aberration correction for time-domain ultrasound diffraction tomography

$: Aberration correction for time-domain ultrasound diffraction tomography$
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Aberration correction for time-domain ultrasounddiffraction tomography

T. Douglas Masta)

Applied Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802

~Received 9 August 2001; revised 26 March 2002; accepted 4 April 2002!

Extensions of a time-domain diffraction tomography method, which reconstructs spatiallydependent sound speed variations from far-field time-domain acoustic scattering measurements, arepresented and analyzed. The resulting reconstructions are quantitative images with applicationsincluding ultrasonic mammography, and can also be considered candidate solutions to thetime-domain inverse scattering problem. Here, the linearized time-domain inverse scatteringproblem is shown to have no general solution for finite signal bandwidth. However, an approximatesolution to the linearized problem is constructed using a simple delay-and-sum method analogous to‘‘gold standard’’ ultrasonic beamforming. The form of this solution suggests that the full nonlinearinverse scattering problem can be approximated by applying appropriate angle- andspace-dependent time shifts to the time-domain scattering data; this analogy leads to a generalapproach to aberration correction. Two related methods for aberration correction are presented: onein which delays are computed from estimates of the medium using an efficient straight-rayapproximation, and one in which delays are applied directly to a time-dependent linearizedreconstruction. Numerical results indicate that these correction methods achieve substantial qualityimprovements for imaging of large scatterers. The parametric range of applicability for thetime-domain diffraction tomography method is increased by about a factor of 2 by aberrationcorrection. © 2002 Acoustical Society of America.@DOI: 10.1121/1.1481063#

PACS numbers: 43.20.Fn, 43.80.Qf, 43.60.Pt@LLT #

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I. INTRODUCTION

This paper concerns time-domain diffraction tomogphy methods for solution of the time-domain inverse scating problem, in which an unknown inhomogeneous mediis determined from its far-field acoustic scattering. This prolem is of interest for medical ultrasonic imaging, sinceverse scattering methods such as diffraction tomographyprovide quantitative reconstruction of tissue propertiescluding sound speed, density, and absorption.

Most practical inverse scattering methods to date hbeen based on linearization of the inverse problem usingBorn or Rytov approximation.1,2 These are weak scatterinapproximations, in which the variation of medium propertis assumed to be a small perturbation from a uniform baground. Nonlinear inverse scattering methods,3,4 which con-sider contributions of strong and multiple scattering, amuch more complex and computationally intensive. Hoever, since large-scale tissue structures cannot be considweak scatterers at diagnostic ultrasound imagfrequencies,5,6 linearized inverse scattering methods arelimited use for medical ultrasonic imaging.

A similar problem arises in conventional B-scan asynthetic-aperture imaging,7,8 which form the basis for cur-rent diagnostic ultrasound scanners. Current scannerssynthetic images based on the assumption of a uniform bground sound speed, which is essentially the Born apprmation. The invalidity of this assumption is associated w

a!Current address: Ethicon Endo-Surgery, 4545 Creek Rd., ML 40, Cinnati, OH 45242. Electronic mail: [email protected]

J. Acoust. Soc. Am. 112 (1), July 2002 0001-4966/2002/112(1)/

ution subject to ASA license or copyright; see http://acousticalsociety.org/c

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image artifacts and focus aberration.5,9 Considerable efforthas been devoted to methods for aberration-corrected iming, which is analogous to nonlinear inverse scattering. Aproaches to aberration correction for pulse-echo imaghave been designed to correct distortion associated witheral simplified propagation models, including refractionhomogeneous layers,10,11 phase aberration close to the tranducer aperture,12–14 and aberration caused by a hypotheticphase screen away from the aperture.15–17All of these aber-ration correction methods require indirect estimation ofmedium-induced distortion based on the received scattedata.

A time-domain diffraction tomography method has beintroduced recently.18,19 This method provides tomographireconstructions of unknown scattering media from scatterdata measured on a surface surrounding the region of inest, using the entire available bandwidth of the signals eployed. The reconstruction algorithm is derived as a simdelay-and-sum formula similar to synthetic-aperture algrithms employed in conventional clinical scanners.7,8 How-ever, unlike current clinical scanners, the present metprovides quantitative images of the spatially dependent tissound speed. These quantitative sound speed maps offersiderable potential for aberration correction, sincemedium-induced distortion can be estimated directly frothe image data.

