Correction of ultrasonic wavefront distortion using backpropagation and a reference waveform method for time-shift compensation

Correction of ultrasonic wavefront distortion using backpropagation and a reference waveform method for time-shift compensation

Dong-Lai Liu Department of Electrical Engineering, University of Rochester, Rochester, New York 14627

Robert C. Waag Departments of Electrical Engineering and Radiology, University of Rochester, Rochester, New York 14627

(Received 24 September 1993; accepted for publication 30 April 1994)

A model is introduced to describe ultrasonic pulse amplitude and shape distortion as well as arrival time fluctuation produced by propagation through specimens of human abdominal wall. In the model, amplitude and shape distortion develops as the wavefront propagates in a uniform medium after passing through a phase screen that only causes time shifts. This distortion is compensated by a backpropagation of the wavefront using the angular spectrum method. The compensation employed waveforms emitted by a pointlike source and measured after propagation through the tissue. The waveforms were first corrected for geometric path and then were backpropagated over a sequence of increasing distances. At each distance, a waveform similarity factor was calculated to find the backpropagation distance at which the waveforms were most similar. A new method was devised to estimate pulse arrival time for geometric correction as well as to perform time-shift compensation. The method adaptively derives a reference waveform that is then cross correlated with all the waveforms in the aperture to obtain a surface of arrival times. The surface was smoothed iteratively to remove outlying points due to waveform distortion. The mean (+s.d.) of the waveform similarity factor for 14 specimens was found to be 0.938 (+0.025) initially. After backpropagation of waveforms to the distance of maximum waveform similarity for each specimen, the waveform similarity factor improved to 0.967 (+0.015). The corresponding energy level fluctuation in the wavefront was 4.2 (_+0.4) dB initially and became 3.3 (+0.3) dB after backpropagation. For wavefronts focused at 180 mm, the -30 dB mean (_+s.d.) effective radius of the focus was 4.2 (_+1.2) mm with time-shift compensation in the aperture and became 2.5 (_+0.5) mm with backpropagation followed by time-shift compensation. These results indicate that a phase screen placed some distance away from the aperture is an improved model for the description of wavefront distortion produced by human abdominal wall and that wavefront backpropagation followed by time-shift estimation and compensation is an effective method to compensate for such distortion.

PACS numbers: 43.80.Vj, 43.80.Qf, 43.80.Ev

INTRODUCTION

Progress in ultrasonic transducer fabrication and control electronics has resulted in diagnostic imaging instruments with lateral resolution near the theoretical limit set by aperture size in experimental settings when the propagation medium is ideal. Development is toward using bigger linear apertures • or two-dimensional apertures 2 to obtain higher spatial resolution. In this development, consideration of wavefront distortion is necessary because investigations 3'4 have shown that beams are degraded by propagation through inhomogeneous tissue paths.

Interest in correction of ultrasonic beam degradation has led to a variety of studies. Early studies 5'6 considered cross correlation of signals in the receiving aperture to estimate arrival time. In another early study, 7 brightness was maximized by adjustment of focusing delays. A recent study 8 that treats an annular geometry and compares the cross- correlation technique with the maximum brightness method has found the maximum brightness method to be better. All of these studies have restricted their means of compensation to time shifts in the receiving aperture.

Time-shift compensation in the receiving aperture is, however, not a complete solution of the compensation problem. An investigation 9 of focusing through abdominal wall has shown that, although time-shift compensation in the receiving aperture can improve the focus between the -5 and -20 dB levels, the compensated focus below the -20 to -25 dB level still differs appreciably from the ideal focus. In addition, measurements •ø'• of wavefront distortion produced by abdominal wall show waveform amplitude variations and shape changes neither of which are affected by time-shift compensation in the measurement aperture. The inability of time-shift compensation in the receiving aperture to correct for the effect of an aberration that is not near the receiving aperture has also been shown experimentally in an investigation •2 of focusing by a time-reversal method. Al- though this method provides a way to eliminate distortion caused by transmission through a distributed, lossless inhomogeneous medium when a point target (such as may exist in lithotripsy) is available, the approach as described cannot be used in a general imaging application because a suitable point target is not often present, •2 the image is available only

649 J. Acoust. Soc. Am. 96 (2), Pt. 1, August 1994 0001-4966/94/96(2)/649/12/$6.00 ¸ 1994 Acoustical Society of America 649

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in the medium itself, •3 and means for compensation during reception are not included. TM In summary, present time-shift techniques for the compensation of ultrasonic wavefront distortion must be extended to achieve diffraction-limited reso-

lution through inhomogeneous tissue paths. The idea of time-shift compensation is based on a model

that lumps the aberrations caused by the nonuniform medium into a thin layer of material of varying propagation delay placed immediately in front of the receiving aperture. Such a layer is commonly known as a phase screen. Phase screen models have been studied in optics and radio science. •5-•7

Although a phase screen causes only aberration in propagation delay, amplitude and waveform distortion can develop as the aberrated wavefront continues to propagate. This observation has led to the use of an appropriately positioned random phase screen to model radio wave propagation through an extended random medium. 18 The utility of such an approach in radio science has stimulated the investigation reported here.

In this investigation, wavefront distortion produced by propagation through the abdominal wall is modeled using a phase screen placed some distance away from the receiving aperture. Then, backpropagation of the received signal is employed to reduce the amplitude and waveform distortion that develops in the model during propagation from the phase screen to the receiving aperture. Time shifts are estimated at the position of the phase screen and these estimates are used to compensate the backpropagated waveform. To estimate the time shifts, an adaptive reference waveform method is introduced.

The rest of this paper, which extends a previous study 9 of ultrasonic focussing through a tissue path, is comprised of six sections. In Sec. I, a measure of waveform similarity is defined and a procedure based on the angular spectrum method is described for the backpropagation of a wavefront in a uniform medium. Next, in Sec. II, an adaptive reference waveform method for delay estimation is described. Then, in Sec. Ill, the computational procedure is explained. Results obtained from backpropagation and focusing of both model and measured waveform data are presented in Sec. IV. The results are discussed in Sec. V. Finally, in Sec. VI, conclu- sions are drawn.

