. INTRODUCTIONhotoacoustic imaging (PAI), which is also often referredo as thermoacoustic imaging, is a new fast-developingechnique of noninvasive medical imaging [1–6]. In theypical PAI of biological tissues, a short-pulsed electro-agnetic wave (usually laser or microwave) is employed
o irradiate the tissue. Absorbing the electromagnetic en-rgy, the tissue radiates photoacoustic waves (normallyltrasound waves), which carry the electromagnetic ab-orption property of the tissue, by thermoelastic expan-ion. Then an ultrasonic transducer (or sometimes aransducer array for fast imaging [3]) is employed to ac-uire photoacoustic signals, from which the electromag-etic absorption distribution image of the tissue can beeconstructed. Combining the high contrast of optical im-ging and the high resolution of ultrasound imaging, PAIs very efficient in the applications of breast tumor detec-ion [7] and blood vessel imaging [8,9]. It has also beenpplied to in vivo imaging of rat brain [10] and in vitroow measurements [5].Photoacoustic tomography (PAT) and photoacoustic mi-
roscopy (PAM) are two main techniques of PAI. Theormer normally uses an unfocused ultrasonic transducero scan around the tissue and provides a high-resolutionmage of the whole tissue in the detection region, whilehe latter uses a focused ultrasonic transducer and fo-uses on imaging the microvascular structure of body sur-ace [9]. This paper studies mainly the PAT reconstructionlgorithm, which aims at reconstructing the electromag-etic absorption distribution from detected photoacousticaves, a key problem of PAT.By now, there already exist many exact and approxi-
ate algorithms for PAT when the ultrasonic transducerollects signals along a sphere, plane, or cylinder in thehree-dimensional (3-D) case or along a circle or line inhe two-dimensional (2-D) case [11–19]. In all exact algo-ithms, it is assumed that photoacoustic signals are de-ected from a full view, 4� sr in the 3-D case or 2� rad inhe 2-D case. However, photoacoustic signals cannot beetected from a full view in many applications, such ashe detection of human breast. In the limited-view PAT,pplying algorithms of the full-view PAT will causeoundaries and sharp details to become blurred [20]. Soar, to the best of our knowledge, no exact and stable al-orithm for the limited-view PAT has been proposed.
In this paper, a novel reconstruction algorithm, the de-onvolution reconstruction (DR), is proposed for PAThen the ultrasonic transducer scans along a sphere in
he 3-D case or a circle in the 2-D case. The key step of theR algorithm is the Fourier-based deconvolution. As a
ast approximate algorithm, the DR is proved to haveood precision and low noise sensitivity for both the full-iew and limited-view cases. Simulation studies are con-ucted, and the results confirm our theoretical prediction.
. THEORY. DR Algorithm for the Full-View PAThe DR algorithm is derived in the 3-D case. Assuming
hat the biological tissue is irradiated by the short-pulsedlectromagnetic wave homogeneously and the soundpeed in the tissue is invariable, the relation between thexcited acoustic pressure p�r , t� at position r and the spa-ial distribution of the electromagnetic absorption A�r� isiven by [11,12]
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C. Zhang and Y. Wang Vol. 25, No. 10 /October 2008 /J. Opt. Soc. Am. A 2437
p�r,t� =�
4�Cp��� d3r�
�r − r��A�r��I��t −
�r − r��
c � , �1�
here t is the time, I�t� is the electromagnetic pulse func-ion, � is the coefficient of volumetric thermal expansion,is the sound speed, Cp is the specific heat, r� is the in-
egral variable, and I��·� is the derivative of I�·�. The PATlgorithm is a typical inverse problem of deriving A�r�rom p�r , t�.
