IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 4, APRIL 2008 1327

Truncated Total Least Squares: A NewRegularization Method for the

Solution of ECG Inverse ProblemsGuofa Shou, Ling Xia*, Mingfeng Jiang, Qing Wei, Feng Liu, and Stuart Crozier, Member, IEEE

Abstract—The reconstruction of epicardial potentials (EPs)from body surface potentials (BSPs) can be characterized as anill-posed inverse problem which generally requires a regularizednumerical solution. Two kinds of errors/noise: geometric errorsand measurement errors exist in the ECG inverse problem andmake the solution of such problem more difficulty. In particular,geometric errors will directly affect the calculation of transfermatrix A in the linear system equation = . In this paper,we have applied the truncated total least squares (TTLS) methodto reconstruct EPs from BSPs. This method accounts for thenoise/errors on both sides of the system equation and treatsgeometric errors in a new fashion. The algorithm is tested using arealistically shaped heart–lung–torso model with inhomogeneousconductivities. The h-adaptive boundary element method [h-BEM,a BEM mesh adaptation scheme which starts from preset meshesand then refines (adds/removes) grid with fixed order of interpo-lation function and prescribed numerical accuracy] is used for theforward modeling and the TTLS is applied for inverse solutionsand its performance is also compared with conventional regular-ization approaches such as Tikhonov and truncated single valuedecomposition (TSVD) with zeroth-, first-, and second-order. Thesimulation results demonstrate that TTLS can obtain similarresults in the situation of measurement noise only but performsbetter than Tikhonov and TSVD methods where geometric errorsare involved, and that the zeroth-order regularization is theoptimal choice for the ECG inverse problem. This investigationsuggests that TTLS is able to robustly reconstruct EPs from BSPsand is a promising alternative method for the solution of ECGinverse problems.

Index Terms—ECG inverse problem, truncated total leastsquares (TTLS) method, h-adaptive BEM, Tikhonov, truncatedsingle value decomposition (TSVD).

Manuscript received April 17, 2007; revised September 7, 2007. This workwas supported in part by the 973 National Key Basic Research & DevelopmentProgram under Grant 2003CB716106, the 863 High-tech Research & Develop-ment Program under Grant 2006AA02Z307, the National Natural Science Foun-dation of China under Grant 30570484, the Program for New Century ExcellentTalents in University under Grant NCET-04-0550 and in part by the AustralianResearch Council. Asterisk indicates corresponding author.

G. Shou is with the Department of Biomedical Engineering, Zhejiang Uni-versity, Hangzhou 310027, China (e-mail: shouguofa@hotmail.com).

*L. Xia is with the Department of Biomedical Engineering, Zhejiang Univer-sity, 28 Zheda Road, Hangzhou 310027, China (e-mail: xialing@zju.edu.cn).

M. Jiang is with the Department of Biomedical Engineering, Zhejiang Uni-versity, Hangzhou, 310027, China, and the College of Electronics and Infor-matics, Zhejiang Sci-Tech University, Hangzhou 310018, China (e-mail: peter-jiang0517@163.com).

Q. Wei, F. Liu, and S. Crozier are with the School of Information Technologyand Electrical Engineering, University of Queensland, Brisbane QLD 4072,Australia (e-mail: tracy@itee.uq.edu.au; feng@itee.uq.edu.au; stuart@itee.uq.edu.au).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TBME.2007.912404

I. INTRODUCTION

I N the study of inverse problems related to Electrocardio-graphy (ECG) [1]–[3], typical target reconstruction objec-

tives generally include: 1) equivalent sources [2] (for example,moving dipole [4], multiple dipoles [5]); 2) cardiac potentialdistributions (such as epicardial potentials (EPs) [3], [6]–[8],and transmembrane potentials [7], [9]); 3) cardiac activationmaps/isochrones [8], [10]–[12]. For all, the main task is to in-terpret the electrical activities of the heart from the recordedbody surface potentials (BSPs), which involves the so-called“forward” and “inverse” problems [13]. The “forward” problemdeals with the modeling of the potential distribution on the bodysurface from equivalent cardiac sources or EPs and is an im-portant component for the model-based “inverse” problem, inwhich the cardiac information are estimated based on forwardsolution including the transfer matrix and solution criteria.

