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A Model of Distributed Phase Aberration for Deblurring Phase

Estimated from Scattering

Jason C. Tillett, Jeffrey P. Astheimer, and Robert C. Waag [Life Fellow, IEEE]

The authors are with the Department of Electrical and Computer Engineering, University of

Rochester, Rochester, NY

Jason C. Tillett: tillett@ece.rochester.edu

Abstract

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.

I. Introduction

Ultrasound b-scan images are commonly used for medical diagnosis. However, the quality

of these images can be degraded by aberration from tissue in the propagation path between

the transmitter-receiver combinations and the anatomic features that are being imaged.

Consequently, considerable research has been aimed at ways to characterize or measure the

aberration and to then use the characterization of aberration to compensate for the aberration

[1]–[8].

Interest in describing and correcting aberration in optical imaging predates the concern for

these issues in ultrasound imaging applications. The optical literature, see e.g., [9] and [10],

suggests use of point-response measurements to characterize aberration. In optical systems,

point-responses are usually obtained by imaging point sources. Stars in telescopic images

are a noteworthy example. In ultrasound imaging systems, point responses may also be

obtained from point-like reflectors when they are available.

Coherent point responses are particularly useful in ultrasound systems that are comprised of

extended arrays of transducer elements. Signals that arrive at the array from a point reflector

in a medium without excessive attenuation may be time-reversed and retransmitted to

© 2010 IEEE

R. C. Waag is also with the Department of Imaging Sciences, University of Rochester, Rochester, NY.

NIH Public Access

Author Manuscript

IEEE Trans Ultrason Ferroelectr Freq Control. Author manuscript; available in PMC 2010 July

26.

Published in final edited form as:

IEEE Trans Ultrason Ferroelectr Freq Control. 2010 January ; 57(1): 214–228. doi:10.1109/TUFFC.

2010.1400.

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produce a focus at the reflector that is uncorrupted by the medium aberrations. These

retransmitted signals appear to use the intervening medium as a lens that produces a better

focus than could be obtained with homogeneous transmission [11]–[13]. Frequency

decomposition of point-reflector responses also yields phase that produces uncorrupted

focuses of monochromatic transmissions at the reflector. Furthermore, geometric adjustment

of the phase permits a high quality focus to be repositioned at neighboring locations as long

as the aberration experienced en route to the new positions remains substantially the same as

the aberration for the original point reflector. A neighborhood of points that share the same

aberration is called an isoplanatic patch or isoplanatic region. Isoplanatic regions are related

to tissue characteristics as well as measurement geometry and have been found to be larger

along the axis of a transducer array than in lateral directions [14].

Point reflectors are usually not available in medical applications. Fortunately, point-reflector

responses can also be obtained from cross-spectral measurements of random media by using

a statistical method that is described in [15]. In this method, echo signals are acquired from a

collection of focuses in random-medium patches that are within a single isoplanatic region.

The focuses for these measurements must be far enough apart to ensure a degree of

statistical independence but are constrained in space by the isoplanatic requirement.

Thus, isoplanatic regions are limiting factors for aberration correction in 2 respects: they

limit the region over which retransmission of point-response measurements can be used to

recover an uncorrupted focus, and they limit the span of focal points needed in statistical

measurements to estimate impulsive responses. The purpose of this study is to reduce these

restrictions by allowing aberration to vary in a deterministic manner. Specifically,

monochromatic aberration is modeled by a phase screen that is located at an intermediate

distance between the transducer array and the focal region. The offset location of the phase

screen introduces variations in the aberration for neighboring focal points because rays from

the transducer elements to different focal points intercept the phase screen at different

locations. This paper shows that by placing the phase screen at an appropriate depth, the

variations that are introduced can model the true aberration across an area that is

significantly larger than the isoplanatic region. In fact, the usual definition of isoplanatic

region may be viewed as the range of accuracy for the special case of the model in which the

phase screen is located in the same plane as the transducer array.

Offset phase screens have been shown to be an applicable model for improving focus

compensation in human abdominal wall [8]. In that work, backpropagation of the received

wavefront to the phase screen location using the angular spectrum method provided better

compensation calculated using the single, backpropagated wavefront. In this work,

backward and forward propagation is along rays to correct for lateral shifts in the phase

screen as the focus is steered to generate the many wavefronts that are required to implement

a statistical estimation of the compensation factors.

The theory section in this paper explains how to incorporate the depth of the phase screen

when correcting for aberration, how the statistical procedure for point-response estimation

can be modified to accommodate an offset phase screen, and how the depth of the phase

screen is estimated. In the section that follows, the theory is validated by measurements

made with a custom 2-D transducer array in a temperature-controlled water-tank. The results

of the measurements are summarized and their significance discussed in succeeding

sections. Finally, a set of conclusions is drawn from the investigations.

