wideband quantitative ultrasonic imaging by time-domain diffraction tomography

$: Wideband quantitative ultrasonic imaging by time-domain diffraction tomography$
Redistrib

Wideband quantitative ultrasonic imaging by time-domaindiffraction tomography

T. Douglas MastApplied Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802

~Received 3 April 1999; revised 27 August 1999; accepted 30 August 1999!

A quantitative ultrasonic imaging method employing time-domain scattering data is presented. Thismethod provides tomographic images of medium properties such as the sound speed contrast; theseimages are equivalent to multiple-frequency filtered-backpropagation reconstructions using allfrequencies within the bandwidth of the incident pulse employed. However, image synthesis isperformed directly in the time domain using coherent combination of far-field scattered pressurewaveforms, delayed and summed to numerically focus on the unknown medium. The time-domainmethod is more efficient than multiple-frequency diffraction tomography methods, and can, in somecases, be more efficient than single-frequency diffraction tomography. Example reconstructions,obtained using synthetic data for two- and three-dimensional scattering of wideband pulses, showthat the time-domain reconstruction method provides image quality superior to single-frequencyreconstructions for objects of size and contrast relevant to medical imaging problems such asultrasonic mammography. The present method is closely related to existing synthetic-apertureimaging methods such as those employed in clinical ultrasound scanners. Thus, the new method canbe extended to incorporate available image-enhancement techniques such as time-gaincompensation to correct for medium absorption and aberration correction methods to reduce errorassociated with weak scattering approximations. ©1999 Acoustical Society of America.@S0001-4966~99!04612-3#

PACS numbers: 43.20.Fn, 43.60.Rw, 43.80.Vj, 43.20.Px@ANN#

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INTRODUCTION

Quantitative imaging of tissue properties is a potentiauseful technique for diagnosis of cancer and other patholcal conditions. Inverse scattering methods such as diffractomography can provide quantitative reconstruction of tisproperties including sound speed, density, and absorpHowever, although previous inverse scattering methods hachieved high resolution and quantitative accuracy, smethods have not yet been incorporated into commercisuccessful medical ultrasound imaging systems.

Current inverse scattering methods are lacking in sevrespects with respect to conventional B-scan and synthaperture imaging techniques. Previous methods of diffractomography, including methods based on the Born andtov approximations,1,2 and higher-order nonlineaapproaches,3,4 have usually been based on single-frequenscattering, while current diagnostic ultrasound scannersploy wideband time-domain signals. The use of widebainformation in image reconstruction is known to provide icreased point and contrast resolution,5,6 both of which areimportant for medical diagnosis.5,7,8

Several approaches have been used to incorporate wband scattering information into quantitative ultrasonic iaging. One group of methods employs time-domain tomraphy based on Radon-transform relationships that h~under the assumption of weak scattering! between scatteredacoustic fields and the reflectivity or scattering strengththe medium. Pioneering work in this area9,10 employed mea-surements of reflectivity in pulse-echo mode, while lastudies have incorporated aberration correction11,12 andmultiple-angle scattering measurements.13,14 A limitation of

3061 J. Acoust. Soc. Am. 106 (6), December 1999 0001-4966/99/10

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

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these methods, however, is that the Radon transform rtionship strictly holds only when the medium is insonifiedan impulsive~infinite bandwidth! wave. When pulses of fi-nite bandwidth are employed, image quality can degrasignificantly.15

A number of linear and nonlinear diffraction tomogrphy methods have been implemented using scatteringfor a number of discrete frequencies~e.g., Refs. 16–19!. Al-though use of multiple-frequency data provides improvments in image quality, computational requirementsmultiple-frequency imaging are typically large becausecomputational cost is proportional to the number of frequcies employed. To achieve image quality competitive wpresent diagnostic scanners, together with quantitative iming of tissue properties, present frequency-domain methmay require solution of the inverse scattering problemmany frequencies within the bandwidth of the transduemployed. This approach thus demands a high computaticost, so that high-quality real-time imaging may not be prently feasible using current frequency-domain inverse stering methods.

Very few previous workers have investigated direct uof time-domain waveform data for inverse scattering meods analogous to frequency-domain diffraction tomograpSeveral methods20,21 have used frequency decompositionscattered pulses to construct a wideband estimate of thetial Fourier transform of an unknown medium; after apprpriate averaging and interpolation, this transform can beverted to obtain a wideband Born reconstruction of tmedium. A study reported in Ref. 22 has showed that broband synthetic aperture imaging using linear arrays is clos

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related to inverse scattering using filtered backpropagatA related method, suggested in Ref. 23, provides a timdomain reconstruction algorithm that employs filtered bapropagation of scattered waveforms measured on a circboundary. However, the time domain reconstruction formof Ref. 23 yields reconstructions that are less general tmultiple-frequency reconstructions obtained using the sasignal bandwidth.

Another approach, related both to multiple-frequenmethods and direct time-domain methods, has recently bpresented.24 This work extends the eigenfunction methodRef. 19 to use the full bandwidth of the incident pulse wavform. In the extended method, eigenfunctions and eigenues of a scattering operator are computed to obtaifrequency-dependent representation of the scatteringdium. Fourier synthesis is then applied to obtain a timdependent estimate of the medium. A cross-correlationeration removes the time dependence of the estimate asas its dependence on the waveform employed.