I. WAVEFORM DISTORTION AND WAVEFRONT BACKPROPAGATION

The physical arrangement of source, aberration, and receiving aperture considered in this paper is diagrammed in Fig. 1 along with the model used to represent the physical configuration. The source emits a spherical wave pulse that is received in a two-dimensional aperture after propagation through a specimen of human abdominal wall. In the model, the phase screen only time shifts the wavefront. Changes in waveform amplitude and shape are produced by propagation of the time-shifted wavefront to the receiving aperture. These changes in amplitude and shape may be removed by backpropagation. The time shifts can then be estimated and removed. The distance of backpropagation is defined as the range at which the backpropagated waveforms are most alike in the sense that the waveforms are only time shifted with a

.) Source

Tissue

I Model

Source

Phase Screen

Aperture

Aperture

FIG. 1. Measurement configuration and equivalent phase screen model. In the measurement, a source emits a spherical wave pulse that is received in a two-dimensional aperture after passing through a specimen of human abdominal wall. The distortion effects are modeled by a phase screen placed some distance away from the aperture.

possible amplitude factor. This motivates the definition of a waveform similarity factor that can be computed during the backpropagation process to measure waveform similarity and to find the appropriate distance at which the waveforms are most alike.

A. Waveform similarity and energy level fluctuation

The measure of the waveform similarity needed to monitor the effect of backpropagation on waveform shape should ideally be unaffected by arbitrary time delays or amplitude factors. Thus the similarity of N waveforms s•(t), s2(t),..., s•v(t) is defined in the present study to be maximum if

si(t)=ais(t--ri), i= 1,2,...,N. (1)

When Eq. (1) is true, si(t+ ri)=ais(t), so that N N

• Si(t+ •'i)-- • ais(t). (2) i=1 i=1

Squaring and integrating Eq. (2) over time yields

•+•( N )2 (•)2•+• 2(t)dt. (3) • Si(t+ ri) dt- a i s -oc i= 1 i=1

From Eq. (1),

ai •= s•(t)dt s2(t)dt. (4)

Substituting Eq. (4) into Eq. (3) gives

• Si(t + ri) dt = si2(t)dt . -o• .= i=1

(5)

650 J. Acoust. Soc. Am., Vol. 96, No. 2, Pt. 1, August 1994 D.-L. Liu and R. C. Waag: Correction for wavefront distortion 650


This leads to the definition

/ r= si(t+Ti) dt .__

• si2(t)dt i=1

(6)

as the waveform similarity factor. The square of this definition may be viewed as a generalized cross-correlation coefficient of N waveforms.

If Eq. (1) is true and the ri's have been selected appropriately, then r-1. The value of r is readily shown to be always between 0 and 1. A nearly alike measure of waveform similarity has been independently proposed •9 as a focus criterion to evaluate the performance of phase aberration correction techniques.

Since the similarity measure in Eq. (6) is immune to amplitude changes or energy fluctuations [i.e., variations in a i when Eq. (1) is true], another measure is needed to describe energy variations, in this case, from a two-dimensional surface that is obtained by integrating over time the square of pulse amplitudes. The energy variation measure used here is the rms value (in dB) of the difference between the logarithm of the energy surface and the fifteen-term, fourth-order polynomial that is a least-mean-square error fit to the surface. In this measure, the difference of the logarithms corresponds to a division that compensates for spherical spreading as well as low spatial-frequency fluctuations such as those caused by gradual changes in thickness across the specimen. A division rather than a subtraction is used to eliminate the influence of

changes in amplitude scale factor. The calculation employs a polynomial fit rather than a spherical fit because the polynomial fit removes low spatial-frequency variations that are not of interest here. The conversion of energy values into decibel units eliminated negative fitted values that very low energy occasionally produced with data on a linear scale. As already noted elsewhere, • no significant error from uncertainty in the energy data is introduced in this process because the ratio of signal power to noise power in the energy data averaged more than 40 dB and the dynamic range of the logarithmic data was restricted to 20 dB below the maximum value to

minimize the effects of extremely low values on the ensuing least-mean-square error fit.

B. Wavefront backpropagation

The theory of decomposing a single temporal-frequency wavefront into a summation or angular spectrum ,of plane waves and propagating them over a given distance in a uni-

20 21 form medium can be found in textbooks ' so only the basic relations are noted here. The appropriate temporal harmonics are obtained for the pulse waveforms in this study by a temporal Fourier transform.

The angular spectrum A(fx,fy;O ) of a particular temporal-frequency component at the plane z=0 is defined as the two-dimensional Fourier transform of the complex wave field U(x,y;O) in that plane and can be written

A(fx,fy;O) = f • U(x,y;O)e-J2•(Xœx+Yœy ) dx dy, (7) where fx and fy•are spatial frequencies and have the dimension of length-. After propagation over a distance z, the angular spectrum becomes

A (fx ,fy ;0 )e j(2rr/x)z x/1 -(kfx)2-(kfY )2, A(fx,fy;z)= if fx2+ fy2<l/)• 2 (8)

0, otherwise,

where )• is the wavelength corresponding to the temporal frequency and evanescent waves have been neglected. The complex field U(x,y;z) in the plane at z is then obtained by an inverse two-dimensional Fourier transform that may be written

U(x,y;z)= f f A(fx,fy;z)eJ2•(Xœx+Yfy ) dfx dfy. (9) After each harmonic component has been backpropa-

gated (by using a negative propagation distance), an inverse temporal Fourier transform is used to synthesize the time- domain signals.