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II. Theory

A. Offset Phase-Screen Model for Aberration

A measurement geometry in which the z-axis coincides with the axis of a 2-D transducer

array located in the z = zT plane as shown in Fig. 1 is considered. In this geometry, let φ0 =

(x, z; x0, z0) denote the monochromatic field that propagates homogeneously with

wavenumber k from a point reflector at (x0, z0) and let φ(x, z; x0, z0) be the corresponding

field that propagates through aberration. Measurement of the inhomogeneous field across

the transducer array gives values for φ(xT, zT; x0, z0) in a rectangle of the z = zT plane. The

aberration within that rectangle can be characterized by the phase term

(1)

This phase can be used to produce a high-quality monochromatic focus at (x0, z0) by

applying the complex transmission amplitude

(2)

to the array. If the point (x1, z1) is in the same isoplanatic patch as (x0, z0), then

(3)

and a high-quality focus can also be produced at (x1, z1) by using the transmission amplitude

(4)

The phase of the first factor in (4) provides geometric focusing that is used to steer the

focus, and the second factor compensates for aberration.

Unfortunately, the accuracy of the approximation in (3) diminishes quickly as the separation

between points (x1, z1) and (x0, z0) increases, particularly in the lateral direction. However,

aberration can be modeled in a way that is more tolerant of the separation between source

points by assuming that the aberration takes place in a plane that lies between the focal

region and the transducer array. Thus, aberration is represented by a phase screen

e j θ(x sc,zsc in a plane z = zsc, where zT ≤ zsc ≤ z0. If the approximations of geometric optics

are applied to the model, then the phase factors at each point in the screen only influence

rays that pass through that point. This geometry is illustrated in Fig. 1.

Ray approximations permit determination of the screen phase θ(xsc, zsc) from the aberration

phase α(xT, zT; x0, z0) given by (1). The result is

(5)

where Pscreen[xsc, zsc; x0, z0] is the projection

(6)

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of the point (xsc, zsc) in the phase screen onto the transducer plane along the ray that

originates at (x0, z0) (i.e., along the dashed line in Fig. 1).

The inverse of this projection maps points on the transducer plane onto the phase screen.

This inverse is given by

(7)

Substitution of the inverse projection in place of the (xsc, zsc) arguments in (5) gives

(8)

Eq. (8) determines the phase compensation for the focal point (x0, z0) from the phase screen.

The factors e−jα(xT,zT;x0,z0) and e−jα(xT,zT;x1,z1) that are needed to compensate the focus at

the neighboring points (x0, z0) and (x1, z1) are given by

(9)

and

(10)

and are no longer equal except in the special case in which the phase screen is in the plane of

the transducer.

B. Statistical Estimation of Aberration for an Offset Phase Screen

As noted in the introduction, point reflectors are usually not available in ultrasound imaging

but can be simulated by a statistical process that uses focused measurements from a

collection of patches of random media sharing the same aberration. The original

development of this technique in [15] required the aberration to be the same in the sense that

the focal points (x1, z1), (x2, z2), …, (xn, zn) for all the measurements had to belong to the

same isoplanatic region. This is a restrictive condition that may also be expressed as

(11)

The offset phase screen model can be used to lessen this constraint. This is accomplished by

applying the statistical computations to measurements that have been back-propagated from

the face of the transducer array to the plane of the phase screen. According to the model, the

aberration for the measurements is different in the transducer plane but is the same in the

plane of the phase screen. The condition given in (10) can then be replaced by the relaxed

requirement that the aberration for all the focal points is accurately modeled by a single

offset phase screen.

In practice, this approach can be further simplified by only backpropagating the phase of the

measured fields and by using (5) to backpropagate along rays, consistent with the

assumptions that the phase information is of paramount importance in aberration correction

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and that phase variations are accumulated along rays. Although the details of the statistical

computations are not important here, it is noteworthy that aberration estimates are computed

iteratively so that improved estimates can be used to produce better focuses and more

accurate beam-steering in successive computations. The offset phase-screen model can help

to accelerate the convergence of these iterations by employing aberration phase corrections

that vary with the focal point rather than adhering to a single fixed correction for all the

focal points.

C. Determination of the Offset Phase-Screen Depth

The previous subsections have described methods for assigning phase variations to the phase

screen. They have also explained how the offset phase screen model can be used to

compensate for aberration when the focus is directed toward different locations. This

subsection explains how the depth of the phase screen can be established from experimental

measurements.

The depth of the phase screen can be derived from a pair of point-reflector measurements

provided that the locations (x0, z0) and (x1, z1) of the reflectors are known, that x0 and x1 are

close to one another, and that the 2 points are at roughly the same depth (i.e., z1 ≈ z1).

Factoring the homogeneous field from these measurements and extracting the phase, as in

(1), leaves the terms e jα(xT,zT;x0,z0) and e jα(xT,zT;x1,z1). The offset phase screen model

predicts that these terms are given by

(12)

and

(13)

where λ = (zsc – z0)/(zT – z0) = (zsc – z1)/( zT – z1).

The expressions on the right side of (12) and (13) can be used to write the correlation of the

measurements as

(14)

The integral on the right side of (14) is clearly maximum when

(15)

Taking the norm of both sides of (15) and solving for zsc gives

(16)

Eq. (16) determines the depth of the phase screen from the magnitude ||y|| of the offset for

the peak value of the correlation function for the measurements from the 2 point reflectors.