The image reconstruction algorithm of Ref. 18 was drived from the frequency-domain exact solution to the lineized inverse scattering problem, i.e., diffraction tomograpemploying the Born approximation. Inverse scattering aproaches based on the Born approximation form adeq-

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images only for relatively small, weakly scatterinobjects,18,20 so that this approximation has limited utility folarge-scale imaging problems such as ultrasonic mammophy. In the present paper, an aberration correction approwhich significantly extends the range of validity of the timdomain diffraction tomography method, is introduced. Treconstruction method of Ref. 18 is shown to result inapproximate solution to the time-domain linearized invescattering problem; application of aberration correctionsults in reconstructions that better approximate the soluto the full nonlinear time-domain inverse problem.

Two related methods for aberration correction are psented here. The first, suggested by the synthetic-apenature of the reconstruction algorithm, employs a focus crection approach in which delays are computed from emates of the medium using an efficient straight-ray appromation. The second approach is suggested by examinatiothe reconstruction itself in the time domain, as in Ref. 21.this approach, delays are applied directly to a time-depenlinearized reconstruction. Numerical results show that bmethods increase the parameter range for which valid imacan be obtained and illustrate differences in performancetween the two.

II. THEORY

The imaging problem considered here concerns recstruction of an unknown medium from far-field, time-domascattering measurements. Solutions of this inverse probare quantitative images of scattering media such as biolcal tissue. Below, the linearized inverse scattering prob~e.g., quantitative ultrasonic imaging without aberration crection! is considered and shown to have no general solutHowever, approximate solutions to the nonlinear inveproblem result in useful aberration correction methodsquantitative imaging.

A. The linearized time-domain inverse scatteringproblem

The time-domain inverse scattering problem analyzbelow is defined as follows. A quiescent, inhomogeneofluid medium is subjected to an incident plane wave pupropagating in the directiona,

pi~r ,t !5u~ t2r•a/c0!, ~1!

wherec0 is a reference or ‘‘background’’ sound speed. Tmedium is assumed to have spatially varying sound spconstant density, and no absorption, and to be complecharacterized by a contrast functiong(r ), defined as

g~r !5c0

2

c~r !221, ~2!

wherec(r ) is the local sound speed at positionr . The inversescattering problem is the determination of the medium ctrast g(r ) from time-domain measurements of the scattefield ps(u,a,t) for all measurement directionsu, incident-wave directionsa, and timest. The implicit neglect of den-sity variations is not severely limiting, since the contragiven by Eq.~2! typically dominates reconstructed imageven in the presence of density variations.18

56 J. Acoust. Soc. Am., Vol. 112, No. 1, July 2002


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A general time-domain solution for the scattered acotic pressure at a far-field measurement radiusR, valid fortwo-dimensional~2D! or three-dimensional~3D! scattering,is then

ps~u,a,t !5F21@ ps~u,a, f !#[E2`

`

ps~u,a, f !e2 i2p f t d f ,

~3!

where ps(u,a, f ) is a single frequency component of thscattered wavefield, given in the far field by

ps~u,a, f !5F@ps~u,a,t !#d

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ps~u,a,t !ei2p f t dt

5k2G~R, f !EV0

e2 iku•rg~r0! pu~r ,a,v! dV0 .

~4!

In Eq. ~4!, k is the wave numberv/c0 andpu(r0 ,a,v) is thetotal frequency-domain acoustic pressure associated witincident plane waveu( f )eika•r0 @i.e., one frequency component of the plane wave pulseu(t2a•r /c0)#. The integral inEq. ~4! is taken over the entire support ofg in R2 for 2Dscattering or inR3 for 3D scattering. The termG(R, f ), as-sociated with the far-field forms of the free-space Greefunctions for the Helmholtz equation,22 is

G~R, f !52A i

8pkRfor 2D scattering,

~5!

G~R, f !51

4prfor 3D scattering.

The time-domain inverse scattering problem is giventhe Fourier inverse of Eq.~4!:

ps~u,a,t !5EVL FpuS r ,a,t2

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u•r

c0D Gg~r ! dV, ~6!

where pu(r ,a,t) is the total time-domain acoustic pressuassociated with the incident plane waveu(t2r•a/c0) andthe linear operatorL is defined as

L @p~r ,a,t !#

51

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8pkRF@ p~r ,a,t !#G for 2D scattering,

~7!

L @p~r ,a,t !#521

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p~r ,a,t ! for 3D scattering.

Equation~6! defines a nonlinear inverse problem for the cotrastg(r ); the nonlinearity is associated with the dependenof p(r ,a,t) on g(r ).

The nonlinear time-domain inverse scattering probldefined by Eq.~6! can be linearized by invoking the Borapproximation, in which the total acoustic pressure isproximated by the incident wave. The resulting linearizequation is

T. Douglas Mast: Abberation correction for diffraction tomography


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ps~u,a,t !5EVL @u„t2t~u,a,r !…#gL~r ! dV, ~8!

where the true potentialg(r ) has been replaced bygL(r ), ahypothetical solution to the linearized inverse problem, athe propagation delay termt(u,a,r ) is defined

t~u,a,r ![R

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c0. ~9!

