Image reconstruction of fluorescent molecular tomography ...ira.lib.polyu.edu.hk/bitstream/10397/7066/1/Zou_Image_Fluorescent_Molecular.pdfImage reconstruction of fluorescent molecular

Image reconstruction of fluorescent moleculartomography based on the simplified

matrix system

Wei Zou,1,2,3 Jiajun Wang,1,2,3,* and David Dagan Feng2,3,4

1School of Electronic and Information Engineering, Soochow University, Suzhou 215006, China2Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong, China

3School of Information Technologies, The University of Sydney, Sydney, NSW 2006, Australia4Med-X Research Institute, Shanghai Jiao Tong University, Shanghai 200030, China

*Corresponding author: [email protected]

Received February 19, 2013; revised June 6, 2013; accepted June 10, 2013;posted June 10, 2013 (Doc. ID 185546); published July 3, 2013

Fluorescent molecular tomographic image reconstruction usually involves repeatedly solving large-scale matrixequations, which are computationally expensive. In this paper, a method is proposed to reduce the scale of thematrix system. The Jacobian matrix is simplified by deleting the columns or the rows whose values are smallerthan a threshold. Furthermore, the measurement data are divided into two groups and are used for iteration ofimage reconstruction in turn. The simplified system is then solved in the wavelet domain to further accelerate theprocess of solving the inverse problem. Simulation results demonstrate that the proposedmethod can significantlyspeed up the reconstruction process. © 2013 Optical Society of America

OCIS codes: (100.3190) Inverse problems; (170.3010) Image reconstruction techniques; (170.3880) Medicaland biological imaging; (170.6960) Tomography; (260.2510) Fluorescence.http://dx.doi.org/10.1364/JOSAA.30.001464

1. INTRODUCTIONOptical molecular imaging has been receiving much attentiondue to its nonionizing, low cost, and high sensitivity. It iswidely utilized for drug discovery and tumor detection, as wellas intraoperative navigation [1]. Among the optical molecularimaging modalities, fluorescence molecular tomography(FMT) plays an extremely important role because of its abil-ities to reconstruct the spatial distribution of optical parame-ters, the fluorescent yield, the fluorescent lifetime, etc. In thisimaging modality, a fluorescent biochemical marker used ascontrast agent is injected into the biological system and con-sequently accumulates in diseased tissue as a result of leakyvasculature, hypermetabolism, and angiogenesis [2,3]. Duringthe imaging process, light at the fluorophore’s excitationwavelength is used to irradiate the tissue, and then it is ab-sorbed by fluorophore that presents in the tissue. The fluoro-phore is elevated to an excited state and then decays to theground state while releasing the energy. This creates fluores-cence, which can be separated from the excitation light viainterference filters [4]. The fluorescent photon has a longerwavelength than that of the excitation photon, resulting in acolor shift. Images of the fluorescent yield and lifetime param-eters are reconstructed from several optical measurements onthe surface of the tissue [5,6]. At present, FMT has been suc-cessfully applied to in vivo monitoring inflammation [7],evaluating treatment [8], and investigating breast cancer [9].

Reconstruction of tomographic data involves the generationof a forward model that predicts the observable states as wellas an inverse problem to calculate the internal optical and fluo-rescent properties with the given measured data and sources.

One of the major challenges in the reconstruction of FMT is itshigh computational complexity resulted from extremely large-scale matrix manipulations. In [10], a model-order reductiontechnique is adopted to reduce the system complexity, wherethe unknowns are expressed in the subspace with a reduceddimension. However, in this method, a transformation matrixneeded to be constructedwith theWilson–Yuan–Dickens basisvectors or the Lanczos basis vectors in the Krylov subspace.Actually, during the process of reconstruction, besides thenumber of unknowns, the scale of Jacobian matrix and thenumber of measurement data are two main factors determin-ing the scale of matrix equation and thus determining thespeed of the whole process of reconstruction. The manipula-tion of Jacobian matrix is usually computationally intensiveespecially for the FMT reconstruction problem, where thereare two coupled diffusion equations describing the forwardproblem. It is well known that the Green’s function methodis an effective tool for accelerating the Jacobian matrix com-puting process [11]. However, due to the increasing scale ofthe matrix, the Green’s function method will not work verywell. The problem mainly lies in the fact that the Green func-tion method cannot reduce the scale of the Jacobian matrixitself. Therefore, a new method is proposed in this paper totackle such a problem and accelerate the reconstruction proc-ess of FMT. In this method, the Jacobian matrix is simplifiedby deleting those columns or the rows with values smallerthan a predefined threshold. Although the increment of thenumber of measurement data can improve the quality ofreconstruction results, the scale of the matrices involved inthe reconstruction of FMT will also become larger, whichmay slow down the process of tomographic image

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http://dx.doi.org/10.1364/JOSAA.30.001464

reconstruction. To solve that problem, the measurement dataare divided into two groups, which are used for iteration ofimage reconstruction in turn. With this strategy, both the scaleof Jacobian matrix and the number of measurement data arereduced during the iteration process of reconstruction. Themost important feature of the wavelet transforms lies in thefact that most information of the signal is contained in a smallnumber of entries with other entries being very small andtherefore can be neglected [12]. Based on the above simplifi-cation in spatial domain, the inverse problem is solved inwave-let domain to improve the efficiency of image reconstruction.Simulation results demonstrate that our proposed algorithmcan significantly speed up the reconstruction process at theexpense of a small reduction in reconstruction accuracy.