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When point reflectors are unavailable, the measurements needed to estimate the phase-

screen depth can also be obtained from statistical estimates formed from random media

(simulated point responses) [15]. Simulation of point responses can utilize subsets of focal

points as well as iteration for improving the quality of the estimated phase. Experience

indicates that screen-depth calculations do not seem to require high-quality simulated-point-

response measurements. Thus, the screen depth can generally be established using fewer

focal points and fewer (or no) iterations of the simulated-point-response estimates.

III. Methods

Data sets were acquired using a large 2-D array system [16] comprised of 80 × 80 transducer

elements with a 0.6 × 0.6 mm2 pitch. The system operates at a center frequency of 3 MHz

and has a −6-dB transmit-receive bandwidth of 1.8 MHz. The received signals are digitized

with 12 bits of resolution at a sampling rate of 20 MHz. The transmitted waveforms are

programmable. The array elements are independently compensated with gain factors that

were determined using the calibration procedure described in [17].

A. Aberration-Phase Maps and Time-Shift Maps

1) Aberration-Phase Maps and Time-Shift Maps for Point-Reflector

Measurements—Point-reflector experiments were used to form ideal profiles of

aberration phase that serve as standards for comparison with the aberration phases derived

from random scattering. The point reflections were produced by the tip of a 0.2-mm

polyvinylidene fluoride (PVDF) needle probe hydrophone placed at a transmit focus located

55 mm below the center of the transducer array. The initial approximate positioning of the

hydrophone was adjusted using high-precision stepper motors that were guided by pulse-

echo measurements. This allowed the depth and the lateral location of the tip of the

hydrophone to be assigned with an accuracy of ±0.01 mm. A hydrophone was used as the

point reflector so that measurements of compensated focuses could be performed without

having to move anything that could cause additional shifts in measured phase.

The wavefronts of measured reflections from the hydrophone have curvatures characteristic

of highly localized scattering. Reflections that were received through the water path had

perfectly spherical wavefronts. However, reflections that were received through aberrators

deviated from the ideal spherical shape. Cancellation of these phase deviations in the

Green’s function by using compensating phases is necessary to produce a high-quality focus.

A tissue-mimicking aberrator was inserted between the transducer and the hydrophone and

pulse-echo measurements were acquired. Phases of the Fourier coefficients of the received

signals at each temporal frequency form a map across the transducer array of the wavefront

for the monochromatic reflections at that frequency. Phases of the perfectly spherical

wavefronts that occur in the absence of aberration (i.e., in the water path) were subtracted

from the phase maps from measurements that were formed through aberration to isolate the

phase deviations directly attributable to aberration. This computation produced aberration-

phase maps for each temporal frequency. Compensation for aberration was accomplished by

applying phases that are the conjugate of the aberration-phase maps for each Fourier

coefficient of each received signal. This provided a distinct compensating factor for each

frequency and each transducer element. However, the aberration-phase maps were very

nearly linear in frequency and, hence, could be approximated using a single time-shift map

in which the time shift at each element was the slope of the phase variation with frequency.

Such time-shift maps across the transducer array represent arrival-time fluctuations (ATF)

from an expected spherical geometry and were derived from the phase maps by forming

linear fits to the unwrapped phases at each transducer element. The phase unwrapping

employed in this time-shift estimation was performed in 3 dimensions (2 dimensions

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spanning the transducer array and the third dimension spanning temporal frequency) rather

than separately unwrapping the phase variations with temporal frequency at each element

because unwrapping is more robust in higher dimensions.

2) Aberration-Phase Maps and Time-Shift Maps for Random-Scattering

Measurements—The experimental configuration for random-scattering experiments was

the same as that for the point-reflector experiments except that a random-scattering phantom

was used in place of the hydrophone. Transmit beams were geometrically focused at 11

points that were a subset of 75 points defined by the center and vertices of the 3 concentric

Platonic figures that were used as focuses in [15]. These 75 points were chosen as focuses in

the earlier study to optimize the statistical independence of the scattering samples while

staying within a designated isoplanatic region [18]. The subset of vertices selected for the

focuses in the current study consists of the center of the solids together with the 10 vertices

with depths that are closest to the depth of the center focus. (See Fig. 2.) These vertices were

chosen because they are all located at roughly the same depth and because reflections from

these locations produced aberration-phase maps that are maximally shifted relative to one

another. Spherical geometry associated with propagation in a water path was removed from

the signals received from each focus. A temporal window determined from average sound

speeds for the aberrator and the phantom was applied to the signals to select random-

medium reflections that were arriving from the focus location. Estimates of aberration phase

are relatively insensitive to the placement of this window because the axial size of an

isoplanatic patch is typically much larger than its lateral extent. The windowed and

geometrically corrected signals were used to form statistical estimates of the aberration

phase as described in [15]. Again, time-shift maps were formed from the phase estimates.

B. Estimation of Phase-Screen Depth

The depth of the phase screen determines the way predicted values for aberration-phase

factors vary for focuses formed at different locations. Screen-depth estimates are based on

detecting parallax in the phase maps or time-shift maps of neighboring sources. This effect

can be observed in any pair of phase maps or time-shift maps for neighboring sources.

Consequently, depth estimates can be formed from any pair of point-reflector measurements.

However, each phase map or time-shift map obtained from random-scattering measurements

must be statistically estimated from measurements for an entire set of focuses. Thus,

detection of parallax in random-scattering measurements requires statistical estimates from 2

distinct sets of focuses.