The present paper offers a new approach to widebquantitative imaging: a time-domain inverse scattermethod that overcomes some of the limitations of previofrequency-domain and time-domain quantitative imagmethods. The new method provides tomographic reconsttions of unknown scattering media using the entire availabandwidth of the signals employed. Reconstructions areformed using scattering data measured on a surfacerounding the region of interest, so that the method is wsuited to ultrasonic mammography. The reconstruction arithm is derived as a simple delay-and-sum formula simto synthetic-aperture algorithms employed in conventioclinical scanners. However, unlike current clinical scannethe present method can provide quantitative images of tisproperties such as the spatially dependent sound speedconstructions obtained in this manner are equivalent toconstructions obtained by combining conventionfrequency-domain diffraction tomography reconstructiofor all frequencies within the signal bandwidth of intereThe current method, however, can be even more efficthan single-frequency diffraction tomography. The methodapplicable both to two-dimensional and three-dimensioimage reconstruction. The direct time-domain nature ofreconstruction algorithm allows straightforward incorpotion of depth- and frequency-dependent amplitude correcto compensate for medium absorption as well as aberracorrection methods to overcome limitations of the Born aproximation.

I. THEORY

A. The time-domain reconstruction algorithm

An inverse scattering algorithm, applicable to quantitive imaging of tissue and other inhomogeneous mediaderived below. For simplicity of derivation, the mediummodeled as a fluid medium defined by the sound speedtrast function

g~r !5c0

2

c~r !221, ~1!

3062 J. Acoust. Soc. Am., Vol. 106, No. 6, December 1999


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wherec0 is a background sound speed andc(r ) is the spa-tially dependent sound speed defined at all pointsr . For thescope of the initial derivation, the medium is assumedhave constant density, no absorption, and weak scattecharacteristics; extensions to the reconstruction algorithat overcome these limiting assumptions are discussethe following section.

For the model of the scattering medium representedEq. ~1!, the time-domain scattered acoustic pressureps(r ,t)obeys the wave equation25

¹2ps~r ,t !21

c02

]2ps~r ,t !

]t25

g~r !

c02

]2p~r ,t !

]t2, ~2!

wherep(r ,t) is the total acoustic pressure in the mediumThe scattering configuration considered here is sketc

in Fig. 1. The medium is subjected to a pulsatile plane wapropagating in the direction of the unit vectora,

pinc~r ,a,t !5 f ~ t2r–a/c0!, ~3!

where f is the time-domain waveform andc0 is the back-ground sound speed. The scattered wavefieldps(u,a,t) ismeasured at a fixed radiusR in the far field, whereu corre-sponds to the direction unit vector of a receiving transduelement.~Alternatively, if scattering measurements are main the near field, the far-field acoustic pressure can be cputed using exact transforms that represent propagathrough a homogeneous medium.16!

A general time-domain solution for the wave equati~2!, valid for two-dimensional~2D! or three-dimensiona~3D! scattering, is then

ps~u,a,t !5E2`

`

ps~u,a,v!e2 ivtdv, ~4!

where ps(u,a,v) is a single frequency component of thscattered wavefield,

ps~u,a,v![1

2p E2`

`

ps~u,a,t !eivtdt, ~5!

given exactly by25

ps~u,a,v!5k2 f ~v!E G0~Ru2r0 ,v!

3g~r0! p~r ,a,v!dV0 . ~6!

In Eq. ~6!, k is the wave numberv/c0 and p(r0 ,a,v) is thetotal acoustic pressure associated with the unit-amplitudecident plane waveeika–r0. The integral in Eq.~6! is takenover the entire support ofg in R2 for 2D scattering or inR3

FIG. 1. Scattering configuration. An incident pressure pulsef (t2a•r /c) isscattered by an inhomogeneous medium and the time-domain scatteredsureps(u,a,t) is measured at a radiusR in the far field.

3062T. Douglas Mast: Time-domain diffraction tomography


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for 3D scattering. The free-space Green’s function, repsented by G0 in Eq. ~6!, is26

G0~r ,v!5i

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and ~7!

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where H0(1) is the zeroth-order Hankel function of the fir

kind andr is the magnitude of the vectorr .The far-field scattered pressure, when specified for

incident-wave directionsa, measurement directionsu, andtimes t, comprises the data set to be used for reconstrucof the unknown medium. The inverse scattering problespecified by Eq.~6! for a single frequency component, isreconstruct the unknown medium contrastg(r ) using themeasured dataps(u,a,v).

The starting point for the present time-domain invescattering method is conventional single-frequency diffrtion tomography. Under the assumption of weak scatterone can make the Born approximation, in which the topressurep(a,v) in Eq. ~6! is replaced by the plane waveikr–a. For scattering measurements made at a radiusR in thefar field, the linearized inverse problem of Eq.~6! can bethen solved for any frequency component using filtebackpropagation,2,16,27 i.e.,

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

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

3 ps~u,a,v!eik(u2a)•rdSadSu , ~8!

where

m~v!52A kR

8ip3,

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

and ~9!

m~v!5kR

4p3, F~u,a!5uu2a u in 3D.