2. FORWARD MODELA. Light Transport and Fluorescence ModelBasically, the forward model of FMT can be depicted by twocoupled diffusion equations as [13]

−∇ · �De∇Φe� � keΦe � Qe; (1)

−∇ · �Dm∇Φm� � kmΦm � ϕμaef1 − iωτ

Φe; (2)

where quantities with subscript e and m represent thosecorrespond to the excitation and emission wavelength andke;m and De;m can be expressed as

ke;m � μae;mi � μae;mf �iωc; (3)

De;m � 13�μae;mi � μae;mf � μ0se;m�

�4�

with μae;mf and μae;mi being, respectively, the absorption coef-ficient due to fluorophore and nonfluorescing chromophore,μ0se;m being the isotropic scattering coefficient, and c repre-senting the speed of light in the medium; Φe;m is the photondensity, Qe is the excitation light source, ϕ and τ represent thefluorescence quantum efficiency and fluorescence lifetime.

In our case, the Robin type boundary condition is applied tosolve the forward equations.

B. Finite Element DiscretizationThe forward equations can be solved using the finite elementmethod (FEM). In the framework of FEM, the domain Ω isdiscretized into P elements connected by N vertex nodes.Upon the discretization procedure [14], Eqs. (1) and (2)can be expressed under the FEM framework as follows:

AeΦe � Qe; (5)

AmΦm � Qm: (6)

3. RECONSTRUCTION OF FMTA. Inverse ProblemThe task of the inverse problem is to derive the distribution oftissue parameters x with the known distribution of the lightsources and the measurements y, i.e.,

x � F−1�y�; (7)

where F is the forward operator.To linearize the inverse problem, F can be expanded

in the vicinity of x0 in a Taylor series [15]. The Tikhonovregularization term is introduced to improve the ill-posedness of the inverse problem [16]. Therefore, thesolution to the inverse problem in a matrix form can beformulated as

Δx � �JTJ� λI�−1JTΔy; (8)

where Δx is the perturbation in optical or fluorescent prop-erties, Δy is residual data between the measurements andthe predicted data, J is the Jacobian matrix, I is the identitymatrix, and λ is a regularization parameter.

B. Simplification of the Jacobian MatrixAn M × N Jacobian matrix can be defined as

J �

2666664

∂y1∂x1

� � � ∂y1∂xN

..

. . .. ..

.

∂yM∂x1

� � � ∂yM∂xN

3777775: (9)

In fact, each element in the Jacobian matrix is just ameasure of the rate of change in measurement with respectto the optical parameter [17]. During the iterative process ofEq. (8) for FMT reconstruction, the Jacobian matrix Jneeds to be repeatedly updated from iteration to iteration.Therefore, the Jacobian matrix is also a key factor determin-ing the implementation speed of the reconstructionalgorithm.

Conventionally, the direct derivation method and theperturbation method are two straightforward methods toobtain the Jacobian matrix. However, the above two meth-ods are computing intensive and are not suitable forobtaining Jacobian matrices with large-scale. Therefore, itis of critical importance to develop an effective Jacobianmatrix calculation method for a fast implementation ofthe reconstruction of FMT. Methods based on the Green’sfunction are typical ones for such a purpose. Althoughthe theory of using the Green’s function in the derivationof the Jacobian matrix has been well-established [11], theefficiency of the Green’s function method decreases rapidlywith the increment of the dimension of the matrix.Therefore, if the scale of the Jacobian matrix can be re-duced, the computation efficiency of iterative process willbe improved and hence the reconstruction of FMT can alsobe accelerated.

Each column in the Jacobian matrix corresponds to onenode and each element in this column represents the sen-sitivity of one measurement with respect to that node. Whenthe sum of the absolute values of all the elements in onecolumn is very small (e.g., being smaller than a threshold),it means that the sensitivity for the corresponding node islow. It can be proved [in Appendix A(1)] that the column Ji

in the Jacobian matrix can be deleted during the process of

Zou et al. Vol. 30, No. 8 / August 2013 / J. Opt. Soc. Am. A 1465

solving Eq. (8) when Ji approaches 0. In such a case, thescale of the Jacobian matrix can be reduced at an expenseof little accuracy loss in reconstruction upon deleting thosecolumns. Therefore, the computing efficiency of the iterativeprocess can be improved due to the simplification ofthe Jacobian matrix. Based on the above idea, an M × NJacobian matrix can be simplified by deleting the jthcolumn if the following condition of the parameter SUMis satisfied:

SUM �XMi�1

jJijj < a; (10)

where a is a threshold and Jij denotes the element ofthe Jacobian matrix. In this way, the scale of the Jacobianmatrix is reduced and hence the process of imagereconstruction can be accelerated.

