Continuous transmit focusing method and apparatus for ultrasound imaging system

[0001] This application is a continuation of U.S. patent application Ser. No. 11/492,471, entitled “Continuous Transmit Focusing Method and Apparatus For Ultrasound Imaging System”, filed on Jul. 24, 2006, now U.S. Pat. No. 8,002,705, issued Aug. 23, 2011, which claims priority to U.S. Provisional Patent Application No. 60/701,812, filed Jul. 22, 2005, the entire disclosures of which are incorporated herein by reference in their entirety for all purposes.

[0002] This invention relates generally to ultrasound imaging systems and, more particularly, to continuous transmit focusing for ultrasound imaging systems.

[0003] Conventional ultrasound imaging systems generally form an image in the following manner. A short acoustic pulse is transmitted into a region of interest from a subset of transducer elements on an array, focused at a particular depth and direction. The acoustic wavefront formed by the superposition of the transmitted pulses propagates along the selected direction and upon backscattering from structures contained within the region of interest, propagates back towards the transducer array (refer to FIG. 1). These echos received from different transducer elements are subsequently amplified and combined using delay, phase, and apodization in such a manner as to provide a dynamic receive focus which changes as a function of time/depth along the transmitted wavefront direction. The combined signal is then log detected and further processed prior to being stored. This process is repeated many times as the transmit and receive directions are changed in such a way as to sweep through the region of interest, i.e., steered, translated, or both. Upon collecting the desired number of line acquisitions, this acoustic data is then scan converted for display to form the resulting ultrasound image. The rate at which these images are formed and displayed is referred to as the frame rate.

[0004] For an ultrasound imaging system to produce high quality images, the region of interest must be properly sampled acoustically, both in the range and azimuth (lateral) dimensions in order to prevent aliasing artifacts, which can arise in all sampled systems. In the range dimension, the Nyquist sampling theorem requires that an adequate number of samples in range be acquired based upon the combined round-trip transmit/receive pulse bandwidth. In the azimuth, or lateral, dimension, the Nyquist sampling theorem requires that 1) the region of interest be laterally insonified by a sufficient number of transmit beams and 2) an adequate number of combined round-trip transmit/receive beams laterally sample the region of interest. Stated another way, the Nyquist sampling theorem imposes a transmit acquisition lateral sampling criteria, as well as a round-trip transmit/receive lateral sampling criteria. The Nyquist transmit beam spacing Δx_xmt is dependent upon the transmit aperture size A_xmt, focusing location r_xmt, and the carrier wavelength of the acoustic radiation λ₀. It is given by
Δx_xmt=λ₀F_xmt  (1)
where the transmit F-number F_xmt=r_xmt/A_xmt. The Nyquist round-trip beam spacing Δx is dependent upon both transmit and receive F-numbers F_xmt and F_rcv respectively. It is given by

[0005] $\begin{array}{cc}\Delta x={\lambda }_0\frac{F_{\mathrm{xmt}}{F}_{\mathrm{rcv}}}{\sqrt{F_{\mathrm{xmt}}^2+F_{\mathrm{rcv}}^2}}=\frac{\Delta x_{\mathrm{xmt}}{\Delta }_{\mathrm{rcv}}}{\sqrt{\Delta x_{\mathrm{xmt}}^2+\Delta x_{\mathrm{rcv}}^2}}& \left(2\right)\end{array}$
where F_rcv=r_rcv/A_rcv for focusing location r_rcv and Δx_rcv=λ₀F_rcv. Note that when the receive F-number is much lower than the transmit F-number, the Nyquist round-trip beam spacing Δx is dominated by the receive beam characteristics. This is the direct result of the round-trip transmit/receive beampattern S(ω,x,r) being equal to the multiplication of the individual transmit and receive beampatterns S_xmt(ω,x,r,r_xmt) and S_rcv(ω,x,r) respectively, at a particular frequency f and range r, and is given by

[0006] $\begin{array}{cc}s\left(t,x,r\right)=\frac{1}{2\pi }{\int }_{}^{}S\left(\omega ,x,r\right)e^{\mathrm{j\omega }t}d\omega S\left(\omega ,x,r\right)=S_{\mathrm{xmt}}\left(\omega ,x,r,r_{\mathrm{xmt}}\right)S_{\mathrm{rcv}}\left(\omega ,x,r\right)s_{\mathrm{xmt}}\left(t,x,r,r_{\mathrm{xmt}}\right)=\frac{1}{2\pi }{\int }_{}^{}{S}_{\mathrm{xmt}}\left(\omega ,x,r,r_{\mathrm{xmt}}\right)e^{\mathrm{j\omega }t}d\omega s_{\mathrm{rcv}}\left(t,x,r\right)=\frac{1}{2\pi }{\int }_{}^{}{S}_{\mathrm{rcv}}\left(\omega ,x,r\right)e^{j\omega t}d\omega & \left(3\right)\end{array}$
where x is the lateral spatial coordinate, t is the time coordinate, ω=2πf, and the transmit, receive, and round-trip point spread functions (PSF) are the inverse Fourier transforms of their respective beampatterns. If the receive beam is much narrower than the transmit beam, then the receive beam dominates the round-trip beampattern. FIG. 2 depicts the situation where the receive beam F-number is substantially lower than the transmit F-number and as such, the receive beam, which is typically dynamically focused, i.e., the aperture size and focal point are increased as a function of time/range so as to maintain constant lateral resolution Δx_rcv, dominates the round-trip beam pattern. The transmit beam is generally focused at a user specified depth. Note the reduction in the round-trip sidelobe clutter energy outside of the round-trip beampattern's mainlobe at the transmit focus location, as compared to away from the transmit focus location. This is due to the multiplicative influence of the transmit beam on the receive beam. The transmit beam can have a more significant influence on the receive beam if lower transmit F-numbers are employed. However, since the transmit beam is not dynamically focused for all depths, the round-trip beampattern will display much better lateral resolution at the transmit focus location compared to away from the transmit focus location. For example, if F_xmt F_rcv, Δx=Δx_rcv/√2 at the transmit focus location, and Δx=Δx_rcv away from the transmit focus location—a 41% degradation in lateral resolution, as well as increased sidelobe clutter. This leads to lateral image non-uniformity, which is undesirable for high quality ultrasound imaging.

[0007] Conventional ultrasound imaging systems which form a single receive beam for each transmit beam as depicted in FIG. 2, necessarily couple the Nyquist transmit beam spacing Δx_xmt to the Nyquist round-trip beam spacing Δx. Instead of having to fire a minimum of L/Δx_xmt transmit beams to adequately insonify the region of interest, the single beam conventional system fires L/Δx transmit beams to cover the region of interest, where L is the lateral extent of the region of interest to be imaged. Using (1) and (2), the potential acoustic acquisition rate is reduced by the factor

[0008] $\begin{array}{cc}\eta =\sqrt{1+\frac{F_{\mathrm{xmt}}^2}{F_{\mathrm{rcv}}^2}}=\sqrt{1+\frac{\Delta x_{\mathrm{xmt}}^2}{\Delta x_{\mathrm{rcv}}^2}}& \left(4\right)\end{array}$
Using a typical single focus example of F_xmt=2.5 and F_rcv=1.0, the reduction in the potential acoustic acquisition rate using (4) is ˜2.7. Almost three times as many transmit beams are being fired than required by Nyquist sampling requirements.

[0009] If the transmit and receive F-numbers are made equal, i.e., F_xmt=F_rcv=1.0, then the reduction in potential frame rate using (4) is ˜1.4, which is not as dramatic as when the receive F-number is much lower than the transmit F-number; however, an additional problem is introduced, namely the limited depth of field of the transmit beam. While the receive beam is dynamically focused, the transmit beam is focused only at a single depth. The range r over which the transmit beam can be considered “in focus” is given by the depth of field expression

[0010] $\begin{array}{cc}{R}_{\mathrm{DOF}}^{\mathrm{xmt}}\sim \mathrm{\beta \lambda }{F}_{\mathrm{xmt}}^2,4\le \beta \le 8r_{\mathrm{xmt}}-\frac{R_{\mathrm{DOF}}^{\mathrm{xmt}}}{2}\le r\le r_{\mathrm{xmt}}+\frac{R_{\mathrm{DOF}}^{\mathrm{xmt}}}{2}& \left(5\right)\end{array}$
where the choice of β depends upon what phase error is assumed at the end elements of the transmit aperture in the depth of field (DOF) derivation. Note the DOF dependence on the square of the transmit F-number. As the transmit F-number decreases, the transmit beam's lateral resolution increases linearly as given by (1); however, the range over which the transmit beam will be in focus, and have influence on the round-trip lateral resolution given by (2), decreases quadratically. This introduces image lateral non-uniformity and higher clutter away from the transmit focus and is undesirable in high quality ultrasound images. The conventional approach to mitigate this behavior is to transmit multiple times along each transmit beam, changing the transmit focus location r_xmt on each firing and performing a conventional receive beam formation on each. The resulting multiple round-trip signals are then typically combined following the detection process, with the DOF image region around each transmit focus r_xmt being retained, with the rest discarded. Thus, each detected line is the composite of multiple lines, each having a different transmit focus. The number of transmit firings along the same transmit beam direction that are required is dependent upon the desired transmit F-number to be supported, and the display range of the region of interest, with the number of transmit firings going up as the square of the transmit F-number reduction. This produces a significant loss in acoustic acquisition rate due to the multiple transmit firings along the same line.

[0011] The reduction in potential acoustic acquisition rate, and ultimately frame rate, due to lateral sampling requirements, increased transmit resolution, and reduced round-trip sidelobe clutter, has led to many techniques being developed which seek to provide a true dynamic transmit focus and provide a better performance/frame rate tradeoff. The techniques developed generally fall into several classes, namely,

I. Composite,

II. Multiple beams,

III. Transmit sub-apertures,

IV. Element dependent transmit waveform generation/receive filtering, and

V. Deconvolution

[0012] I. Composite techniques seek to approximate a dynamic focused transmit beam by compositing multiple transmit beams of different foci together along the same direction, either fired sequentially in time as discussed above, or simultaneous in time through linear superposition. If fired sequentially in time, the compositing of transmit beams of different foci can be performed following detection by simply discarding receive data beyond the transmit depth of field and stitching together the remaining detected receive data, which is the conventional sequential transmit approach. Or the compositing can be performed using pre-detected, i.e., coherent, receive data and combined using a weighted superposition of the coherent receive data from the multiple transmit firings. See Synthetic Dynamic Transmit Focus, Brent Robinson & Cliff Cooley, 2000 IEEE Ultrasonics Symposium, or Synthetic Transmit Focusing for Ultrasound Imaging, Bruno Haider, 2000 IEEE Ultrasonics Symposium. Compositing the transmit beams of different foci sequentially in time is not very efficient since progressively less and less information from each component transmit firing is being retained, while the number of transmit firings increases, decreasing the frame rate accordingly. Another technique composites the transmit beams of different foci simultaneous in time by linearly superimposing each transmit channel's apodized and delayed transmit waveforms together, i.e., compound focusing. See U.S. Pat. No. 5,581,517. While the multiple transmit firings in time and the commensurate decrease in frame rate are avoided, it requires a linear transmitter (which is generally more costly and power inefficient), round-trip sidelobe clutter is increased around each foci compared to a single transmit foci due to the simultaneous transmit beam's acoustic interference (which limits how close the compound transmit foci can be), and the technique does not address the reduction in potential frame rate due to transmit lateral sampling issues. A simpler, older technique simply used a transmit delay profile which was a composite of multiple transmit delay profiles each focused at different depths. The outer transmit channel's delay was dominated by the deeper transmit foci, and the inner transmit channel's delay was dominated by the shallower transmit foci. However this technique, while not reducing the frame rate, improves transmit beamformation performance away from the focus, i.e., round-trip sidelobe clutter, at the expense of beamformation performance at the focus, i.e., poorer resolution, and it still does not address the reduction in potential frame rate due to transmit lateral sampling issues.