It is usually assumed that the electromagnetic pulseunction is a Dirac delta function and the radiation beginst t=0, i.e., I�t�=��t�. Moreover, the ultrasonic transduceroves along a spherical surface whose radius is r0 and
enter of the sphere is the origin. So it satisfies that rr0 in p�r , t�. Then Eq. (1) can be simplified to [12,16]
p�r0,t� =�
4�Cpc
�
�t��r�−r0�=ct
A�r��
td2r�. �2�
quation (2) can be rewritten as
��0
t
p�r0,t�dt · t = ���r�−r0�=ct
A�r��d2r�, �3�
here �=� / �4�Cpc�.Defining
S�r0,t� = ��0
t
p�r0,t�dt · t = ���r�−r0�=ct
A�r��d2r�. �4�
hen a new function C�r� is constructed as
C�r� = S� r
�r�· r0,
� − �r�
c � , �5�
here � is an arbitrary constant, satisfying that ��2r0.bviously S�r0 , t� is not equal to zero only if �r0�r��max� /c� t� �r0+ �r��max� /c, so C�r� has nonzero valuesnly if
� − �r��max − r0 � �r� � � + �r��max − r0. �6�
ubstituting Eq. (4) into Eq. (5), it is obtained that
C�r� = ���
A�r��d2r�, �7�
here � is the integral surface: �r�−r ·r0 / �r��=�− �r�. Forhe tissue that is included in the scanning sphere, it sat-sfies that �r��max�r0 (�r��max is the furthest distance be-ween the tissue and the center of scanning circle). Basedn Taylor series expansion, it can be obtained that
r� −r
�r�· r0 =��r��2 + r0
2 − 2r� · r
�r�· r0
= r0 ·�1 − 2r� · r
�r� · r0+
�r��2
r02
= r0 · �1 −r� · r
�r� · r0+ �r�,r,r0��
= r0 −r� · r
�r�+ r0 · �r�,r,r0�, �8�
here �r� ,r ,r0� is a small positive quantity and ap-roaches zero if �r��maxr0. If a large � is chosen in Eq. (5)r the detected tissue is small, thereby satisfying that�r��max��−r0, it can be obtained that �r�� �r��max by us-ng Eq. (6). So
�r − r�� = ��r�2 + �r��2 − 2r · r�
= �r� ·�1 − 2r · r�
�r�2+
�r��2
�r�2
= �r� · �1 −r · r�
�r�2+ ��r,r���
= �r� −r · r�
�r�+ �r� · ��r,r��, �9�
here ��r ,r�� is a small positive quantity and approachesero if �r��max �r�. Based on Eqs. (8) and (9), the integralurface � can be rewritten as
he second term on the left side of Eq. (10), which is theifference of two small positive quantities, can be approxi-ated to zero. (However, the error will be enlarged when
he tissue is large or � is set to a small value approachingr0). Then Eq. (7) can be rewritten as
C�r� = ���r−r��=�−r0
A�r��d2r�. �11�
f A�r� and C�r� are considered the input and output of aystem, respectively, the system is easily shown to be ainear shift-invariant one. So
A�r� � h�r� = C�r�, �12�
here h�r� is the impulse response of the system and � ishe continuous convolution. When the input of the systems the 3-D Dirac delta function, the output is h�r�. So itan be obtained that
h�r� = ���r−r��=�−r0
�3�r��d2r� = ����r� − � + r0�. �13�
Accordingly, given that h�r� and C�r� are alreadynown, A�r� can be calculated by a deconvolution ap-roach, which is normally carried out in the frequency do-ain. Appling 3-D Fourier transform to A�r�, h�r�, and�r�, we get A���, h��� and C���, respectively. So
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2438 J. Opt. Soc. Am. A/Vol. 25, No. 10 /October 2008 C. Zhang and Y. Wang
A��� =C���
h���. �14�
n fact, h��� is not an elementary function and cannot beiven by any algebraic expression. Nevertheless, in theractical application C�r� is constructed from the sampledutput of ultrasonic transducer at a series of spatial posi-ions and is therefore in the discrete form. Also, A�r� isiscrete, which represents the pixel values of the recon-tructed image. So Eq. (14) could be carried out by nu-erical methods.Additionally, there may exist some problems if one is
irectly using Eq. (14) to calculate A���. First, h��� maye zero at some points. Second, the error of C��� will benlarged at those points where the amplitude of h��� isather small. As it turns out, an efficient solution is to cal-ulate A��� by [21,22]
A��� =C���
h��� · �1 + /�h����2�, �15�
here is a constant.In sum, the DR algorithm consists of the following
teps: to construct C�r� from p�r0 , t� based on Eqs. (4) and5); to apply 3-D Fourier transform to C�r�, and then C���s obtained; to calculate A��� based on Eqs. (13) and (15);o apply 3-D inverse Fourier transform to A���, and then�r� is obtained.The key step of the DR algorithm is the Fourier-based
iscrete deconvolution. Using fast fourier transformFFT) and inverse fast Fourier transform (IFFT) algo-ithms, the DR algorithm has fast calculation speed.oreover, according to Eq. (6), the smaller � is, the
maller the size of C�r� is. Accordingly, the smaller � is,he faster the DR algorithm is. However, the error of theR algorithm will be increased when � is set to a very
mall value or the tissue is quite large (with fixed r0).herefore, the choice of � confronts a trade-off betweenpeed and accuracy, which is included in the simulationtudies.