Similarly, it can also be proved [in Appendix A(2)] thatthe row Ri of the Jacobian matrix can be deleted as Ri

approaches 0. Therefore, in order to further improve thereconstruction efficiency, the row of the Jacobian matrixcan also be deleted in a similar manner. Suppose that thesum of the absolute values of the ith row of anM × N Jacobianmatrix is smaller than a threshold, i.e.,

XNj�1

jJijj < a: (11)

This inequality implies that sensitivity for the correspond-ing measurement node is low and hence that row can bedeleted. Additionally, since the ith row of the Jacobianmatrix relates to the ith component of Δy, the deletionof the ith row of the Jacobian matrix implies that theith element of Δy should also be deleted so that the prod-uct of JTΔy can be obtained. Therefore, the dimension re-duction in the row direction of the Jacobian matrix canlead to the dimension reduction of the residual data Δy.In this way, the computational requirement can be signifi-cantly reduced.

However, it is possible that there are some prominent el-ements in the column, where the sum of the absolute valuesof all the elements in this column is very small. Thus, someuseful information will be deleted, which will lead to thepoor reconstruction results. To tackle such a problem, an-other parameter, namely PROPORTION, is introduced inaddition to the above mentioned condition in Eq. (10) fordetermining whether a column in the Jacobian matrixshould be deleted. That is, if both of the inequality inEq. (10) and that in Eq. (12) are satisfied, the columncan be deleted:

PROPORTION � maxMi�1 jJijjPMi�1 jJijj

< k: (12)

In such a way, some useful information may be preserved,especially when the predefined threshold for SUM is large.In our simulations, comparisons will be performed amongthe results from the algorithm with and without consideringthe condition in Eq. (12).

In summary, the process for Jacobian matrix simplificationcan be described in Algorithm 1:

(1) Given x, compute Δy and J at x;(2) for (j � 1;…; N) do

if ��PMi�1 jJijj < a� & &��maxMi�1 jJij j∕

PMi�1 jJij j� < k�� then

delete the jth column of Jacobian matrix;delete the jth element of Δx;end

end(3) for (i � 1;…M do

if �PNj�1 jJijj < a� then

delete the ith row of Jacobian matrix;delete the ith element of Δy;end

end

C. Iterative Method Based on the Grouped MeasurementDataAlthough the increment of the number of measurement datacan obviously improve the quality of reconstructed results, thescale of the matrices involved in the reconstruction of FMTwill also become larger and larger, which may slow downthe process of tomographic image reconstruction. In orderto tackle the above problem, we propose to partition themeasurement data into two groups that are then used in turnduring the iteration process of the reconstruction. That is, onegroup is used in the first iteration of reconstruction and theother in the next iteration, etc. In this manner, the numberof rows in the Jacobian matrix is reduced to one half of thatin the Jacobian matrix with all the measurement data in-volved. Hence both the computational complexity and thecomputational requirements for the Jacobian matrix are re-duced, which will be helpful for reducing the computationalburden of our reconstruction problem. One additional superi-ority of this strategy is that the iterative results from one groupof measurement data can provide a good initial guess for thenext iteration for the other group of data. Figure 1 gives aschematic illustration of our reconstruction strategy based

Fig. 1. Scheme of the iteration method based on grouped measure-ment data: (a) measurement data without grouping, (b) measurementdata of red group, and (c) measurement data of black group.


on two groups of measurement data used in turn in the iter-ations of the reconstruction. Figure 1(a) illustrates the meas-urement data without being grouped. In Figs. 1(b) and 1(c),the two groups of measurement data are illustrated with blackcircles and red circles, respectively.

D. Image Reconstruction Based on the WaveletTransformIt is well known that the most important feature of the wavelettransforms lies in the fact that most information of the signal iscontained in a small number of entries with other entriesbeing very small and therefore can be neglected [12]. Somerelated research on wavelet-based image reconstruction hasbeen conducted. In [18], a time-resolved forward model andits projection onto wavelet basis functions have been imple-mented. Ducros et al. apply compression techniques to themeasurements acquired with structured illuminations [19].That method is based on the exploitation of the wavelettransform of the measurements acquired after wavelet-patterned illuminations. In [20], a multiresolution techniquethat uses wavelet decomposition is chosen to reduce compu-tation complexity in the modeling stage. All the matrices areprojected onto an orthonormal wavelet basis and reducedaccording to the wavelet’s properties.

To exploit the multiresolution property of the wavelet andfurther reduce the time of reconstruction process, the simpli-fied system is solved in the wavelet domain upon performingwavelet transform in this paper. For such a purpose, Eq. (8)can be rewritten as

KΔx � b (13)

with K � �JTJ� λI� and b � JTΔy.In Eq. (13), upon connecting all the rows in a head to tail

manner, the perturbations of the two-dimensional opticalparameters is represented with a one-dimensional vectorΔx. To solve the simplified system in the wavelet domain,we perform the wavelet transform in both sides of Eq. (13)and have

K̂Δx̂ � b̂; (14)

where K̂ � WbKWTx , Δx̂ � WxΔx, b̂ � Wbb, Wx, and Wb are,

respectively, the wavelet transform matrix of Δx and b; Wx isan orthonormal matrix.

From Eq. (14), it can be seen that the wavelet transformconducted in Eq. (13) leads to the multiresolution represen-tation of the original signals, such as one-dimensional dis-crete signal Δx and b as well as two-dimensional signal K.Basically, the signal Δx and b with N components can bedescribed as

Δx̂N×1 ��A−1ΔxN∕2×1D−1ΔxN∕2×1

�; b̂N×1 �

�A−1bN∕2×1D−1bN∕2×1

�: (15)

It can be seen from this equation that the original signal canbe decomposed into two parts of the approximation compo-nent as A−1ΔxN∕2×1 or A−1bN∕2×1 and the detail component asD−1ΔxN∕2×1 or D−1bN∕2×1.

