Estimation and correction of ultrasonic wavefront distortion using pulse-echo data received in a two-dimensional aperture

Three-dimensional ultrasound matrix imaging

Matrix imaging paves the way towards a next revolution in wave physics. Based on the response matrix recorded between a set of sensors, it enables an optimized compensation of aberration phenomena and multiple scattering events that usually drastically hinder the focusing process in heterogeneous media. Although it gave rise to spectacular results in optical microscopy or seismic imaging, the success of matrix imaging has been so far relatively limited with ultrasonic waves because wave control is generally only performed with a linear array of transducers. In this paper, we extend ultrasound matrix imaging to a 3D geometry. Switching from a 1D to a 2D probe enables a much sharper estimation of the transmission matrix that links each transducer and each medium voxel. Here, we first present an experimental proof of concept on a tissue-mimicking phantom through ex-vivo tissues and then, show the potential of 3D matrix imaging for transcranial applications.

Aberration correction in diagnostic ultrasound: A review of the prior field and current directions

Medical ultrasound images are reconstructed with simplifying assumptions on wave propagation, with one of the most prominent assumptions being that the imaging medium is composed of a constant sound speed. When the assumption of a constant sound speed are violated, which is true in most in vivoor clinical imaging scenarios, distortion of the transmitted and received ultrasound wavefronts appear and degrade the image quality. This distortion is known as aberration, and the techniques used to correct for the distortion are known as aberration correction techniques. Several models have been proposed to understand and correct for aberration. In this review paper, aberration and aberration correction are explored from the early models and correction techniques, including the near-field phase screen model and its associated correction techniques such as nearest-neighbor cross-correlation, to more recent models and correction techniques that incorporate spatially varying aberration and diffractive effects, such as models and techniques that rely on the estimation of the sound speed distribution in the imaging medium. In addition to historical models, future directions of ultrasound aberration correction are proposed.

Coherent Multi-Transducer Ultrasound Imaging

This work extends the effective aperture size by coherently compounding the received radio frequency data from multiple transducers. As a result, it is possible to obtain an improved image, with enhanced resolution, an extended field of view (FoV), and high-acquisition frame rates. A framework is developed in which an ultrasound imaging system consisting of N synchronized matrix arrays, each with partly shared FoV, take turns to transmit plane waves (PWs). Only one individual transducer transmits at each time while all N transducers simultaneously receive. The subwavelength localization accuracy required to combine information from multiple transducers is achieved without the use of any external tracking device. The method developed in this study is based on the study of the backscattered echoes received by the same transducer and resulting from a targeted scatterer point in the medium insonated by the multiple ultrasound probes of the system. The current transducer locations along with the speed of sound in the medium are deduced by optimizing the cross correlation between these echoes. The method is demonstrated experimentally in 2-D for two linear arrays using point targets and anechoic lesion phantoms. The first demonstration of a free-hand experiment is also shown. Results demonstrate that the coherent multi-transducer ultrasound imaging method has the potential to improve ultrasound image quality, improving resolution, and target detectability. Compared with coherent PW compounding using a single probe, lateral resolution improved from 1.56 to 0.71 mm in the coherent multi-transducer imaging method without acquisition frame rate sacrifice (acquisition frame rate 5350 Hz).

Physical-geometric optics method for large size faceted particles

A new physical-geometric optics method is developed to compute the single-scattering properties of faceted particles. It incorporates a general absorption vector to accurately account for inhomogeneous wave effects, and subsequently yields the relevant analytical formulas effective and computationally efficient for absorptive scattering particles. A bundle of rays incident on a certain facet can be traced as a single beam. For a beam incident on multiple facets, a systematic beam-splitting technique based on computer graphics is used to split the original beam into several sub-beams so that each sub-beam is incident only on an individual facet. The new beam-splitting technique significantly reduces the computational burden. The present physical-geometric optics method can be generalized to arbitrary faceted particles with either convex or concave shapes and with a homogeneous or an inhomogeneous (e.g., a particle with a core) composition. The single-scattering properties of irregular convex homogeneous and inhomogeneous hexahedra are simulated and compared to their counterparts from two other methods including a numerically rigorous method.