Statistical methods that rely on time-varying media rather than neighboring focuses [19] can

be used to form estimates of offset time-shift screens and offset-phase screens without

having to backpropagate the signals to the plane of the screen. The screen depth is only used

in these cases to adjust the compensation factors as the beam is steered to different locations.

However, estimating the depth of the phase screen still requires a parallax calculation based

on statistical estimates from at least 2 neighboring locations.

1) Phase-Screen Depth Estimates from Point-Reflector Measurements—Point-

reflector experiments were performed with the hydrophone located at each of the 11 focuses

that were selected for the random scattering experiments. Aberration-phase maps were

computed for each of these focuses as described in Section III-A. A total of 55 ways exist to

pair measurements from different locations. However, in 27 of these pairs, the lateral

separation of the reflectors is less than 1.5 mm. This makes the parallax effect difficult to

discern. Therefore, estimates of the screen depth were only computed for the other 28 pairs.

For each pair, the 2.5-MHz (near center frequency of the system) phase maps for the focuses

were cross correlated, as in (14), and the spatial shift of maximum correlation was found.

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The screen depth was estimated by applying (16) to this spatial shift. Estimates of the screen

depth were also obtained from the positions of the cross correlation peaks for each of the 28

pairs of time-shift maps.

2) Phase-Screen Depth Estimates from Random-Scattering Measurements—

Random-scattering measurements were obtained from 2 sets of focuses with each set

surrounding one of the 2 simulated source locations as shown in Fig. 2. A more localized

cluster of focuses was used than was used in estimating the phase screen to avoid blurring of

the parallax effect that could result from larger lateral displacements. Each set of focuses

consisted of the center and vertices of the smallest of the 3 Platonic solids (an icosahedron)

whose vertices were identified in [18] as optimal locations for the focuses within an

isoplanatic patch with an expected size of 3 mm. Use of this more concentrated cluster of

focuses permitted the formation of 2 disjoint sets of scattering volumes that were separated

by less than the size of the isoplanatic patch.

Random-scattering experiments contrast with point-source experiments in that the precise

location of the source is unknown and, consequently, the exact source geometry cannot be

removed from the estimated phase maps. Instead, the methods detailed in [20] were used to

fit an apparent focus location and the geometry for this fitted location was removed. An

alternative approach would be to use iterative methods like those detailed in Section III-D to

make successive refinements in the source location. However, because the objective was to

estimate a single parameter, i.e., the screen depth, a more expeditious computation was

employed. After the fitted geometry was removed, mean positions of the 2 source locations

were calculated. Time-shift maps were formed from the statistical phase maps and depth

estimates were derived in the same way as described for point-source experiments.

C. Compensation of Focused Beams Using Estimated Phase

The tip of the PVDF hydrophone was positioned under the 2-D array using the alignment

technique described in Section III-A. Aberration-phase maps from this point reflector were

measured. These phases were used to compensate a transmit beam focused at the same

location. The hydrophone was then used to sample values of the amplitude of the focus at

different locations. These amplitudes were initially scanned in the axial direction of the

transducer array to find the location of the peak intensity. The focus was then scanned along

the 3 axes of an orthogonal coordinate system in which one of the axes was perpendicular to

the transducer array (i.e., depth) and the other 2 lateral axes were oriented along directions

defined by the grid of transducer elements. The position of the origin of the lateral axes was

directly below element (40, 40), the center element of the 2-D array. Samples of the focus

amplitude in the axial direction were taken in 0.5-mm increments over a span of 10 mm

centered at the origin. Samples of the focus in the lateral direction were taken in 0.2-mm

increments over a span of 10 mm, also centered at the origin. Measurements were obtained

from transmit subapertures and the total field at each hydrophone position was computed by

summing all subaperture measurements. Corresponding measurements of the focus were

obtained for focused beams that were steered away from the center-focus position without

changing the transmit compensation.

In Section II, ray-propagation arguments were described to obtain phase compensations

from the offset phase-screen model for focused beams that are steered to different locations.

These compensations are effective at focal points that are outside of the isoplanatic patch

used to determine the phase screen. The compensating phases for each focal point are

theoretically obtained from a projective transformation of the phase screen onto the surface

of the transducer array along rays that emanate from the focal point as illustrated in Fig. 3.

However, projective transformations of a discretely sampled phase screen produce values in

the plane of the transducer array that are sampled at locations that do not coincide with the

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transducer elements so phase compensation values at the element locations must be

interpolated. To perform this assignment, Fourier interpolation was employed. These

compensations were applied to beams steered to other locations and the focus was measured

at these locations to quantify focus improvement.

D. Estimation of the Offset Phase Screen from Random Scattering

Random scattering was produced by a tissue-mimicking random-scattering phantom

positioned beneath the aberrator in place of the hydrophone used for the point-reflector

experiments. Measurements were acquired for the scattering from each of the 11 transmit

focuses identified in Section III-A.

Propagation of the transmit beams through the aberrator can result in unknown shifts of the

focuses away from their intended locations. Shifts of this type are not important in point-

reflector measurements because the phase variations for these measurements only depend on

the location of the reflector. However, the focal points for random scattering measurements

are carefully chosen to balance the need for statistical independence and the need for the

aberration associated with all the focuses to be accurately modeled by a single phase screen.