Similarly, the two-level wavelet-based multiresolutionrepresentation of K sized N × N can be expressed with thefollowing formula:

K̂N×N ��A−1KN∕2×N∕2 D1

−1KN∕2×N∕2D2

−1KN∕2×N∕2 D3−1KN∕2×N∕2

�: (16)

Four elements in the matrix of the right-hand side ofEq. (16) are, respectively, the approximation imageA−1KN∕2×N∕2 and three detail images D1

−1KN∕2×N∕2,

D2−1KN∕2×N∕2 and D3

−1KN∕2×N∕2.The wavelet transform can be successively performed level

by level with respect to the approximation components inboth sides of Eq. (14) to obtain a multiresolution representa-tion of the reconstruction problem.

E. Algorithm DescriptionActually, the solution to the reconstruction problem can beobtained by minimizing the residue between the measuredand predicted data as follows:

ψ�x� � 12�y − F�x��T �y − F�x��; (17)

where ψ�x� is the objective function measuring the discrep-ancy between the measured and predicted data. y is a vectorof the measured data on the boundary, and F�x� is a vector ofthe predicted data on the boundary according to a forwardmodel. Therefore, the reconstruction problem can beregarded as an optimization problem minimizing ψ�x� with re-spect to x. In terms of this, the whole reconstruction algorithmcan be summarized in Algorithm 2:

(1) Set x0 to an initial guess, x � x0; Set i � 2;(2) if (i%2⩵0) then

ComputeΔy and J at xwith Algorithm 1 based on the first groupof measurement data;

elseCompute Δy and J at x with Algorithm 1 based on thesecond group of measurement data;

endi � i� 1;

(3) for (l � −1;…;−L) doPerform wavelet transform for K and b to obtain K̂l and b̂l,which are the wavelet transform at the lth level for K and b,respectively;

endSet l � −L and Δx̂�0�

−L � 0;(4) while (l < 0) do

Obtain a solution from K̂lΔx̂l � b̂l using the CGD method withan initial value Δx̂�0�l ;Prolongate Δx̂l through padding zeros to obtain an initial guessfor Δx̂l�1 at the next higher resolution, i.e., Δx̂�0�l�1 � �Δx̂Tl ; 0T �T ;l � l� 1;

endSolve KΔx � b with Δx�0� � WT

xΔx̂�0�0 as an initial guess;

Update x using Eq. (13) and compute the correspondingobjective function ψ�x� according to Eq. (17);

(5) if (ψ�x� < δ) thenStop and output x;

elseGo to step (2);

end

Based on the proposed algorithms, the matrix system ofreconstruction is simplified in both of the spatial domainand wavelet domain, which can improve the efficiency ofreconstruction.


4. SIMULATION RESULTS ANDDISCUSSIONTo evaluate the performance of our proposed algorithm, sim-ulation experiments are implemented. The simulated forwarddata are obtained from the diffusion equations as illustrated inEqs. (1) and (2), where Gaussian noise with a signal-to-noiseratio of 10 dB is added for evaluating the noise robustness ofthe algorithms. Figure 2 illustrates the simulated phantomswith one anomaly in Fig. 2(a) and two anomalies of differentshapes in Fig. 2(b). Four sources and 30 detectors are equallydistributed around the circumference of the simulatedphantom.

The accuracy of FEM solutions to the partial differentialequations depends on the mesh size. To reduce the

computation requirements without significantly reducing theimage resolution, the image reconstruction is implementedbased on an adaptively refined mesh. During the process of

Fig. 2. Simulated phantoms for reconstruction of FMT (a) with oneanomaly and (b) with two anomalies.

Fig. 3. Prior images used for (a) one-anomaly phantom and(b) two-anomaly phantom.

Fig. 4. Adaptively refined meshes for reconstruction of FMT for(a) one-anomaly phantom and (b) two-anomaly phantom.

Table 2. Optical and Fluorescent Propertiesof Two-Anomaly Phantom

Excitation Light μaef �mm−1� μaei �mm−1� μ0se �mm−1� ϕ τ �ns�Anomalies 0.2, 0.3 0.03 4.0 0.2 0.6Background 0.05 0.03 4.0 0.2 0.6

Fluorescent light μamf �mm−1� μami �mm−1� μ0sm �mm−1� ϕ τ �ns�Anomalies 0.02, 0.03 0.02 3.0 0.2 0.6Background 0.003 0.02 3.0 0.2 0.6

Table 1. Optical and Fluorescent Propertiesof One-Anomaly Phantom

Excitation Light μaef �mm−1� μaei �mm−1� μ0se �mm−1� ϕ τ �ns�Anomaly 0.3 0.03 4.0 0.2 0.6Background 0.05 0.03 4.0 0.2 0.6

Fluorescent light μamf �mm−1� μami �mm−1� μ0sm �mm−1� ϕ τ �ns�Anomaly 0.2 0.02 3.0 0.2 0.6Background 0.005 0.02 3.0 0.2 0.6

Fig. 5. Distributions of SUMs and PROPORTIONs.