II. REFERENCE WAVEFORM METHOD FOR ARRIVAL TIME ESTIMATION

The calculation of the waveform similarity factor defined by Eq. (6) and the process of time-shift compensation require a method to find the arrival time of the waveforms. In Ref. 9, a least-mean-square error approach that uses arrival time differences of neighboring waveforms was described. Disadvantages of the approach are that a large number of equations must be solved to obtain arrival time throughout the aperture and that arrival time differences must be estimated to maintain the structure of the equations when cross correlation of waveforms at neighboring points yields unsatisfactory arrival time differences. The adaptive reference waveform method described here avoids the need to solve a

large number of equations and provides an alternative way to estimate arrival time when cross correlation of waveforms is

unsatisfactory. The reference waveform method is comprised of three

steps. In the first step, an initial reference pulse is either selected from the measured data or defined in some other

way. The reference pulse is modified continually to reduce the effect of the arbitrariness in the selection of the initial

reference pulse as waveforms in the aperture are cross correlated with it. The modification is accomplished by time shifting and adding a new waveform to the reference pulse if the peak value p of the cross-correlation function between the new waveform and the current reference pulse is greater than some value. The value of p for comparison may be chosen to depend on the number n of pulses already incorporated. In the present study, the test value of p was varied according to 0.8-O.le -n/•ø. With this definition, the test value increases from 0.7 to 0.8 as n increases, making modification of the reference pulse easier at first and more difficult later. The second step is to cross correlate the final reference pulse again with the original waveforms and then to



estimate the peak position of each cross-correlation function by spline interpolation. The third step is to smooth the obtained peak positions because some of the waveforms may not be highly correlated with the reference pulse and the corresponding position of the peak may be erroneously far away from neighboring arrival times.

Two methods have been developed to smooth outlying arrival times. The methods are applied according to the type of wavefront being processed. The first type of wavefront contains a geometric delay component and has a curvature large compared to the fluctuations caused by tissue. The second type of wavefront, for which the arrival times are roughly equal and contain only a small fluctuation, is obtained from geometric correction of the first type of wavefront. Such wavefronts may be backpropagated but remain essentially flat over distances of interest here.

For the first type of wavefront, the purpose of processing is to estimate and remove the geometric delay component. Smoothing in this case is performed to reduce the effect of erroneously large delays on a fitted surface that is interpreted as the geometric delay. The procedure is the following. A 15-term, fourth-order polynomial is fitted to the arrival time surface. The rms error of fit is calculated and denoted by 0-o. Next, those points with an absolute fit error greater than 30- o are removed. A second fitting is performed to the remaining points. Similarly, 0-• is calculated and the points with an absolute fit error greater than 30-• are removed. This process goes on until no new points are removed in a round of fitting and removing. The final fit obtained in this way is interpreted as the geometric delay. By shifting the waveforms according to the fitted arrival times, a geometrically corrected wavefront is obtained.

For the second type of wavefront, the purpose of processing is to estimate arrival times that can be used for time- shift compensation so waveforms are accurately aligned after compensation. The smoothing in this case is performed to replace arrival times that differ from their neighbors by a significant amount compared to the period of the pulse since the variation in arrival time between neighboring p9ints is known to be small. The procedure is the following. A local average is first calculated for each point using its 24 neighboring points in a 5 x 5 region. (For points on the borders and at the corners, the number of neighboring points is smaller.) The delay at each point is next compared to the corresponding local average. A point is not included in the later local average calculation if the difference is greater than 0.5 T o , where T o is the period corresponding to the peak spectral amplitude in the reference waveform. New local averages are calculated again for each point. If fewer than three neighbors are available for the calculation of the local average, a value determined from linear interpolation of the neighboring local averages is used as the local average. Comparison of the arrival time with the newly obtained local average is made to identify new points that need to be removed. The process goes on until no new points are removed.

A feasible substitute arrival time is found for the points with outlying arrival times in the second type of wavefront. Although the value of the local average could be used, the local average position does not depend directly on the wave-

form itself. Therefore, satisfactory alignment of the waveform is not likely. In order to connect the substitute arrival time directly with the waveform, the corresponding cross- correlation function of the waveform and the reference wave-

form is utilized once more, and the nearest peak position is found starting from the local average position. If the normal- ized correlation coefficient at that local peak is greater than 0.5, the local peak position is used as the substitute. Other- wise, the local average is used. This approach limits the possibility that an erroneous peak position of a cross-correlation function will be used. In other words, if the peak value of the cross correlation is unacceptably low, smoothness of the wavefront becomes the guideline.

III. COMPUTATIONAL PROCEDURE

A. Wavefront backpropagation and description of wavefront characteristics

The computational procedure in this study was designed for pulse waveforms measured in a two-dimensional aperture after emission from a pointlike source and propagation through a tissue path as shown in Fig. 1. The measurement was made by mechanically translating a one-dimensional, 128-element array over 32 elevations. The pitch of the array elements was 0.72 mm. Their effective dimension in the el-

evation direction was 1.44 mm, the same as the elevation step size. The size of the aperture was thus 92.16x46.08 mm 2. The source-aperture distance was 180 mm. The waveforms had a nominal center frequency of 3.75 MHz and a -6-dB bandwidth of about 2.2 MHz and were sampled over an 11.8/as interval at a 20-MHz rate that yielded 236 values. Details of the measurement process along with descriptions of the wavefront distortion are in Ref. 11.

The wavefront in the receiving aperture may be backpropagated directly. However, this conceptually straightforward approach was not used because accounting for wave spreading or converging complicates the processing. Instead, the waveforms in the receiving aperture were first corrected for geometric delay before backpropagation. Correction for geometric delay effectively reduces wave convergence during backpropagation and simplifies the processing.

The arrival time estimation procedure described in Sec. II was applied to remove geometric delay in preparation for backpropagation. After the removal of geometric delay, the pulses exist only in a small time interval so that a shorter signal length of 40 sampling points (instead of 236 points) could be employed. The signals were then zero-padded to 64 points before temporal Fourier transform%i_mn•-and backpropagation. The backpropagation process used harmonics 1 through 32. (Harmonic number 0, which is the dc component, cannot propagate.) Array sizes used in the calculation were 256x 64 to avoid cyclic convolution artifacts.