Thus, drift in the transmit focuses can have undesirable effects on aberration estimates

derived from random scattering.

The iterative procedure summarized in Fig. 4 was employed to reduce drift in the transmit

focus. The initial transmissions were geometrically focused, without any compensation, at

each of the 11 focuses. Successive transmissions were focused with compensation derived

from the offset phase screen estimated from the measurements of the previous step. Offset

phase screens were obtained by first backpropagating the Fourier phase of the received

signals along rays from the transducer array to the phase screen. The backpropagated signals

were then employed to form a statistical estimate of the phase screen by using the estimation

procedure described in [15]. Iterations of the procedure were repeated until the differences

in the phase screen estimates for successive iterations were no longer significant. Iteration

was evaluated for some of the aberration estimates formed from random scattering. In these

cases, almost all the improvement was realized in the first iteration.

Successful correction of focus dislocation was verified using the procedure detailed in [20].

This method requires the selection of time-gates to localize the geometry. Choosing an

appropriate time-gate for point-reflector measurements is straightforward and may be based

on visual inspection of the signals, but the echos from the random media focus are not as

easy to identify. The time gates for the random medium reflections were, therefore,

determined by using the known average sound speeds of the aberrator and phantom to

calculate the arrival time of echoes from the focus. When a known object, like a scatterer-

free sphere, is in the focal region this method produces time gates consistent with time gates

that would be assigned using visual inspection of the received echoes.

E. Experiments

Four aberrators were studied. Their properties are summarized in Table I. These aberrators

were designed to simulate effects of ultrasound propagation in tissues. They were made by

Computerized Imaging Reference Systems, Inc., Norfolk, VA (CIRS) and were selected to

represent a variety of aberration geometries and aberration strengths. Magnetic resonance

images of the aberrators are shown in Fig. 5. These vertical cross sections are near the center

of the field through which the ultrasound signals were propagated. The magnetic resonance

images show differences in the way that the aberration is distributed. In Aberrators 4491 and

6277, the inclusions are generally closer to the transducer array. In Aberrators 6289 and

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4064, the inclusions are farther away. There are also differences in the amount of variation

in the depth of the inclusions.

Aberrator strengths were characterized by computing standard deviations of the arrival time

fluctuations of signals from point sources, as detailed in [20]. In Table II, the aberrators are

categorized by aberration strength estimates for point sources that were positioned near the

lateral centers of the aberrators.

1) Measurement of Improvement in Phase Predictions and Time-Shift

Predictions—Aberration-phase maps at 2.5 MHz were obtained from point-reflector

experiments using each of the 11 reflector positions described in Section III-A. A phase

screen was then produced by back-propagating the phase map for the center focus along

rays, as described in Section II-C, to the screen depth that was estimated and described in

Section III-B. Aberration-phase maps were then computed for each of the focuses as phase-

screen projective transformations (Eq. 8).

Videos were produced for each of the 4 aberrators (

scale images of measured aberration-phase maps together with aberration-phase maps that

were projected from the phase screen. In these videos, the position of the focus changes

from frame to frame. The (unchanging) aberration-phase map for the center focus also

appears in a separate panel. This stationary phase map may be interpreted as the projection

of a phase screen that is located at the depth of the transducer array because all projective

transformations are identity maps when the screen depth is zero.

). The videos show gray-

The videos provide visual comparisons of the aberration modeled by an offset phase screen

with the aberration modeled by a phase screen that is not offset. Two of the lower panels in

the video contain gray-scale images of the differences between measured aberration phase

and the aberration that is predicted by each of the 2 phase screens (i.e., the offset phase

screen and the phase screen with no offset). A mean and a standard deviation are reported

for each of these images. The standard deviations are averaged over the focus positions (i.e.,

over the frames of the video) to obtain a single parameter for each of the 2 phase-screen

models that measures how well the model is able to predict the aberration-phase maps for

the different focuses.

Time-shift maps were also computed from point-reflector experiments for each of the

reflector positions using the method described Section III-A. The time shifts for the center

focus were backpropagated along rays to the screen depth that was estimated in Section III-

B to produce an offset time-shift screen. Time-shift maps were then computed for each of

the focuses as projective transformations of the time-shift screen. The accuracy of these

predictions was compared with the accuracy of time shifts that were predicted by a time-

shift screen with no offset (i.e., the unchanging time shifts of the center focus) using the

same standard-deviation calculation that was described above for comparison of phase

predictions.

2) Measurement of Improvement in Focus—Phase screens for these experiments

were estimated from point-reflector experiments. The phase-screen depth was determined as

described in Section III-B. The 2.5-MHz phase map for the reflection from the center focus

was used as a zero-offset phase screen. A second phase screen at the estimated depth was

obtained by applying the statistical estimation procedure described in [15] to signals from

the point-reflector experiments that were backpropagated along rays to that depth.

Both phase screens were used to obtain phase compensations for transmit beams that were

focused at the location of the center focus and also at an offset location that was 1.5 mm

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away from the center focus. A hydrophone was used to sample the amplitudes of these

focuses in the vicinity of the focal points. Plots of the effective radii of the focuses were

found, as described in [21], from these measurements.