. DR Algorithm for the Limited-View PATn the limited-view case, the PAT algorithm is an incom-lete data problem of deriving A�r� from p�r , t� when r of�r , t� does not cover a whole sphere in the 3-D case or ahole circle in the 2-D case.Without loss of generality, we analyze a simple example
here � is the discrete convolution. In the incompleteata deconvolution, we want to derive ai from all of hi andart of ci. It is obviously that ai can be calculated, if c0c is known, by solving the following equations:
2
�a0 · h0 = c0
a0 · h1 + a1 · h0 = c1
a0 · h2 + a1 · h1 + a2 · h0 = c2� , �17�
hen h0�0.In the DR algorithm, incomplete C�r� is constructed
rom the limited-view p�r , t�. Also, Eq. (12) is solvable byisting equations like Eq. (17), the only difference beinghat the image reconstruction is a 3-D or a 2-D case. Ashown by the following simulation studies, in the 2-D caseetected signals over 90° are enough to reconstruct an ap-roximate image by the DR algorithm.However, in the DR algorithm the discrete deconvolu-
ion had better not be solved in the time domain like Eq.17). On the one hand, in practical applications the reso-ution of C�r� and A�r� is always quite large, which willreate difficulty in solving the equations. On the otherand, it is well known that the time-domain deconvolu-ion is very sensitive to data noise. Therefore, an efficientpproach is to carry out the deconvolution in the fre-uency domain, described as follows.Assume that the resolution of A�r� is M1�M2�M3 and
he resolution of the incomplete C�r� is N1�N2�N3, sat-sfying that N1�M1, N2�M2, N3�M3. Obviously only
1�N2�N3 points of h�r� are used in the incomplete con-olution, which is defined as h��r�. The convolution of�r� and h��r� has the size �N1+M1−1�� �N2+M2−1��N3+M3−1�, and its partial region with the first N1N2�N3 values is exactly C�r�. Then the incomplete�r� is filled with zeros (sometimes gradually smoothing
he data to zero at the boundary in order to reduce arti-acts in the reconstruction) to the size �N1+M1−1�� �N2M2−1�� �N3+M3−1�, defined as C��r�. So
A�r� � h��r� � C��r�. �18�
hen A�r� can be calculated by FFT and IFFT algorithmss in Eq. (15). The accuracy and robustness to noise ofhis approach will be validated in the simulation studies.
. Two Popular PAT Algorithmsn the following simulation studies, the DR algorithm isompared with two popular PAT algorithms, the time-omain reconstruction (TDR) [12,15] and the filtered backrojection (FBP) [11,17]. The TDR is an exact algorithmroposed in 2002; and the FBP, proposed as early as in995, is an approximate algorithm based on the Radonransform and is shown to work well for the limited-viewAT [20].These two algorithms were both originally proposed for
he full-view PAT. In the limited-view PAT, p�r , t� is con-atenated with zeros in both algorithms. If one point ofhe scanned object is detected over the solid angle � inhe 3-D case (angle � in the 2-D case), a useful corrections to compensate that point’s value by multiplying the re-onstructed function by a factor 4� /� (2� /� in the 2-Dase) [20].
. RESULTS AND DISCUSSION. Simulation Methodhe computer simulation is conducted with Matlab pro-rams. First, the model of the scanned tissue is estab-
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C. Zhang and Y. Wang Vol. 25, No. 10 /October 2008 /J. Opt. Soc. Am. A 2439
ished. Second, excited photoacoustic signals are simu-ated by the finite-difference time-domain method [23].inally, the tissue image is reconstructed from the simu-
ated signals (with and without noise).The simulation is conducted in the 2-D case to reduce
omputational complexity. The extension of conclusions ofhe 2-D case to a 3-D one is straightforward. However, inhe 3-D case the attenuation of the acoustic signal origi-ating from a point is proportional to 1/r, while in the 2-Dase the attenuation is proportional to 1/�r [16]. Accord-ngly, in the 2-D DR algorithm, Eq. (4) should be rewrit-en as
S�r0,t� = ��0
t
p�r0,t�dt · �t, �19�
hile other steps are the same.