The delay specified by Eq.~9! is precisely that required torefocus scattered waves through a homogeneous (c5c0)medium onto each image point.

In the asymptotic weak scattering limit, the linearizinverse scattering problem~8! is equivalent to the originanonlinear problem~6!, so that an exact solution for anwaveformu(t) is given bygL(r )→g(r ) asg(r )→0. How-ever, unlike the frequency-domain linearized inverse scaing problem, the inverse problem of Eq.~8! has no generasolution for nonzerog(r ). To prove this, one may examinthe Fourier transform of Eq.~8!, which is simply the linear-ization of Eq.~4!:

ps~u,a, f !5G~R, f !u~ f !EVe2 ik~u2a!•rgL~r ! dV, ~10!

where k is the wave number 2p f /c0 . Thus, any generatime-independent solution of Eq.~8! must also be afrequency-independent solution to the linearized frequendomain inverse scattering problem~10!.

For u5a ~the forward scattering case!, Eq. ~10! leads tothe condition

ps~u,u, f !e2 ikR

k2u~ f !5

1

4pR EVgL~r !dV5const~; f ! ~11!

for existence of a general solution to Eq.~8!. This require-ment is easily seen by counterexample to be impossible.example, Eq.~11! requires that, for all frequenciesf, themagnitude of the forward scattered pressure should~for aunit-amplitude incident wave! be proportional tof 2. A coun-terexample is given by any high-contrast scatterer~e.g., g;1!, for which this f 2 dependence occurs only at very lofrequencies, such that the scatterer’s dimensions are msmaller than the wavelengthc0 / f .23 Thus, although the nonlinear time-domain inverse scattering problem has an esolution @equal to the true contrastg(r )#, the correspondinglinearized problem has no general solution for arbitrary snal bandwidth except in the limiting caseg→0.

B. Approximate linearized solutions by Fouriersynthesis

Although no solutiongL(r ) to the quantitative imagingproblem of Eq.~8! exists in general, one can still obtaapproximate solutions by applying Fourier synthesis towell-known exact solution of the frequency-domain lineaized inverse scattering problem. For any frequency comnent of ps(u,a,t), the frequency-domain linearized inversproblem ~10! has an exact, frequency-dependent solutgiven by the frequency-domain filtered backpropagatformula.2,24

J. Acoust. Soc. Am., Vol. 112, No. 1, July 2002 T. D


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gB~r , f !5m~ f !e2 ikR

u~ f !E E F~u,a! ps~u,a, f !

3eik~u2a!•r dSa dSu , ~12!

where

m~ f !5A kR

8ip3, F~u,a!5usin~u2a!u in 2D,

~13!

m~ f !5kR

4p3 , F~u,a!5uu2au in 3D.

Each surface integral in Eq.~12! is performed over the entiremeasurement circle for the 2D case and over the entire msurement sphere for the 3D case.

Fourier inversion of Eq.~10! into the time domain canbe performed using the convolution theorem.25 The result,with the hypothetical linearized solutiongL(r ) replaced bythe Born reconstructiongB(r , f ), is

ps~u,a,t !521

4pc02R

EVu~ t2t~u,a,r !! ^ gB~r ,t ! dV,

~14!

where gB(r ,t) is the inverse Fourier transform of thfrequency-domain solutiongB(r , f ). The time-domain re-constructiongB(r ,t) is an exact solution of the integral eqution ~14!, which is similar but not equivalent to the linearizetime-domain inverse scattering problem of Eq.~8!. BecausegB(r , f ) is conjugate symmetric, the time-domain potentgB(r ,t) is purely real.21

Comparison of Eqs.~8! and~14! shows that, in the weakscattering limit,

gB~r ,t !→g~r !d~ t !1c~r ,t !, ~15!

wherec(r ,t) is a ‘‘nonradiating source’’26 that satisfies theconstraint

EVu„t2t~u,a,r !…^ c~r ,t ! dV50. ~16!

The presence of the nonradiating source termc(r ,t) is con-sistent with the nonuniqueness of solutions to Eq.~14!.27 Forexample, additional solutions to Eq.~14! include the class offunctionsgB(r ,t)1f(r ), wheref(r ) is the inverse Fouriertransform of any functionf(k) that is zero inside the Ewaldsphere,1 defined for the upper frequency limit of the incidepulse ask<4p f h /c0 , where f h is the upper limit of thepulse frequency content.