3) Measurement of Improvement in B-Scan Contrast—A scatterer-free cyst-

mimicking region was imaged with no compensation and also with compensation derived

from both an offset phase screen and a phase screen with no offset. The offset phase screen

was estimated from random scattering using the iterative process described in Section III-D.

The screen with no offset was estimated from random scattering using the statistical

estimation procedure described in [15]. To scan the cyst-mimicking region, the 2-D array

was linearly translated in 0.2-mm increments across the area of interest. This area contained

the cyst-mimicking region that was 4 mm in diameter with its center located approximately

55 mm from the transducer. The array was translated a total of 3 mm to acquire 15 scan

lines. Transmissions were geometrically focused before phase compensation was applied.

Signals received in the aperture were beamformed using coherent summation of aperture

waveforms that were shifted in time to focus at the transmit focus position. The set of scan

lines formed a 2-D image spanning the region of interest. Quantitative comparisons were

made based on contrast ratios. These contrast ratios are expressed as differences, in decibels,

between the mean intensities in subregions of the cyst-mimicking region and the background

regions. These regions were 1.7 mm in diameter, a little over half the lateral extent of the

scanned lines and smaller than the known 4-mm diameter of the cyst-mimicking scatterer-

free region. The reference background regions were above and below the cyst-mimicking

region to avoid bias from a time-gain compensation that was not perfectly flat.

IV. Results

A. Estimates of Phase-Screen Depth

The results of screen-depth estimates derived from the spatial shifts using (16) as described

in Section III-B are reported in Fig. 6 and Fig. 7. Variations in the sizes of the error bars

indicate that screen-depth estimates from the point-reflector phases are more reliable than

estimates based on point-reflector time shifts. The screen-depth estimates derived from

random scattering are only slightly shallower than the estimates from point-reflector

experiments. This demonstrates that reliable estimates for the screen depth can be

determined when point-reflector information is not available. All of the estimates reported in

Fig. 6 are consistent with one another and also agree with visual impressions of where the

phase screens should be located that are suggested by the magnetic resonance images of the

aberrator cross sections in Fig. 5.

The effect of changes in the temporal frequency on the phase maps is shown in Fig. 7.

Variations in temporal frequency appear to have little effect except at the lowest frequencies

where the estimates have more variability. The frequency-dependent screen-depth estimates

for Aberrators 6277 and 4064 generally have the smallest standard deviations. Because

Aberrator 4064 has inclusions at a well-defined depth, the larger-than-expected standard

deviation spread is because aberration is close to the focuses so that ray spreading that

occurs between the aberration and the transducer array causes small depth variations to

produce exaggerated spatial shifts. Thus, more error may be expected in screen-depth

estimates for inclusions that are close to the focuses.

B. Accuracy of the Offset Phase-Screen Model

The phase maps projected from the offset phase screen are in closer agreement with the

phase maps for the other focuses than the phase map for the unshifted center focus. To

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quantify this improvement, standard deviations were computed for the images (that can also

be viewed at the web site given by [22]) of the differences between the stationary phase map

for the center focus and the phase maps of other focuses (i.e., the images shown in the lower

center panel of each video). These standard deviations were averaged together to obtain a

single number as a measure of the disparity between the phase map of the unshifted center

focus and the phase maps for the other focuses. This calculation was repeated for the images

of the differences between the phase map projected from the offset phase screen and the

phase maps of other focuses (i.e., the images shown in the lower right panel of each video).

The results of these calculations for each of the 4 aberrators are tabulated in Table III. Small

portions of the difference images for Aberrator 6289 where the aberration was particularly

severe were eliminated from the standard deviation calculations because aberration from

these portions of the aberrator were not attributed to lateral shifts.

The standard deviations in Table III show that predicting aberration phase with an offset

phase screen is better than relying on a phase screen in the plane of the transducer that does

not change as the focus is relocated. The offset does not account for all the variations in

aberration that occur as the focus is moved. However, a significant portion of the phase

variability is clearly attributable to the offset.

These results indicate that each aberrator may be modeled by a set of phase screens (one for

each frequency) or by a single time-shift screen at the prescribed depth. However, the

estimated depth of the phase screen for a given experimental configuration is not the same as

the intrinsic depth that locates the screen relative to the top surface of the aberrator. The

intrinsic depth differs from the estimated depth by a flying distance to avoid contact between

the shifting array and the stationary aberrator. The gap was kept as small as possible but

large enough to ensure that shifts in the transducer across the surface of the aberrator would

not cause the transducer to rub against the aberrator surface and damage the aberrator. The

intrinsic phase-screen depths reported in Table III are obtained from the estimated depths by

subtracting the flying distance for each set of experiments.