The geometrically corrected wavefront was backpropagated with the propagation distance z equal to -30, -10, 20, 30, 40, 50, 60 mm. (A negative z in this case is a forward propagation.) A sound speed of 1.5 mm//zs was assumed throughout this study. At each distance and in the receiving aperture (z-0), the arrival time was estimated using the reference waveform method as described in Sec. II and the



Original Peak Pos. Peak Pos. Geom. WF after Wavefront (Unsm'd) (Smoothed) Delay Geom. Corr.

WF after

Backprop. Peak Pos. Peak Pos. WF after Focussed (Unsm'd) (Smoothed) TSC Waveform

FIG. 2. Flow of processing. The processing sequence employed to focus wavefronts is shown with the inclusion of a backpropagation step before time-shift compensation.

arrival time estimates were used for time-shift compensation at the corresponding distance. The wavefront characteristics were described by the number of pulses incorporated in the formation of the reference waveform, the waveform similarity factor, and the energy level fluctuation to show the effects of backpropagation. The distance at which the waveform similarity factor peaked was used as the position of the phase screen in our model. (See Fig. 1.)

B. Focusing method and description of focus characteristics

The time-shift compensated wavefront at each distance was focused at 180 mm using the Fourier transform method described in Ref. 9. This consisted of finding the complex amplitudes of the temporal Fourier harmonics across the aperture, calculating the Fraunhofer diffraction pattern at each harmonic, and summing the patterns. Details of the process and definitions of the parameters that describe the focus are in Ref. 9.

The focus obtained with waveforms at each backpropagation distance after time-shift compensation was described by three parameters to show the effects of backpropagation on the focus. These are the effective radius, the peripheral energy ratio, and the level at which the effective radius differs by 10% from that of the focus produced by ideal waveforms. The ideal waveforms in this case were obtained by replicating throughout the aperture a pulse that was the result of averaging the time-shift compensated wavefront as was done in Ref. 9.

The flow of processing is shown in Fig. 2 for the case that includes backpropagation before time-shift estimation and compensation. The calculation of the focus obtained with uncompensated data omitted backpropagation as well as time-shift estimation and compensation. The calculation of the focus obtained with time-shift estimation and compensation in the receiving aperture omitted only backpropagation.

C. Generation of model waveform data

Model waveforms were generated to test the computational procedure. These waveforms were obtained using the phase screen model depicted in Fig. 1. A spherical wave

pulse was first modified by passage through a phase screen and then propagated to the receiving aperture. The geometry is shown in Fig. 3.

A spherical wavefront with a virtual origin d= 300 mm from the phase screen was generated to occupy the central 160x 40 points of the 256 x 64 array. (See Fig. 3.) The selection of d, which is large compared to the corresponding distance of 180 mm in the measurements, was made to reduce

the effect of undersampling in space. The spherical wavefront, which included both geometric delay and spherical spreading, was then passed through the random phase screen. The time shifts used to define the phase screen were a real- ization of a zero-mean Gaussian process with an rms value of 50 ns and a radially symmetric correlation function that was also Gaussian with a half-maximum full width of 3 mm. The

wavefront was next forward propagated over a distance of z = 30 mm and the central 128x32 waveforms were consid-

ered to be the received data. This selection of parameters avoided significant edge diffraction.

The pulses in the wavefront had the time dependence of

e -It-tl')-'/2'• sin[2rrf•.(t-to)], with f, = 3.75 MHz and rrt=0.15 /xs. The -6-dB bandwidth of this pulse is 2.50 MHz. The simulated waveforms were sampled at 20 MHz and were each 236 points long, the same as the experimental data. During forward propagation, the waveforms were zero padded to 256 points and Fourier transformed. Forward propagation was performed for harmonic numbers I through 128. Time-domain results in the receiving aperture were synthesized using an inverse Fourier transform.

IV. RESULTS

A. Model waveform data

Model waveforms at representative elevations after geometric correction at four stages in the processing are shown in Fig. 4. Amplitude and shape distortion that has resulted from propagation of the wavefront from the phase screen to the receiving aperture is apparent in the receiving aperture both before and after time-shift compensation. After backpropagation to the position of the phase screen, the distortion is practically not visible in either uncompensated or time- shift compensated data. Backpropagation followed by time-

256

d•:300mm • z--$Omm •-- IA4mm

.40X•

•--- (• 8 0.72mm Sam.,,.. Increments

Phase Screen

*mm/

FIG. 3. Simulation geometry and parameters.



FIG. 4. Model wavefronts after correction for geometric path delay at four stages in the processing. Each panel shows temporal waveforms as a shade of gray on a linear scale in a plane of constant elevation. In each panel, the horizontal coordinate is time and spans 2.0 •s while the vertical coordinate is in the array direction and spans 92.16 min. The (zero-origin) number below the panels identifies the elevation being displayed from the available total of 32 that sequentially span in numerical order an interval of 46.08 mm at increments of 1.44 min. First row: Before time-shift compensation in the receiving aperture. Second row: After backpropagation to the position of maximum waveform similarity, i.e., the position of the phase screen. Third row: After time-shift estimation and compensation in the receiving aperture. Fourth row: After backpropagation followed by time-shift estimation and compensation at the position of the phase screen.

is reduced from 4.11 to 2.04 dB as a result of backpropagation. Also in Fig. 5, the -10-dB peripheral energy ratio of the focus of the time-shift compensated wavefront decreases from 0.708 before backpropagation to 0.357 after backpropagation of 30 mm while the -30-dB effective radius of the focus decreases from 3.35 to 1.46 mm. Simultaneously, the 10% deviation level drops from -27.0 dB for time-shift compensation in the receiving aperture, to -41.2 dB for backpropagation followed by time-shift compensation.

