higher noise image, both of which are better than the best median, α-TMF and MTMF filtering results.

[00453] In this disclosure, a new filtering algorithm for removing additive noise from a corrupted image is presented. In comparison to standard filtering schemes, the DWSAF algorithm employs two concentric moving windows to determine the gray level value used to replace that of the center pixel. Employing the larger window does not lead to significant blurring of the image because of the effect of the smaller window. The GTMF is used in the smaller window for obtaining the criterion for standard selective averaging in the larger window. Numerical examples show that the proposed filtering algorithm is both efficient and robust for recovering blocked signals contaminated with additive noise.

[none]

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[00462] The distributed approximating functional (DAF) approach to the time evolution of wave packets is a powerful, new method aimed at taking advantage of the local nature of the potential in the coordinate representation for ordinary chemical collisions, and the localized nature of the kinetic energy in this same representation. These features of the Hamiltonian lead to a highly band-pass expression for the propagator, specific to a restricted class of wave packets (DAFs). With control of the parameters which determine it, the DAF class can be made to include the wave packets of interest in any particular time-dependent quantum problem. Because the propagator is banded, its application to the propagation on the grids (1) scales like the number of grid points, N, in any dimension (which is the ultimate for any grid methods; the scaling constant depends on the band width); (2) requires reduces communication time when implemented on massively parallel computer; and (3) minimizes the storage requirement for the propagator.

[00463] In general, DAF can be regarded as a radial interpolating basis, as well as the scaling function. The corresponding DAF wavelets can be implemented using generalized DAFs. As a kind of radial basis functional (RBF), DAF neural networks are constructed and applied in signal processing.

[00464] We presented the derivation of the DAFs independent of the choice of coordinates and show how the Hermite functions becomes generalized for arbitrary numbers of degrees of freedom and choice fo orthogonal coordinates. We discuss the properties of the generalized “DAF polymonals” associate with different coordinate and numbers of degrees of freedom, and how angular momentum conservation leads to a radial propagator for spherical waves. Symmetric quincunx-DAF Dirichlet classes are carefully designed for hyper spherical surface representation.

[00465] The theory of fundamental distributed approximating functional is based on the Hermite function representation of the 1D δ function which is $\begin{array}{cc}\delta \left(x❘\sigma \right)=\frac{1}{\sigma \sqrt{2\pi }}\exp \left(\frac{-x^2}{2{\sigma }^2}\right)\sum _{n=0}^{\infty }{\left(-\frac{1}{4}\right)}^n\frac{1}{n!}{H}_{2n}\left(\frac{x}{\sqrt{2\sigma }}\right)& \left(85\right)\end{array}$

[00466] The function H_2n is the Hermite polynomial of even order, 2n. The quantity σ is the width of the Gaussian window of the Hermite polynomial. The qualitative behavior of one particular Hermite DAF is shown in FIG. 1(a). The Hermite polynomial H is generated by the usual recursion $\begin{array}{cc}{H}_n\left(x\right)=\{\begin{array}{cc}1,& n=0\\ 2x,& n=1\\ 2{\mathrm{xH}}_{n-1}\left(x\right)-2\left(n-1\right)H_{n-2}\left(x\right),& n>1\end{array}& \left(86\right)\end{array}$

[00467] The predominant advantage of the Hermite polynomial approximation is its high-order derivative preservation (which leads to a smooth approximation). This disclosure expands and tests the approach of Madan and Milne (1994) for pricing contingent claims as elements of a separable Hilbert space. We specialize the Hilbert space basis to the family of Hermite polynomials and use the model to price options on Eurodollar futures. Restrictions on the prices of Hermite polynomial risk for contingent claims with different times to maturity are derived. These restrictions are rejected by our empirical tests of a four-parameter model. The unrestricted results indicate skewness and excess kurtosis in the implied risk-neutral density. These characteristics of the density are also mirrored in the statistical density estimated from a time series on LIBOR. The out-of-sample performance of the four-parameter model is consistently better than that of a two-parameter version of the model.

[00468] The Hermite polynomial are given in terms of their generator by $\begin{array}{cc}{H}_n\left(x\right)\exp \left(-x^2\right)={\left(-1\right)}^n\frac{d^n}{d{x}^n}\exp \left(-x^2\right)={\left(-1\right)}^n{\nabla }_x^n\exp \left(-x^2\right)& \left(87\right)\end{array}$
where ∇_x is the x component of the gradient operator. This equation provides two equivalent and useful ways in which the above function can be expressed as $\begin{array}{cc}\delta \left(x-x^{\prime }|\sigma \right)=\frac{1}{\sigma \sqrt{2\pi }}\sum _{n=0}^{\infty }{\left(-\frac{1}{4}\right)}^n{\frac{1}{n!}\left[2{\sigma }^2\right]}^n{\nabla }_{x^{\prime }}^{2n}\exp \left[-{\left(x-x^{\prime }\right)}^2/2{\sigma }^2\right]& \left(88\right)\end{array}$
or $\begin{array}{cc}\delta \left(x-x^{\prime }|\sigma \right)=\frac{1}{\sigma \sqrt{2\pi }}\sum _{n=0}^{\infty }{\left(-\frac{1}{4}\right)}^n{\frac{1}{n!}\left[2{\sigma }^2\right]}^n{\nabla }_{x^{\prime }}^{2n}\exp \left[-{\left(x-x^{\prime }\right)}^2/2{\sigma }^2\right]& \left(89\right)\end{array}$
where in the second equation ∇_x²ⁿ has been replaced by ∇_x′²ⁿ. The two are equivalent because the derivative acts on an even function of (x−x′).