[0013] II. Multiple beam techniques, either transmit, receive, or both, seek to improve frame by reducing the number of transmit firings it takes to laterally sample the region of interest. In the case of multiple simultaneous transmit beams, along with receive beams aligned along each transmit beam, less time is taken to sweep the region of interest to form the image for a given receive line density and thus, higher frame rate is achieved. However, the interference between the simultaneous transmit beams leads to increased acoustic clutter and degrades image quality. Using multiple receive beams formed from the same transmit beam can also improve frame rates since less time is required to sweep the region of interest for a given receive line density. However, receive beams which are spatially displaced from the transmit beam center can lead to image distortion, more so around the transmit focus, higher acoustic clutter, and results in spatial variance which is undesirable for high quality imaging. A way to mitigate this distortion was described in U.S. Pat. No. 6,016,285, using two receive beams from each transmit firing and coherently (pre-detection) combining receive beams from two neighboring transmit firings, along with combining receive beams from the same firing as a way to re-center the receive beams along transmit firings. While eliminating geometric distortion, line-to-line gain variations are introduced, which if not removed through spatial filtering (which reduces lateral resolution) lead to imaging artifacts. A multiple beam approach, Parallel Beamforming using Synthetic Transmit Beams by T. Hergum, T. Bjastad, and H. Torp, 2004 IEEE International Ultrasonics, Ferroelectrics, and Frequency Control Joint 50^th Anniversary Conference, pages 1401-1404, uses a weighted sum of multiple coherent receive beams from different transmit beams, forming an interpolated, or synthesized transmit beam, aligned along each receive beam. This addresses the lateral spatial sampling component of frame rate reduction, however dynamic transmit focusing is not addressed. Therefore, use of multiple beams partially addresses the lateral spatial sampling component of frame rate reduction; however, these classes of techniques have not addressed the transmit beam focusing aspect and as such, if increased transmit resolution is desired, multiple beams combined with some class of compositing is required, with the advantages and disadvantages described previously.

[0014] III. Transmit sub-aperture techniques seek to improve frame rates by trying to eliminate the process of compositing, or sequential transmit focusing, altogether by creating a dynamic transmit focus, analogous to dynamic receive focusing. The difficulty is that once the transmit wavefront is launched and propagation/diffraction begins, its characteristics cannot be directly altered. However, the effective transmit focus can be altered after the fact during receive processing by breaking up the transmit aperture into several, or many, sub-apertures (in the limit, each transducer element is fired separately) where each sub-aperture is fired sequentially in time. See Multi-Element Synthetic Transmit Aperture Imaging using Temporal Encoding by K. L. Gammelmark and J. A. Jensen, 2002 SPIE Medical Imaging Meeting, Ultrasonic Imaging and Signal Processing, 2002; pages 1-13. On receive, the echoes from each sub-aperture firing can be delayed, phased, apodized, and summed prior to, or after, receive beam formation, varying the delay, phase, and apodization values with range, to form a dynamic transmit focus. This has several limitations/complexities—first, it requires linearity in the transmit-receive processing path. While this is true for fundamental imaging, it is not strictly true for 2^nd harmonic and higher order non-linear imaging modes. For example, in 2^nd harmonic imaging, the transmit beam becomes essentially “squared” during the transmit propagation process, effectively reducing sidelobes and clutter in the transmit beam. The subsequent squaring that will occur during the propagation/diffraction of each transmit sub-aperture and summing on receive will not be equivalent to transmitting the entire original transmit aperture, which is the sum of the transmit sub-apertures, and squaring the result, i.e.,

[0015] $\begin{array}{cc}\sum _{i=1}^{N_{\mathrm{sub}}}{\left\{G\left[z_i\left(n\right)\right]\right\}}^2\ne {\left\{G\left[\sum _{i=1}^{N_{\mathrm{sub}}}{z}_i\left(n\right)\right]\right\}}^2\therefore \sum _{i=1}^{N_{\mathrm{sub}}}{\left\{G\left[z_i\left(n\right)\right]\right\}}^2\ne {\left\{\sum _{i=1}^{N_{\mathrm{sub}}}G\left[z_i\left(n\right)\right]\right\}}^2& \left(6\right)\end{array}$
where n is the transmit element, N_sub is the number of transmit sub-apertures, z_i(n) represents the apodized, phased, and delayed transmit aperture function, and G is the linear propagation/diffraction operator. In other words, the sum of the squares is not equal to the square of the sums. Also, the signal-to-noise ratio (SNR), already a problem in non-linear imaging modes, will be an even greater problem using transmit sub-apertures due to the sum of the squares being less than the square of the sums. More importantly, to the extent these techniques seek to break up the transmit aperture into smaller and smaller sub-apertures, thus improving the ability to form an effective dynamic focused transmit beam, the broader and less focused each sub-aperture's individual transmit beampattern becomes, approaching that of a point source as the sub-aperture shrinks to a single element, which will have a fairly uniform amplitude distribution. Any non-linear imaging mode which reduces transmit beam sidelobe clutter through squaring, or higher order non-linearity, will no longer be as effective, i.e., squaring a uniform beam produces a uniform beam. Attempts to improve the SNR situation by transmitting on multiple elements, delayed and phased in such a way as to create point source wavefront, analogous to a single element, does increase the transmitted signal level; however, the transmit beampattern is still fairly uniform and as such, it will not see much improvement in sidelobe clutter from non-linear propagation/diffraction. Motion will be an issue in both fundamental and non-linear imaging modes since to form the entire desired transmit aperture, many firings may be required, i.e., 128, etc., which will require a fair amount of time. For example, assuming a carrier frequency of 3.5 MHz, a depth of 240 mm with an assumed sound speed of 1540 m/s, the round-trip propagation time is 311 μsec, which for 128 elements/firings is 40 msec. This is quite a long time for phase coherency to be maintained across the elements to a fraction of the 3.5 MHz wavelength, which is 0.44 mm. Motion during that time will introduce transmit beamformation delay/phase errors due to the many firings it takes to construct the effective transmit beam, which will degrade the quality of the effective transmit beam characteristics and introduce increased acoustic clutter and other motion artifacts. Lastly, depending upon how many transmit sub-apertures are employed, the processing requirements can be quite high, which can be even higher if motion compensation techniques are considered, making some of these techniques impractical for real-time imaging given the current state of the art technology.

[0016] IV. Techniques that use element dependent transmit waveform generation, possibly combined with receive filtering, seek to employ different transmit waveforms on each element in such a way that during propagation and diffraction, the individual waveforms combine which, when filtered on receive, an effective dynamic focused transmit beam is formed. U.S. Pat. No. 6,108,273 describes starting with a traditional short pulse at the center of the transmit aperture, whose higher spectral frequency components are progressively advanced in time compared to the lower frequency components, as the transmit element location approaches either end of the transmit aperture. This produces a chirp-like waveform of increasing length away from the transmit aperture center. On receive, a bandpass filter is employed whose center frequency is decreased with depth. The transmitted pulse's higher frequency components that are selectively passed by the depth dependent receive bandpass filter have diffracted from a transmit delay arc of higher curvature, which effectively focuses the transmit wavefront at shallower depths. The transmitted pulse's lower frequency components that are selectively passed by the depth dependent receive bandpass filter have diffracted from a transmit delay arc of lower curvature, which effectively focuses the transmit wavefront at deeper depths. This approximates a dynamically focused transmit beam. The limitations of this approach is that the potential round-trip pulse bandwidth and hence, axial resolution, is compromised at the expense of the dynamic transmit focusing effect. In addition, this technique alone does not address the lateral spatial sampling component of frame rate reduction, although some of the other techniques outlined could be used in conjunction with this technique.

[0017] V. Techniques that use a deconvolution approach to improve the transmit focusing characteristics also have various limitations. For example, Retrospective Dynamic Transmit Focusing by S. Freeman, Pai-Chi Li, and M. O'Donnell, Ultrasonic Imaging 17, pages 173-196, (1995), used a single receive beam system along with a spatial filter of a finite number of taps, operating across receive beams. The design of the spatial filter was designed with the intent to deconvolve out the effects of the defocused transmit beam from the round-trip point spread function. As such, this approach will be limited by the receive beam's influence on the round-trip point spread function, reducing the extent to which the transmit focusing characteristics can be improved. In addition, it does not address the lateral spatial sampling component of frame rate reduction.

[0018] The present invention seeks to provide high quality imaging using a minimum of transmit firings by decoupling the Nyquist transmit and round-trip spatial sampling requirements, and at the same time maintaining a continuous transmit focus throughout the region of interest, which reduces or eliminates the need for transmit focus compositing, i.e., sequential transmit focus, both of which dramatically improve the acoustic acquisition rate. Reduction of the number of transmit firings becomes especially important for ultrasound systems which are battery powered in order to conserve energy and reduce power consumption. Elimination of transmit focus user controls is also a desirable feature, especially for portable ultrasound systems given their compact user interface. The invention works in linear or non-linear imaging modes, works with standard 1-D and 2-D transducer arrays, can be extended to three spatial dimensions, i.e., 3-D imaging, is compatible with all scan formats, and is robust in presence of motion.

[0019] An aspect of the present invention is directed to an ultrasound imaging method for achieving transmit and receive focusing at every echo location within a region of interest. The method comprises providing a probe that includes one or more transducer elements for transmitting and receiving ultrasound waves; generating a sequence of spatially distinct transmit beams which differ in one or more of origin and angle; determining a transmit beam spacing substantially based upon a combination of actual and desired transmit beam characteristics, thereby achieving a faster echo acquisition rate compared to a transmit beam spacing based upon round-trip transmit-receive beam sampling requirements; storing coherent receive echo data, from two or more transmit beams of the spatially distinct transmit beams; combining coherent receive echo data from at least two or more transmit beams to achieve a substantially spatially invariant synthesized transmit focus at each echo location, i.e., to achieve transmit synthesis; and combining coherent receive echo data from each transmit firing to achieve dynamic receive focusing at each echo location.