Once the phase-screen depth has been estimated, better agreement between phase maps at

different focuses is obtained by propagating the aberration-phase maps backward along rays

from the transducer array to the plane of the phase screen. This is because the offset phase-

screen model is based on the assumption that backpropagated phase maps from different

focuses will coincide at the depth of the phase screen. Fig. 8 shows the extent to which this

assumption is satisfied for Aberrator 4064, which is the aberrator with the deepest phase

screen. The upper-left panel is a gray-scale image of the time shifts that results from

averaging together the time-shift maps for each of the focuses, and the upper-right panel is a

gray-scale image of standard deviations for these averages. The bottom panels show

corresponding images of the averages and standard deviations of the time-shifts obtained

from phase maps that were backpropagated to the plane of the phase screen. Arrival-time

fluctuations for the time-shifts derived from backpropagated phase maps were increased by

almost 5 ns and the standard deviations of the arrival-time fluctuation for these time-shifts

were reduced by more than 10 ns. The same analysis was performed for the other 3

aberrators and the results are summarized in Table IV.

The entries in the screen-depth column of Table IV give the depths used for

backpropagation of the phase maps for the different aberrators. The aberrators in the table

are in order of increasing phase-screen depth. The amounts of decrease in the standard

deviations of the arrival time fluctuations that appear in the standard deviation decrease

column are also increasing. This confirms the expectation that offset phase screens are more

helpful for modeling aberration that is far away from the transducer array.

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C. Improvement in Focusing

Fig. 9, obtained using Aberrator 4064, shows the effective-radius improvement that results

in a focus obtained using an offset phase screen to compensate for aberration when the focus

is steered to different locations. The curve labeled D (compensated with center-focus phase)

shows the effective radial profile of a beam that is focused at the center focus with

aberration compensation based on phase measurements from a point reflector at the same

location. This curve may be viewed as the optimal focus that can be obtained through the

aberrator. The curve labeled A (uncompensated) is an effective radial profile of a beam that

is focused at the center focus without any compensation for aberration. The other 2 curves

are effective radial profiles of focused beams that are steered to a neighboring location offset

by 1.5 mm from the center focus. The curve labeled B (steered and compensated with the

center-focus phase) is compensated for aberration by phase measurements from the point

reflector at the center focus (i.e., by the same compensation as curve D). The radial profile

labeled C (steered and compensated with the shifted center-focus phase) is compensated for

aberration using phases that are derived from the offset phase screen that is obtained by

backpropagating the point-reflector phases from the center focus to the screen depth that was

estimated for the aberrator. These plots demonstrate that aberration corrections obtained

from an offset phase screen can improve the focus of beams that are steered to different

locations.

D. Improvement in Aberration Estimation

Improvement realized by statistical estimation of aberration by using offset phase screens is

summarized in Fig. 10. The effective radial profiles of the focuses that are formed through

the shallow screens of Aberrators 6277 and 4491 did not vary significantly for different

compensations. In each case, the focus was restored to near water-path levels. However,

compensation using offset-screen-model statistical estimates performed better than

compensation using statistical estimates when focusing through the deep screens of

Aberrators 6289 and 4064. The effective radial profile of the focus through Aberrator 6289

was nearly fully compensated down to the 30-dB level. Similarly, the offset-screen-model

compensation restored water-path performance in Aberrator 4064 down to around the 20-dB

level, and matched the performance of the center-focus compensation down to the 30-dB

level.

E. Improvement in B-Scan Contrast

Aberrator 6289 was selected for investigating the improvement in b-scan images that can be

realized by use of an offset phase-screen model to compensate for aberration. The inclusions

in this aberrator are deeply situated and are modeled by a phase screen with a large offset

that produces significant variations in the predicted aberration at different focuses.

Furthermore, the compensated transmit focus for this aberrator that is shown in Fig. 10 has

low sidelobe levels that should help to produce better image quality.

The 6 gray-scale images in Fig. 11 are created using the methods described in Section III-D.

They are all b-scans that are 3-mm wide and are formed from 15 vertical scan lines. The b-

scan on the left side of the figure (labeled uncomp) is obtained through the aberrator without

any compensation on transmit or receive. The b-scan on the right side of the figure (labeled

water) is obtained through a water path. The 4 b-scans in the middle of the figure are

obtained through the aberrator with compensation. The b-scans labeled tx are formed using

compensation for only the transmit focus and the b-scans labeled tx/rx are formed using

compensation on both transmit and receive. The compensation for the scans labeled a is

derived from a zero-offset phase screen that is statistically estimated from the phase maps of

random scattering measurements. The compensation for the scans labeled b is derived from

an offset phase screen that is statistically estimated from backpropagated phase maps of

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random scattering measurements. Images formed with compensation factors predicted by the

offset phase screen use new compensation factors for each scan line.

The improved performance of the offset phase-screen model evident in the images is

quantified by computing contrast ratios for regions of special interest. The vertically-

centered region in each image possesses the minimum mean intensity over each image and

identifies the scatterer-free cyst-like region in the phantom. Regions for comparison are

chosen above and below the vertically-centered region offset by the same amount in each

image to avoid bias from time-gain compensation that is not perfectly flat. The offset is

chosen also to avoid the specular reflections visible just above and below the vertically-

centered region. The contrast ratio for the tx/rx image that is compensated using the offset

phase screen is within 6 dB of the contrast ratio for the water-path image, whereas use of the

zero-offset phase screen yields a contrast ratio that is about 14 dB away from the contrast

ratio of the water-path image. The vertically-centered region can appear at different depths

due to compensation. Also the average sound speed, that changes significantly when the

aberrator is removed, can cause the vertically-centered region to shift as seen in the water-

path b-scan.