Each surface integral in Eq.~8! is performed over the entiremeasurement circle for the 2D case and over the entire msurement sphere for the 3D case. Equation~8! provides anexact solution to the linearized inverse scattering problema single frequency component of the scattered wavefips(u,a,t). The resulting reconstruction,gB(r ,v), has spatialfrequency content limited by the ‘‘Ewald sphere’’ of radiu2k in wavespace.1

To improve upon the single-frequency formulas spefied by Eq.~8!, one can extend the spatial-frequency contof reconstructions by exploiting wideband scattering infmation. The method outlined here synthesizes a ‘‘multipfrequency’’ reconstructiongM(r ) by formally integratingsingle-frequency reconstructionsgB(r ,v) over a range offrequenciesv. A generalized formula for this approach cabe written



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where g(v) is an appropriate frequency-dependent weiging function. In practice, the weighting functiong(v) is cho-sen to be bandlimited because~for a given set of physicascattering measurements! the frequency-dependent contragB(r ,v) can only be reliably reconstructed for a finite ranof frequenciesv associated with the spectra of the incidewaves employed. Thus, the integrands in Eq.~10! are non-zero only over the support ofg(v) and the correspondingintegrals are finite.

Using Eq.~8!, and making the definition

N[2E0

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Eq. ~10! can be written in the form

gM~r !52

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g~v!m~v!e2 ikR

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

3 ps~u,a,v!eik(u2a)•rdSadSudv. ~12!

If the frequency weightg(v) is now specified to incor-porate the incident-pulse spectrumf (v) and to compensatefor the frequency- and dimension-dependent coefficim(v),

g~v!5f ~v!

m~v!, ~13!

Eq. ~12! reduces to the form

gM~r !52

N E E F~u,a!E0

`

ps~u,a,v!

3e2 ik[R1(a2u)•r ]dvdSadSu . ~14!

The choice of frequency weight from Eq.~13! allows themultiple-frequency reconstruction formula of Eq.~12! to begreatly simplified. Specifically, the inner integral of Eq.~14!resembles a weighted inverse Fourier transform offrequency-domain scattered fieldp(u,a,v). To obtain anexplicit time-domain expression forgM(r ), Eq. ~14! can berewritten using the definition ofps(u,a,v) from Eq. ~5! toyield

gM~r !51

N E E F~u,a!

3L FpsS u,a,R/c01~a2u!•r

c0D GdSadSu , ~15!

whereL denotes the linear operator

L @c~ t !#52E0

`

c~v!e2 ivtdv ~16!

and c(v) is the Fourier transform ofc(t) using the defini-tion from Eq.~5!.

Using the conjugate symmetry ofc(v) @i.e.,c(u,a,v)5c* (u,a,2v) for any realc(t)], the real part of



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L @c(t)# is shown to be simplyc(t). Similarly, using theconvolution theorem as well as the conjugate symmetryc(t), the imaginary part ofL @c(t)# is seen to be an inversHilbert transform28 of c(t),

Im@L @c~ t !##521

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t2tdt5H21@c~ t !#. ~17!

This transform, also known as a quadrature filter, appliephase shift ofp/2 to each frequency component of the inpsignal.

Thus, the time-domain reconstruction formula cannally be written

gM~r !51

N E E F~u,a!S ps~u,a,t!

1 iH21@ps~u,a,t!# DdSadSu , ~18!

where

t5R/c01~a2u!•r

c0. ~19!

The direction-dependent weightF(u,a), which is the sameas the ‘‘filter’’ employed in single-frequency filtered bacpropagation, is given for the 2D and 3D cases by Eq.~9!.

Equation~18! is notable in several respects. First, it prvides a linearized reconstruction that employs scatteringformation from the entire signal bandwidth without any frquency decomposition of the scattered wavefield. Secothe delay termt corresponds exactly to the delay requiredconstruct a focus at the pointr by delaying and summing thscattered wavefieldps(u,a,t) for all measurement directionu and incident-wave directionsa. Thus, the time-domainreconstruction formula given by Eq.~18! can be regarded aa quantitative generalization of confocal time-domain sthetic aperture imaging, in which signals are syntheticadelayed and summed for each transmit/receive pair to foat the image point of interest.22,29,30

A reconstruction formula similar to, although less geeral than, Eq.~18! was independently derived in Ref. 23 fothe two-dimensional inverse scattering problem. In viewthe present derivation, the method of ‘‘probing by plapulses’’ in Ref. 23 can be regarded to yield a multipfrequency reconstruction of Re@gM(r )#, while the presentmethod yields the complex functiongM(r ). In Ref. 23, thismethod was proposed as a more convenient way to imment narrow-band diffraction tomography. However, the nmerical results given below show that the reconstructionmula of Eq. ~18!, when directly implemented usinwideband signals, provides considerable improvement inage quality over narrow-band reconstructions.

Reconstructions using Eq.~18! can be performed usingany pulse waveform. However, the frequency compounddefined by Eq.~10! is most straightforwardly interpretedthe frequency weightg(v) has a phase that is independentfrequency. This criterion can be met, for instance, if thecident pulse waveformf (t) is even in time,

f ~ t !5 f ~2t !, ~20!