[0020] In some embodiments, the probe is a 1-D, 2-D array with varying degrees of elevation beamforming control, including aperture, delay, and phase, or a general 2-D scanning array, or a sparse 1-D or 2-D array. The spatially distinct transmit beams are generated electronically, mechanically, or any combination thereof; to scan a 2-D plane or 3-D volume. Coherent receive echo data is receive channel data from different transducer elements. Combining coherent receive echo data from each transmit firing to achieve dynamic receive focusing is performed prior to combining coherent receive echo data from at least two or more transmit beams to achieve transmit synthesis. Combining coherent receive echo data from each transmit firing to achieve dynamic receive focusing is performed prior to storing coherent receive echo data. Alternatively, combining coherent receive echo data from each transmit firing to achieve dynamic receive focusing is performed subsequent to combining coherent receive echo data from at least two or more transmit beams to achieve transmit synthesis. In other embodiments, combining coherent receive echo data from each transmit firing to achieve dynamic receive focusing and combining coherent receive echo data from at least two or more transmit beams to achieve transmit synthesis are performed in combination.

[0021] In specific embodiments, combining coherent receive echo data from each transmit firing to achieve dynamic receive focusing and combining coherent receive echo data from at least two or more transmit beams to achieve transmit synthesis are performed at arbitrary echo locations, i.e., area/volume formation, or along multiple ray-like paths, i.e., multiple receive beamformation. One or more non-linear components of the transmitted ultrasound waves are included in the steps of storing coherent receive echo data, combining coherent receive echo data from each transmit firing to achieve dynamic receive focusing, and combining coherent receive echo data from at least two or more transmit beams to achieve transmit synthesis. The method may further comprise forming an image with an imaging mode that includes one or more of B, color Doppler (velocity, power), M, spectral Doppler (PW), with or without using contrast agents. Velocity imaging modes including color velocity and spectral Doppler (PW) are achieved using motion compensation in combining coherent receive echo data from at least two or more transmit beams to achieve transmit synthesis. The spectral Doppler (PW) mode involves processing one or more sample volumes along a synthesized transmit beam obtained by the transmit synthesis.

[0022] In some embodiments, combining coherent receive echo data from at least two or more transmit beams to achieve transmit synthesis includes combining using one or more of delay, phase, amplitude, and convolution. Combining coherent receive echo data from at least two or more transmit beams to achieve transmit synthesis may be responsive to echo location, as well as to the spatial and temporal characteristics of the actual and/or desired transmit beam characteristics, which may include one or more non-linear components of the transmitted ultrasound waves. The spatially distinct transmit beams may differ in one or more of delay, phase, apodization, amplitude, frequency, or coding. The spatially distinct transmit beams may have a single focus at a predetermined range, with an F-number ranging from 0.5 to 10. Two or more of the spatially distinct transmit beams are fired simultaneously, or with a time gap less than a round-trip propagation time, for faster echo acquisition.

[0023] Another aspect of the invention is directed to an ultrasound imaging system for achieving transmit and receive focusing at every echo location within a region of interest. The system comprises a probe that includes one or more transducer elements for transmitting and receiving ultrasound waves; and a processor. The processor generates a sequence of spatially distinct transmit beams which differ in one or more of origin and angle; determines a transmit beam spacing substantially based upon a combination of actual and desired transmit beam characteristics, thereby achieving a faster echo acquisition rate compared to a transmit beam spacing based upon round-trip transmit-receive beam sampling requirements; stores coherent receive echo data, from two or more transmit beams of the spatially distinct transmit beams; combines coherent receive echo data from at least two or more transmit beams to achieve a substantially spatially invariant synthesized transmit focus at each echo location, i.e., to achieve transmit synthesis; and combines coherent receive echo data from each transmit firing to achieve dynamic receive focusing at each echo location.

[0035] The present invention seeks to provide high quality imaging using a minimum of transmit firings by decoupling the Nyquist transmit and round-trip spatial sampling requirements, and at the same time maintaining a continuous transmit focus throughout the region of interest, which reduces or eliminates the need for transmit focus compositing, i.e., sequential transmit focus, both of which dramatically improve the acoustic acquisition rate. Reduction of the number of transmit firings becomes especially important for ultrasound systems which are battery powered in order to conserve energy and reduce power consumption. Elimination of transmit focus user controls is also a desirable feature, especially for portable ultrasound systems given their compact user interface. The invention works in linear or non-linear imaging modes, works with standard 1-D and 2-D transducer arrays, can be extended to three spatial dimensions, i.e., 3-D imaging, is compatible with all scan formats, and is robust in presence of motion.

[0036] FIG. 3 shows a block diagram of the system architecture according to an embodiment of the present invention. However, this is but a representative example and many other system architectures can be used. Referring to FIG. 3, following receive front-end TGC, analog-to-digital conversion and demodulation from RF to baseband is performed. The resulting in-phase and quadrature, i.e., I/Q, data is then stored into a random access channel domain memory. The memory can be SDRAM, MRAM, or any other suitable memory device. Equivalently, magnitude and phase data could be stored. The channel domain memory holds Nt time samples for all Nch system channels, for an arbitrary number of transmit firings, i.e., the number of transmit firings required to insonify the entire region of interest—a frame, or more. Next, the channel domain I/Q data is read by the digital signal processor (DSP) and employing software algorithms, the DSP performs receive reconstruction, detection, and scan conversion for display. Alternatively, the DSP may consist of a single DSP or multiple DSP's, or a combination of programmable logic, i.e., FPGA's, and DSP's. The receive reconstruction process is analogous to receive beamformation; however, it is more general and, in conjunction with the random access channel domain memory and DSP/software programmable architecture, provides arbitrary spatial sampling, allowing it to take advantage of spatially dependent sampling requirements and reduce computation time.

[0037] Reconstruction represents the transformation of channel data into pre-detection image data over an area or volume, from the insonification by a focused, broad, or defocused transmit beam, defined as a transmit zone. The area is defined by a collection of range and azimuth coordinates, where the azimuth coordinate is defined by a suitable lateral coordinate, i.e., horizontal position x, steering angle θ, etc. This is shown in FIG. 4 which depicts several representative examples. The transducer can be planar, curved, or of some other form, and the scan geometry, i.e., transmit/receive beam origins and angles, can be arbitrary. The random access channel domain I/Q data memory allows the DSP to form an arbitrary I/Q point from any transmit firing, or zone. Alternatively, the DSP can use channel magnitude and phase data to form a magnitude and phase point. The reconstruction software algorithm in the DSP may include channel delay, phase, apodization, and summation processing as is well known in the art to form an image I/Q point, but it is not limited to this particular image formation technique and can include non-linear processing elements. In general, the collection of range and azimuth coordinates defining a reconstructed I/Q point within the area formation region, from any given transmit zone, can be described by
r_ij=ƒ_r(r,x,i,j, . . . ) or r_ij=ƒ_r(r,θ,i,j, . . . )
x_ij=ƒ_x(r,x,i,j, . . . ) or θ_ij=ƒ_θ(r,θ,i,j, . . . )  (7)
where ƒ_{r,x,θ} describes the functional dependence of the specified coordinate on its input parameters, (i,j) are suitable indexing parameters, and . . . indicates any other parameter dependence for the range and azimuth coordinates, i.e., transmit focus location, etc. If the collection of range and azimuth coordinates defining the reconstructed area are independent and assume a more standard dependence on the indexing parameters, i.e.,
r_i=r₀+i·Δr i={0, 1, . . . N_r−1}
x_j=x₀+j·Δx j={0, 1, . . . N_x−1}
or
θ_j=θ₀+j·Δθj={0, 1, . . . N_θ−1}  (8)
then area formation reduces to standard receive beamformation, i.e., ray-like paths. In the case where more than one of these receive beams are formed simultaneously from a single transmit firing, or zone, then it is considered multiple receive beamformation, also known in the art as parallel beamformation. In three spatial dimensions, i.e., 3-D imaging, area formation becomes volume formation and can be described by a collection of suitable range, azimuth, and elevation coordinates. The examples shown in FIG. 4 can be easily extended into three spatial dimensions, i.e., if y represents the elevation coordinate, then
r_ijk=ƒ_r(r,x,y,i,j,k, . . . )
x_ijk=ƒ_x(r,x,y,i,j,k, . . . )
y_ijk=ƒ_y(r,x,y,i,j,k, . . . )  (9)

[0038] If angular coordinates are preferred in describing the desired scan geometry, then equation (9) can be suitably modified. If the collection of range and azimuth coordinates defining the reconstructed volume are independent and assume a more standard dependence on the indexing parameters, i.e.,
r_i=r₀+i·Δr i={0, 1, . . . N_r−1}
x_j=x₀+j·Δx j={0, 1, . . . N_x−1}
y_k=y₀+k·Δy k={0, 1, . . . N_y−1}  (10)
If angular coordinates are preferred in describing the desired scan geometry, then (10) can be suitably modified.

[0039] It is important to note that if the transmit and round-trip spatial sampling requirements are decoupled using the system architecture shown in FIG. 3, thereby reducing or eliminating the potential acoustic acquisition reduction factor η given by (4), then the frame rate is essentially decoupled from the acoustic acquisition frame rate. While the acoustic acquisition rate is determined by the number of transmit firings and acoustic propagation time, the frame rate will be determined by the DSP and software algorithm's computation time required to process the acquired channel I/Q data and form image I/Q data, i.e., area formation, perform detection, scan conversion, etc., and display the image, up until the acoustic acquisition rate becomes the limiting factor. This characteristic provides the system the opportunity to produce high quality images at much higher frame rates than conventional systems, provided sufficient processing performance is utilized. Alternatively, the system could expend the additional time available performing more sophisticated image formation algorithms to further improve image quality, reduce system power consumption by maintaining the frame rate at conventional system frame rates, etc. The channel domain data memory serves as a buffer between the acoustic acquisition rate and the DSP and software algorithm processing rate.

[0040] FIG. 5 shows a comparison between conventional single-beam line-by-line acoustic acquisition where the transmit and round-trip spatial sampling requirements are coupled, and zone acquisition of the present invention where the transmit and round-trip spatial sampling requirements are decoupled. Depending upon the transmit F-number, the improvement in acoustic acquisition rate compared to conventional systems can be substantial as described previously. The process by which the region of interest is insonified based upon transmit sampling requirements alone is termed zone acquisition, from which a relatively large image I/Q area can be reconstructed from channel I/Q data such that round-trip Nyquist spatial sampling requirements are met or exceeded. To the extent that area formation reduces to the special case of multiple receive beamforming, then the receive beam density must meet or exceed Nyquist round-trip sampling requirements.