B. Measured waveform data

Representative measured waveforms that correspond to the model waveforms at the same four stages in the processing are shown in Fig. 6. Amplitude and shape distortion are evident both before and after time-shift compensation in the receiving aperture. This distortion is visibly reduced after backpropagation of the wavefront over 40 mm, the distance at which the waveform similarity factor is maximum. Al- though amplitude and shape distortion in the measured data after backpropagation is greater than in the model waveform data, backpropagation has reduced amplitude and shape distortion and backpropagation followed by time-shift estimation and compensation has produced waveforms more uniform in amplitude and shape than those obtained with time- shift compensation in the receiving aperture.

The effects of backpropagation on the same representative measured data wavefront and on the corresponding fo-

(x10"2) Incorp. Pulses -10 dB PER 0.8

40 0.7

36 0.6 0.5

32 0.4 ,

28 0.3 3o to to ,o -,o

shift estimation and compensation at the phase screen has restored the uniformity in the original model wavefront data before passage through the phase screen.

The effects of backpropagation and time-shift compensation on the characteristics of the model data wavefront and

the corresponding focus are shown in Fig. 5 for a range of backpropagation distances. The waveform similarity factor and the number of pulses incorporated in the reference waveform are maximum at the position of the phase screen while the energy level fluctuation is simultaneously a minimum. The -10-dB peripheral energy ratio, the -30-dB effective radius, and the 10% deviation level are also minimized for the focus obtained with wavefronts that have been back-

propagated to the position of the phase screen and then time- shift compensated.

In Fig. 5, the number of pulses incorporated in the reference waveform rises from 3350 at the receiving aperture to 4020 (out of a total of 4096 pulses) after a backpropagation of 30 mm. The number of smoothed points in the calculation of the arrival time surface simultaneously drops from 184 to eight. At the same time, the waveform similarity factor increases from 0.957 to 0.992 and the energy level fluctuation

½x10"-3) I•SF

lOOO [ 980

94o

920 -30 -10 10 30 50

-30 dB Radius

4.0

3.0

2.0

1.0 ' -30 -iO 10 30 50

• of' ELF (dB) 4..5

4..0

3.5

3.0

2.5

2.0 -30 -10 10 30 50

10• !)ev. Lev. (dB)

-26

-30

-34

-38

-42 ' -30 -10 10 30 50

FIG. 5. Wavefront characteristics (left) and focus characteristics (right) as a function of backpropagation distance (in ram) for model waveform data. WSF=waveform similarity factor. ELF=energy level fluctuation. PER =peripheral energy ratio.

654 d. Acoust. Soc. Am., Vol. 96, No. 2, Pt. 1, August 1994 D.-L. Liu and R. C. Waag: Correction for wavefront distortion 654


FIG. 6. Wavefronts of representative measured waveform data (run number r56) after correction for geometric path delay at four stages in the processing. The format and scales are the same as in Fig. 4.

Focal plane time histories at representative instants of time are shown in Fig. 8 for uncompensated waveforms, waveforms that have been time-shift compensated in the aperture, waveforms that have been backpropagated and then time-shift compensated, and ideal data. The main effect in the ideal data is the limitation imposed by a finite aperture and the pulse bandwidth. The uncompensated tissue path data show high levels and large fluctuations in the spatial and temporal distributions of peripheral signals relative to those in the ideal data. Time-shift estimation and compensation in the receiving aperture reduces the level and the fluctuations of the low signals relative to those of the uncompensated tissue path data. Backpropagation followed by time-shift estimation and compensation further reduces the level and fluctuations of the low peripheral signals and results in a focus much closer to that of the ideal data. Comparing the four time histories shows that the use of time-shift compensation in the receiving aperture improves the focus but that time-shift compensation after backpropagation yields a focus closer to the ideal.

The effective radius curves obtained from uncompensated, time-shift compensated, backpropagated, and time- shift compensated, and ideal data are presented in Fig. 9. These curves show that the uncompensated focus deviates by 10% from the ideal at -7.9 dB while the focus from each of

the two compensations deviates at -24.5 and -28.8 dB, respectively. Backpropagation of the wavefront improves, in

cus are shown in Fig. 7. The number of pulses incorporated in the reference waveform peaks at the position that is defined by maximum of the waveform similarity factor while the energy level fluctuation is about minimum at that position. The -10-dB peripheral energy ratio, the -30-dB effective radius, and the 10% deviation level are all about minimum for the focus obtained with wavefronts that have been

backpropagated to the position of the phase screen and then time-shift compensated. Although the reduction in amplitude and shape variation resulting from backpropagation of measured waveforms is not as great as in the model waveform data, the backpropagated measured wavefront is substantially more uniform and the focus characteristics are nearer those

of the ideal focus as well as substantially better than the characteristics obtained with time-shift compensation in the receiving aperture.

In Fig. 7, the number of incorporated pulses rises from 3141 at the receiving aperture to 3848 (out of a total of 4096 pulses) after backpropagation over 40 mm. The number of smoothed points in the calculation of the arrival time surface drops from 287 to 46. At the same time, the waveform similarity factor increases from 0.939 to 0.977 and the energy level fluctuation is reduced from 4.2 to 2.9 dB. Also in Fig. 7, the -10-dB peripheral energy ratio of the focus decreases from 0.63 to 0.41 after backpropagation of 40 mm while the -30-dB effective radius of the focus decreases from 6.41 to

2.09 mm. Simultaneously, the 10% deviation level drops from -24.5 dB for time-shift compensation in the receiving aperture, to -28.8 dB for backpropagation followed by time- shift compensation.

(xt0"2) Incorp. Pulses -10 dB PER

34

30

26 ' -t0 t0

.70

.60

.50

.40 -30 -10 l0 30 50

(xlO"-3) I•SF

•70

930

-30 -10 10 30 50

-30 dB Radius (era) 7

-30 -10 10 30 50

RHS of ELF (dB)

4.0

3.6

3.2

2.8 -30 -10 10 30 50

log Dev. Lev. (dB)

-22

-24

-26

-2:8

-30 -30 -10 10 30 50

FIG. 7. Wavefront characteristics (left) and focus characteristics (right) as a function of backpropagation distance (in ram) for representative measured waveform data (r56). The abbreviations are the same as in Fig. 4.