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so that f (v) is purely real.@Similarly, if the incident pulsewaveform is odd in time,f (v) is purely imaginary and Eq~18! can still be employed.#

However, supposition of a frequency-independent phfor f (v) does not result in any loss of generality. For alinear-phase signal, such that the Fourier transform hasform

f ~v!5u f ~v!ueivz, v.0, ~21!

an additional delay term of magnitudez can be applied to allscattered signals to obtain the signals associated withpurely-real spectrumu f (v)u. In general, the scattered fielassociated with a desired waveformf (t) can be determinedfor an arbitrary waveformu(t) from the deconvolution op-eration

@ps~u,a,t !# f (t)5F21F f ~v!

u~v!@ps~u,a,t !#u(t) G . ~22!

For stable deconvolution using Eq.~22!, the desiredf (v)should not have significant frequency components outsthe bandwidth ofu(v).

B. Extensions to the reconstruction algorithm

For large tissue structures at high ultrasonic frequencweak scattering approximations such as the Born approxition are of limited validity. Thus, for problems of interestmedical ultrasonic imaging, reconstructed image quality cbe improved by aberration correction methods that incorrate higher-order scattering and propagation effects.present time-domain reconstruction formula~18! provides anatural framework for quantitative imaging with aberratiocorrection. In general, if the background medium is knoor can be estimated, the received scattered signals caprocessed to provide an estimate of the scattered fieldwould be measured for the same scatterer within a homoneous background medium. This approach essentiallymoves higher-order scattering effects from the measuredfield scattering, so that a Born inversion can be performedthe modified data; similar processes occur implicitly in manonlinear inverse scattering methods.31

For example, a simple implementation of aberration crection can be derived if one makes the assumptionbackground inhomogeneities result only in cumulative dlays ~or advances! of the incident and scattered wavefrontThis crude model does not include many propagationscattering effects important to ultrasonic aberration, butbeen shown to provide a reasonable first approximationlocal delays in wavefronts propagating through large-sctissue models.32,33 Given this approximation, the total delafor an anglef and a point positionr is given by

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

R

c0, ~23!

where the integral is performed along the line that joinsspatial pointsr and Rf, Aberration-corrected reconstructions can then be performed using Eq.~18! with t replacedby the corrected delay term



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t→R/c01~a2u!•r

c01dt~a,r !1dt~u,r !. ~24!

Improved approximations could be obtained by applicatof the delay functiondt(f,r ) after numerical backpropagation of the far-field scattered wavefronts through a homoneous medium34,35 or by compensation for both delay anamplitude variations.36,37More general, although much morcomputationally expensive, aberration correction could abe performed by synthetic focusing using full-wave numecal computation of acoustic fields within an estimated reization of the unknown medium. A method of this kind hbeen implemented, within the context of a frequency-domdiffraction tomography method, in Ref. 19.

The present imaging method has been derived ussimplifying assumptions including zero absorption and cstant density for the scattering medium. However, thesesumptions do not substantially restrict the validity of tmethod. For example, the effect of absorption can be reduusing time-gain compensation, with or without frequencdependent corrections,38 of received scattered signals foeach transmit/receive pair. Such time-gain compensacould be performed either using an estimated bulk attention for the medium~as with current clinical ultrasound scanners!, or by implementation of an adaptive attenuation moin a manner similar to the time-shift compensation schediscussed above.

Inclusion of density variations as well as sound spevariations adds additional complication to the time-domdiffraction tomography algorithm derived here. For singfrequency diffraction tomography in the presence of souspeed and density variations, the quantitygB(r ,v) recon-structed by Eq.~8! can be shown39 to provide an estimate oa physical quantity that depends both on sound speed vtions and density variations. In the notation used here,quantity can be written

g8~r !5g~r !2g~r !gr~r !11

2k2¹2gr~r !, ~25!

where the density variation is definedgr512r0 /r(r ).Thus, for time-domain reconstructions of media with densvariations, the reconstruction formula of Eq.~18! will pro-vide the estimate

gM~r !'g~r !2g~r !gr~r !11

2k02

¹2gr~r !, ~26!

where k0 is the wave number corresponding to the cenfrequency of the pulse employed. For media such as hutissue, where density variations are fairly small and abrdensity transitions are rare, the last two terms of Eq.~26! aresmall compared tog(r ), so that the reconstruction algorithmderived above can still be regarded to provide an imagethe sound-speed variation functiong(r ). However, if de-sired, a reconstruction employing pulses with two distincenter frequencies could allow separation of sound speeddensity variations by techniques similar to those describeRef. 16 or 39.



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II. COMPUTATIONAL METHODS

The time-domain inverse scattering method describabove has been tested with 2D and 3D synthetic datapared using three numerical methods: a Born approximamethod for point scatterers and 3D slabs, an exact sesolution for cylindrical inhomogeneities, and ak-spacemethod for arbitrary 2D inhomogeneous media.