[0041] The zone acquisition process can be easily extended to three dimensions to yield even larger improvements, where the large number of transmit firings typically required by conventional systems results in very low frame rates. The zone acquisition process has advantages in other applications. For example, in contrast agent imaging where micro-bubbles are injected into the body and imaged with ultrasound, either at essentially the same insonification frequency or suitable harmonic, reducing the number of transmit firings and increasing the lateral spacing of the transmit beams results in less bubble destruction where minimum or no bubble destruction is desired.

[0042] The procedure to decouple transmit and round-trip spatial sampling requirements, as well as provide a continuous transmit focus, is now described for an embodiment of the present invention. The motivation for this technique is based upon the following line of reasoning. A conventional fixed-focused transmit beam produces high quality images around the focus, and works well in harmonic imaging modes, etc., however lateral resolution and uniformity degrade away from the focus. Rather than reinventing the transmit beam, i.e., decomposing the transmit aperture into many pieces, only to sum the resulting component beam responses back up on receive as outlined in some of the other techniques, suffering the many disadvantages described, the present embodiment simply corrects the transmit beam away from the focus as needed. This can be accomplished using the following insight. Consider a single receive element on a transducer array receiving the backscattered signal from the transmit pulse wavefront impinging on a point scatterer in the region of interest. As the transmit beam is swept through the region of interest, the received backscattered signal from the point scatterer, on a given receive element, will sample the transmit beam from unique spatial locations. In principle, if the transmit beam can be adequately sampled in time as well as in space, then a spatial-temporal filter can be designed that can transform the actual transmit beam into that of another transmit beam of differing spatial and temporal characteristics, provided that the transmit beam's spatial and temporal characteristics are sufficiently predictable. The receive element on the transducer array provides that sampling. For example, if a point scatterer in the region of interest lies in the unfocused region of the transmit beam, i.e., at a depth shallow to the transmit focus, then the evolution of signals received on a given transducer element, as the transmit beam is swept through the region of interest, will provide a representation of the defocused transmit beam.

[0043] FIG. 6 shows an example of a planar transducer array employing a linear scan format, where a single point scatterer is located at either r=r_a, (a depth shallow to the transmit focus), r=r_b (a depth at the transmit focus), or r=r_c (a depth deeper than the transmit focus), are shown. The transmit apodization and delay profiles are translated across the transducer array so as to translate the transmit beam across the region of interest. The transmit beam's pulse waveform, after having been subjected to frequency dependent attenuation, 1/r falloff, the transducer's receive spectral response for the element shown, as well as any other filter response in the receive signal path, produces the receive waveforms shown. The evolution of these receive waveforms as the transmit beam is translated across the region of interest are shown at times corresponding to a point scatterer depth of either r_a, r_b, or r_c. For a given point scatterer depth, the receive waveform is shifted in time according to how long the transmit beam's wavefront takes to propagate from the transducer face, to the point scatterer, and back to a given receive element, referenced to a suitable time origin, as the transmit beam is translated across the region of interest. For a point scatterer at r=r_a, the receive waveform arrives earlier at the edge of the transmit beam, and later for the center of the transmit beam due to the focusing curvature. For a point scatterer at either r=r_b or r=r_c, the receive waveforms display the behavior shown for analogous reasons.

[0044] Now consider the demodulated receive waveforms. For a given time/range sample lying within the evolution of receive waveforms from a given point scatterer depth, plotting the magnitude and phase, or equivalently, real and imaginary, i.e., I/Q, values of the given time/range sample as the transmit beam is swept through the region of interest will result in the beam patterns shown in FIG. 6. At range locations r=r_a or r=r_c, away from the transmit focus depth, the beam pattern is defocused and broad, displaying the classic quadratic phase error. At range location r=r_b, the beam pattern is focused. The real and imaginary beam pattern components display the same behavior. One skilled in the art will recognize the beam patterns away from the transmit focus as that of a chirp waveform, albeit a spatial chirp as opposed to a temporal chirp, and is the result of the transmit beam's approximately parabolic delay profile. Diffraction provides the spatial compression necessary to transform the defocused transmit beam in the near field to a focused beam at the focus location, however only at the focus location. Therefore, in order to produce a focused transmit beam for all ranges, a spatial filter is required to correct the transmit beam away from the transmit focus, correcting for diffraction effects, and perform spatial compression throughout the region of interest.

Transmit Synthesis—Spatial Diffraction Transform

[0045] The following analysis will consider several examples which are intended to illustrate the transmit synthesis technique; however, the invention is not meant to be limited by the examples provided. Note that in the analysis, continuous time is used for simplicity unless otherwise noted. Sampled time can also be used in an analogous manner.

[0046] Consider the architecture topologies shown in FIG. 7 (Image/Volume Reconstruction I) and FIG. 8 (Image/Volume Reconstruction II). Received channel data from each transmit beam firing n is acquired and stored in channel domain memory. This channel domain data can either be received RF, demodulated RF to IF, or demodulated to baseband I/Q signals from the transducer elements, or other suitable representation. The sequence of transmit beam firings can be swept through the region of interest in an arbitrary fashion, either in two or three dimensions. The transmit beam can be swept through the region of interest either electronically, mechanically, or any combination thereof. The transmit beams from spatial location to spatial location may differ in characteristics such as amplitude, phase, delay, carrier frequency, focusing characteristics, waveforms, coding, etc. The transducer employed can be either conventional 1-D arrays, 1½-D arrays, or 2-D matrix arrays. In FIG. 7, the Transmit Synthesis block precedes the Area/Volume Formation block, whereas in FIG. 8 the Transmit Synthesis block follows the Area/Volume Formation block. In both cases, the Reconstruction block contains both Transmit Synthesis and Area/Volume Formation Blocks. The Transmit Synthesis block in FIG. 7 consists of transmit beam n, time t, channel ch, and spatial coordinate r_ijk dependent filters h^syn whose outputs are summed together across N_syn(r_ijk) transmit beam firings. This is followed by the Area/Volume Formation block, which can form an output point s_out(r_ijk) at location r_ijk using time, channel, and spatially dependent delay, apodization, phase, and summing across N_ch channels, or some other suitable algorithm. In FIG. 8, the Area/Volume Formation block operates across channel data acquired from N_syn(r_ijk) transmit beam firings, either applying a time, channel, and spatially dependent delay, apodization, phase, and summing across N_ch channels, or other suitable algorithm, producing N_syn(r_ijk) outputs. Each of these outputs in turn are applied to the Transmit Synthesis block, consisting of N_syn(r_ijk) spatial coordinate r_ijk dependent filters h^syn whose outputs are then summed together across the N_syn(r_ijk) transmit beam firings to form an output point s_out(r_ijk) at location r_ijk. Now since the received signals on each of the transducer elements provide a representation of the transmit beam pattern over the evolution of transmit beam spatial firings, a spatial filter h^syn, whose region of support operates on either the channel signals, or area/volume formed signals, across a finite number of the sequence of transmit beam spatial firings which cover the region of interest, provides the desired spatial compression through a properly delayed, phased, weighted, and filtered superposition. Therefore, the Transmit Synthesis block which performs this filtering process can be viewed as a spatial diffraction transform since it transforms the actual transmit diffraction response into a desired transmit diffraction response, whose characteristics, both temporal and spatial, can be substantially different.

[0047] While the transmit synthesis technique will be covered in more detail, at a high level, the basic idea is the following. Consider an expression for the round-trip beampattern formed from a sequence of transmit beams whose apertures are centered at nΔx
S_n(ω,x)=S_xmt(ω,x−nΔx)S_rcv(ω,x)  (11)
where S_xmt is the transmit beampattern, S_rcv is the receive beampattern, ω is the carrier frequency, x is the target's lateral position, and

[0048] $\begin{array}{cc}{s}_n\left(t,x\right)=\frac{1}{2\pi }{\int }_{}^{}{S}_n\left(\omega ,x\right)P\left(\omega \right)e^{\mathrm{j\omega }t}d\omega & \left(12\right)\end{array}$
where s_n(t,x) is the round-trip point spread function response for pulse spectrum P(ω). Forming a linear combination of the sequence of roundtrip beampatterns, yields

[0049] $\begin{array}{cc}\begin{array}{c}{S}^{\mathrm{syn}}\left(\omega ,x\right)=\sum _{n=1}^{N_{\mathrm{syn}}}{H}_n\left(\omega \right)S_n\left(\omega ,x\right)\\ =\sum _{n=1}^{N_{\mathrm{syn}}}{H}_n\left(\omega \right)S_{\mathrm{xmt}}\left(\omega ,x-n\Delta x\right)S_{\mathrm{rcv}}\left(\omega ,x\right)\end{array}& \left(13\right)\end{array}$

[0050] This can be rewritten as

[0051] $\begin{array}{cc}{S}^{\mathrm{syn}}\left(\omega ,x\right)=S_{\mathrm{xmt}}^{\mathrm{syn}}\left(\omega ,x\right)S_{\mathrm{rcv}}\left(\omega ,x\right)\begin{array}{c}{S}_{\mathrm{xmt}}^{\mathrm{syn}}\left(\omega ,x\right)=\sum _{n=1}^{N_{\mathrm{syn}}}{H}_n\left(\omega \right)S_{\mathrm{xmt}}\left(\omega ,x-n\Delta x\right)\\ =\sum _{n=1}^{N_{\mathrm{syn}}}\underset{H_n\left(\omega \right)}{\underbrace{H_n\left(\omega \right)e^{{\mathrm{j\psi }}_n\left(\omega \right)}}}{S}_{\mathrm{xmt}}\left(\omega ,x-n\Delta x\right)\end{array}& \left(14\right)\end{array}$
where H_n(ω) can be thought of as a diffraction correction filter (complex in general, i.e., magnitude/phase, real/imaginary, etc.), the filtering operation termed diffraction transform. Thus, the operation attempts to transform, or correct, a sequence of actual transmit diffraction patterns S_xmt (ω,x−nΔx), into the desired transmit diffraction pattern S_xmt^syn using N_syn zones.