4000 f Incorporated Pulses 0.8 -I0 dB PER 2000' 0 ß

BP& Ideal At After TSC TSC Aperture Backprop.

I-01 WSF 5.0 -:50 dB r e .95 E 2.5

.90 0 At After TSC BP& Ideal

Aperture Backprop. TSC

5.0 ELF -20 I0% Der. Lev.

O' -$0' At After At After

Aperture Backprop. Aperture Backprop.

FIG. 8. Focal plane time histories of representative measured data (r56). Each panel shows the bipolar distribution of signal amplitude as a shade of gray on a 50-dB log scale for each polarity in the focal plane at an instant of time. In each panel, the horizontal coordinate is elevation and spans 37.504 mm while the vertical coordinate is in the array direction and spans 56.256 mm. The number beneath each panel identifies the (zero-origin) instant of time in the 128-point interval employed in the temporal Fourier transform. First row: Uncompensated data. Second row: After time-shift estimation and compensation in the aperture. Third row: after backpropagation of 40 mm followed by time-shift estimation and compensation. Fourth row: Ideal data.

the range of -30 to -40 dB, the focus obtained with time- shift compensation in the receiving aperture.

The average and standard deviation of the wavefront and focus characteristics determined for fourteen sets of wave-

forms measured after propagation through different specimens of human abdominal wall are shown in Fig. 10 for time-shift compensation in the receiving aperture and for

FIG. 10. Statistics of wavefront characteristics (left) and focus characteristics (right) obtained from sets of 14 measured waveform data. re=effective radius. The other abbreviations are the same as in Fig. 5.

time-shift compensation after backpropagation to the distance at which the waveform similarity factor is maximized. The a,,)erage (_+s.d.) number of pulses incorporated in the final reference waveform increases from 3132+379 to 3605

+259 (out of a total of 4096 pulses). The corresponding average of waveform similarity factors increases from 0.938 _+0.025 to 0.967_+0.015 while the energy level fluctuation decreases from 4.2_+0.4 to 3.3+_0.3 dB. Although not included in Fig. 10, the average number of smoothed points in the calculation of the arrival time surfaces drops from 354 +_209 to 135 -+92 while the average peak value of the cross- correlation coefficient rises from 0.891_+0.030 to 0.930

_+0.022 as a result of backpropagation. The -10-dB

or• . Unc I .... TSC F •x,,• --'-- BP EtTSC

• -40

0 2 4 6 8 Effective Radius (mm)

FIG. 9. Effective radius of the focus obtained with representative measured waveform data (r56). UNC=uncompensated data. TSC=time-shift compensation in the receiving aperture. BP and TSC=backpropagation followed by time-shift compensation. Ideal=replication of a single average waveform throughout the receiving aperture.

6O

4O

2O

-E•- Optimum BP Distance ß -O-. Specimen Thickness

I • I i

i ß •)\\ % / ß

52 54 55 56 60 65 67 69 72 74 78 79 80 81 Run Number

FIG. 11. Optimum distance of backpropagation and specimen thickness.



peripheral energy ratio at the focus decreases from t}.63 +0.12 for time-shift compensation alone to {}.47__+{}.{}6 after backpropagation and followed by time-shift compensation. Similarly, the -30-dB effective radius decreases from 4.2 _+ 1.2 to 2.5+0.5 mm. The corresponding statistics for these two characteristics with ideal data are 0.34_+t}.{}2 and 1.9

-+0.1 mm. A comparison of the effective radius curves of each focus with those of the corresponding ideal focus shows that the 10% deviation levels decrease from -23.5_+3.3 to

-26.3_+3.1 dB after backpropagation. The optimum distance of backpropagation fi•r each

specimen is plotted in Fig. 11 along with the nominal thickness of the specimen. This figure shows that the two vari- ables are closely related. A thick specimen usually requires a big distance of backpropagation to correct for amplitude and waveform distortion. From the data in Fig. 11, the average of the optimum backpropagation distance is found to be 31.4 _+9.1 mm and the average specimen thickness is found to be 20.7_+5.3 mm.

v. DISCUSSION

The backpropagation process outlined in Sec. I for spatially continuous signals was implemented with a discrete summation. Thus a possibility exists that the frequency domain may be undersampled because the propagation function

2 2 2 eJ(2rdX)zx/1-- h. (f.• + f ?) in Eq. (8)oscillates infinitely fast as x/f.[+f• approaches 1/1•. (See Ref. 22.) However, undersampling was not a problem in this study for two reasons. First, the angular spectrum A(fx,f•, ) in the current application has a bandwidth much narrower than 1/1•. For example, the angular spectrum of a uniform wavefront across a square aperture is the product of two sinc functions whose first zeros are at 1/W, where W is

the spatial width of the aperture. When W is much larger than 1• as in this study, this shows that the angular spectrum

A (fx ,f•,) has no significant value when x/f• + f• approaches 1/1•. Second, the integration of a function oscillating rapidly about zero has little contribution to the final result. This is

similar to the idea of stationary phase. 23'24 In order to con- firm that undersampling in the frequency domain was not a problem in this study, the same calculation was made with the array sizes zero padded to 512x128, double the usual values. This effectively halved the sampling interval in the frequency domain but virtually no difference appeared in the computed backpropagation results.