The time-domain waveform employed for all the computations reported here was

f ~ t !5cos~v0t !e2t2/(2s2), ~27!

wherev052p f 0 for a center frequency off 0 and s is thetemporal Gaussian parameter. This waveform has theeven Fourier transform

f ~v!5As2

8p~e2s2(v2v0)2/21e2s2(v1v0)2/2!. ~28!

Values used for the computations reported here weref 0

52.5 MHz ands50.25 ms, so that the26 dB bandwidthof the signal was 1.5 MHz. These parameters correspclosely to those of an existing 2048-element ritransducer.40

For the case of point scatterers, the contrast functiogwas assumed to take the form

g~r !5(1

M

m jd~r2r j !. ~29!

Using the far-field form of the 2D Green’s function and nglecting multiple scattering, Eq.~6! for the scattered far fieldcan be rewritten as

ps~u,a,v!52k2A i

8pkRf ~v!(

jm je

ik(a2u)•r j ~30!

for each frequency component of interest. Time-domwaveforms were synthesized by using Eq.~30! for each fre-quency with f (v).1023 and inverting the frequencydomain scattered wavefield by a fast Fourier transform~FFT!implementation of Eq.~4!. The temporal sampling rate employed was 10 MHz. An analogous formula, with a differemultiplicative constant, was also employed for the 3D ca

The Born approximation was also used to compthree-dimensional scattering for slab-shaped objects defiby the equation

g~r !5g0H~ax2uxu!H~ay2uyu!H~az2uzu!. ~31!

For this object, the linearized forward problem can be solvanalytically. Under the Born approximation, the frequencdomain scattered far field has the form

ps~u,a,v!52 f ~v!g0axayazeikR/~pR!

3sin@kLx~a2u!•ex#

kLx~a2u!•ex

sin@kLy~a2u!•ey#

kLy~a2u!•ey

3sin@kLz~a2u!•ez#

kLz~a2u!•ez, ~32!

whereex , ey , andez represent unit vectors in thex, y, andzdirections. The time domain scattered pressureps(u,a,t) is



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obtained, as for the point scatterer case described abovinverse transformation of the frequency-domain wavefifor all frequencies within the bandwidth of interest.

For 2D cylindrical inhomogeneities, an analogous pcedure was followed, except that the frequency-domain stered wavefieldps(u,a,v) was computed using an exact sries solution25 for each frequency component of interest.implementation of the series solution, summations were trcated when the magnitude of a single coefficient dropbelow 10212 times the sum of all coefficients.

Solutions were also obtained for arbitrary 2D inhomogneous media using a time-domaink-space method.41 Gridsizes of 2563256 points, a spatial step of 0.0833 mm, andtime step of 0.02734ms were employed. Scattered acouspressure signals on a circle of virtual receivers were recorat a sampling rate of 9.144 MHz. The receiver circle, whhad a radius of 3.0 mm in these computations, complecontained the inhomogeneities used. 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,16 and performinginverse Fourier transformation to yield time-domain far-fiewaveforms. All forward and inverse temporal Fourier tranforms, as well as angular transforms occurring in the nefield-far-field transformation,16 were performed by FFT.

The time-domain imaging method was directly implmented using Eq.~18!, evaluated using straightforward numerical integration over all incident-wave and measuremdirections employed. The reconstruction formula employcan be explicitly written as

gM~r !51

N2DE

0

2pE0

2p

usin~a2u!uS ps~u,a,t!

1 iH21@ps~u,a,t!# D dadu,

~33!

t5R/c01~cosa2cosu!•x1~sina2sinu!•y

c0

for the 2D case, wherea andu are the angles correspondinto the direction vectorsa andu, and as

gM~r !51

N3DE

0

2pE0

pE0

2pE0

p

ua2uuS ps~u,a,t!

1 iH21@ps~u,a,t!# D sin~Fa!sin~Fu!dFa

3dQadFudQu ,

t5R/c01~a2u!•r

c0, ~34!

a2u5~cosQa sinFa2cosQu sinFu!•ex

1~sinQa sinFa2sinQu sinFu!•ey

1~cosFa2cosFu!•ez

for the 3D case, whereQa andFa are direction angles fothe incident-wave directiona and Qu and Fu are direction



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angles for the measurement directionu. For each case, thenormalization factorN was determined from Eq.~11! withg(v)5 f (v)/m(v) and m(v) given by Eq. ~9!. Beforeevaluation of the argumentt for each signal, the time-domain waveforms were resampled at a sampling rate otimes the original rate. This resampling was performed usFFT-based Fourier interpolation. The inverse Hilbert traform was performed for each signal using an FFT implemtation of Eq.~16!. Values of the pressure signals at the timt were then determined using linear interpolation betwesamples of the resampled waveforms. The integrals of E~33! and ~34! were implemented using discrete summatiover all transmission and measurement directions emplo

Computations were also performed using the timdomain diffraction tomography algorithm for limitedaperture data. For these reconstructions, the integrals of~33! were evaluated only for angles corresponding to tramitters and receivers within a specified aperture of anguwidth fap, i.e.,

uau<fap/2, uu2pu<fap/2. ~35!

Use of a small value forfap corresponds to use of a smaaperture in pulse-echo mode.