A straightforward approach to estimategB(r ,t) @andthusg~r !# is to perform inverse Fourier transformation on tfrequency-domain Born inversiongB(r , f ). A natural esti-mate of the medium contrast is a reconstruction employinformation from multiple frequencies contained in the incdent pulse, e.g.,

57ouglas Mast: Abberation correction for diffraction tomography


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gv~r ,t !5E2`

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gB~r , f !v~ f !e22p i f t d fY E2`

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v~ f ! d f

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5gB~r ,t ! ^ v~ t !/v~0! ~18!

'g~r !v~ t !/v~0!, ~19!

where the final expression results from Eq.~15!. The fre-quency weightv( f ) must be integrable and have no suppoutside the support ofu( f ), but is otherwise arbitrary. Thetime dependence of the reconstructed contrast can bemoved from Eq.~19! by settingt50 ~called the ‘‘imagingcondition’’ in Ref. 28!.

If the incident waveform is sinusoidal, so that, for istance,u( f )5d( f 2 f 0)1d( f 1 f 0), the reconstructed potential gv(r ,0) is equal to the real part of the frequency-domasolution gB(r , f 0). Thus,gv(r ,0) is an exact solution of thelinearized inverse problem in the single-frequency limHowever, as proven above, no time-independent reconstion can solve the general linearized time-domain invescattering problem, so thatgv(r ,0) is only anapproximatesolution for any nonzero-bandwidth incident waveformu(t).

The Fourier inversion of Eq.~17! can be performed either numerically or analytically. Numerical inversion, usinfrequency-domain reconstructions at a number of discfrequencies within the bandwidth of the incident pulse, wthe approach employed by Lin, Nachman, and Waa21

~However, the frequency-domain inversions of Ref. 21 wperformed using eigenfunctions of the far-field scatteroperator29 instead of filtered backpropagation.! Alternatively,particular choices of the weightv( f ) allow analytic inver-sion of the frequency-domain reconstructiongB(r , f ) intothe time domain, resulting in a simple delay-and-sum fmula. For the weightv( f )5u( f )/m( f )H( f ), whereH( f )is the Heaviside step function, the resulting formula is

gv~r ,t !5ReF 1

N E E F~u,a!„ps~u,a,t!

1 iH21@ps~u,a,t!#… dSa dSuG , ~20!

where

N52E0

` m~ f !

m~ f !d f , ~21!

t is given by Eq.~9!, andH21 is the inverse Hilbert transform operator~quadrature filter!, which results from limitingfrequency integration to the interval~0, `!.18

The reconstruction formula of Eq.~20! is identical tothat derived in Ref. 18 and similar to that derived in Ref. 3In view of the present derivation, these previous methodsunderstood to provide approximate solutions to the linearitime-domain inverse scattering problem~8!.

C. Aberration-corrected solutions

The form of the approximate linearized solution derivabove suggests possible approaches to improvement oages beyond the limits of the Born approximation.



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First, one may observe that the reconstruction formulaEq. ~20! synthetically focuses the time-domain scattered fiback onto each point in the medium.18 This observation leadsto the idea of aberration correction by iterative refinementhe focus quality. Since the reconstruction provides an emate of the medium itself, this refinement is fairly straighforward. One simple implementation employs an assumpthat background inhomogeneities result only in cumulatdelays ~or advances! of the incident and scattered wavefronts, so that the total delay for an anglef and a pointposition r is given by

dt~f,r !5Ejc~j!21dj2

R

c0, ~22!

where the integral is performed along the line that joinsspatial pointsr and Rf, Aberration-corrected reconstructions can then be performed using Eq.~20! with t replacedby the corrected delay term

t→R/c01~a2u!•r

c01dr ~a,r !1dt~u,r ! ~23!

and by then computinggv(r ,0) using Eq.~20!.An alternative approach to aberration correction is m

tivated by the observation, made in Ref. 21, that tempodelays from wave propagation in the inhomogeneous mdium result in corresponding delays to the time-domainconstruction of Eq.~17!. That is, the reconstructed waveforms gv(r ,t) may be delayed or advanced relative to twaveformv(t). In Ref. 21, correction for this temporal aberration was implemented by adaptive demodulationgv(r ,t) from the weighting waveformv(t). Here, envelopedetection is applied togv(r ,t) and the time of maximumenvelope amplitudetmax is found for each pointr , resultingin the aberration-corrected reconstruction

g~r !'gv„r ,tmax~r !…. ~24!

Envelope detection can also be applied to iterative recstructions obtained using the focus correction given by~22!.