Fig. 6. Reconstructed results of absorption coefficient μaef (a) with-out the condition for PROPORTION and (b) with the condition forPROPORTION.


mesh generation, some a priori information derived fromother imaging modalities, such as the structural imagingcan be incorporated. The reconstructed domain is first uni-formly discretized according to the Delaunay triangulationscheme, after which the uniform mesh is then adaptively re-fined in combination with the a priori information. The areaswith fine details should be reconstructed with high resolution,whereas other areas composed mainly of the low-frequencycomponent with little variation can be reconstructed withlow resolution to reduce the computational requirementsand also to improve the ill-posedness. Based on this idea,the a priori images with a resolution of 100 × 100 pixels, asshown in Figs. 3(a) and 3(b) corresponding to Figs. 2(a)and 2(b), respectively, are used to generate the adaptively re-fined meshes. In Fig. 4, (a) shows the adaptively refined meshwith 122 nodes and 212 triangular elements, while (b) givesthe one with 148 nodes and 264 triangular elements, bothof which are generated with the prior information, as shownin Figs. 3(a) and 3(b) incorporated, respectively.

Tables 1 and 2 outline the optical and fluorescent parame-ters in different areas of the simulated phantoms correspond-ing to Figs. 2(a) and 2(b), respectively. To quantitativelyassess the accuracy of the different algorithms, the meansquare error (MSE) is introduced as a measure:

MSE �XNi�1

�1N�xcal − xact�2i

�; (18)

where N is the total number of nodes in the domain. Thesuperscript cal denotes the values obtained using re-construction algorithms and act denotes the actual distribu-tion of the optical or fluorescent parameters, which are usedto generate the synthetic image data set.

In this work, assuming that the scattering coefficients areknown, we focus on the reconstruction of the absorption co-efficient μaef . As Daubechies 1 (Haar wavelet) has advantages,such as orthogonality and symmetry, the Daubechies 1 wave-let is used in the simulations [21]. For a convenience of the

Fig. 7. Reconstructed results of absorption coefficient μaef for phan-tom with one anomaly based on (a) method without simplified matrixsystem and (b) proposed method.

Fig. 8. Reconstructed results of absorption coefficient μaef for phan-tom with two anomalies based on (a) method without simplified ma-trix system and (b) proposed method.

Table 4. Performance Comparison of ReconstructionMethods for Phantom with Two Anomalies

MethodsOur

AlgorithmMethod without Simplified

Matrix System

Computation time (s) 211 290MSE 2.845 × 10−4 2.814 × 10−4

Table 3. Performance Comparison of ReconstructionMethods for Phantom with One Anomaly

MethodsOur


Matrix System


Fig. 9. Reconstructed results of absorption coefficient μaef based on (a) simplification of the Jacobian matrix, (b) combined simplification of theJacobian matrix and grouped measurement data, and (c) proposed algorithm.

Table 5. Performance Comparison of ReconstructionMethods for Three Techniques

MethodsMethod inFig. 9(a)

Method inFig. 9(b)

Method inFig. 9(c)

Computation time (s) 241 235 211MSE 2.986 × 10−4 2.981 × 10−4 2.845 × 10−4


following discussion, the threshold a is selected so that thefollowing equation is satisfied:

a � c ×XMi�1

XNj�1

jJijj; (19)

where Jij denotes the element of an M × N Jacobian matrix.To determine a proper threshold for PROPORTION, we

analyze the distribution of the elements in the columns ofJacobian matrix for one-anomaly phantom with one source.Based on this distribution, the distributions of SUMs andPROPORTIONs can be calculated whose results are shown

in Fig. 5. It can be seen from this figure that most of PROPOR-TION are less than 50%, which can be set as the value of k instep (2) of Algorithm 1. Figure 6 shows the reconstructed re-sults of μaef with c � 0.2. It can be seen that the reconstructionquality can be improved especially for the area of anomaly ifthe condition of PROPORTION is considered. Therefore, allthe subsequent simulation results in this paper are fromour proposed algorithm with the condition of PROPORTIONconsidered.

The reconstructed images of μaef for the phantom with oneanomaly using different algorithms are shown in Fig. 7, whereFig. 7(a) shows the reconstructed result from the methodwithout using the simplified matrix system, and Fig. 7(b) de-picts the reconstructed result from the proposed method withc being selected as 0.05. The reconstructed images of μaef forphantom with two anomalies using different algorithms areillustrated in Fig. 8. The reconstruction result from the methodwithout using the simplified matrix system and thereconstruction result from our method with c � 0.05 areshown in Figs. 8(a) and 8(b), respectively.

To further evaluate the reconstruction quality, Tables 3and 4 summarize the performance of reconstructions interms of the computation time and MSE as defined inEq. (18) for phantoms with one anomaly and two anomalies,respectively. From these tables, it can be seen that ourproposed algorithm can significantly speed up thereconstruction process at the expense of a small reductionin reconstruction accuracy.

Fig. 10. Reconstructed results of absorption coefficient μaef for phantom of two anomalies with the value of c being (a) 0.05, (b) 0.1, and (c) 0.2,respectively.