FIG. 2. Point-spread functions for time-domain and single-frequencyfraction tomography methods. In each panel, the vertical scale correspto the relative amplitude of the reconstructed contrastg(r ), while the hori-zontal scale corresponds to number of wavelengths at the center frequ~a! Two-dimensional case.~b! Three-dimensional case.



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III. NUMERICAL RESULTS

Two-dimensional and three-dimensional point-sprefunctions ~PSF! for the present time-domain diffraction tomography method are illustrated in Fig. 2. The time-domreconstructions shown here, like the other time-domainconstructions shown in this paper, were obtained usinincident pulse of center frequency 2.5 MHz and a Gaussenvelope corresponding to a26 dB bandwidth of 1.5 MHz.Point-spread functions were determined by reconstructinpoint scatterer located at the origin. For the 2D casewhich the point scatterer can be regarded as a thin wsynthetic scattering data was obtained using the Bornproximation method outlined above for 16 incident-waverections and 64 measurement directions. The 3D timdomain reconstruction was obtained using Born data forincident-wave directions and 288 measurement directioeach evenly spaced on a rectangular grid defined byanglesQ and F. For comparison, analogous point-sprefunctions are also shown for standard frequency-domainfraction tomography reconstructions using single-freque~2.5 MHz! data.

For the 2D case illustrated in Fig. 2, the time-domaPSF has a slightly narrower peak, indicating that point relution has been slightly improved by the increased bawidth employed in the time domain method. More signicantly, sidelobes of the time-domain PSF are significansmaller than those for the single-frequency PSF~the firstsidelobe is reduced by 7 dB, while the second is reduced19 dB!, so that contrast resolution for time-domain diffration tomography is seen to be much higher than for singfrequency diffraction tomography. For the 3D case, the timdomain reconstruction shows a much more dramimprovement over the single-frequency reconstruction.this case, the time-domain solution shows significantcreases in both the point resolution~PSF width at half-maximum reduced by 27%! and contrast resolution~firstsidelobe reduced by 13 dB and second sidelobe reduce18 dB!. Furthermore, a comparison of the PSFs for 2D a3D time-domain reconstruction indicates that much higimage quality is achievable for 3D time-domain imagithan for the 2D case. This increase in image quality suggthat the time-domain diffraction tomography method pposed here may benefit from the overdetermined naturthe general wideband 3D inverse scattering problem.42,43

The effect of transmit and receive aperture charactetics on image quality is illustrated in Fig. 3. Panels~a! and~b! of Fig. 3 show the point-spread function for a numberaperture configurations, each employing 64 measuremenrections. Figure 3~a! shows the point-spread function for reconstructions obtained using 1, 4, 8, and 16 incident-wdirections. The point scatterer is clearly imaged even forreconstruction using one incident-wave direction. Optimimage quality~indistinguishable from reconstructions wit64 incident-wave directions! is obtained for 16 incident-wave directions, so that scattering data obtained usingincident-wave direction for each group of four measuremdirections appears to be sufficient for the present reconsttion method.

The effect of limited view range on the point spre



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function is also illustrated in Fig. 3. Panel~b! shows thepoint-spread function for four differently limited aperturewhile panel~c! shows reconstructions of a homogeneous cinder (a51.0 mm,g50.02) for the same apertures. In eacase, limitation of the transmit and receive aperturesangles near the backscatter direction~aperture sizep/2) re-sults in images that resemble a conventional B-scans. Usapertures corresponding to pulse-echo mode in the laaperture limit~aperture sizep) yield higher resolution in alldirections. Using three-fourths of a circular aperture~size3p/2) yields image quality close to that for the full apertu(2p) case. The characteristics of all these images result fthe set of spatial-frequency vectors interrogated by egroup of scattering measurements.1 Apertures with only alimited range of transmit and receive [email protected]., the‘‘b-scan’’ apertures shown in the first column of panels~b!and ~c!# provide only information corresponding to largspatial frequency vectors oriented nearly on-axis, so tsuch images mainly show those edges that are nearly perdicular to the axis of the aperture.

Reconstructions performed using exact solutionsscattering from cylindrical inhomogeneities providestraightforward means to assess the accuracy of the tdomain scattering method for a range of object sizescontrasts. A number of example reconstructions are showFigs. 4 and 5. The number of measurement directions forcylinder reconstructions was chosen based on an empitest of the number required for a satisfactory image ohomogeneous cylinder; for a cylinder of radius 1 mm, trequired number of measurement directions was determto be approximately 96. Based on spatial-frequency sampconsiderations, the number of measurement directionsincreased in proportion to the size of the inhomogene

FIG. 3. Effect of aperture characteristics on image quality. Each pashows the real part of a time-domain reconstruction, Re@gM#, on a lineargrayscale with white representing the maximum amplitude ofugM(r )u andblack represents21 times the maximum amplitude.~a! Point-spread func-tions for the same waveform parameters as Fig. 2. Each panel shows anof 0.630.6 mm2, corresponding to one square wavelength at the cefrequency. Left to right: 1, 4, 8, and 16 incident-wave directions.~b! Point-spread functions for aperture sizes ofp/2, p, 3p/2, and 2p radians, formatas in previous panel.~c! Real parts of reconstructions for a homogeneocylinder (a51.0 mm,g50.02). The area shown in each panel is 2.032.0mm2.Left to right: aperture sizes ofp/2, p, 3p/2, and 2p radians.