Feasibility of Swept Synthetic Aperture Ultrasound Imaging

Ultrasound image quality is often inherently limited by the physical dimensions of the imaging transducer. We hypothesize that, by collecting synthetic aperture data sets over a range of aperture positions while precisely tracking the position and orientation of the transducer, we can synthesize large effective apertures to produce images with improved resolution and target detectability. We analyze the two largest limiting factors for coherent signal summation: aberration and mechanical uncertainty. Using an excised canine abdominal wall as a model phase screen, we experimentally observed an effective arrival time error ranging from 18.3 ns to 58 ns (root-mean-square error) across the swept positions. Through this clutter-generating tissue, we observed a 72.9% improvement in resolution with only a 3.75 dB increase in side lobe amplitude compared to the control case. We present a simulation model to study the effect of calibration and mechanical jitter errors on the synthesized point spread function. The relative effects of these errors in each imaging dimension are explored, showing the importance of orientation relative to the point spread function. We present a prototype device for performing swept synthetic aperture imaging using a conventional 1-D array transducer and ultrasound research scanner. Point target reconstruction error for a 44.2 degree sweep shows a reconstruction precision of 82.8 μm and 17.8 μm in the lateral and axial dimensions respectively, within the acceptable performance bounds of the simulation model. Improvements in resolution, contrast and contrast-to-noise ratio are demonstrated in vivo and in a fetal phantom.

Acoustic reciprocity of spatial coherence in ultrasound imaging

A conventional ultrasound image is formed by transmitting a focused wave into tissue, time-shifting the backscattered echoes received on an array transducer, and summing the resulting signals. The van Cittert-Zernike theorem predicts a particular similarity, or coherence, of these focused signals across the receiving array. Many groups have used an estimate of the coherence to augment or replace the B-mode image in an effort to suppress noise and stationary clutter echo signals, but this measurement requires access to individual receive channel data. Most clinical systems have efficient pipelines for producing focused and summed RF data without any direct way to individually address the receive channels. We describe a method for performing coherence measurements that is more accessible for a wide range of coherence-based imaging. The reciprocity of the transmit and receive apertures in the context of coherence is derived and equivalence of the coherence function is validated experimentally using a research scanner. The proposed method is implemented on a commercial ultrasound system and in vivo short-lag spatial coherence imaging is demonstrated using only summed RF data. The components beyond the acquisition hardware and beamformer necessary to produce a real-time ultrasound coherence imaging system are discussed.

Equivalence of time and aperture domain additive noise in ultrasound coherence

Ultrasonic echoes backscattered from diffuse media, recorded by an array transducer and appropriately focused, demonstrate coherence predicted by the van Cittert-Zernike theorem. Additive noise signals from off-axis scattering, reverberation, phase aberration, and electronic (thermal) noise can all superimpose incoherent or partially coherent signals onto the recorded echoes, altering the measured coherence. An expression is derived to describe the effect of uncorrelated random channel noise in terms of the noise-to-signal ratio. Equivalent descriptions are made in the aperture dimension to describe uncorrelated magnitude and phase apodizations of the array. Binary apodization is specifically described as an example of magnitude apodization and adjustments are presented to minimize the artifacts caused by finite signal length. The effects of additive noise are explored in short-lag spatial coherence imaging, an image formation technique that integrates the calculated coherence curve of acquired signals up to a small fraction of the array length for each lateral and axial location. A derivation of the expected contrast as a function of noise-to-signal ratio is provided and validation is performed in simulation.

Equivalence of time and aperture domain additive noise in ultrasound coherence

Ultrasonic echoes backscattered from diffuse media, recorded by an array transducer and appropriately focused, demonstrate coherence predicted by the van Cittert-Zernike theorem. Additive noise signals from off-axis scattering, reverberation, phase aberration, and electronic (thermal) noise can all superimpose incoherent or partially coherent signals onto the recorded echoes, altering the measured coherence. An expression is derived to describe the effect of uncorrelated random channel noise in terms of the noise-to-signal ratio. Equivalent descriptions are made in the aperture dimension to describe uncorrelated magnitude and phase apodizations of the array. Binary apodization is specifically described as an example of magnitude apodization and adjustments are presented to minimize the artifacts caused by finite signal length. The effects of additive noise are explored in short-lag spatial coherence imaging, an image formation technique that integrates the calculated coherence curve of acquired signals up to a small fraction of the array length for each lateral and axial location. A derivation of the expected contrast as a function of noise-to-signal ratio is provided and validation is performed in simulation.