III. COMPUTATIONAL METHODS

The present aberration correction methods have btested using simulated scattering data for a number of tdimensional test objects. The computational configuratwas chosen to mimic the characteristics of an available 20element ring transducer.31 The time-domain waveform employed for all the computations reported here was

u~ t !5cos~v0t !e2t2/~2s2!, ~25!

wherev052p f 0 for a center frequency off 0 , taken here tobe 2.5 MHz, ands is the temporal Gaussian parameter. Tvalue of s chosen here was 0.25, which corresponds t26-dB bandwidth of 1.5 MHz.

For 2D cylindrical inhomogeneities, the frequencdomain scattered fieldps(u,a,v) was computed using anexact series solution32 for each frequency component of interest. In implementation of the series solution, summatiwere truncated when the magnitude of a single coeffici



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dropped below 10212 times the sum of all coefficients. Thessingle-frequency solutions, which correspond to Fourierefficients of the time-domain scattered field, were weighand inverted by discrete Fourier transform to obtain the extime-domain scattered field associated with the incidpulse of Eq.~25!. Scattering from cylinders of radius 4.0 mand contrasts ranging fromg50.001 tog50.14 was com-puted on a measurement circle of radius 176 mm for 3incident-wave directions and 96 measurement directioThe sampling rate employed was 9.14 MHz.

Solutions were also obtained for a large-scale bremodel using a time-domaink-space method.33 The breastmodel was obtained by image processing a coronal csection of three-dimensional photographic data from the Vible Human Female data set with a pixel size of 0.333 mHue, saturation, and value were mapped to sound speeddensity using empirically determined relations. Sound spand density were assumed to be linearly proportional;assumption is realistic for mammalian soft tissues.34,35

Sound speed and density maps were smoothed usiGaussian filter to reduce artifacts associated with the slicprocess.~The tissue map employed is shown in Fig. 3.! Thistissue model was scaled down by a factor of 0.6 fromoriginal data set and mapped onto a grid of 5123512 pointswith a spatial step of 0.111 mm. A time step of 0.0546ms,corresponding to a Courant–Friedrichs–Lewy number0.75, was employed. Based on the scaling of the tismodel, the scattered field obtained is equivalent to that offull-scale breast model~largest dimension 75 mm! for a cen-ter frequency of 0.5 MHz.

Scattered acoustic pressure signals were recordedsampling rate of 9.15 MHz for 128 incident-wave directionA circle of 512 simulated point receivers, which had a radof 9.0 mm in these computations~equivalent to a radius o45 mm for a 0.5 MHz center frequency!, completely con-tained the scaled-down breast model. Far-field wavefowere computed by Fourier transforming the time-domwaveforms on the near-field measurement circle, transfoing these to far-field waveforms for each frequency usinnumerically exact transformation method,21,36 and perform-ing inverse Fourier transformation to yield time-domain fafield waveforms at a measurement circle of radius 234 cm~or1170 mm if scaled to a 0.5 MHz center frequency!. All for-ward and inverse temporal Fourier transforms, as wellangular transforms occurring in the near-field–far-field traformation, were performed by fast Fourier transform~FFTs!.37

The time-domain imaging method was directly implmented using Eq.~20!, evaluated using straightforward numerical integration over all incident-wave and measuremdirections employed. In one implementation, similar to thfrom Ref. 18, images were evaluated only for the timet50. In this case, before evaluation of the argumentt foreach signal, the time-domain waveforms were resampledsampling rate of 16 times the original rate. This resamplwas performed using FFT-based Fourier interpolation. Tinverse Hilbert transform was implicitly performed using tsame FFT operation. Values of the pressure signals attime t were then determined using linear interpolation b



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tween samples of the oversampled waveforms. The integof Eq. ~20! were implemented using discrete summation oall transmission and measurement directions employed.

In the implementation of reconstructions for multiptimes, storage and computation time requirements necetated modification of the algorithm implementation. Fmultiple-time reconstructions, a reconstruction ofgv(r ,t) atthe sampling rate of the scattering data was first obtaineddirect integration. Delays of the time-domain scattered waforms were implemented using cubic spline interpolation38

Reconstructions were performed for an interval of lengthms, multiplied by a window with cosine tapers of length 0ms at each end, and upsampled by a factor of 8 using Fouinterpolation. Inverse Hilbert transformation ofgv(r ,t) wasperformed by the same FFT operation used to implementFourier interpolation. Finally, the temporal position of thenvelope peak was found from the zero crossing of thevelope derivative,

]ugv~r ,tpeak!1 iH 21@gv~r ,tpeak!#u]t

50. ~26!

The derivative in Eq.~26! was evaluated using a seconorder-accurate center-difference scheme.