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FIG. 4. Cross sections of reconstructed contrast functionsg(r ) for a cylinder of radius 1 mm, using time-domain~TD! and single-frequency~SF! diffractiontomography. Waveform parameters are as in Fig. 1.~a! g50.02. ~b! g50.04. ~c! g50.06. ~d! g50.08.

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region to be reconstructed. Since the results shown in Fiindicate that considerably fewer incident-wave directiothan measurement directions are needed, the number ofdent directions was chosen to be one-quarter the numbemeasurement directions in each case.

Cross sections of time-domain and single-frequencyconstructions, plotted in Fig. 4, show the relative accuracyeach reconstruction method for a cylinder of 1-mm radand purely real contrast ranging fromg50.02 to g50.08.For the synthetic scattering data in each case, 96 measment directions and 24 incident-wave directions were eployed. The time-domain reconstructions show improvemover the single-frequency reconstructions both in improvcontrast resolution~smaller sidelobes outside the supportthe cylinder! and in decreased ringing~Gibbs phenomenon!artifacts within the support of the cylinder. However, fincreasing contrast values, both methods show similarcreases in phase error, as indicated by increased imagparts of the reconstructed contrast. This error results fromBorn approximation, which is based on the assumptionthe incident wave propagates through the inhomogenemedium without distortion. Perturbations in the local arrivtime of the incident wavefront, which are more severehigher contrasts and larger inhomogeneities, can resultscattered field that is phase shifted relative to the ideal cassumed in the Born approximation; linear inversion of tphase-distorted data naturally results in a phase-distorteconstruction of the scattering medium.~A complementary



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explanation of this phase error, based on the unitarity ofscattering operator, is given in Ref. 19.!

A test of image fidelity for the time-domain reconstrution method is shown in Fig. 5. The real parts of timdomain reconstructions are shown as grayscale imageshomogeneous cylinders with radii between 1 and 4 mmcontrasts betweeng50.02 andg50.08. The number ofmeasurement directions employed for the synthetic scaing data was 96 for the 1-mm radius cylinders, 192 for t2-mm cylinders, 288 for the 3-mm cylinders, and 384 for t4-mm cylinders. In each case, four incident-wave directioper measurement direction were used. The first row offigure corresponds to the time-domain reconstructions shin Fig. 4.

The images shown in Fig. 5 provide a basis for evaluing the ability of the present time-domain diffraction tomoraphy method to image homogeneous objects of varisizes and contrasts. In this figure, images of Re@gM# showuniform quality for small cylinder sizes and contrasts, bpoorer image quality for larger sizes and contrasts. Forlargest size and contrast employed (a54.0 mm,g50.08),the reconstruction primarily shows the edges of the cylinand fails to image the interior. Particularly notable is that t‘‘matrix’’ of images in Fig. 5 is nearly diagonal; that is,linear increase in object contrast causes image degradacomparable to a corresponding linear increase in object sThus, a nondimensional parameter directly relevant to imquality for homogeneous objects iska g, wherek is a domi-



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nant wave number,a is the object radius, andg is the objectcontrast. Using the wave numberk0510.472 rad/mm corre-sponding to the center frequency of 2.5 MHz and a souspeed of 1.5 mm/ms, the reconstructions shown in Fig.indicate that the interior of the cylinder is imaged satisfacrily for the approximate rangeka g,2.5. This result is con-sistent with a previous study of single-frequency diffractitomography, in which adequate Born reconstructions of cinders were obtained for the parameter rangeka g<2.2.44

Reconstructions for several scattering objects withspecial symmetry are shown in Fig. 6. All of these recostructions were performed using synthetic data producedthek-space method described in Ref. 41. Synthetic scattedata were computed for 64 incident-wave directions andmeasurement directions in each case. The first panel shoreconstruction of a cylinder of radius 2.5 mm and contrg520.0295 with an internal cylinder of radius 0.2 mm acontrastg50.0632. These contrast values correspond, baon tissue parameters given in Ref. 32, to the sound-spcontrasts of human skeletal muscle for the outer cylinderof human fat for the inner cylinder. The second panel shoa reconstruction of a 2.5-mm-radius cylinder with randointernal structure. The third reconstruction shown employa portion of a chest wall tissue map from Ref. 45. In thcase, the synthetic data was obtained using a tissue mo45

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FIG. 5. Images of time-domain reconstructions for cylinders of varyradiusa and contrastg. Each panel shows the real part of the reconstruccontrast, Re@gM(r )#, for a pulse of center frequency 2.5 MHz and26 dBbandwidth 1.5 MHz. The area shown in each panel is 2a32a. All imagesare shown on a linear, bipolar gray scale where white represents the mmum amplitude ofugM(r )u and black represents21 times the maximumamplitude.