Refraction correction in 3D transcranial ultrasound imaging

We present the first correction of refraction in three-dimensional (3D) ultrasound imaging using an iterative approach that traces propagation paths through a two-layer planar tissue model, applying Snell's law in 3D. This approach is applied to real-time 3D transcranial ultrasound imaging by precomputing delays offline for several skull thicknesses, allowing the user to switch between three sets of delays for phased array imaging at the push of a button. Simulations indicate that refraction correction may be expected to increase sensitivity, reduce beam steering errors, and partially restore lost spatial resolution, with the greatest improvements occurring at the largest steering angles. Distorted images of cylindrical lesions were created by imaging through an acrylic plate in a tissue-mimicking phantom. As a result of correcting for refraction, lesions were restored to 93.6% of their original diameter in the lateral direction and 98.1% of their original shape along the long axis of the cylinders. In imaging two healthy volunteers, the mean brightness increased by 8.3% and showed no spatial dependency.

Pitch-catch phase aberration correction of multiple isoplanatic patches for 3-D transcranial ultrasound imaging

Having previously presented the ultrasound brain helmet, a system for simultaneous 3-D ultrasound imaging via both temporal bone acoustic windows, the scanning geometry of this system is utilized to allow each matrix array to serve as a correction source for the opposing array. Aberration is estimated using cross-correlation of RF channel signals, followed by least mean squares solution of the resulting overdetermined system. Delay maps are updated and real-time 3-D scanning resumes. A first attempt is made at using multiple arrival time maps to correct multiple unique aberrators within a single transcranial imaging volume, i.e., several isoplanatic patches. This adaptive imaging technique, which uses steered unfocused waves transmitted by the opposing, or beacon, array, updates the transmit and receive delays of 5 isoplanatic patches within a 64° x 64° volume. In phantom experiments, color flow voxels above a common threshold have also increased by an average of 92%, whereas color flow variance decreased by an average of 10%. This approach has been applied to both temporal acoustic windows of two human subjects, yielding increases in echo brightness in 5 isoplanatic patches with a mean value of 24.3 ± 9.1%, suggesting that such a technique may be beneficial in the future for performing noninvasive 3-D color flow imaging of cerebrovascular disease, including stroke.

Aberration compensation of an ultrasound imaging instrument with a reduced number of channels

Focusing and imaging qualities of an ultrasound imaging system that uses aberration correction were experimentally investigated as functions of the number of parallel channels. Front-end electronics that consolidate signals from multiple physical elements can be used to lower hardware and computational costs by reducing the number of parallel channels. However, the signals from sparse arrays of synthetic elements yield poorer aberration estimates. In this study, aberration estimates derived from synthetic arrays of varying element sizes are evaluated by comparing compensated receive focuses, compensated transmit focuses, and compensated b-scan images of a point target and a cyst phantom. An array of 80 x 80 physical elements with a pitch of 0.6 x 0.6 mm was used for all of the experiments and the aberration was produced by a phantom selected to mimic propagation through abdominal wall. The results show that aberration correction derived from synthetic arrays with pitches that have a diagonal length smaller than 70% of the correlation length of the aberration yield focuses and images of approximately the same quality. This connection between correlation length of the aberration and synthetic element size provides a guideline for determining the number of parallel channels that are required when designing imaging systems that employ aberration correction.

A k-space Green's function solution for acoustic initial value problems in homogeneous media with power law absorption

An efficient Green's function solution for acoustic initial value problems in homogeneous media with power law absorption is derived. The solution is based on the homogeneous wave equation for lossless media with two additional terms. These terms are dependent on the fractional Laplacian and separately account for power law absorption and dispersion. Given initial conditions for the pressure and its temporal derivative, the solution allows the pressure field for any time t>0 to be calculated in a single step using the Fourier transform and an exact k-space time propagator. For regularly spaced Cartesian grids, the former can be computed efficiently using the fast Fourier transform. Because no time stepping is required, the solution facilitates the efficient computation of the pressure field in one, two, or three dimensions without stability constraints. Several computational aspects of the solution are discussed, including the effect of using a truncated Fourier series to represent discrete initial conditions, the use of smoothing, and the properties of the encapsulated absorption and dispersion.

Application of a discontinuous Galerkin time domain method to simulation of optical properties of dielectric particles

We applied a discontinuous Galerkin time domain (DGTD) method, using a fourth-order Runga-Kutta time stepping of the Maxwell equations, to the simulation of the optical properties of dielectric particles in two-dimensional (2D) geometry. As examples of the numerical implementation of this method, the single-scattering properties of 2D circular and hexagonal particles are presented. In the case of circular particles, the scattering phase matrix was computed using the DGTD method and compared with the exact solution. For hexagonal particles, the DGTD method was used to compute single-scattering properties of randomly oriented 2D hexagonal ice crystals, and results were compared with those calculated using a geometric optics method. We consider both shortwave (visible) and longwave (infrared) cases, with particle size parameters 50 and 100. In the hexagonal case, scattering results are also presented as a function of both incident and scattering angles, revealing a structure apparently not reported before. Using the geometric optics method, we are able to interpret this structure in terms of contributions from varying numbers of internal reflections within the crystal.