Focus correction was implemented using a straight-approximation, which is based on the assumption that baground inhomogeneities result only in cumulative delays~oradvances! of the incident and scattered wavefronts. In thapproximation, the total delay for an image positionr and adirection f is given by Eq.~22! and aberration-correctereconstructions are performed using Eq.~20! with t replacedby the corrected delay term of Eq.~23!. The path integrals ofEq. ~23! were performed using an algorithm based ondigital differential analyzer ~DDA! image processingmethod.39 This method very efficiently finds the neareneighbors to a line of specified starting position and slothus, the integrals can be evaluated by simple summawithout any need for interpolation. To account for variabstep size along the integration path, this summation is nmalized by multiplication withL/N, whereL is the length ofthe specified line andN is the number of points employed ithe summation. Since the reconstruction process acts inas a low-pass filter, the integral performed using neaneighbors to the line of interest is sufficiently accurate.

Iterative focus correction was performed by first costructing an uncorrected image, either fort50 or t5tpeak.The reconstructed sound speed was then employed to eate the delay corrections of Eq.~23! using the DDA imple-mentation of the integrals from Eq.~22!. To avoid spuriousmodification of image points outside the support of the scterer, the delay term of Eq.~23! was multiplied by the factor

A5H 1, ugv~r !u>gmax/2,

~12cos@2pugv~r !u/gmax# !/2, ugv~r !u,gmax/2,~27!

wheregmax is the maximum value ofugv(r )u for the previousreconstruction and the temporal criterion~t50 or t5tpeak!employed.

Iteration proceeded as follows. A new reconstructiwas compared to the previous reconstruction; if the rela



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rms error between the two was greater than 5%, furtheerations were carried out up to a prescribed maximum nber of iterations, taken here to be 20. The criterion of 5% wchosen because image quality was not substantially enhaby use of lower error thresholds. Due to the efficiency ofdelay computation, each iteration required about the sacomputation time as the original reconstruction.

IV. NUMERICAL RESULTS

The performance of aberration-corrected time-domdiffraction tomography imaging, using the two approachintroduced above, is illustrated by the numerical examppresented in this section.

Figure 1 shows reconstructions of a homogeneous

FIG. 1. Cross sections of time-domain reconstructions with adaptive focorrection for both imaging criteria. Reconstructions are of a homogenecylinder with a radius of 4 mm (ka541.2) and a contrastg50.08. In eachcase, the ‘‘0’’ curve refers to an uncorrected reconstruction, while curlabeled ‘‘1’’ and higher correspond to subsequent iterations of focus cortion. ~a! t50. ~b! t5tpeak.



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inder with a radius of 4 mm and a contrastg50.08. For thecenter frequency of 2.5 MHz, this corresponds to a nonmensional radiuska541.2. Panel~a! shows cross sections oreconstructions obtained using thet50 criterion. The ‘‘0’’curve refers to an uncorrected~Born approximation! recon-struction, while curves labeled ‘‘1’’ and higher correspondsubsequent iterations of focus correction performed usingdelay correction of Eq.~23! as described in Sec. III. Panel~b!shows corresponding cross sections obtained using tht5tpeakcriterion. One may observe that iterative focus corretion greatly improves reconstructions for thet50 criterion.The initial ~Born! reconstruction shows mainly the edgesthe cylinder; further iterations improve the accuracy withthe cylinder interior. This process somewhat resemblesinverse scattering method of layer stripping,40,41 in which anunknown medium is iteratively reconstructed with each itetion probing further into the medium interior.

In contrast, iterative focus correction provides little,any, improvement to the reconstructions obtained usingt5tpeak criterion @Fig. 1~b!#. In this case, the initial reconstruction captures the cylinder interior very well. Furthererations slightly increase the reconstructed contrast neaedges, but also introduce artifacts not present in the inreconstruction. After convergence, the reconstructed valumore accurate than thet50 image for the cylinder edges buless accurate for the interior.

For the reconstructions shown in Fig. 1, images of s1283128 pixels were computed from time-domain scatting data for 96 incident-wave directions and 384 measument directions. The computation time required on a 6Mhz Athlon processor was about 6 CPU min per iterationthe t50 image criterion~about 38 min total for the six iterations performed! and about 45 CPU min per iteration for tht5tpeak criterion.

The relative performance of iterative focus correctiusing the two image criteria is illustrated in Fig. 2. Herreconstructions were based on exact scattering data f4-mm cylinder with contrast 0.01<g<0.12. Since previousstudies have shown that the accuracy of diffraction tomogphy reconstructions is roughly a function of the nondimesional parameterka•g,18,20 the relative error is plotted asfunction of this nondimensional parameter. The Born aproximation is considered to provide useful images for cinders up toka•g;2;18,20 by this standard, the iterative focus correction implemented here increases the upper limvalidity for t50 images toka•g;4. As in Fig. 1, iterativefocus correction is seen to provide little improvement in acuracy for images obtained using thet5tpeak criterion. Thequantitative accuracy of reconstructions is slightly increaby iteration for large values of the parameterka•g, but canbe slightly diminished for smaller values. Also notable is thiteration using thet50 criterion fails completely aboveka•g;4, while thet5tpeak criterion reaches a comparable eror level aroundka•g;4 and then increases graduallyerror with increasing scatterer contrast.