V. Discussion

Screen-depth estimate differences that were obtained from point-source data and from the

corresponding random medium data can be largely attributed to differences in experimental

conditions. In point-source experiments, the aberrator was kept from 1 to 3 mm below the

transducer to avoid deforming the aberrator. The additional offset was measured and the

screen-depth estimates were adjusted accordingly to obtain depths that were relative to the

top of the aberrator. However, in random-medium experiments, the presence of supporting

material below the aberrator (i.e., the random medium phantom) allowed the transducer to

be placed directly in contact with the aberrator. The transducer was, therefore, assumed to be

positioned at the top of the frame that contains the aberrating medium (plastic retaining form

open on the top and bottom) so no adjustments were made to the estimated screen depths.

However, depressions in the surface of the aberrator that dip below the top surface of the

retaining form still introduced small errors into the assumed height of the transducer. These

errors translated into shallower relative screen depths for the estimates obtained from

random media. Error can arise in the screen-depth estimates from random media data when

only a single iteration is used for estimating phase because the change in focus depth caused

mainly by refracting surfaces may not be adequately corrected. Furthermore, error in depth

estimates can arise from incomplete convergence of the statistical estimates that result in

broader cross-correlation peaks used to determine the screen depth.

Although improvement in representation of aberration achieved by using offset models is

shown in Fig. 8, regions in the upper-right and lower-right panels have not been corrected by

the offset. The lack of improvement in these regions is attributed mostly to 2 factors. One

factor is that sources of aberration are located at different depths. These differences

confound attempts to assign a single depth to the model. Some of the aberrators contain

sources of aberration that are very close to the upper surface where the transducer is

positioned. These features appear to be motionless in measurements from focuses at

different locations. However, deep sources of aberration produce measurement features that

shift as the location of the focus varies. Screens that model deep sources of aberrations are

not able to accommodate the motionless features caused by shallow sources of aberration.

Shallow sources of aberration can be seen in the video available at the web page given in

[22]. The other factor is violation of the isoplanatic patch assumption. If the phase changes

associated with shifting the focus location are not projective transformations of an offset

phase screen, they cannot be corrected. This is the main reason for the uncorrected areas in

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Fig. 8. The corruption that appears in Fig. 8 near the edges of the array occurs because the

projection of a phase or time-shift screen onto the transducer array along rays from an offset

focus subtend a rectangle that omits portions of the transducer array. These elements can be

omitted from compensated transmissions, included in the transmissions without

compensation, or populated with other factors. Compensations for these elements in this

work result from the projective transformation of the phase screen that is tapered to reduce

artifact that can arise because the projective transformation is performed in the spatial-

frequency domain.

Fig. 10 shows that statistical models of aberration formed by phase and time-shift screens

that are estimated at offset depths can produce effective compensation for focusing through

complex aberration. However, if the sources of aberration are very deep, as in Aberrator

4064, then the effects of the aberration may not be accurately represented as phase variations

along rays. Use of angular spectra may be more effective in these cases as a method for

propagating fields from the transducer array to the surface of the model screen and vice

versa. The statistical estimate of the screen should then be derived from measured fields that

are more precisely backpropagated to the screen from the transducer array. Compensation

coefficients for a given focal point can also determined by more precise forward propagation

from the focal point through the phase or time-shift screen and then on to the transducer

array.

The b-scans of Fig. 11 further illustrate the benefits of using an offset model for aberration.

Some improvement in b-scan imaging can be realized from statistical models of aberration

that are not offset. This simpler approach may be justified in clinical scanners that do not

require beam steering in lateral directions, because lateral beam steering benefits most from

corrections that are associated with the screen depth. However, b-scans formed with

compensation from models of aberration that are not offset will generally have contrast

ratios that are not as good as the corresponding contrast ratios in b-scans formed with

compensation from offset models because the side-lobes of focuses that employ the offset

models are lower than the sidelobes of focuses that employ models that are not offset. This

can be seen from the focus measurements in Fig. 10.

A region is generally said to be isoplanatic if the same aberration is experienced along the

propagation paths from a single transducer element to all locations in the region. However,

the exact size of the region depends on the stringency of the condition of equivalence for

aberration along different paths. A practical way to define an isoplanatic region is to

measure the quality of the focuses that are produced by a single set of compensation factors

as the beam is steered to different locations. The isoplanatic region is then defined by the

range of locations where the focus is of sufficient quality.

The notion of an isoplanatic region described is linked to aberration models that consist of a

filter-bank at the transducer array. Regions are isoplanatic if the aberration in all signals that

propagate between the region and the transducer array can be modeled by assigning a fixed

(possibly different) filter to each transducer element. However, this interpretation of

isoplanatic is not appropriate when considering more general models of aberration. A useful

extension of the foregoing definition is to call a region isoplanatic when the aberration in all

signals that propagate between the region and the transducer array can be accurately

represented by a single set of model parameters for a given model. The model parameters

can then be estimated from reflections that originate in an extended isoplanatic region and

used to compensate focuses that are steered to locations in the same extended region. Thus,

the significance of the traditional isoplanatic region is preserved by the extended definition.

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