A model of distributed phase aberration for deblurring phase estimated from scattering

Correction of aberration in ultrasound imaging uses the response of a point reflector or its equivalent to characterize the aberration. Because a point reflector is usually unavailable, its equivalent is obtained using statistical methods, such as processing reflections from multiple focal regions in a random medium. However, the validity of methods that use reflections from multiple points is limited to isoplanatic patches for which the aberration is essentially the same. In this study, aberration is modeled by an offset phase screen to relax the isoplanatic restriction. Methods are developed to determine the depth and phase of the screen and to use the model for compensation of aberration as the beam is steered. Use of the model to enhance the performance of the noted statistical estimation procedure is also described. Experimental results obtained with tissue-mimicking phantoms that implement different models and produce different amounts of aberration are presented to show the efficacy of these methods. The improvement in b-scan resolution realized with the model is illustrated. The results show that the isoplanatic patch assumption for estimation of aberration can be relaxed and that propagation-path characteristics and aberration estimation are closely related.

The cross algorithm for phase-aberration correction in medical ultrasound images formed with two-dimensional arrays

Common-midpoint signals in the near-field signal-redundancy (NFSR) algorithm for one-dimensional arrays are acquired using three consecutive transducer elements. An all-row-plus-two-column algorithm has been proposed to implement the one-dimensional NFSR algorithm on two dimensional arrays. The disadvantage of this method is that its ambiguity profile is not linear and a timeconsuming iterative method has to be used to linearize the ambiguity profile. An all-row-plus-two-column-and-a-diagonal algorithm has also been proposed. Its ambiguity profile is linear, but it is very sensitive to noise and cannot be used. In this paper, a novel cross algorithm is proposed to implement the NFSR algorithm on two-dimensional arrays. In this algorithm, common-midpoint signals are acquired using four adjacent transducer elements, which is not available in one-dimensional arrays. Its advantage includes a linear ambiguity profile and a higher measurement signal-to-noise ratio. The performance of the cross algorithm is evaluated theoretically. The region of redundancy is analyzed. The procedure for deriving the phaseaberration profile from peak positions of cross-correlation functions between common-midpoint signals is discussed. This algorithm is tested with a simulated data set acquired with a two-dimensional array, and the result shows that the cross algorithm performs better than the all-row plus-twocolumn NFSR algorithm.

Phase aberration correction using ultrasound radiation force and vibrometry optimization

We describe a phase aberration correction method that uses dynamic ultrasound radiation force to harmonically vibrate an object using amplitude modulated continuous wave ultrasound. The phase of each element of an annular array transducer is adjusted to maximize the radiation force and obtain optimal focus of the ultrasound beam. The maximization of the radiation force is performed by monitoring the velocity of scatterers in the focus region. We present theory that shows focal optimization with radiation force has a well-behaved cost function. Experimental validation is shown by correction of manual defocusing of an annular array as well as correcting for a lens-shaped aberrator placed near the transducer. A Doppler laser vibrometer and a pulse-echo Doppler ultrasound method were used to monitor the velocity of a sphere used as a target for the transducer. By maximizing the radiation force-induced vibration of scatterers in the focal region, the resolution of the ultrasound beam can be recovered after aberration defocusing.

Adaptive imaging on a diagnostic ultrasound scanner at quasi real-time rates

Constructing an ultrasonic imaging system capable of compensating for phase errors in real-time is a significant challenge in adaptive imaging. We present a versatile adaptive imaging system capable of updating arrival time profiles at frame rates of approximately 2 frames per second (fps) with 1-D arrays and up to 0.81 fps for 1.75-D arrays, depending on the desired near-field phase correction algorithm. A novel feature included in this system is the ability to update the aberration profile at multiple beam locations for 1-D arrays. The features of this real-time adaptive imaging system are illustrated in tissue-mimicking phantoms with physical near-field phase screens and evaluated in clinical breast tissue with a 1.75-D array. The contrast-to-noise ratio (CNR) of anechoic cysts was shown to improve dramatically in the tissue-mimicking phantoms. In breast tissue, the width of point-like targets showed significant improvement: a reduction of 26.2% on average. Brightness of these targets, however, marginally decreased by 3.9%. For larger structures such as cysts, little improvement in features and CNR were observed, which is likely a result of the system assuming an infinite isoplanatic patch size for the 1.75-D arrays. The necessary requirements for constructing a real-time adaptive imaging system are also discussed.