Quantitative images of a large-scale 2D breast modused to generate simulated scattering data in the mannescribed in Sec. IV, are shown in Figs. 3 and 4. Panel~a! ofFig. 3 shows the 2D model used to generate the synth

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data. For this model, the parameterka•g is about 9.3 ifestimated using the sound speed of fat, the center frequof 0.5 MHz, and the largest half-dimension of 37.5 mHowever, a more conservative estimate employing the a

FIG. 2. The rms error for reconstructions of a 4.0-mm-radius cylinder wboth imaging criteria, with adaptive focus correction~solid lines! and with-out ~dashed lines!. ~a! t50. ~b! t5tpeak.

FIG. 3. Reconstruction of a large-scale two-dimensional breast modelsimulated scattering data.~a! Model. ~b! Initial time-domain reconstructionusing t5tpeak criterion.



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age contrastg within the scatterer yieldska•g;2.8, whichmeets the accuracy criterionka•g,4 determined from thecylinder simulations. Panel~b! of Fig. 3 shows the imagereconstructed using thet5tpeak criterion without any focuscorrection. In this case, the reconstructed image appeabe artifactually sharpened compared to the original modAlthough there is a close correspondence between mosttures of the model and the reconstruction, some differenexist. For example, the reconstructed skin thickness isnificantly smaller than that of the actual model in sevelocations.

Reconstructions of the 2D breast model, obtained usthe t50 criterion and iterative focus correction, are shownFig. 4. In this case, the initial~Born! reconstruction rendersthe skin layer fairly well, but the interior of the breast modis reconstructed poorly. Subsequent iterations improverendering of the connective and glandular tissue strucwithin the breast. Both focus quality and quantitative accracy of the reconstructions improve with iteration. The coverged reconstruction~iteration 5! resembles a low-pass filtered version of the original model@Fig. 3~a!# except for asmall area of spuriously high reconstructed contrast witthe interior glandular tissue.

Both reconstruction criteria successfully image tsound speed variation of the 2D breast model, even thothe model also included realistic density variations. This

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FIG. 4. Images of the large-scale breast model obtained using thet50criterion with adaptive focusing. Panel 0 shows the initial~linear! recon-struction and panels 1–5 show the subsequent iterations up to converg



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sult is expected, since diffraction tomography imagessound speed are not greatly degraded by any density vtions that are small and fairly smooth.18 These criteria aremet by the breast model employed here, in which the denvariations were of comparable magnitude to the small~maxi-mum about 6%! sound speed variations.

For the large-scale 2D breast model, computation timrequired for 2563256 pixel images, 128 incident-wave drections, and 512 measurement directions were aboutCPU h per iteration for thet50 image criterion~8.0 h for thesix iterations up to convergence! and about 4.6 CPU h for theinitial reconstruction using thet5tpeak criterion.

V. DISCUSSION

The two abberation correction methods considered hmay be compared as follows. Both methods have the efof improving the alignment of the time-domain reconstrution gv(r ,t). In the case oft50 images with adaptive focucorrection, the time-domain reconstruction is implicitaligned by compensation for propagation delay withininhomogeneous medium. Thet5tpeak criterion can bethought of as an explicit alignment of the time-domainconstruction.

Previous qualitative studies of the validity of the Boapproximation18,20 have established a threshold for valBorn reconstructions atka•g;2, which corresponds to anormalized rms error of about 0.5~Fig. 2!. Given this some-what arbitrary threshold for the maximum allowable errboth aberration correction methods employed here havsimilar range of validity, up to aboutka•g;4. Thus, eitherapproach extends the parametric range of validity for timdomain diffraction tomography by about a factor of 2.

Each image criterion also introduces characteristic afacts. Thet50 criterion with adaptive focusing acts in paas a low-pass filter to reconstructions, consistent withwell-known low-pass filtering effect of conventional diffraction tomography.1 The t5tpeakcriterion introduces edge artifacts that have the qualitative effect of artifactually sharping images. More robust methods of delay estimation, sas cross-correlation between the time-domain reconstrucgv(r ,t) and the modulating waveformv(t),21 may providebetter reconstruction quality than thet5tpeak criterion, par-ticularly for scattering data corrupted by noise or measument imprecision.