[0052] The symbols in the above equations are defined as follows:

• [0053] r_xmt=transmit focus range along the transmit beam direction
• [0054] r=point scatterer location in 2-D/3-D space
• [0055] r′=transducer array receive element location in 2-D/3-D space, either electronically swept, mechanically swept, or a combination thereof
• [0056] r_ijk=image/volume point location in 2-D/3-D space respectively
• [0057] φ(t)=demodulation phase
• [0058] s_xmt(t)=transmitted RF pulse waveform from transducer element
• [0059] {tilde over (s)}_xmt(t, r, . . . )=transmit beam RF pulse wavefront at point scatterer location r due to transmit excitation s_xmt(t), where ˜ indicates that the pulses transmitted by each element on the array will interfere at the location r and therefore, the temporal response of the pulse may be altered due to transmit diffraction
• [0060] {tilde over (s)}_ch(t, . . . )=receive channel signal from transducer element following TGC, A/D conversion, demodulation, baseband filtering, etc. ˜ indicates that the pulses received by each element on the array may have been altered by transmit diffraction
• [0061] T_xmt( . . . )=effective time delay of the transmit beam RF pulse wavefront {tilde over (s)}_xmt(t, r, . . . ) to a point scatterer located at r, i.e., the propagation delay plus the differential delay arc of the transmit pulse wavefront, ΔT_xmt, due to diffraction
• [0062] T_rcv( . . . )=time delay of the backscattered pulse from a point scatterer located at r to a transducer receive element
• [0063] T(ch,r_ijk), φ(ch,r_ijk), a(ch,r_ijk)=receive channel delay and phase correction, and apodization, used in area/volume formation in 2-D/3-D space
• [0064] h_rcv(t)=receive filter which includes the effects of frequency dependent attenuation, 1/r falloff, element response, the transducer's one-way receive response, etc.
• [0065] h^syn( . . . )=transmit synthesis filter
• [0066] N_syn( . . . ), N_syn( . . . )=number/list of transmit beams in 2-D/3-D space used in transmit synthesis

[0067] Consider the demodulated receive signal from a given transducer element as shown in FIG. 6. The signal {tilde over (s)}_ch received by a transducer element located at r′=x′_p, from a point scatterer at r=(r,x), insonified by the n-th transmit beam wavefront {tilde over (s)}_xmt(t−T_xmt,x−x_n,r,r_xmt), due to a transmit excitation of s_xmt(t), where x_n is the transmit beam origin location on the array, can be mathematically expressed as given by (15). The receive signal s_ch is simply the backscattered signal from the point scatterer at (r,x), attenuated and delayed by the propagation time T_rcv from (r,x) back to the transducer receive element at x′_p.

[0068] $\begin{array}{cc}{T}_{\mathrm{xmt}}=T_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}}\right)=\frac{r}{c}+\Delta T_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}}\right)T_{\mathrm{rcv}}=T_{\mathrm{rcv}}\left(x_p^{\prime }-x,r\right)=\frac{\sqrt{r^2+{\left(x_p^{\prime }-x\right)}^2}}{c}{\tilde{s}}_{\mathrm{ch}}\left(t-T_{\mathrm{xmt}}-T_{\mathrm{rcv}},x_p^{\prime },x-x_n,r,r_{\mathrm{xmt}}\right)=e^{-{\mathrm{j\phi }}_d\left(t\right)}{\int }_{}^{}{h}_{\mathrm{rcv}}\left(t-t^{\prime },x_p^{\prime }\right){\tilde{s}}_{\mathrm{xmt}}\left(t^{\prime }-T_{\mathrm{xmt}}-T_{\mathrm{rcv}},x-x_n,r,r_{\mathrm{xmt}}\right)d{t}^{\prime }& \left(15\right)\end{array}$

[0069] If the transmit excitation by the transducer is represented by
s_xmt(t)=p(t)e^jωxt  (16)
where p(t) is the transmit pulse envelope, potentially complex, and ω_x is the transmit carrier frequency, then without loss of generality the transmit pulse wavefront can be expressed as
{tilde over (s)}_xmt(t−T_xmt,x−x_n,r,r_xmt)={tilde over (p)}(t−T_xmt,x−x_n,r,r_xmt)e^{jωx(t−Txmt)}S_xmt(x−x_n,r,r_xmt,ω_x)  (17)
where ˜ indicates that the pulse envelope has been modified due to range and position dependent transmit diffraction effects, ω_x is the effective transmit carrier frequency which may be depth dependent due to frequency dependent attenuation effects, and S_xmt represents the transmit beampattern. Upon substitution into equation (15), the receive channel signal becomes
{tilde over (s)}_ch(t−T_xmt−T_rcv,x′_p,x−x_n,r,r_xmt)={tilde over (p)}_rcv(t−T_xmt−T_rcv,x−x_n,r,r_xmt)·e^{−jφd(t)+ωc(t−Txmt−Trcv)}S_xmt(x−x_n,r,r_xmt,ω_c)  (18)
where {tilde over (p)}_rcv(t−T_xmt−T_rcv,x−x_n,r,r_xmt) includes the effects of transmit diffraction, receive propagation back to the transducer, attenuation, filtering, etc., and ω_c is the depth dependent receive carrier frequency. Matching the demodulation and receive carrier frequencies results in the baseband channel signal
{tilde over (s)}_ch(t−T_xmt−T_rcv,x′_p,x−x_n,r,r_xmt)={tilde over (p)}_rcv(t−T_xmt−T_rcv,x−x_n,r,r_xmt)·e^−jωcTreve^−jωcTxmtS_xmt(x−x_n,r,r_xmt,ω_c)  (18)

[0070] Image/Volume Reconstruction I shown in FIG. 7, creates a single channel domain dataset from which the Area Formation block operates on after the Transmit Synthesis block has effectively compressed the transmit beam. Let the transmit synthesis and area formation outputs be given by

[0071] $\begin{array}{cc}T\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)=\frac{r_{\mathrm{ij}}+\sqrt{r_{\mathrm{ij}}^2+{\left(x_p^{\prime }-x_{\mathrm{ij}}\right)}^2}}{c}{\tilde{s}}_{\mathrm{ch}}^{\mathrm{syn}}\left(t-T_{\mathrm{xmt}}-T_{\mathrm{rcv}},x_p^{\prime },{\underset{\_}{r}}_{\mathrm{ij}},r,x,r_{\mathrm{xmt}}\right)=\underset{\mathrm{Spatial}\mathrm{Diffraction}\mathrm{Transform}}{\underbrace{\sum _{n={\underset{\_}{N}}_{\mathrm{syn}}\left({\underset{\_}{r}}_{\mathrm{ij}}\right)}^{}{\int }_{}^{}{h}_n^{\mathrm{syn}}\left(t-t^{\prime },x_p^{\prime },{\underset{\_}{r}}_{\mathrm{ij}}\right){\tilde{s}}_{\mathrm{ch}}\left(\begin{array}{c}{t}^{\prime }-T_{\mathrm{xmt}}-T_{\mathrm{rcv}},x_p^{\prime },\\ x-x_n,r,r_{\mathrm{xmt}}\end{array}\right)d{t}^{\prime }}}\tilde{s}\left(r_{\mathrm{ij}},x_{\mathrm{ij}},x\right)=\underset{\mathrm{Area}\mathrm{Formation}}{\underbrace{\sum _p^{}a\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)\delta \left[\begin{array}{c}t-T\\ \left(\begin{array}{c}{x}_p^{\prime }-\\ x_{\mathrm{ij}},r_{\mathrm{ij}}\end{array}\right)\end{array}\right]e^{\mathrm{j\varphi }\left(r_{\mathrm{ij}},x_p^{\prime }-x_{\mathrm{ij}}\right)}{\tilde{s}}_{\mathrm{ch}}^{\mathrm{syn}}\left(\begin{array}{c}t-T_{\mathrm{xmt}}-\\ T_{\mathrm{rcv}},x_p^{\prime },{\underset{\_}{r}}_{\mathrm{ij}}\\ r,x,r_{\mathrm{xmt}}\end{array}\right)}}& \left(20\right)\end{array}$

[0072] Inserting equation (19) into equation (20) yields an expression for the synthesized channel signals, namely

[0073] $\begin{array}{cc}{\tilde{s}}_{\mathrm{ch}}^{\mathrm{syn}}\left(t-T_{\mathrm{xmt}}-T_{\mathrm{rcv}},x_p^{\prime },{\underset{\_}{r}}_{\mathrm{ij}},r,x,r_{\mathrm{xmt}}\right)=\sum _{n={\underset{\_}{N}}_{\mathrm{syn}}\left({\underset{\_}{r}}_{\mathrm{ij}}\right)}^{}{\int }_{}^{}{h}_n^{\mathrm{syn}}\left(t-t^{\prime },x_p^{\prime },{\underset{\_}{r}}_{\mathrm{ij}}\right){\tilde{p}}_{\mathrm{rcv}}\left(t^{\prime }-T_{\mathrm{xmt}}-T_{\mathrm{rcv}},x-x_n,r,r_{\mathrm{xmt}}\right)e^{-{\mathrm{j\omega }}_c{T}_{\mathrm{rcv}}}{e}^{-{\mathrm{j\omega }}_c{T}_{\mathrm{xmt}}}{S}_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}},{\omega }_c\right)d{t}^{\prime }& \left(21\right)\end{array}$

[0074] Consider three different functional forms for the transmit synthesis filters h_n^syn, namely

[0075] $\begin{array}{cc}{h}_n^{\mathrm{syn}}\left(t,x_p^{\prime },{\underset{\_}{r}}_{\mathrm{ij}}\right)=\{\begin{array}{c}{h}^{\mathrm{syn}}\left(x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\right)\delta \left(t\right)\\ \begin{array}{c}{h}^{\mathrm{syn}}\left(x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\right)\delta \\ \left[t+\Delta T_{\mathrm{xmt}}\left(x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\right)\right]\end{array}\\ h^{\mathrm{syn}}\left[\begin{array}{c}t+\Delta T_{\mathrm{xmt}}\left(x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\right),\\ x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\end{array}\right]\end{array}& \left(22\right)\end{array}$

[0076] These transmit synthesis filter forms: a) are lateral convolution filters, and b) depend upon image range coordinate r_ij relative to transmit focus range r_xmt so as to provide range dependent transmit beam diffraction correction. The N_syn filters of h^syn are approximately centered around the n^th transmit beam closest to the lateral image coordinate x_ij, and are designed in such a manner as to correct for the amplitude variations and range r dependent approximate quadratic delay/phase error experienced by the received backscattered signals as the transmit beam is translated across the region of interest, i.e., a spatial chirp compression filter. The number of filters necessary to provide complete spatial compression varies with depth r and hence, image coordinate r_ij, being dependent upon how wide the defocused transmit beam is, which is based upon how far r is from the transmit focus range r_xmt. The wider the defocused transmit beam, the greater the number of filters required for full spatial compression. To the extent that a limited number of filters are used due to fixed computational resources or otherwise, the transmit beam only be partially compressed, which in many cases can suppress receive sidelobe clutter dramatically. For image coordinate locations r_ij around the transmit focus range r_xmt, not many filters are required since the transmit beam is nearly focused.

[0077] The lateral convolution form of these transmit synthesis filters provides an effective lateral spatial shifting of the transmit beam, from the combination of spatial transmit beam firings, to center the synthesized transmit beam over the image coordinate location x_ij. This property effectively decouples transmit and round-trip spatial sampling requirements, while creating a transmit beam centered over the desired image location. This property is achievable provided that the transmit beams satisfy spatial Nyquist sampling requirements. Optimal design of the transmit synthesis filters requires sufficiently predictable temporal and spatial pre-detection, i.e., coherent, characteristics of both the actual transmit beams and the desired synthesized transmit beams as they are swept through the region of interest, whether electronically or mechanically scanned, or any combination thereof, as well as taking into account the number of synthesis filters N_syn used at a given image coordinate location. In general, the filtering of signals can yield different results depending upon whether the signal of interest is coherent, i.e., backscattered signals from the region of interest, or incoherent in origin, i.e., random noise, and produce changes in the signal to noise ratio (SNR). Since these transmit synthesis filters will process channel domain data from multiple transmit beams, each of which will contain receive front-end random noise, optimal design of the transmit synthesis filters may also include these effects.