Phase-aberration correction with a 3-D ultrasound scanner: feasibility study

We tested the feasibility of using adaptive imaging, namely phase-aberration correction, with two-dimensional (2-D) arrays and real-time, 3-D ultrasound. Because of the high spatial frequency content of aberrators, 2-D arrays, which generally have smaller pitch and thus higher spatial sampling frequency, and 3-D imaging show potential to improve the performance of adaptive imaging. Phase-correction algorithms improve image quality by compensating for tissue-induced errors in beamforming. Using the illustrative example of transcranial ultrasound, we have evaluated our ability to perform adaptive imaging with a real-time, 3-D scanner. We have used a polymer casting of a human temporal bone, root-mean-square (RMS) phase variation of 45.0 ns, full-width-half-maximum (FWHM) correlation length of 3.35 mm, and an electronic aberrator, 100 ns RMS, 3.76 mm correlation, with tissue phantoms as illustrative examples of near-field, phase-screen aberrators. Using the multilag, least-squares, cross-correlation method, we have shown the ability of 3-D adaptive imaging to increase anechoic cyst identification, image brightness, contrast-to-speckle ratio (CSR), and, in 3-D color Doppler experiments, the ability to visualize flow. For a physical aberrator skull casting we saw CSR increase by 13% from 1.01 to 1.14, while the number of detectable cysts increased from 4.3 to 7.7.

Spatial and temporal aberrator stability for real-time adaptive imaging

Reported real-time adaptive imaging systems use near-field phase correction techniques, which are desired because of their simple implementation and their compatibility with current system architectures. Aberrator stability is important to adaptive imaging because it defines the spatial and temporal limits for which the near-field phase estimates are valid. Spatial aberrator stability determines the required spatial sampling of the aberrator, and temporal aberrator stability determines the length of time for which the aberration profile can be used. In this study, the spatial and temporal stability of clinically measured aberrations is reported for breast, liver, and thyroid tissue. Cross correlations between aberration estimates revealed aberrators to have azimuthal isoplanatic patch sizes of 0.44, 0.28, and 0.20 mm for breast, liver, and thyroid tissue, respectively, at 80% correlation. Axial isoplanatic patch sizes were 1.26, 0.76, and 1.80 mm for the same tissue, respectively, at 80% correlation. Temporal stability at 80% correlation was determined to be greater than 1.5 seconds for breast and thyroid tissue, and 0.65 seconds for the liver. The effects of noise, motion, and target nonuniformity on aberrator stability are characterized by simulations and experiments in tissue mimicking phantoms.

Comparisons of lesion detectability in ultrasound images acquired using time-shift compensation and spatial compounding

The effects of aberration, time-shift compensation, and spatial compounding on the discrimination of positive-contrast lesions in ultrasound b-scan images are investigated using a two-dimensional (2-D) array system and tissue-mimicking phantoms. Images were acquired within an 8.8 x 12-mm2 field of view centered on one of four statistically similar 4-mm diameter spherical lesions. Each lesion was imaged in four planes offset by successive 45 degree rotations about the central scan line. Images of the lesions were acquired using conventional geometric focusing through a water path, geometric focusing through a 35-mm thick distributed aberration phantom, and time-shift compensated transmit and receive focusing through the aberration phantom. The views of each lesion were averaged to form sets of water path, aberrated, and time-shift compensated 4:1 compound images and 16:1 compound images. The contrast ratio and detectability index of each image were computed to assess lesion differentiation. In the presence of aberration representative of breast or abdominal wall tissue, time-shift compensation provided statistically significant improvements of contrast ratio but did not consistently affect the detectability index, and spatial compounding significantly increased the detectability index but did not alter the contrast ratio. Time-shift compensation and spatial compounding thus provide complementary benefits to lesion detection.