. Full-View PAThe given electromagnetic absorption of the simulatedissue is shown in Fig. 1(a). The image size is 15 mm15 mm. The ultrasonic transducer with a sampling rate
f 20 MHz detects signals along a circle whose radius is.5 mm, with the angle step of 2.25°.The reconstructed images using the TDR, FBP, and DR
lgorithms are shown in Figs. 1(b)–1(j), with different lev-ls of zero-mean Gaussian noise added to simulated pho-oacoustic signals. As shown by Fig. 1, the accuracy of theR algorithm is as good as that of the TDR and the FBP,o matter what level of noise is added. That is, the DR
lgorithm is precise and not sensitive to data noise. The c
ifference among three algorithms’ accuracy and robust-ess to noise is so small that it could hardly be assessedy the naked eye.In the DR algorithm, the parameter � in Eq. (5) is set
o the minimum value: 2r0. Although the error of the DRlgorithm will be increased theoretically when � is set tosmall value or the tissue is large (compared with the
canning radius), the results in Fig. 1 demonstrate thathe DR algorithm has good precision, nearly equivalent tohat of the TDR and the FBP, for a large tissue with theinimum �. Moreover, the minimum � brings the fastest
peed. Therefore, the optimum value of � is 2r0 in theull-view PAT.
A further study is conducted to quantitatively comparehe precision of the three algorithms for tissues of variousizes. Square tissues are simulated, and their recon-tructed images are evaluated by the peak signal-to-noiseatio (PSNR). Results are shown in Fig. 2, where r0 is thecanning radius and r� is the farthest distance betweenhe tissue and the center of scanning circle. Different lev-ls of noise are added to simulated photoacoustic signals.t can be seen in Fig. 2 that when no noise is added, theSNR of the DR is a bit lower than that of the TDR and ait higher than that of the FBP if r� /r0 is smaller than6%. If the tissue is so large that r� /r0 exceeds 76%, theSNR of the DR is the lowest, which is in accordance with
he theoretical prediction. When 6% noise is added, theSNR of the TDR declines more than those of the FBPnd DR. As a result, the PSNR of the DR is the highesthen r� /r0 is smaller than 60%. This is because the cal-
ulations in the TDR are based on the differential of p, in
ig. 1. (a) Original image and (b)–(j) reconstructed images of the simulated tissue with different levels of Gaussian noise added (therayscale denotes the value of the electromagnetic absorption). The three rows from top to bottom, except (a), correspond to reconstructedmages with no noise, 6% noise, and 12% noise, respectively. (b), (e), and (h) TDR results, (c), (f), and (i) FBP results, (d), (g), and (j) DResults.
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2440 J. Opt. Soc. Am. A/Vol. 25, No. 10 /October 2008 C. Zhang and Y. Wang
hich the noise is increased, yet the FBP and DR are bothased on the integral of p, in which the noise is counter-cted.The calculation time costs by the TDR, FBP, and DR al-
orithms are compared in a computer (Intel Pentium 4rocessor with a clock frequency of 2.0 GHz) with Matlabrograms. Results of various reconstruction resolutionsre shown in Table 1. It can be found that the DR algo-ithm is more than ten times faster than the TDR and theBP, the chief reason being that the key step of the DRlgorithm is the discrete deconvolution which can be cal-ulated very quickly by FFT and IFFT algorithms.
. Limited-View PATn the DR algorithm of the limited-view PAT, the reso-ution of C�r� is proportional to the scanning angle andhe parameter u as shown in Eq. (5). While the resolutionf C�r� decreases with the loss of p�r , t�, u should be set tolarger value than in the full-view case ��=2r0� in order
o strike a balance, since the extremely low resolution of�r� would result in large errors. It is shown by simula-
ion studies that �=3r0 is a good choice.The given electromagnetic absorption distribution of
he simulated tissue is shown in Fig. 3. The image size is5 mm�15 mm. The ultrasonic transducer detects sig-als along a circle whose radius is 7.5 mm, with the angletep of 2.25°. We define the angle with the coordinate (x,) as follows: (7.5, 0) is 0°, (0, −7.5) is 90°, (−7.5, 0) is 180°,nd (0, 7.5) is 270°.The reconstructed images of the TDR, FBP, and DR al-
orithms with different scanning angles are shown in Fig.
ig. 2. PSNR of reconstructed images of square tissues by threelgorithms with various r� /r0, with (a) no noise added and (b) 6%oise added.