[0078] The first form of h_n^syn=h^syn(x_n−x_ij,r_ij,r_xmt)δ(t) represents a range dependent, complex lateral filter, i.e., magnitude/phase, real/imaginary, etc., which is independent of time and simply provides an amplitude weighted phase adjustment of the transmit beam dependent channel domain data sets prior to transmit synthesis summation. Referring to FIG. 7 which shows a representation of the receive channel signals across transmit firings for a point scatterer which lies shallow to the transmit focus range, i.e., r=r_a<r_xmt, the backscattered receive signals arrive earlier as the transmit beam is laterally translated away from the position of the point scatterer at (r,x) due to the converging transmit pulse wavefront. The weighted phase adjustment is sufficient provided that the delay excursion of the transmit pulse wavefront, for the spatial portion of the defocused transmit beam to be compressed, or refocused, is on the order of the round-trip pulse length or less.

[0079] The second form of h_n^syn=h^syn(x_n−x_ij,r_ij,r_xmt)δ[t+ΔT_xmt(x_n−x_ij,r_ij,r_xmt)] represents a range dependent, complex lateral filter as well; however, in addition to an amplitude weighted phase adjustment of the transmit beam dependent channel domain data sets prior to transmit synthesis summation, it provides a differential time delay correction ΔT_xmt(x_n−x_ij,r_ij,r_xmt) (refer to equation (15)) to the backscattered receive channel signals as the transmit beam is laterally translated away from the position of the point scatterer at (r,x) due to the converging/diverging transmit pulse wavefront. This additional time correction will be required when the delay excursion of the transmit pulse wavefront, for the spatial portion of the defocused transmit beam to be compressed, or refocused, begins to exceed the round-trip pulse length.

[0080] The third form of h_n^syn=h^syn[t+ΔT_xmt(x_n−x_ij,r_ij,r_xmt),x_n−x_ij,r_ij,r_xmt] represents a range dependent, complex lateral filter as well; however, in addition to an amplitude weighted phase adjustment and differential time delay correction ΔT_xmt(x_n−x_ij,r_ij,r_xmt) applied to the receive channel signals, the temporal filter portion also provides a range and lateral position dependent pulse shape correction. The intent is to compensate the receive channel signals {tilde over (s)}_ch(t−T_xmt−T_rcv,x′_p,x−x_n,r,r_xmt), and through extension, the transmit signal {tilde over (s)}_xmt(t−T_xmt,x−x_n,r,r_xmt), for diffraction effects away from the transmit focus, receive filtering, etc., in addition to differential transmit delay correction and transmit refocusing.

[0081] Substitution of the first form of h_n^syn=h^syn(x_n−x_ij,r_ij,r_xmt)δ(t) into the synthesized channel data given by equation (21), along with equation (15) yields

[0082] $\begin{array}{cc}{\tilde{s}}_{\mathrm{ch}}^{\mathrm{syn}}\left(t-T_{\mathrm{xmt}}-T_{\mathrm{rcv}},x_p^{\prime },x_{\mathrm{ij}},r_{\mathrm{ij}},x,r,r_{\mathrm{xmt}}\right)=e^{-{\mathrm{j\omega }}_c\left(\frac{r}{c}+T_{\mathrm{rcv}}\right)}\sum _{n={\underset{\_}{N}}_{\mathrm{syn}}\left({\underset{\_}{r}}_{\mathrm{ij}}\right)}^{}{h}^{\mathrm{syn}}\left(x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\right){\tilde{p}}_{\mathrm{rcv}}\left(t-T_{\mathrm{xmt}}-T_{\mathrm{rcv}},x-x_n,r,r_{\mathrm{xmt}}\right)e^{-{\mathrm{j\omega }}_c\Delta T_{\mathrm{xmt}}}{S}_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(23\right)\end{array}$

[0083] In order to determine a solution for the transmit synthesis filter coefficients, they may be derived based upon theoretical considerations, i.e., deriving a spatial compression filter analogous to linear FM chirp pulse compression, or an error function of the following form can be minimized at each image location (r_ij,x_ij), over point scatterer lateral locations x, using the list of potential range/position dependent N_syn(r_ij,x_ij) transmit beams, generally centered about the transmit beam closest to x_ij, i.e., x₀(x_ij):

[0084] $\begin{array}{cc}{\varepsilon }^2=\sum _x^{}|\sum _{n={\underset{\_}{N}}_{\mathrm{syn}}\left({\underset{\_}{r}}_{\mathrm{ij}}\right)}^{}\left\{h^{\mathrm{syn}}\left(x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\right){\tilde{p}}_{\mathrm{rcv}}\left[-\Delta T_{\mathrm{xmt}}\left(x-x_n,r_{\mathrm{ij}},r_{\mathrm{xmt}}\right),x-x_n,r,r_{\mathrm{xmt}}\right]·e^{-{\mathrm{j\omega }}_c\Delta T_{\mathrm{xmt}}\left(x-x_n,r_{\mathrm{ij}},r_{\mathrm{xmt}}\right)}{S}_{\mathrm{xmt}}\left(x-x_n,r_{\mathrm{ij}},r_{\mathrm{xmt}},{\omega }_c\right)\right\}-p^{\mathrm{desired}}\left(0,x-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)S_{\mathrm{xmt}}^{\mathrm{desired}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}},{\omega }_c\right){|}^2\mathrm{where}{p}^{\mathrm{desired}}\left(t-\frac{2r_{\mathrm{ij}}}{c},x-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)& \left(24\right)\end{array}$
and s_xmt^desired( . . . ) are defined as the desired transmit pulse response and desired beampattern respectively, or taken together—the desired transmit point spread function, which is the targeted response for the spatial diffraction transform. t is evaluated at the nominal receive time t=2r/c (see FIG. 6), r is evaluated at the image range coordinate r_ij, receive channel location x′_p is evaluated at the point scatterer location x. If S_xmt^desired( . . . ) is that of a continuous focused transmit beam, then the subsequent transmit synthesis filter coefficients derived through error minimization or other means, will provide spatial compression of the transmit beam away from the focus, in addition to the spatial shifting property. Multiple time locations t can also be used in evaluating {tilde over (p)}_rcv( . . . ), S_xmt( . . . ), and e^{−jωcΔTxmt} in equation (23) in forming the error function, as well as p^desired( . . . ). Other error functions are possible. For one skilled in the art, there are many ways in which to perform this error minimization, for example a least-squares, weighted least-squares, constrained least-squares error minimization, etc., where a set of simultaneous linear equations are created from the error function, and the unknown coefficients h_n^syn=h^syn(n) are solved. For a fixed number of transmit synthesis filters N_syn(r_ij,x_ij), error minimization provides for a flexible solution. Evaluation of the receive pulse waveform(s) {tilde over (p)}_rcv( . . . ), transmit beampattern S_xmt( . . . ), desired pulse response p^desired( . . . ) and beampattern S_xmt^desired( . . . ), and transmit differential time delay ΔT_xmt( . . . ) in the error minimization can be performed either through experimental measurement, simulation, or a combination thereof, provided they are sufficiently predictable. Applying the optimized transmit synthesis filter coefficients to the receive channel signals produces synthesized channel data given by

[0085] $\begin{array}{cc}{\tilde{s}}_{\mathrm{ch}}^{\mathrm{syn}}\left(t-T_{\mathrm{xmt}}-T_{\mathrm{rcv}},x_p^{\prime },x_{\mathrm{ij}},r_{\mathrm{ij}},x,r,r_{\mathrm{xmt}}\right)\approx e^{-{\mathrm{j\omega }}_c\left(\frac{r}{c}+T_{\mathrm{rcv}}\right)}{\tilde{p}}_{\mathrm{rcv}}\left\{t-\Delta T_{\mathrm{xmt}}\left[x_{\mathrm{ij}}-x_0\left(x_{\mathrm{ij}}\right)\right]-\left(\frac{r}{c}+T_{\mathrm{rcv}}\right),x-x_0\left(x_{\mathrm{ij}}\right),r,r_{\mathrm{xmt}}\right\}{S}_{\mathrm{xmt}}^{\mathrm{syn}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(25\right)\end{array}$
where the synthesized transmit beampattern is given by

[0086] $\begin{array}{cc}{S}_{\mathrm{xmt}}^{\mathrm{syn}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,r_{\mathrm{xmt}},{\omega }_c\right)=\sum _{n={\underset{\_}{N}}_{\mathrm{syn}}\left({\underset{\_}{r}}_{\mathrm{ij}}\right)}{h}^{\mathrm{syn}}\left(x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\right)e^{-{\mathrm{j\omega }}_c\Delta T_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}}\right)}{S}_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(26\right)\end{array}$
Note that the synthesized transmit beampattern will be centered over the image coordinate location x_ij as described previously.

[0087] Upon substituting equation (25) into the expression for the Area Formation block output given by equation (20), an expression is obtained for the round-trip point spread function

[0088] $\begin{array}{cc}\tilde{s}\left(r_{\mathrm{ij}},r,x_{\mathrm{ij}},x\right)\approx \sum _{p}a\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)e^{\mathrm{j\varphi }\left(r_{\mathrm{ij}},x_p^{\prime }-x_{\mathrm{ij}}\right)}{e}^{-{\mathrm{j\omega }}_c\left(\frac{r}{c}+T_{\mathrm{rcv}}\right)}·{\tilde{p}}_{\mathrm{rcv}}\left\{\frac{2\left(r_{\mathrm{ij}}-r\right)}{c}-\Delta T_{\mathrm{xmt}}\left[x_{\mathrm{ij}}-x_0\left(x_{\mathrm{ij}}\right)\right],x-x_0\left(x_{\mathrm{ij}}\right),r,r_{\mathrm{xmt}}\right\}{S}_{\mathrm{xmt}}^{\mathrm{syn}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(27\right)\end{array}$
where the area formation element dependent delay T(x′_p−x_ij,r_ij) has effectively cancelled the element dependent receive delay T_rcv(x′_p−x,r) given in equation (15), for r_ij˜r. Then,