Sound speed estimation using automatic ultrasound image registration

A mismatch between the sound speed assumed for beamforming and scan conversion and the true sound speed in the tissue to be imaged can lead to significant defocusing and some geometric distortions in ultrasound images. A method is presented for estimating the average sound speed based on detection of these distortions using automatic registration of overlapping, electronically steered images. An acrylamide gel phantom containing vaporized dodecafluoropentane droplets as point targets was constructed to evaluate the technique. Good agreement (rms deviation <0.4%) was found between the sound speeds measured in the phantom using a reference pulse-echo technique and the image-based sound speed estimates. A significant improvement in accuracy (rms deviation <0.1%) was achieved by including the simulated sound field of the probe rather than assuming straight acoustic beams and propagation according to ray acoustics.

Spectral estimation for characterization of acoustic aberration

Spectral estimation based on acoustic backscatter from a motionless stochastic medium is described for characterization of aberration in ultrasonic imaging. The underlying assumptions for the estimation are: The correlation length of the medium is short compared to the length of the transmitted acoustic pulse, an isoplanatic region of sufficient size exists around the focal point, and the backscatter can be modeled as an ergodic stochastic process. The motivation for this work is ultrasonic imaging with aberration correction. Measurements were performed using a two-dimensional array system with 80 x 80 transducer elements and an element pitch of 0.6 mm. The f number for the measurements was 1.2 and the center frequency was 3.0 MHz with a 53% bandwidth. Relative phase of aberration was extracted from estimated cross spectra using a robust least-mean-square-error method based on an orthogonal expansion of the phase differences of neighboring wave forms as a function of frequency. Estimates of cross-spectrum phase from measurements of random scattering through a tissue-mimicking aberrator have confidence bands approximately +/- 5 degrees wide. Both phase and magnitude are in good agreement with a reference characterization obtained from a point scatterer.

Eigenfunction analysis of stochastic backscatter for characterization of acoustic aberration in medical ultrasound imaging

Presented here is a characterization of aberration in medical ultrasound imaging. The characterization is optimal in the sense of maximizing the expected energy in a modified beamformer output of the received acoustic backscatter. Aberration correction based on this characterization takes the form of an aberration correction filter. The situation considered is frequently found in applications when imaging organs through a body wall: aberration is introduced in a layer close to the transducer, and acoustic backscatter from a scattering region behind the body wall is measured at the transducer surface. The scattering region consists of scatterers randomly distributed with very short correlation length compared to the acoustic wavelength of the transmit pulse. The scatterer distribution is therefore assumed to be delta correlated. This paper shows how maximizing the expected energy in a modified beamformer output signal naturally leads to eigenfunctions of a Fredholm integral operator, where the associated kernel function is a spatial correlation function of the received stochastic signal. Aberration characterization and aberration correction are presented for simulated data constructed to mimic aberration introduced by the abdominal wall. The results compare well with what is obtainable using data from a simulated point source.

Errors in ultrasonic scatterer size estimates due to phase and amplitude aberration

Current ultrasonic scatterer size estimation methods assume that acoustic propagation is free of distortion due to large-scale variations in medium attenuation and sound speed. However, it has been demonstrated that under certain conditions in medical applications, medium inhomogeneities can cause significant field aberrations that lead to B-mode image artifacts. These same aberrations may be responsible for errors in size estimates and parametric images of scatterer size. This work derives theoretical expressions for the error in backscatter coefficient and size estimates as a function of statistical parameters that quantify phase and amplitude aberration, assuming a Gaussian spatial autocorrelation function. Results exhibit agreement with simulations for the limited region of parameter space considered. For large values of aberration decorrelation lengths relative to aberration standard deviations, phase aberration errors appear to be minimal, while amplitude aberration errors remain significant. Implications of the results for accurate backscatter and size estimation are discussed. In particular, backscatter filters are suggested as a method for error correction. Limitations of the theory are also addressed. The approach, approximations, and assumptions used in the derivation are most appropriate when the aberrating structures are relatively large, and the region containing the inhomogeneities is offset from the insonifying transducer.

Simulation of ultrasonic focus aberration and correction through human tissue

Ultrasonic focusing in two dimensions has been investigated by calculating the propagation of ultrasonic pulses through cross-sectional models of human abdominal wall and breast. Propagation calculations used a full-wave k-space method that accounts for spatial variations in density, sound speed, and frequency-dependent absorption and includes perfectly matched layer absorbing boundary conditions. To obtain a distorted receive wavefront, propagation from a point source through the tissue path was computed. Receive focusing used an angular spectrum method. Transmit focusing was accomplished by propagating a pressure wavefront from a virtual array through the tissue path. As well as uncompensated focusing, focusing that employed time-shift compensation and time-shift compensation after backpropagation was investigated in both transmit and receive and time reversal was investigated for transmit focusing in addition. The results indicate, consistent with measurements, that breast causes greater focus degradation than abdominal wall. The investigated compensation methods corrected the receive focus better than the transmit focus. Time-shift compensation after backpropagation improved the focus from that obtained using time-shift compensation alone but the improvement was less in transmit focusing than in receive focusing. Transmit focusing by time reversal resulted in lower sidelobes but larger mainlobes than the other investigated transmit focus compensation methods.