The t50 image criterion can provide faster reconstrutions, since the reconstructed contrastgv(r ,t) needs only tobe evaluated for one time. However, for large or higcontrast scatterers, iterative aberration correction is nesary to obtain high-quality reconstructions. Thet5tpeak cri-terion requires longer computation time for eareconstruction; however, because this criterion implicitlycorporates a form of aberration correction, subsequent ittions provide little additional benefit. As a result, compution times required for a given level of accuracy cancomparable for either image criterion.

Notable is that reconstruction quality, as characterizby criteria such as the point-spread function of a quantitaimage, can be improved by optimization of the weig



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v( f ).21 Although the delay-and-sum reconstruction formu~20! depends on a frequency weight determined by the indent waveformu(t), any desired weightv( f ) can still beapplied by preprocessing of the scattering data. That is,inverse problem associated with an arbitrary incident waform w(t) ~such as the impulse response of a particular etroacoustic transducer! can be transformed into the inversproblem associated with a desired waveformu(t) by apply-ing the deconvolution operation

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where F denotes temporal Fourier transformation, to tmeasured scattering data. This operation transforms the msured data into the corresponding data that would be msured using an optimal incident pulseu(t). For reasons ofstability, the effective bandwidth ofu( f ) should be compa-rable to that ofw( f ) ~as determined, for instance, by thnoise floor of a given measurement!.

The adaptive focusing implemented here employedsimple straight-ray approximation for wavefront aberratiincurred in tissue. However, the principle of aberration crection by adaptive focusing should allow greater improvments to be gained using more complete distortion modFor example, the distortion caused by a strongly scattemedium can be accurately modeled using a full-wave coputational method such as that of Ref. 33. In principle, apropriate deconvolution could be employed to removeeffects of the intervening medium for each incident-wadirection, measurement direction, and image location, soan aberration-corrected reconstruction could then beformed by applying Eq.~20! to the corrected scattering datIn some cases,a priori information on the scattering mediummay be exploited to improve the convergence of such adtive focusing algorithms. This basic approach, in whichlinearized reconstruction is performed on scattering datahas been transformed to remove higher-order scatteringfects, is common to a number of existing nonlinear invescattering methods.42

The methods of aberration correction proposed herefer from most adaptive imaging methods for pulse-echotrasound~e.g., Refs. 12 and 16! because adaptive focusingperformed using a direct reconstruction of the medium ratthan a simpler distortion estimate. Thus, aberration corrtion using quantitative imaging methods could be of grinterest for pulse-echo systems such as current clinical sners. However, the limited spatial-frequency informatiprovided in pulse-echo mode1,18 reduces the quality of quantitative images of this kind. One possible approach tocreasing the spatial-frequency content of pulse-echo quatative images could be to apply deconvolution to tscattered signals.43–45 If such deconvolution methods coulincrease the spatial-frequency coverage sufficiently to obaccurate ~although possibly low-resolution! quantitativesound-speed maps, such maps could be employed direfor adaptive focusing in pulse-echo images.



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VI. CONCLUSIONS

Two related approaches to aberration correctionquantitative ultrasonic imaging have been presented. Thmethods are based on approximate solutions to the lineartime-domain inverse scattering problem, implemented usadaptations of two previous time-domain diffraction tomoraphy methods.18,21One approach, based on a delay-and-sreconstruction formula, applies adaptive focusing basedestimates of the scattering medium. The other approimplements aberration correction by applying appropridelays to a time-dependent reconstruction.

Numerical results show that each of the considerederration correction approaches increases the parametric rof validity for time-domain diffraction tomography by aboua factor of 2. The extended range of validity is sufficientallow effective quantitative imaging of large-scale scattermedia, such as the 75-mm breast model imaged here aMHz. Adaptive focusing correction based on more complscattering models could further increase this range of vaity. Given sufficienta priori information on the unknownmedium, the principle of focus correction may allow accrate quantitative images to be obtained for strongscattering media at larger scales and higher frequencies

The approaches presented here may also be usefuaberration correction in pulse-echo imaging. If sufficienbroadband information can be extracted from pulse-escattering data, the time-domain diffraction tomograpmethods considered here may allow quantitative tissue cacterization using clinically convenient measurement cfigurations. Quantitative maps obtained in this manner woalso be useful as medium models for aberration correctioconventional B-scan and synthetic-aperture imaging.

ACKNOWLEDGMENTS

This work was supported by the Breast Cancer ReseProgram of the U.S. Army Medical Research and MateCommand under Grant No. DAMD17-98-1-8141. Any opiions, findings, conclusions, or recommendations expressethis publication are those of the author and do not necessreflect the views of the U.S. Army. The author is grateful fhelpful discussions with James F. Kelly, Adrian I. NachmaFeng Lin, and Robert C. Waag.

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