[0089] $\begin{array}{cc}\tilde{s}\left(r_{\mathrm{ij}}-r,x_{\mathrm{ij}}-x,r_{\mathrm{ij}},x_{\mathrm{ij}}\right)\approx e^{-{\mathrm{j\omega }}_c\frac{2r}{c}}{\tilde{p}}_{\mathrm{rcv}}\left\{\frac{2\left(r_{\mathrm{ij}}-r\right)}{c}-\Delta T_{\mathrm{xmt}}\left[x_{\mathrm{ij}}-x_0\left(x_{\mathrm{ij}}\right)\right],x-x_0\left(x_{\mathrm{ij}}\right),r,r_{\mathrm{xmt}}\right\}·S_{\mathrm{xmt}}^{\mathrm{syn}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,r_{\mathrm{xmt}},{\omega }_c\right)S_{\mathrm{rcv}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,{\omega }_c\right)& \left(28\right)\end{array}$
where the round-trip point spread function's parameters are meant to indicate that it is in the form of a convolution kernel which could vary slowly throughout the imaging region defined by (r_ij,x_ij). The receive beampattern S_rcv( . . . ) is given by

[0090] $\begin{array}{cc}{S}_{\mathrm{rcv}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,{\omega }_c\right)=\sum _{p}a\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)e^{\mathrm{j\varphi }\left(r_{\mathrm{ij}},x_p^{\prime }-x_{\mathrm{ij}}\right)}{e}^{-{\mathrm{j\omega }}_c\left(T_{\mathrm{rcv}}-\frac{r}{c}\right)}& \left(29\right)\end{array}$

[0091] In order to correct for receive element phase errors due to the differing propagation paths from the point scatterer back to the transducer array, the receive phase is given by

[0092] $\begin{array}{cc}\varphi \left(r_{\mathrm{ij}},x_p^{\prime }-x_{\mathrm{ij}}\right)={\omega }_c\left[T\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)-\frac{2r_{\mathrm{ij}}}{c}\right]={\omega }_c\left[\frac{\sqrt{r_{\mathrm{ij}}^2+{\left(x_p^{\prime }-x_{\mathrm{ij}}\right)}^2}-r_{\mathrm{ij}}}{c}\right]& \left(30\right)\end{array}$
Substituting equation (30) into equation (29) yields an expression for the receive beampattern

[0093] $\begin{array}{cc}{S}_{\mathrm{rcv}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,{\omega }_c\right)=\sum _{p}a\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)e^{{\mathrm{j\omega }}_c\left[\frac{\sqrt{r_{\mathrm{ij}}^2+{\left(x_p^{\prime }-x_{\mathrm{ij}}\right)}^2}-r_{\mathrm{ij}}}{c}-\frac{\sqrt{r^2+{\left(x_p^{\prime }-x\right)}^2}-r}{c}\right]}{S}_{\mathrm{rcv}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},{\omega }_c\right)\approx e^{-{\mathrm{j\omega }}_c\frac{1}{2c}\frac{{\left(x-x_{\mathrm{ij}}\right)}^2}{r}}\sum _{p}a\left(u_p,r_{\mathrm{ij}}\right)e^{-{\mathrm{j\omega }}_c\frac{u^2}{2c}\left(\frac{1}{r}-\frac{1}{r_{\mathrm{ij}}}\right)}{e}^{{\mathrm{j\omega }}_c\frac{\left(x-x_{\mathrm{ij}}\right)u}{r_{\mathrm{ij}}c}}& \left(31\right)\end{array}$
where a Taylor series expansion of the phase was used in the approximation and yields the well known Fourier transform of the receive apodization for r_ij=r, i.e., in the focal plane. Note the expression for the round-trip point spread function given by equation (28) is in the form of a linear convolution kernel in both image coordinates, with a slow variation in characteristics due to the remaining parameter dependencies. The parameter dependencies uncorrected by the first form of the transmit synthesis filters equation (22) are: a) the uncorrected spatially dependent differential transmit delay, and b) the transmit diffraction and receive filtering effects on the transmit pulse. These are addressed with the second and third forms of the transmit synthesis filters given by equation (22) respectively.

[0094] Substituting the second form of h_n^syn=h^syn(x_n−x_ij,r_ij,r_xmt)δ[t+ΔT_xmt(x_n−x_ij,r_ij,r_xmt)] into the synthesized channel data given by equation (21), along with equation (15) yields

[0095] $\begin{array}{cc}{\tilde{s}}_{\mathrm{ch}}^{\mathrm{syn}}\left(t-T_{\mathrm{rcv}},x_p^{\prime },x_{\mathrm{ij}},r_{\mathrm{ij}},x,r,r_{\mathrm{xmt}}\right)\approx e^{-{\mathrm{j\omega }}_c\left(\frac{r}{c}+T_{\mathrm{rcv}}\right)}\sum _{n={\underset{\_}{N}}_{\mathrm{syn}}\left({\underset{\_}{r}}_{\mathrm{ij}}\right)}{h}^{\mathrm{syn}}\left(x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\right){\tilde{p}}_{\mathrm{rcv}}\left(t-T_{\mathrm{rcv}},x-x_n,r,r_{\mathrm{xmt}}\right)e^{-{\mathrm{j\omega }}_c\Delta T_{\mathrm{xmt}}}{S}_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(32\right)\end{array}$
where the differential transmit delay is effectively cancelled for x_ij˜x. Solving for the transmit synthesis filter coefficients, the following error equation is minimized for each image location (r_ij,x_ij), over point scatterer lateral locations x as before.

[0096] $\begin{array}{cc}{\varepsilon }^2=\sum _x|\sum _{n={\underset{\_}{N}}_{\mathrm{syn}}\left({\underset{\_}{r}}_{\mathrm{ij}}\right)}\left\{h^{\mathrm{syn}}\left(x_n-x_{\mathrm{ij}},r_{\mathrm{xmt}}\right){\tilde{p}}_{\mathrm{rcv}}\left(0,x-x_n,r_{\mathrm{ij}},r_{\mathrm{xmt}}\right)e^{-{\mathrm{j\omega }}_c{\mathrm{\Delta T}}_{\mathrm{xmt}}\left(x-x_{n,}{r}_{\mathrm{ij}},r_{\mathrm{xmt}}\right)}{S}_{\mathrm{xmt}}\left(x-x_n,r_{\mathrm{ij}},r_{\mathrm{xmt}},{\omega }_c\right)\right\}-p^{\mathrm{desired}}\left(0,x-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)S_{\mathrm{xmt}}^{\mathrm{desired}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}},{\omega }_c\right){|}^2& \left(33\right)\end{array}$
where time t is evaluated at the nominal receive time t=2r/c (see FIG. 6). Note that since the differential transmit delay ΔT_xmt ( . . . ) has been corrected, the pulse signals on a given receive channel are aligned in time, i.e., sampling along the transmit delay arc.

[0097] As before, multiple time locations can be used in constructing the error function. Applying the optimized transmit synthesis filter coefficients to the receive channel signals produces synthesized channel data given by

[0098] $\begin{array}{cc}{\tilde{s}}_{\mathrm{ch}}^{\mathrm{syn}}\left(t-T_{\mathrm{rcv}},x_p^{\prime },x_{\mathrm{ij}},r_{\mathrm{ij}},x,r,r_{\mathrm{xmt}}\right)\approx e^{-{\mathrm{j\omega }}_c\left(\frac{r}{c}+T_{\mathrm{rcv}}\right)}{\tilde{p}}_{\mathrm{rcv}}\left[t,x-x_0\left(x_{\mathrm{ij}}\right),r,r_{\mathrm{xmt}}\right]S_{\mathrm{xmt}}^{\mathrm{syn}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(34\right)\end{array}$
where the synthesized transmit beampattern is given by

[0099] $\begin{array}{cc}{S}_{\mathrm{xmt}}^{\mathrm{syn}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,r_{\mathrm{xmt}},{\omega }_c\right)=\sum _{n={\underset{\_}{N}}_{\mathrm{syn}}\left({\underset{\_}{r}}_{\mathrm{ij}}\right)}{h}^{\mathrm{syn}}\left(x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\right)e^{-{\mathrm{j\omega }}_c\Delta T_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}}\right)}{S}_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(35\right)\end{array}$

[0100] Note that even though the differential transmit delay has been corrected, a phase error due to the transmit delay still exists due to demodulation being performed prior to the correction. Note further that the synthesized transmit beampattern is now centered over the image coordinate location as before.

[0101] Upon substituting equation (34) into the expression for the Area Formation block output given by equation (20), an expression is obtained for the round-trip point spread function

[0102] $\begin{array}{cc}\tilde{s}\left(r_{\mathrm{ij}},r,x_{\mathrm{ij}},x\right)\approx \sum _{p}a\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)e^{\mathrm{j\varphi }\left(r_{\mathrm{ij}},x_p^{\prime }-x_{\mathrm{ij}}\right)}{e}^{-{\mathrm{j\omega }}_c\left(\frac{r}{c}+T_{\mathrm{rcv}}\right)}·{\tilde{p}}_{\mathrm{rcv}}\left[\frac{2\left(r_{\mathrm{ij}}-r\right)}{c},x-x_0\left(x_{\mathrm{ij}}\right),r,r_{\mathrm{xmt}}\right]S_{\mathrm{xmt}}^{\mathrm{syn}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(36\right)\end{array}$
where the Area Formation element dependent delay T(x′_p−x_ij,r_ij) has effectively cancelled the element dependent receive delay T_rcv(x′_p−x,r) given in equation (15), for r_ij˜r. Then,

[0103] $\begin{array}{cc}\tilde{s}\left(r_{\mathrm{ij}}-r,x_{\mathrm{ij}}-x,r_{\mathrm{ij}},x_{\mathrm{ij}}\right)\approx e^{-{\mathrm{j\omega }}_c\frac{2r}{c}}{\tilde{p}}_{\mathrm{rcv}}\left[\frac{2\left(r_{\mathrm{ij}}-r\right)}{c},x-x_0\left(x_{\mathrm{ij}}\right),r,r_{\mathrm{xmt}}\right]·S_{\mathrm{xmt}}^{\mathrm{syn}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,r_{\mathrm{xmt}},{\omega }_c\right)S_{\mathrm{rcv}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,{\omega }_c\right)& \left(37\right)\end{array}$
where the receive beampattern s_rcv( . . . ) is the same as before, given by equations (29)-(31).