Modeling of nonlinear ultrasound propagation in tissue from array transducers

A computationally efficient model capable of simulating finite-amplitude ultrasound beam propagation in water and in tissue from phased linear arrays and other transducers of arbitrary quasiplanar geometry is described. It is based on a second-order operator splitting approach [Tavakkoli et al., J. Acoust. Soc. Am. 104, 2061-2072 (1998)], with a fractional step-marching scheme, whereby the effects of diffraction, attenuation, and nonlinearity can be computed independently over incremental steps. This approach is an extension to that of Christopher and Parker [J. Acoust. Soc. Am. 90, 507-521; 90, 488-499 (1991)], wherein linear and nonlinear effects are propagated separately over incremental steps, and the computation of the diffractive substeps are based on an angular spectrum technique with a modified sampling scheme for accurate and efficient implementation of diffractive propagation from nonradially symmetric sources. Results of the model are compared with published data. Predicted field profiles for nonlinear propagation in tissue from realistic array transducers using the pulse inversion method are presented.

Spatial coherence analysis applied to aberration correction using a two-dimensional array system

Complex degree of coherence functions are computed using synthetic and measured ultrasound data to demonstrate noteworthy aspects of coherence analysis in the context of aberration correction. Coherence functions calculated from synthetic data illustrate the importance of proper normalization of the constituent cross-correlation integrals when weak elements and receiver directivity are significant factors. The synthetic data also show that a spike can occur at the zero-lag position of the coherence function when the signal-to-noise ratio is reduced by element directivity near the edges of a large aperture. The latter observation is confirmed by experimental data acquired through tissue-mimicking distributed aberration phantoms using a low f-number two-dimensional array system. The coherence of data acquired at neighboring elements is not changed by time-shift compensation of transmit and receive focusing, but time-shift compensation does improve the coherence of echoes measured over larger separations. The resulting increase in coherence widths evaluated at levels between 0.2 and 0.5 is correlated with narrower -10 dB and -20 dB effective widths in focuses visualized using single-transmit images. Iterative focus compensation methods may benefit from aberration estimation algorithms that take advantage of these longer-range correlations in random-scattering waveforms.

Analysis and correction of ultrasonic wavefront distortion based on a multilayer phase-screen model

A model is introduced that incorporates the cumulative wavefront distortion effects caused by spatial heterogeneities along the path of propagation, and a corresponding model-based wavefront distortion-correction method is presented. In the proposed model, a distributed heterogeneous medium is lumped into a series of parallel phase screens. The distortion effects can be compensated--without a priori knowledge of the distorting structure--by backpropagation of received wavefronts through hypothetical multiple phase screens located between the imaging system and targets, while each pointwise time shift is adjusted iteratively to maximize a specified image quality factor at the final layer. Theoretical analyses indicate that the mean speckle brightness decreases monotonically with the root-mean-square value of distributed phase distortions; therefore, the speckle brightness can be used as an image quality factor. Experimental one-dimensional (1-D) array data with simulated distortion effects based on a real 2-D abdominal-tissue map were used to evaluate the performance of the proposed method and existing aberration-correction techniques. The simulated characteristics of wavefront distortion and relative performance of existing correction techniques were similar to reports based on abdominal-wall data and breast data. This investigation shows that the proposed method provides better compensation for wavefront distortion.

Element size effect on phase aberration correction

The distortion effect of tissue on ultrasonic beamforming is simplified as a 2-D time-shifting screen placed against the array surface. The ideal correction of such distortion requires a 2-D array with infinitesimally small elements. However, elements of finite sizes must be utilized in practice, causing the so-called residual phase error (RPE). As element size is reduced, the magnitude of the RPE is reduced, and its spatial feature becomes finer. Analyses have been performed to reveal that the magnitude of the RPE is proportional to the imaging frequency, the rms magnitude of the original time-delay error, and the diagonal size of individual rectangular elements, and is inversely proportional to the correlation length of the original time-delay error. Simulations have been performed to study the peak sidelobe level caused by the RPE as the element sizes are reduced. The sidelobe is defined here as the difference between the ideal beam (with no phase error) and the beam obtained in the presence of the RPE. For a multi-row array in which a conventional 1-D array is divided into N rows of independent elements in the elevation direction, the peak sidelobe level is found to vary approximately as N(-1) instead of the anticipated N(-2). The reduction is caused by the reduced magnitude of the RPE, and the finer spatial feature of the RPE, although apparent in the reduced spatial correlation length, does not result in additional reduction of the sidelobe level. The reason for this has been analyzed. The results of this study provide guidance for designing multi-row arrays suitable for phase aberration correction.