Table 6. Impact of the Threshold on theReconstruction for Phantom with Two Anomalies

c 0.05 0.1 0.2

Computation time (s) 211 185 160MSE 2.845 × 10−4 3.982 × 10−4 6.015 × 10−3

Fig. 11. Simulated phantom for 3D reconstruction. The phantom ofradius 10 mm and height 40 mm with a uniform background ofμaef � 0.005 mm−1, which is positioned at x � 10 mm, y � 0 mm,and z � 20 mm. The small cylindrical anomaly has a radius of2 mm and height 6 mm with μaef � 0.01 mm−1.The anomaly is posi-tioned at x � 5 mm, y � 0 mm, and z � 20 mm. The dashed curvesrepresent the measurement planes, at z � 15 mm, z � 20 mm,z � 25 mm. Fig. 12. 3D mesh for image reconstruction.


Actually, three techniques are included in the proposedalgorithm as described in Subsections 3.B, 3.C, and 3.D. Toillustrate the contribution to the performance from each ofthese three techniques, Fig. 9 gives the reconstructed resultsfor phantom with two anomalies under different combinationof the above mentioned techniques. Quantitative comparisonsof the reconstructed results are listed in Table 5. From thistable, it can be seen that, in this simulated reconstruction,the method of simplification of the Jacobian matrix contrib-utes most to about 62% of the reconstruction acceleration.The wavelet method and strategy of grouped measurementdata contribute to about 30% and 8% of the reconstruction ac-celeration, respectively. Furthermore, the simplification of theJacobian matrix may increase MSE while the wavelet methodcan improve the quality of reconstruction by decreasing about5% of MSE.

To investigate the impact of the threshold on the MSE andthe computation time, the reconstructions are implementedusing our proposed algorithm with different settings of c.The reconstructed results under different settings of c forphantom with two anomalies are shown in Figs. 10(a)–10(c) with c being set as 0.05, 0.1, and 0.2, respectively. Table 6lists the corresponding performance of reconstruction underdifferent settings of the threshold (i.e., c). It can be seen that,when the threshold becomes larger, the reconstructionspeed can be further improved, but it results in a relativelylarger MSE. A similar conclusion can also be drawn forthe case of one anomaly. This means that our algorithmcan significantly speed up the process of reconstructionat the expense of a small reduction in reconstructionaccuracy.

To further validate the proposed algorithm for 3Dreconstruction, we extend the methods previously defined fortriangular elements to tetrahedral elements. The integration ofproducts of shape functions over the volume of the elements,and surface integrals over a side of the element, as requiredfor the computation of element stiffness and mass matrices, isperformed by a numerical integration rules.

In the 3D case, a cylindrical phantom as illustrated sche-matically in Fig. 11 is used for simulations. Within this phan-tom, a small cylindrical object is suspended. The dashedcurves represent the planes of measurement. Four sourcesand 16 measurements are used for each plane in the simula-tions. The data are collected in all three measurement planes.The mesh for reconstructing the 3D image as shown in Fig. 12contains 858 nodes and 3208 tetrahedral elements. Figures 13and 14 depict the 3D reconstructed images using the proposedalgorithm and the method without using the simplified matrixsystem, respectively. These are 2D cross sections through thereconstructed 3D images. Table 7 lists the performance of theabove two methods for a quantitative comparison. From thistable, we can conclude that our proposed algorithm can also

Fig. 13. Reconstructed images using the proposed algorithm, which are 2D cross sections through the reconstructed 3D volume. The right-handside corresponds to the top of the cylinder (z � 40 mm) and the left corresponds to the bottom of the cylinder (z � 0 mm), with each slicerepresenting a 10 mm increment.

Fig. 14. Reconstructed images using the method without simplified matrix system, which are 2D cross sections through the reconstructed 3Dvolume. The right-hand side corresponds to the top of the cylinder (z � 40 mm), and the left corresponds to the bottom of the cylinder (z � 0 mm),with each slice representing a 10 mm increment.

Table 7. Performance Comparison of 3DReconstruction Methods

MethodsOur


Matrix System



significantly speed up the reconstruction process for the3D case.

5. CONCLUSIONIn this paper, a new method based on a simplified matrix sys-tem is proposed for image reconstruction of FMT. In thismethod, the traditional computationally intensive Jacobianmatrix is simplified by deleting the rows or columns whosevalues are smaller than the threshold. In order to further re-duce the scale of Jacobian matrix as well as the number ofmeasurement data involved in each iteration, the measure-ment data are divided into two groups that are used in turnin the iterations of reconstruction. The simplified system issolved in wavelet domain to further accelerate process ofsolving the inverse problem. Simulation results show thatour algorithm can significantly accelerate the reconstructionprocess.

APPENDIX A(1) Proof. □

Let

Δx � �Δx1 Δx2 � � � Δxi � � � ΔxN �T �Δx ∈ RN �

and

J � � J1 J2 � � � Ji � � � JN � �J ∈ RM×N�

denote a vector of the perturbation in optical parameters andthe Jacobian matrix, respectively. Substituting them intoEq. (8), we have

266666666666664

0BBBBBBBBBBBBB@

JT1

JT2

..

.

JTi

..

.

JTN

1CCCCCCCCCCCCCA

· � J1 J2 � � � Ji � � � JN � � λI

377777777777775

·

0BBBBBBBBBBBBB@

Δx1Δx2

..