[0104] Substituting the third form of h_n^syn=h^syn[t+ΔT_xmt(x_n−x_ij,r_ij,r_xmt),x_n−x_ij,r_ij,r_xmt] into the synthesized channel data given by equation (21), using variable substitution along with equation (15) yields

[0105] $\begin{array}{cc}{\tilde{s}}_{\mathrm{ch}}^{\mathrm{syn}}\left(t-T_{\mathrm{rcv}},x_p^{\prime },x_{\mathrm{ij}},r_{\mathrm{ij}},x,r,r_{\mathrm{xmt}}\right)\approx e^{-{\mathrm{j\omega }}_c\left(\frac{r}{c}+T_{\mathrm{rcv}}\right)}·\sum _{n={\underset{\_}{N}}_{\mathrm{syn}}\left({\underset{\_}{r}}_{\mathrm{ij}}\right)}\int h^{\mathrm{syn}}\left[t-t^{\prime },x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\right]{\tilde{p}}_{\mathrm{rcv}}\left(t^{\prime }-T_{\mathrm{rcv}},x-x_n,r,r_{\mathrm{xmt}}\right)e^{-{\mathrm{j\omega }}_c\Delta T_{\mathrm{xmt}}}{S}_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}},{\omega }_c\right)d{t}^{\prime }& \left(38\right)\end{array}$

[0106] Solving for the transmit synthesis filter coefficients assuming a finite number of discrete time samples t_m, m=[0, 1, . . . M−1], the following error equation needs to be minimized, for each image location (r_ij,x_ij), over point scatterer lateral locations x and time t

[0107] $\begin{array}{cc}{\varepsilon }^{\prime }=\sum _t\sum _x|\sum _{n={\underset{\_}{N}}_{\mathrm{syn}}\left({\underset{\_}{r}}_{\mathrm{ij}}\right)}\left\{\sum _{m^{\prime }=0}^{M-1}\left\{h^{\mathrm{syn}}\left(t-t_{m^{\prime }},x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\right){\tilde{p}}_{\mathrm{rcv}}\left(t_{m^{\prime }},x-x_n,r_{\mathrm{ij}},r_{\mathrm{xmt}}\right)\right\}{e}^{{\mathrm{j\omega }}_c{\mathrm{\Delta T}}_{\mathrm{xmt}}\left(x-x_n,r_{\mathrm{ij}},r_{\mathrm{xmt}}\right)}{S}_{\mathrm{xmt}}\left(x-x_n,r_{\mathrm{ij}},r_{\mathrm{xmt}},{\omega }_c\right)\right\}-p^{\mathrm{desired}}\left(t,x-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)S_{\mathrm{xmt}}^{\mathrm{desired}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}},{\omega }_c\right){|}^2& \left(39\right)\end{array}$
where time t′ is evaluated around the nominal receive time t′=2r/c (see FIG. 6). Note that since the differential transmit delay ΔT_xmt( . . . ) has been corrected, the pulse signals on a given receive channel are aligned in time, i.e., sampling along the transmit delay arc. In addition, the time dependence of the desired transmit pulse response p^desired(t,x−x_ij,r_ij) appears in the error minimization function. A set of simultaneous linear equations are created from the error function, and the unknown coefficients h_mn^syn=h^syn(m,n) are solved. Thus, the transmit synthesis coefficients will attempt to correct the round-trip temporal response for transmit diffraction effects as well as receive filtering. Applying the optimized transmit synthesis filter coefficients to the receive channel signals produces synthesized channel data given by

[0108] $\begin{array}{cc}{s}_{\mathrm{ch}}^{\mathrm{syn}}\left(t-T_{\mathrm{rcv}},x_p^{\prime },x_{\mathrm{ij}},r_{\mathrm{ij}},x,r,r_{\mathrm{xmt}}\right)\approx e^{-j{\omega }_c\left(\frac{r}{c}+T_{\mathrm{rcv}}\right)}{p}^{\mathrm{syn}}\left(t,x-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)S_{\mathrm{xmt}}^{\mathrm{syn}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(40\right)\end{array}$
where the synthesized transmit point spread function is given by

[0109] $\begin{array}{cc}{p}^{\mathrm{syn}}\left(t,x-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)S_{\mathrm{xmt}}^{\mathrm{syn}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,r_{\mathrm{xmt}},{\omega }_c\right)\approx \sum _{n={\underset{\_}{N}}_{\mathrm{syn}}\left({\underset{\_}{r}}_{\mathrm{ij}}\right)}\sum _{m^{\prime }=0}^{M-1}\left\{h^{\mathrm{syn}}\left(t-t_{m^{\prime }},x_n-x_{\mathrm{ij}},r_{\mathrm{ij}},r_{\mathrm{xmt}}\right){\tilde{p}}_{\mathrm{rcv}}\left(t_{m^{\prime }},x-x_n,r,r_{\mathrm{xmt}}\right)\right\}{e}^{-{\mathrm{j\omega }}_c\Delta T_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}}\right)}{S}_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(41\right)\end{array}$

[0110] Upon substituting equation (40) into the expression for the Area Formation block output given by equation (20), an expression is obtained for the round-trip point spread function

[0111] $\begin{array}{cc}s\left(r_{\mathrm{ij}},r,x_{\mathrm{ij}},x\right)\approx \sum _{p}a\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)e^{\mathrm{j\varphi }\left(r_{\mathrm{ij}},x_p^{\prime }-x_{\mathrm{ij}}\right)}{e}^{j{\omega }_c\left(\frac{r}{c}+T_{\mathrm{rcv}}\right)}{p}^{\mathrm{syn}}\left[\frac{2\left(r_{\mathrm{ij}}-r\right)}{c},x-x_{\mathrm{ij}},r_{\mathrm{ij}}\right]S_{\mathrm{xmt}}^{\mathrm{syn}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(42\right)\end{array}$
where the Area Formation element dependent delay T(x′_p−x_ij,r_ij) has effectively cancelled the element dependent receive delay T_rcv(x′_p−x,r) given in equation (15), for r. Then,

[0112] $\begin{array}{cc}s\left(r_{\mathrm{ij}}-r,x_{\mathrm{ij}}-x,r_{\mathrm{ij}},x_{\mathrm{ij}}\right)\approx e^{-j{\omega }_c\frac{2r}{c}}{p}^{\mathrm{syn}}\left[\frac{2\left(r_{\mathrm{ij}}-r\right)}{c},x-x_{\mathrm{ij}},r_{\mathrm{ij}}\right]S_{\mathrm{xmt}}^{\mathrm{syn}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,r_{\mathrm{xmt}},{\omega }_c\right)S_{\mathrm{rcv}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,{\omega }_c\right)& \left(43\right)\end{array}$
where the receive beampattern s_rcv( . . . ) is the same as before, given by equations (29)-(31). Thus, using the more general transmit synthesis filter form, both differential time delay correction and pulse shape correction can be obtained simultaneously.

[0113] Image/Volume Reconstruction II shown in FIG. 8, which is a rearrangement of the Area Formation and Transmit Synthesis blocks, the Area Formation block operates separately on channel domain datasets, from which the Transmit Synthesis block effectively compresses the transmit beam across multiple area formed outputs. Let the area formation and transmit synthesis outputs be given by

[0114] $\begin{array}{cc}T\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)=\frac{r_{\mathrm{ij}}+\sqrt{r_{\mathrm{ij}}^2+{\left(x_p^{\prime }-x_{\mathrm{ij}}\right)}^2}}{c}{\tilde{s}}_n(r_{\mathrm{ij}},r,x_{\mathrm{ij}},x)=\underset{\underset{\mathrm{Area}\mathrm{Formation}}{︸}}{\begin{array}{c}\sum _{p}a\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)\delta \left[t-T\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)\right]e^{\mathrm{j\varphi }\left(r_{\mathrm{ij}},x_p^{\prime }-x_{\mathrm{ij}}\right)}\\ {\tilde{s}}_{\mathrm{ch}}\left(t-T_{\mathrm{xmt}}-T_{\mathrm{rcv}},x_p^{\prime },x-x_n,r,r_{\mathrm{xmt}}\right)\end{array}}\tilde{s}(r_{\mathrm{ij}},r,x_{\mathrm{ij}},x)=\underset{\underset{\mathrm{Spatial}\mathrm{Diffraction}\mathrm{Transform}}{︸}}{\sum _{n={\underset{\_}{N}}_{\mathrm{syn}}\left({\underset{\_}{r}}_{\mathrm{ij}}\right)}\sum _{r_{\mathrm{ij}}^{\prime }}{h}_n^{\mathrm{syn}}\left(x_n-x_{\mathrm{ij}},r_{\mathrm{ij}}-r_{\mathrm{ij}}^{\prime },r_{\mathrm{ij}}\right){\tilde{s}}_n\left(r_{\mathrm{ij}}^{\prime },r,x_{\mathrm{ij}},x\right)}& \left(44\right)\end{array}$

[0115] Inserting equation (19) into equation (44) yields an expression for the area formation outputs, namely

[0116] $\begin{array}{cc}{\tilde{s}}_n(r_{\mathrm{ij}},r,x_{\mathrm{ij}},x)=\sum _{p}a\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)\delta \left[t-T\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)\right]e^{\mathrm{j\varphi }\left(r_{\mathrm{ij}},x_p^{\prime }-x_{\mathrm{ij}}\right)}{\tilde{p}}_{\mathrm{rcv}}\left(t-T_{\mathrm{xmt}}-T_{\mathrm{rcv}},x-x_n,r,r_{\mathrm{xmt}}\right)e^{-{\mathrm{j\omega }}_c{T}_{\mathrm{rcv}}}{e}^{-{\mathrm{j\omega }}_c{T}_{\mathrm{xmt}}}{S}_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(45\right)\end{array}$
Simplifying equation (45), along with equation (15) yields

[0117] $\begin{array}{cc}{\tilde{s}}_n\left(r_{\mathrm{ij}},r,x_{\mathrm{ij}},x\right)\approx \sum _{p}a\left(x_p^{\prime }-x_{\mathrm{ij}},r_{\mathrm{ij}}\right)e^{\mathrm{j\varphi }\left(r_{\mathrm{ij}},x_p^{\prime }-x_{\mathrm{ij}}\right)}{e}^{-j{\omega }_c\left(\frac{r}{c}+T_{\mathrm{rcv}}\right)}{\tilde{p}}_{\mathrm{rcv}}\left[\frac{2\left(r_{\mathrm{ij}}-r\right)}{c}-\Delta T_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}}\right),x-x_n,r,r_{\mathrm{xmt}}\right]e^{-j{\omega }_c\Delta T_{\mathrm{xmt}}}{S}_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(46\right)\end{array}$
where the area formation element dependent receive delay T(x′_p−x_ij,r_ij) has effectively cancelled the element dependent receive delay T_rcv(x′_p−x,r) given in equation (15), for r_ij˜r. Then,

[0118] $\begin{array}{cc}{\tilde{s}}_n\left(r_{\mathrm{ij}},r,x_{\mathrm{ij}},x\right)\approx e^{-{\mathrm{j\omega }}_c\frac{2r}{c}}{\tilde{p}}_{\mathrm{rcv}}\left[\frac{2\left(r_{\mathrm{ij}}-r\right)}{c}-\Delta T_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}}\right),x-x_n,r,r_{\mathrm{xmt}}\right]S_{\mathrm{rcv}}\left(x-x_{\mathrm{ij}},r_{\mathrm{ij}},r,{\omega }_c\right)e^{-j{\omega }_c\Delta T_{\mathrm{xmt}}}{S}_{\mathrm{xmt}}\left(x-x_n,r,r_{\mathrm{xmt}},{\omega }_c\right)& \left(47\right)\end{array}$
where the receive beampattern S_rcv( . . . ) is given by