Time-shift estimation and focusing through distributed aberration using multirow arrays

The effects of element height on time-shift estimation and transmit focus compensation are demonstrated experimentally. Multirow ultrasonic transducer arrays were emulated by combining adjacent elements of a 3.0-MHz, 0.6-mm pitch, two-dimensional array to define larger virtual elements. Pulse-echo data were acquired through tissue-mimicking distributed aberrators, and time-shift maps estimated from those data were used for transmit focus compensation. Compensated beams formed by arrays with fine row pitches were similar, but focus restoration was significantly less effective for "1.75-D" arrays with a coarse row pitch. For example, when focus compensation was derived from strongly aberrated random scattering data [70-ns nominal rms arrival time fluctuation with 7 mm FWHM (full-width at half-maximum) correlation length], the mean -20 dB lateral beamwidths were 5.2 mm for f/2.0 arrays with 0.6- and 1.8-mm row pitches and 9.5 mm for an f/2.0 array with 5.4-mm pitch. Time-shift maps estimated from random scattering data acquired with 5.4-mm pitch arrays included large discontinuities caused by low correlation of signals received on vertically and diagonally adjacent emulated elements. The results indicate that multirow arrays designed for use with aberration correction should have element dimensions much less than 75% of the correlation length of the aberration and perhaps as small as 25 to 30% of the correlation length.

Phase aberration effects in elastography

In sonography, phase aberration plays a role in the corruption of sonograms. Phase aberration does not have a significant impact on elastography, if statistically similar phase errors are present in both the pre- and postcompression signals. However, if the phase errors are present in only one of the pre- or postcompression signal pairs, the precision of the strain estimation process will be reduced. In some cases, increased phase errors may occur only in the postcompression signal due to changes in the tissue structure with the applied compression. Phase-aberration effects increase with applied strain and may be viewed as an image quality derating factor, much like frequency-dependent attenuation or undesired lateral tissue motion. In this paper, we present a theoretical and simulation study of the effects of phase aberration on the elastographic strain-estimation process, using the strain filter approach.

Small element array algorithm for correcting phase aberration using near-field signal redundancy. I. Principles

A near-field, signal-redundancy algorithm for measuring phase-aberration profiles has been proposed previously. It is designed for arrays with a relatively large element size for which relatively narrow beams are transmitted and received. The algorithm measures the aberration profile by cross-correlating signals collected with the same midpoint position between transmitter and receiver, termed common midpoint signals, after a dynamic near-field delay correction. In this paper, a near-field signal-redundancy algorithm for small element arrays is proposed. In this algorithm, subarrays are formed of adjacent groups of elements to narrow the beams used to collect common midpoint signals and steer the beam direction, so that angle-dependent, phase-aberration profiles can be measured. There are several methods that could be used to implement the dynamic near-field delay correction on common midpoint signals collected with subarrays. In this paper, the similarity between common midpoint signals collected with these methods is also analyzed and compared using a so-called corresponding-signal concept. This analysis should be valid for general target distributions in the near field and wide-band signals.

Small element array algorithm for correcting phase aberrations using near-field signal redundancy. Part II: experimental results

For part I see ibid., vol.47, p.29 (2000). A small element array algorithm for phase-aberration correction using near-field signal redundancy was proposed in part I. Using this algorithm, subarrays are formed to narrow the transmitted and received beams when collecting common midpoint signals, so that angle-dependent aberration profiles across the array can be measured. In this paper, this algorithm is tested on data collected from a phantom with a non-isoplanatic aberrator attached to the front surface of a phased array. The aberrator is made from cast room temperature vulcanizing (RTV) silicone rubber, which has a sound velocity of about 1.02 mm.micros(-1). Results show that the subarray technique can be used to measure and correct angle-dependent, phase-aberration profiles. The theoretical results regarding the performance of several implementation methods for dynamic near-field delay correction on subarrays are also experimentally tested using data from a phantom.