The reference waveform method employs parameters in the process of building the reference pulse and smoothing arrival time estimates. However, experience in the present investigation is that the results are generally insensitive to variations in these parameters. The test value of p, including its dependence on n, that determines whether a waveform is incorporated into the reference waveform does not appear to influence the final reference waveform much unless that

value is extremely close to 1.0 and the initial reference waveform is severely distorted. In the first smoothing algorithm, the error threshold (3o' in the reported processing) may influence the number of points that are removed but the two-

dimensional polynomial fit changes little as a result. In the second smoothing algorithm, the parameters of the processing are: The sizes of the neighboring area used in the calculation of local average (5 x5 in the reported processing), the number defining too few neighbors (3 here), the time (0.5 T0 here) that determines if an arrival time is too far away from its neighbors and needs to be removed, and the threshold (0.5 here) that determines whether a peak position or a local average is used to substitute the point to be removed. The first two parameters influence the calculation of local average whereas the third parameter controls which points are to be removed. However, none of these has been found in this

study to have a significant influence on the final result as long as the local averages are not too far off and the multi- plier is not so big that many outlying points are missed in smoothing. The reason is that the final removal is usually made using the nearest peak position of the cross-correlation function from the local average position so that a slight shift in the local average position does not change the result.

The number of incorporated pulses and the number of smoothed points are determined at different stages in the processing and have different meanings. The former describes the similarity of each waveform relative to final reference waveform, while the latter describes the smoothness

of the arrival time surface determined solely by the position of cross-correlation peaks. On average, 76% of the measured waveforms were incorporated into the reference waveform at the position of the receiving aperture and 88% of the waveforms were incorporated into the reference waveform at the position of maximum waveform similarity while 9% of the arrival time points were smoothed in the receiving aperture and 3% of the arrival time points were smoothed at the position of maximum similarity. The data show that the large fraction (about 3 out of 4) of waveforms incorporated into the reference pulse at the position of the receiving aperture becomes still larger (about 9 out of 10) at the position of maximum waveform similarity and that the relatively small number (about I out of I l) of smoothed points in the receiving aperture becomes still smaller (about 3 out of 100) at the position of maximum waveform similarity.

In corresponding representative water path data at the receiving aperture, all the measured waveforms were incorporated into the reference pulse and no arrival time points were smoothed. The peak value of the cross-correlation coefficient between each waveform and the reference wave-

form averaged 0.988 and the waveform similarity factor was 0.996 in that water path data. Thus, the water path data indicate the measurement system nonidealities do not have an appreciable effect.

The interpretation of the fourth-order polynomial fit as the geometric delay is arbitrary in a sense but the geometric delay is difficult to calculate accurately from first principles since, even if the specimen is a uniform slab with sound speed different from the surrounding medium, a spherical incident wavefront becomes nonspherical after passing through. The fourth-order polynomial has greater flexibility (compared to a spherical fit) to fit arbitrary slowly varying components in the wavefront. Under the condition that no accurate knowledge about the shape of the tissue specimen is

657 J. Acoust. Soc. Am., Vol. 96, No. 2, Pt. 1, August 1994 D.-L. Liu and R. C. Waag' Correction for wavefront distortion 657


available, this fit is regarded as a good representation of the geometric delay.

Two limitations have made the backpropagation and subsequent compensation of the model waveform data im- perfect. The first is the removal of the geometric delay from the wavefront. This step is included to keep the size of the wavefront approximately the same before and after backpropagation so that comparison of waveform similarity and other parameters can be conveniently performed. (If the original diverging wavefront is backpropagated, the wavefront becomes smaller in spatial extent because of the focusing effect.) However, this time-shift operation on the wavefront is not accompanied by any amplitude adjustment. As an example, a spherical wavefront can be made planar by the removal of geometric delay, but the amplitude variation as- sociated with spherical spreading remains, and this has an unintended effect when the wavefront is propagated as a plane wave. The second limitation is also connected with the wavefront and aperture sizes, i.e., only part of the waveforms diffracted from the phase screen have been recorded and backpropagated, so that perfect reconstruction of the original wavefront is not possible. In spite of these limitations, which were imposed so that the processing of the model data is parallel to that of the experimental data, the results in Fig. 4 and Fig. 5 show that backpropagation after geometric correction can effectively remove pulse amplitude and shape distortion developed in forwardpropagation.

The initial increase and subsequent decrease of the waveform similarity factor as the distance of backpropagation increases is explained by reference to the model in which the aberration is caused by a phase screen placed some distance away from the aperture. The initial increase of waveform similarity factor results from removal of diffraction that accumulates during propagation to the receiving aperture. However, the uncompensated wavefront still contains time-shift aberration so further backpropagation with- out time-shift compensation introduces diffraction that is accompanied by changes in waveform shape and reduced waveform similarity. This is exactly what happens when a distance larger than the distance between the aperture and phase screen is used in one step for backpropagation. Thus, the maximum of the waveform similarity factor can be employed to identify the position of the phase screen.

The wavefront distortion with which the processing be- gan deserves special comment because the need for a technique to correct for wavefront distortion arising from distributed inhomogeneities in imaging depends on the distortion coming from within the tissue specimens rather than at the boundary between the specimen and the transmitter. Investi- gations described at the time the measurements were published •2 eliminate the possibility that surface irregulari- ties of the specimen, refraction at the smooth interface between media in which sound speeds differ by the same amount as in the tissue path measurements, and diffuse scattering such as produced by a graphite-gel mixture contribute appreciably to the distortion. Another possibility is that a varying speed of sound along the smooth surface of the tissue acts like a phase screen and contributes substantially to the measured distortion. To demonstrate that this is not so,

tissue-path measurements were made in the usual geometric configuration but with two different sound speeds in the surrounding fluid. The surrounding fluid was water (as usual) with a sound speed of 1523 m/s in one measurement. In the other measurement, a mixture of water and 7.8% alcohol with a sound speed of 1556 m/s was used. Care was taken to make sure that the measurements were made at the same

temperature (36.8+0.2 øC). If a measurable interface effect were present, the two measurements would be different. However, the measurement for the water surrounding yielded 30.3 ns and 2.87 dB for the arrival time fluctuations and

energy level fluctuations, respectively, while the corresponding values for the measurement with the water-alcohol surrounding were 30.0 ns and 2.90 dB. The patterns of arrival time fluctuation for the two measurements had an identical

visual appearance as did the patterns of energy level fluctuation. In summary, based on all the tests that have been performed, the conclusion is drawn that wavefront distortion caused by refraction at the boundaries between water and the specimen was not significant.