(with the same coordinates as Fig. 3). The two rowsrom top to bottom correspond to the scanning angles of0° (from 0° to 90°) and 135° (from 0° to 135°). The threeolumns from left to right correspond to the results of theDR, FBP, and DR algorithm, respectively. As shown inig. 4, the reconstructed images of the DR algorithm areuperior to those of the TDR and FBP, with clearer bound-ries and fewer artifacts. The effect of boundary blurrings explained in [20]. Moreover, the smaller the scanningngle, the more obvious the superiority of the DR algo-ithm. This is because the detected signals over 90° arenough to reconstruct an approximate image in the DR al-orithm, but the TDR and FBP both need full-view sig-als. However, the DR algorithm is an approximate one,nd the error will increase with the decrease of scanningngles, as shown in Fig. 4.The profiles through the original image (Fig. 3) and re-
onstructed images (Fig. 4) along two transects (x2.3 mm and y=0 mm) are shown in Fig. 5. Obviously theR algorithm is quantitatively more accurate than theDR and FBP for the limited-view PAT. Generally all
hese algorithms show the accurate contrast level, but theDR and FBP results have more fluctuations and largerrrors at the same level and change more slowly betweenifferent levels compared with the DR results.The robustness of the DR algorithm for the limited-
iew PAT is also verified by adding zero-mean Gaussianoise to simulated photoacoustic signals. Figure 6 showshe reconstructed images of the TDR, FBP, and DR algo-ithms with 6% noise added. Comparing Fig. 6 with Fig., it can be found that the DR algorithm presents strongerobustness to noise than the TDR and FBP in the limited-iew PAT.
The DR algorithm treats PAT as a discrete problem andims at calculating limited pixel values of the recon-tructed image. The plain fact is that most other algo-
Table 1. Time Cost by Three Algorithms withVarious Reconstruction Resolutions
ig. 3. The given electromagnetic absorption distributionshown by grayscale) of the simulated tissue.
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C. Zhang and Y. Wang Vol. 25, No. 10 /October 2008 /J. Opt. Soc. Am. A 2441
ithms, such as the TDR and the FBP, treat PAT as annalytical problem and aim at obtaining the analytical re-ation between the tissue’s electromagnetic absorptionistribution and detected photoacoustic signals, which isuch more difficult than the discrete problem. It has been
hown that although an arbitrarily small scanning arcuffices for the uniqueness of recovery [20,24], the processf computing unmeasured signals from limited-view sig-als is unstable [25,26]. So the exact analytical method isot suitable for the limited-view PAT. Yet the DR algo-ithm starts from sampled photoacoustic signals whoseigh-frequency components are eliminated. Accordingly,nly relatively low-frequency components in the imagere reconstructed on the basis of numerical approaches.n fact, reconstruction of the high-frequency components
ig. 5. Profiles through the original image (as shown in Fig. 3)0° scanning, along x=2.3 mm, (b) 90° scanning, along y=0 mm,
ig. 4. Reconstructed images of the tissue (as shown in Fig. 3).0° (from 0° to 90°) and 135° (from 0° to 135°); the three column
as proved to be the cause of instability in the limited-iew PAT [25]. In other words, the DR algorithm avoidshe difficult problem of exactly reconstructing high-requency components and seeks a near-optimal solution.hese may be the reasons why the DR algorithm is an ef-cient one for the limited-view PAT.
. CONCLUSIONhe DR algorithm is proposed as an approximate algo-ithm for the full-view and limited-view PAT. It is demon-trated theoretically that the DR algorithm is computa-ionally efficient and is suitable for the limited-view PAT,et the error of the DR algorithm increases with the sizef the detected object. Simulation results prove our theory
constructed images (as shown in Fig. 4) along two transects. (a)° scanning, along x=2.3 mm, (d) 135° scanning, along y=0 mm.
o rows from top to bottom correspond to the scanning angles ofleft to right correspond to the results of the TDR, FBP, and DR.
and re
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2442 J. Opt. Soc. Am. A/Vol. 25, No. 10 /October 2008 C. Zhang and Y. Wang
nd show that in the full-view PAT the DR algorithm hasood precision and strong robustness to noise, nearlyquivalent to those of the TDR and the FBP even for rela-ively large objects. Yet the DR is more than ten timesaster than the TDR and FBP due to the available FFTnd IFFT algorithms. In the limited-view case, it is shownhat reconstructed images of the DR algorithm are supe-ior to those of the TDR and FBP with clearer boundaries,ewer artifacts, and stronger robustness to noise. Herehe smaller the scanning angle, the more obvious the su-eriority of the DR algorithm.
CKNOWLEDGMENTShis work was supported by the National Basic Researchrogram of China (No. 2006CB705707), Natural Scienceoundation of China (NSFC) (No. 30570488) and Shang-ai Leading Academic Discipline Project (No. B112).
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noise added. The two rows from top to bottom correspond to thee three columns from left to right correspond to the results of the
ith 6%and th
2
2
2
2
2
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