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good image quality, with high resolution and very little evdence of artifacts. Particularly notable is the accuratelytailed imaging of internal structure for the random cylindand the chest wall cross section. As expected, the denvariations present in the chest wall cross section havegreatly affected the image appearance; there is, howeveslight edge enhancement, associated with the Laplacianin Eq. ~26!, at boundaries between tissue regions. Alsotable is the nearly complete absence of any artifacts outthe scatterer in each case; this result indicates that hightrast resolution has been achieved. However, in each cthe imaginary part of the reconstruction is nonzero, indicing that the Born approximation is not fully applicable. Thimaginary parts of each reconstruction are, however, smcompared to the real parts. Thus, simple aberration cortion methods@of which one example is given by Eq.~24!#could substantially reduce this phase error, as for multipfrequency diffraction tomography in Ref. 19.

Three-dimensional reconstructions of a homogeneslab are shown in Fig. 7. The scatterer is characterized by~31! with g050.01, ax50.5 mm,ay51.0 mm, andaz51.5mm. Synthetic data was computed using Eq.~34! for 288incident-wave directions and 1152 measurement directioeach evenly spaced in the anglesF and Q. Signal param-eters were as for the examples above, except that the insampling rate for the time-domain signals was 9.0 MHz. Isurface renderings of the real part ofgM are shown for thesurfacesgM50.0025. Since the scattering data were otained using a Born approximation for the 3D case,

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FIG. 6. Time-domain reconstructions from full-wave synthetic data for tharbitrary scattering objects. Each row shows the actual~purely real! contrastfunction g together with the real and imaginary parts of the reconstruccontrast functiongM , using the same linear bipolar gray scale for eapanel. Each panel shows a reconstruction area of 535 mm2. ~a! Cylinder,radius 2.5 mm, with an internal cylinder of radius 0.2 mm.~b! Cylinder,radius 2.5 mm, with random internal structure.~c! Tissue structure, withvariable sound speed and density, from chest wall cross section 5Ref. 45.



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imaginary part of each reconstruction is identically zeroboth reconstructions. Consistent with the point-spread futions shown in Fig. 2, the time-domain reconstructionmuch more accurate than the single-frequency reconsttion. While the single-frequency reconstruction shows anroneously rippled surface, the time-domain reconstructiosmooth. The time-domain reconstruction is nearly identito the original object except for some rounding of the shedges due to the limited high-frequency content of the sigemployed. The length scale of the rounded edges is onorder of one-half the wavelength of the highest frequencythe pulse, i.e., about 0.2 mm for the26-dB cutoff of 3.25MHz.

Since three-dimensional inverse scattering is a comptionally demanding problem, comparison of computatioefficiency for single-frequency and time-domain methodsof interest. For both reconstructions shown in Fig. 7, idencal discretizations of the reconstructed medium were eployed. Both computations included solution of the appcable linearized forward problem as well as the inveproblem. Nonetheless, the time-domain method was mefficient than the single-frequency method; the total Ctime required on a 200-MHz AMD K6 processor was 133CPU min for the time-domain method and 287.4 CPU mfor the single-frequency method. This gain in efficiency wpossible because the greatest computational expensecurred in the ‘‘backpropagation’’ of the signals for each rconstruction point. For the single-frequency method, tstep required evaluation of complex exponentials for eincident-wave direction, measurement direction, and spapoint. For the time-domain method, however, the computionally intensive steps~including the forward problem solution and Fourier interpolation of the scattered signa!needed only to be performed once for each transmit/recpair. For the backpropagation step, performed at each pin the 3D spatial grid, the time-domain reconstructimethod required only linear interpolation of the oversampfarfield pressure waveforms.

IV. CONCLUSIONS

A new method for time-domain ultrasound diffractiotomography has been presented. The method provides qtitative images of sound speed variations in unknown mewhen two pulse center frequencies are employed, the meis also capable of imaging density variations. Reconstrtions performed using this method are equivalent to multip

FIG. 7. Three-dimensional reconstructions of a uniform slab with contg50.01. Each reconstruction shows an isosurface rendering of the sugM50.0025. Left: single-frequency reconstruction. Right: time-domainconstruction.



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frequency reconstructions using filtered backpropagation,can be obtained with much greater efficiency.

The time-domain reconstruction algorithm has beenrived as a simple filtered delay-and-sum operation appliefar-field scattered signals. This algorithm is closely relatedtime-domain confocal synthetic aperture imaging, so thacan be considered a generalization of imaging algorithemployed in current clinical instruments. The simplicitythe imaging algorithm allows straightforward addition of fetures such as time-gain compensation and aberration cotion.

Numerical results obtained using synthetic data forand 3D scattering objects show that the time-domain metcan yield significantly higher image quality~and, in somecases, also greater efficiency! than single-frequency diffraction tomography. Quantitative reconstructions, obtaineding signal parameters comparable to those for presentclinical instruments, show accurate imaging of objects wsimple deterministic structure, random internal structure,structure based on a cross-sectional tissue model.method is hoped to be useful for diagnostic imaging prolems such as the detection and characterization of lesionultrasonic mammography.

ACKNOWLEDGMENTS

This research was funded by the Breast Cancer ReseProgram of the U.S. Army Medical Research and MateCommand, under Grant No. DAMD17-98-1-8141. The athor is grateful for helpful discussions with Adrian I. Nachman, Feng Lin, and Robert C. Waag.

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wideband quantitative ultrasonic imaging by time-domain diffraction tomography

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