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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 21, NO. 7, JULY 2002 823
Exact Frequency-Domain Reconstruction forThermoacoustic TomographyI: Planar Geometry
Yuan Xu, Dazi Feng, and Lihong V. Wang*
AbstractWe report an exact and fast Fourier-domain re-construction algorithm for thermoacoustic tomography in aplanar configuration assuming thermal confinement and constantacoustic speed. The effects of the finite size of the detector andthe finite length of the excitation pulse are explicitly included inthe reconstruction algorithm. The algorithm is numerically andexperimentally verified. We also demonstrate that the blurringcaused by the finite size of the detector surface is the primarylimiting factor on the resolution and that it can be compensatedfor by deconvolution.
Index TermsFourier-domain reconstruction, planar, thermo-acoustic tomography.
USING thermoacoustic tomography (TAT) to image bio-logical tissues has two primary advantages. The first isthe high spatial resolution comparable with pure ultrasoundimaging. The second advantage results from the large contrastin microwave absorption that exists between cancerous tissueand the normal tissue . Reviews of TAT and relatedworks  can be found in  and .
Various reconstruction algorithms for TAT , , , , have been reported. Under the approximation that the dis-tance between the detector and the absorbing object is muchlarger than the dimension of the absorbing object, a three-di-mensional (3-D) Radon transform was applied to reconstructthe object in TAT . However, the fact that this approxima-tion may not always hold in real-world situations limits the ap-plication of this method. Further, the spatial resolutions of theimaging system using this reconstruction method are limited byblurs  caused by the finite size of the transducer surface,the finite width of the stimulating pulse, and the finite band-width of the transducers and amplifiers. Among these effects,the blur from the size of the transducer surface is expected to bethe largest contributor to the total blur. The analysis of error is
Manuscript received May 15, 2001; revised May 13, 2002. This work wassupported in part by the U.S. Army Medical Research and Material Commandunder Grant DAMD17-00-1-0455, in part by the National Institutes of Health(NIH) under Grant R01 CA71980 and Grant R21 CA83760, in part by theNational Science Foundation (NSF) under Grant BES-9734491, and in part bythe Texas Higher Education Coordinating Board under Grant ARP 000512-0123-1999. The Associate Editor responsible for coordinating the review ofthis paper and recommending its publication was G. Wang. Asterisk indicatescorresponding author.
Y. Xu and D. Feng are with the Optical Imaging Laboratory, Departmentof Biomedical Engineering, Texas A&M University, College Station, TX77843-3120 USA.
*L. V. Wang is with the Optical Imaging Laboratory, Department of Biomed-ical Engineering, Texas A&M University, 3120 TAMU, College Station, TX77843-3120 USA (e-mail: LWang@tamu.edu).
Publisher Item Identifier 10.1109/TMI.2002.801172.
limited in numerical simulations, and, hence, no analytical formwas available prior to this work. A time-domain beam-formingtechnique was applied in one study to image reconstruction forthe photoacoustic scanning of tissue structures . A weighteddelay-and-sum algorithm was used to account for the near-fieldeffect and to reduce noise. This algorithm is an approximate one,and its lateral resolution is limited by the size of the detector sur-face. The above reconstructions were implemented in the timedomain and consequently are time-consuming, especially in 3-Dtomography. TAT was also obtained in a way similar to that usedin conventional B-scan ultrasonic imaging, but it had difficultydetecting the boundaries of objects that are oblique to the trans-ducer axis . Exact reconstructions have been implementedfor TAT in spherical and cylindrical configurations in the com-panion papers , .