.

Δxi

..

.

ΔxN

1CCCCCCCCCCCCCA

�

0BBBBBBBBBBBBB@

JT1

JT2

..

.

JTi

..

.

JTN

1CCCCCCCCCCCCCA

· Δy: (A1)

Equation (A1) can be further written as

0BBBBBBBBBBBBB@

JT1 J1 � λ JT

1 J2 � � � JT1 Ji � � � JT

1 JN

JT2 J1 JT

2 J2 � λ � � � JT2 Ji � � � JT

2 JN

..

. ... . .

. ... ..

.

JTi J1 JT

i J2 � � � JTi Ji � λ � � � JT

i JN

..

. ... ..

. . .. ..

.

JTNJ1 JT

NJ2 � � � JTNJi � � � JT

NJN � λ

1CCCCCCCCCCCCCA

·

0BBBBBBBBBBBBB@

Δx1Δx2

..

.

Δxi

..

.

ΔxN

1CCCCCCCCCCCCCA

�

0BBBBBBBBBBBBB@

JT1 · Δy

JT2 · Δy

..

.

JTi · Δy

..

.

JTN · Δy

1CCCCCCCCCCCCCA

; (A2)

From Eq. (A2), it can be seen that the above matrix equation iscomposed of N equations, and the ith equation can be ex-pressed as

� JTi J1 JT

i J2 � � � JTi Ji � λ � � � JT

i JN � · Δx � JTi · Δy:

(A3)

Then, we have

Δxi

�JTi �Δy−Δx1J1−Δx2J2 ��−Δxi−1Ji−1−Δxi�1Ji�1 ��−ΔxNJN �

JTi Ji�λ

:

(A4)

The limit of Δxi as Ji approaches 0 can be achieved as

limJi→0

Δxi

� limJi→0

JTi �Δy−Δx1J1−Δx2J2��−Δxi−1Ji−1−Δxi�1Ji�1��−ΔxNJN�

JTi Ji�λ

�0: (A5)

For convenience of the following discussion, Eq. (A1) can berewritten as

0BBBBBBBBBBBBB@

JT1 JΔx

JT2 JΔx

..

.

JTi JΔx

..

.

JTNJΔx

1CCCCCCCCCCCCCA

� λI ·

0BBBBBBBBBBBBB@

Δx1Δx2

..

.

Δxi

..

.

ΔxN

1CCCCCCCCCCCCCA

�

0BBBBBBBBBBBBB@

JT1Δy

JT2Δy

..

.

JTi Δy

..

.

JTNΔy

1CCCCCCCCCCCCCA

: (A6)

For the case when Ji approaches 0, we have


limJi→0

266666666666664

0BBBBBBBBBBBBB@

JT1 JΔx

JT2 JΔx...

JTi JΔx...

JTNJΔx

1CCCCCCCCCCCCCA

� λI ·

0BBBBBBBBBBBBB@

Δx1Δx2...

Δxi...

ΔxN

1CCCCCCCCCCCCCA

377777777777775

� limJi→0

0BBBBBBBBBBBBB@

JT1Δy

JT2Δy...

JTi Δy...

JTNΔy

1CCCCCCCCCCCCCA

. (A7)

Since limJi→0 JTi JΔx � 0, limJi→0 Δxi � 0 and

limJi→0 JTi Δy � 0, we have

limJi→0

26666666666666666664

0BBBBBBBBBBBBBBBBBB@

JT1 JΔx

JT2 JΔx

..

.

JTi−1JΔx

0

JTi�1JΔx

..

.

JTNJΔx

1CCCCCCCCCCCCCCCCCCA

� λI ·


Δx1Δx2

..

.

Δxi−10

Δxi�1

..

.

ΔxN


37777777777777777775

� limJi→0


JT1Δy

JT2Δy

..

.

JTi−1Δy

0

JTi�1Δy

..

.

JTNΔy


.

(A8)

From Eq. (A8), we can obtain

limJi→0

26666666666666664

0BBBBBBBBBBBBBBB@

JT1 JΔx

JT2 JΔx

..

.

JTi−1JΔx

JTi�1JΔx

..

.

JTNJΔx

1CCCCCCCCCCCCCCCA

� λI ·

0BBBBBBBBBBBBBBB@

Δx1Δx2

..

.

Δxi−1Δxi�1

..

.

ΔxN

1CCCCCCCCCCCCCCCA

37777777777777775

� limJi→0

0BBBBBBBBBBBBBBB@

JT1Δy

JT2Δy

..

.

JTi−1Δy

JTi�1Δy

..

.

JTNΔy

1CCCCCCCCCCCCCCCA

:

(A9)

The above equation can be further written as

limJi→0

26666666666666664

0BBBBBBBBBBBBBBB@

JT1

JT2

..

.

JTi−1

JTi�1

..

.

JTN

1CCCCCCCCCCCCCCCA

· �J1Δx1 � J2Δx2 � � � � � JiΔxi � � � � JNΔxN �

�λI ·

0BBBBBBBBBBBBBBB@

Δx1Δx2

..

.

Δxi−1Δxi�1

..

.