The position of an equivalent phase screen, i.e., the optimum distance of backpropagation, is determined in this study by maximizing the waveform similarity factor. Other measures may be investigated to serve the same purpose but may lead to a different optimum distance of backpropagation. However, as illustrated in the plots of Fig. 6, the waveform similarity peaks that determine the optimum distance of backpropagation are relatively broad. This shows that the results are not very sensitive to the exact distance of backpropagation.

The optimum distance of backpropagation found using the waveform similarity factor is, as shown in Fig. 11, usually larger than the specimen thickness. A reason is that a distance of about 8 mm is always present between the top surface of the specimen and the receiving aperture in the measurement. Adding 8 mm to the average thickness of the specimen yields 28.7 mm. This value is smaller than the average optimum distance of backpropagation by 2.7 mm, a value about one-half the standard deviation in the thickness

from specimen-to-specimen and about one-third the standard deviation of the optimum backpropagation distance. Thus the data indicate that the optimum distance of backpropagation is approximately the sum of the specimen thickness and the specimen-receiver distance. However, in view of observa- tions and experiments noted earlier, this circumstance is not caused by a dominating refractive effect at the specimen boundary. Nevertheless, the results do show that the aberration is equivalent to that caused by a phase screen located close to the specimen surface nearest the source. The physical origin of this may be the muscle layer proximal to the peritoneal surface that faced the source, the muscle-fat boundary that was usually near the surface facing the source, or a combination of these. Additional studies are needed to

provide information about the relation between tissue mor- phology and ultrasonic aberration.

A comparison of the focusing results obtained with time- shift compensation in the aperture and with backpropagation to an appropriate distance followed by time-shift estimation and compensation can be made. The main improvement, as



shown in Figs. 8, 9, and 10, is at levels more than 20 dB below the peak level. No significant differences are visible at higher levels. This implies that wavefront backpropagation improves mostly the contrast resolution while the point resolution, mainly controlled by aperture size after time-shift compensation, is not much affected.

The wavefronts and focusing results obtained after backpropagation and then time-shift compensation may also be compared to those of the ideal data. The average results in Fig. 10 show that the waveform similarity factor is 0.967 and the energy level fluctuation is 3.3 dB after backpropagation, indicating that some distortion still remains in the wavefront. This distortion is also reflected in the average statistics of the focusing results. In summary, modeling a spatially extended inhomogeneous medium by a properly positioned phase screen is still an approximation. Multiple phase screens can be employed for more accurate modeling but new measurements and techniques are necessary to define each phase screen.

The application of backpropagation followed by time- shift estimation and compensation to pulse-echo imaging is conceptually straightforward. Initially, a beam is focused as usual using geometric delay and echoes are received. Then, backpropagation followed by time-shift estimation is used to define the position and time shift of the phase screen in the model of the distributed inhomogeneous medium. A pulse waveform that corresponds to the known response of the elements in the aperture is next repeated in space at the position of the phase screen, time shifted with the negative of the phase screen characteristics, and then backpropagated to the receiving aperture. The waveforms in the resulting wavefront are then employed as excitations for the next transmission. In this way, the transmitted waveforms are predistorted and become undistorted after propagation through the medium. In reception, the waves are first backpropagated and then time-shift compensated after which focusing can be car- ried out as usual. If initial estimates of the phase screen position and characteristics are degraded by the focal blur caused by the inhomogeneous medium, the process may be repeated to obtain estimates that will improve as the focus becomes more ideal. The process may also be adapted in principle to image over an extended range either by calculating phase screens that each represent an increment in path or by determining a new single phase screen for the total path.

Vl. CONCLUSION

Wavefront backpropagation using the angular spectrum method was applied in this paper to compensate for pulse amplitude and shape distortion caused by propagation through a human abdominal wall. This method is based on modeling the aberration as the result of propagation in a uniform medium from a phase screen placed some distance away from the receiving aperture. The results obtained for both model and measured data show a significant improvement in waveform similarity and in focusing.

A new technique was developed to perform time-shift compensation in the presence of waveform distortion. In this approach, a reference waveform is first adaptively formed

from the data and next cross correlated with each waveform

to calculate the arrival time of the pulses. The arrival time surface across a two-dimensional aperture is then smoothed by two different schemes, comparing each point either with a fourth-order polynomial fit or with a local average. In the first case, the final polynomial fit is used for geometric correction while in the second case, outlying points are replaced by nearest peak position of cross-correlation function, and the result is utilized for time-shift compensation.

From the results of processing 14 sets of data measured using different specimens of human abdominal wall, the distance of backpropagation that resulted in the greatest improvement in waveform similarity factor was found to be related to the specimen thickness. The waveform similarity in the wavefront was improved by backpropagation and energy level fluctuation was reduced. The results of focusing after backpropagation followed by time-shift compensation were found to be significantly closer to the ideal focus, par- ticularly at levels more than -20 dB from the peak, compared to those obtained by time-shift compensation in the aperture. In an imaging system, this implies significantly improved contrast resolution.

ACKNOWLEDGMENTS

Laura M. Hinkelman is thanked for providing the measurements used in the processing reported here and for help- ing make the new measurement used to show refraction at the specimen boundary is not significant. Gratefully ac- knowledged is support from the University of Rochester Di- agnostic Ultrasound Research Laboratory Industrial Associ- ates as well as from NIH Grants HL 150855 and DK 45533.

The computations in this research were performed at the Cornell National Supercomputing Center, which is supported by the National Science Foundation, New York State, and the IBM Corporation.

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Correction of ultrasonic wavefront distortion using backpropagation and a reference waveform method for time-shift compensation