Next, we present our studies on an exact and fast reconstruc-tion algorithm using a Fourier transform for TAT in a planarconfiguration. The reconstruction of an image by Fourier trans-form has been used in X-ray computed tomography , ul-trasonic reflectivity imaging , and diffraction tomog-raphy  successfully. The computation complexity is reducedgreatly due to the efficiency of the Fourier transform. We devel-oped image reconstruction by Fourier transform for planar TATand obtained an exact reconstruction algorithm for the first time.Furthermore, some limitations from experiments, such as the ef-fects of the finite size of the detectors and the finite length of theexcitation pulse, are included explicitly in the reconstruction al-gorithm. The reconstruction algorithm is verified by both nu-merically simulated and experimental results. Our simulationsalso demonstrate that the blur due to the finite size of the de-tector surface, which is a key limiting factor on the resolutionof images , , can be alleviated by deconvolution with re-spect to the size of the detector surface. Other effects that maycause blurring of images can be treated in a similar way. In ourinitial experiments, an image in good agreement with the realobjects was reconstructed and the deconvolution improved theresolution of the imaging system.
A. Image Reconstruction
Assume that the detector scans within the plane andthat the object is distributed only in the half space .In order to obtain a spatial resolution of about 1 mm, the mi-crowave pulse should be set to less than 1 s because thespeed of sound in soft biological tissue is 1.5 mm/ s. Forthese parameters, the diffusion term in the heat conduction equa-tion is about six orders of magnitude less than the term of the
0278-0062/02$17.00 2002 IEEE
824 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 21, NO. 7, JULY 2002
first-derivative of the temperature . Therefore, heat conduc-tion can be ignored. This is known as the assumption of thermalconfinement. In this case, the acoustic wave is related tomicrowave absorption by the following wave equation:
where , is the acoustic speed, is the specific heat,and is the coefficient of volume thermal expansion. In (1), theacoustic speed is assumed constant, which will be further ad-dressed in the discussion section. Equation (1) can be rewrittenin terms of :
where . The source term can further bewritten as the product of a purely spatial and a purely temporalcomponent, i.e.,
where is a scaling factor proportional to the incident radiationintensity, describes the microwave absorption propertiesof the medium at . describes the shape of the irradiatingpulse and is a nonnegative function whose integration over timeequals the pulse energy. Substituting (3) into (2) results in
We proceed by transforming the time-dependent wave equa-tion into the temporal-frequency domain. Denoting the Fouriertransforms of and by and , we have
Substituting (5)and (6) into (4) results in
(7)If the acoustic signals are collected along a line or in a plane,for example at , following the line of Nortan and Linzerin , it can be shown that for the case and
where , is the signum function
Equation (8) can further be simplified to
The lower limit of the above integration is changed from 0 tobecause when . Equation (11) gives
an exact mapping relation between the spectrum of the collectedsignals and the spectrum of the distribution of microwave en-ergy deposition and is the essence of our reconstruction method.However, (11) stands only if the acoustic detector is a point de-tector. In practice, the detector is of finite size, whose surfaceshape can be described by . The signal from the detector
can be expressed as an integral of the acoustic waveover the detector surface
After transforming (13) into the temporal- and spatial-frequencydomain, we have
where is the temporal and spatial Fourier transformof , and is the spatial Fourier transform of
. Substituting (14) into (11) results in
Mapping the space into the space by therelation
yields an explicit expression for
At last, the distribution of the microwave energy deposition canbe reconstructed from by 3-D inverse Fourier transform.Equation (17) gives an exact reconstruction algorithm for planar
XU et al.: EXACT FREQUENCY-DOMAIN RECONSTRUCTION FOR THERMOACOUSTIC TOMOGRAPHY: I. PLANAR GEOMETRY 825
TAT for the first time. Furthermore, the effects of the finite sizeof the detectors and the finite length of the excitation pulse areincluded explicitly. From (17), it can be inferred that the re-constructed image spectrum from the experimentaldata without the consideration of these two effects, as was pre-sented by previous researchers , , is related to the actualimage spectrum by
Both of the effects result in multiplications of a function to theactual image spectrum in the frequency domain. They are equiv-alent to convolutions in the spatial domain, which blur the re-constructed image. However, given the pulse shape and the sur-face configuration of the detector surface, the two effects can bereduced by deconvolution.
To summarize, the reconstruction procedure consists of thefollowing steps.
1) The signal from the detector is Fourier trans-formed with respect to to yield . Deconvolu-tion with respect to the finite pulse length can be imple-mented immediately after the Fourier transform.
2) is Fourier transformed with respect to and, yielding .
3) According to (16) an