ΔxN

1CCCCCCCCCCCCCCCA

37777777777777775

� limJi→0

0BBBBBBBBBBBBBBB@

JT1Δy

JT2Δy

..

.

JTi−1Δy

JTi�1Δy

..

.

JTNΔy

1CCCCCCCCCCCCCCCA

: (A10)

Since limJi→0 JiΔxi � 0, we have

limJi→0

26666666666666664

0BBBBBBBBBBBBBBB@

JT1

JT2

..

.

JTi−1

JTi�1

..

.

JTN

1CCCCCCCCCCCCCCCA

�J1Δx1 � J2Δx2 � � � � � Ji−1Δxi−1

�Ji�1Δxi�1 � � � JNΔxN� � λI ·

0BBBBBBBBBBBBBBB@

Δx1Δx2

..

.

Δxi−1Δxi�1

..

.

ΔxN

1CCCCCCCCCCCCCCCA

37777777777777775

� limJi→0

0BBBBBBBBBBBBBBB@

JT1Δy

JT2Δy

..

.

JTi−1Δy

JTi�1Δy

..

.

JTNΔy

1CCCCCCCCCCCCCCCA

: (A11)

Then, we can further obtain

limJi→0

26666666666666664

0BBBBBBBBBBBBBBB@

JT1

JT2

..

.

JTi−1

JTi�1

..

.

JTN

1CCCCCCCCCCCCCCCA

· � J1 J2 � � � Ji−1 Ji�1 � � � JN � � λI

37777777777777775

·

0BBBBBBBBBBBBBBB@

Δx1Δx2

..

.

Δxi−1Δxi�1

..

.

ΔxN

1CCCCCCCCCCCCCCCA

� limJi→0

0BBBBBBBBBBBBBBB@

JT1

JT2

..

.

JTi−1

JTi�1

..

.

JTN

1CCCCCCCCCCCCCCCA

· Δy: (A12)

From Eqs. (A1) and (A12), it can be seen that the ith columnJi of the Jacobian matrix can be deleted as Ji approaches 0during solving Eq. (8).


(2) Proof. □Let

Δy � �Δy1 Δy2 � � � Δyi � � � ΔyM �T �Δy ∈ RM�

and

J � �RT1 RT

2 � � � RTi � � � RT

M �T �J ∈ RM×N�

denote the vector for residual data between the measure-ments and the predicted data, and the Jacobian matrix,respectively. Substituting them into Eq. (8), we have

266666666666664

�RT1 RT

2 � � � RTi � � � RT

M � ·

0BBBBBBBBBBBBB@

R1

R2

..

.

Ri

..

.

RM

1CCCCCCCCCCCCCA

� λI

377777777777775

· Δx

� �RT1 RT

2 � � � RTi � � � RT

M � ·

0BBBBBBBBBBBBB@

Δy1Δy2

..

.

Δyi

..

.

ΔyM

1CCCCCCCCCCCCCA

. (A13)

From the above equation, we have

��RT1R1 � RT

2R2 � � � �RTi Ri � � � �RT

MRM� � λI� · Δx� Δy1RT

1 � Δy2RT2 � � � � � ΔyiRT

i � � � �ΔyMRTM: (A14)

In the case when Ri approaches 0, we have

limRi→0

��RT1R1 � RT

2R2 � � � �RTi Ri � � � �RT

MRM � � λI� · Δx

� limRi→0

�Δy1RT1 � Δy2RT

2 � � � � � ΔyiRTi � � � �ΔyMRT

M�:

(A15)

Since limRi→0 RTi Ri � 0 and limRi→0 ΔyiRT

i � 0, we have

limRi→0

��RT1R1 � RT

2R2 � � � �RTi−1Ri−1 � RT

i�1Ri�1 � � �RTMRM� � λI�

· Δx � limRi→0

�Δy1RT1 � Δy2RT

2 � � � � � Δyi−1RTi−1

� Δyi�1RTi�1 � � �ΔyMRT

M�: (A16)

Equation (A16) can be further written as

limRi→0

2666666666666664

� RT1 RT

2 � � � RTi−1 RT

i�1 � � � RTM � ·

0BBBBBBBBBBBBBB@

R1

R2

..

.

Ri−1

Ri�1

..

.

RM

1CCCCCCCCCCCCCCA

� λI

3777777777777775

·Δx

� limRi→0

� RT1 RT

2 � � � RTi−1 RT

i�1 � � � RTM � ·

0BBBBBBBBBBBBBB@

Δy1Δy2...

Δyi−1Δyi�1

..

.

ΔyM

1CCCCCCCCCCCCCCA

.

(A17)

From Eqs. (A13) and (A17), it can be seen that the ith rowRi of the Jacobian matrix can be deleted as Ri approaches 0during solving Eq. (8).

ACKNOWLEDGMENTSThis research is supported by Specialized Research Fundfor the Doctoral Program of Higher Education (SRFDP)No. 20123201120009, Natural Science Foundation of theJiangsu Higher Education Institutions of ChinaNo. 12KJB510029, National Natural Science Foundation ofChina No. 60871086, Natural Science Foundation of JiansuProvince China No. BK2008159, the Natural Science Founda-tion of the City of Suzhou Nos. SYG201113 and SYG201031,Hong Kong Polytechnic University (PolyU), and AustralianResearch Council (ARC) grants.

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