In this paper, we investigate the reconstruction of EPs fromBSPs, which requires the solution of the Laplacian equationwith the Cauchy boundary conditions [3], [13]:

inonon

Find on the (1)

where is the quasi-static potential, and are the poten-tials on the epicardial surface and body surface , whichencloses the volume conductor is the tissue-dependent con-ductivity tensor. The final numerical formulation for andare

(2)

where is the transfer-matrix. The most widely used numericalmethods for the solution of electromagnetic problems includethe boundary element method (BEM) [14]–[16], the finiteelement method (FEM) [17]–[19], the finite difference method(FDM) [20], and the finite volume method (FVM) [21]. Ofthese, the BEM, an integral equation technique, is well suitedfor modeling surfaces; and the latter three are all differentialequation techniques and are typically applied for volume con-ductor problems. All of these methods use meshes to modelthe structure; recently, a meshless method [22] was success-fully employed for cardiac modeling. For heart modeling, it isreasonable to treat the conducting medium as isotropic and theconductivity values as regionally inhomogeneous (change/jump

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1328 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 4, APRIL 2008

at organ interfaces), although the effect of the conductivitydistributions to the bio-electromagnetic problems is still underactive debate [22]–[25]. For the studies herein, multiple regionsof homogeneous media were assumed and the BEM techniqueis applied to formulate the problem.

To achieve high numerical accuracy for both the forwardand inverse solutions, the h-adaptive BEM method is appliedto the forward problem [26]. In previous work, high-orderBEM [14], [16] and adaptive FEM [19] have been applied tothe ECG problem, and we have employed the adaptive BEMfor ECG forward modeling [26]. The adaptive BEM techniquewas introduced in the 1980s and has been widely used in manyfields [27], [28]. Compared with traditional BEM formulation,the adaptive BEM can self-adjust the number and size of theBE meshes guided by an error indicator and thus can reducecomputational time while achieving optimal BE meshes. Theadaptive BEM involves three parts [27]: error estimation,adaptive tactics, and mesh refinement processes. The errorestimation process evaluates the discretization errors of theboundary element solutions and the adaptive tactics select theelements to be refined and the relevant mesh refinement scheme.The mesh refinement can be further classified into h-, p-, andr-schemes and/or combinations therein. In the h-refinementscheme, the total number of elements is increased but the orderof interpolation function remains invariant; in the p-scheme,the initially assumed mesh is unchanged but the order of inter-polation function is increased; while in the r-scheme, both thenumber of elements and the order of interpolation function arekept invariant but the mesh points are relocated. In this paper,the h-refinement scheme is considered.

It is well known that the transfer matrix is ill-conditioned andthat the cardiac inverse problem can be characterized as a dis-crete, ill-posed problem [29], [30]. Small measurement errorsin the body surface potentials or geometry errors in the volumeconductors lead to large perturbations in the EPs. Therefore, theEPs cannot be calculated using a simple least squares methodas follows:

(3)

where symbol denotes the Euclidean norm of vector space.In order to obtain a stable solution, a regularization methodis typically used. For example, Tikhonov regularization withzeroth, first and second orders [1], [3], [30], [31] have beenfrequently used to deal with the ill-posed nature of the problemby imposing constraints on the magnitude or derivatives of theEPs. Truncated singular-value decomposition (TSVD) [1], [3],[13] is another effective method to solve the problem usinga truncation technique which ignores small singular values.These two methods are called direct regularization methods,requiring an accurate choice of a regularization parameter. Avariety of schemes have been developed for the determinationof “smart” regularization parameters [29], including compositeresidual and smoothing operators (CRESO) [31], L-curve [32],discrepancy principle (DP) [33], generalized cross-validation(GCV) [34], and zero crossing methods [35]. It is recognizedthat the above regularization schemes are mostly problemdependent and our comparison study showed that the GCVapproach is the most robust scheme [36] for our applications.

In addition to regularization, there are other techniques avail-able for the solution of ECG inverse problems. Temporal andspatial constraints techniques [9], [37], level-set methods [11],and statistical parameter-based methods [38] are also used forinverse problem solution; Ramanathan et al. [39] used the gen-eralized minimal residual (GMRes) method to reconstruct EPswithout adding any constraints and obtained similar solutions tothose given by Tikhonov regularization. In another study, Modreet al. [12] utilized an iterative algorithm to reconstruct the my-ocardial activation time image.

In the forward and inverse problems of ECG, errors/noiseexist on both sides of the linear system equation .The vector contains the measurement errors, which arise fromthe recording procedure for BSPs. The transfer matrix , how-ever, is contaminated by system noise such as model discretiza-tion error and geometric errors. In general, discretization erroris introduced numerically and might be manageable; however,the geometric errors which are normally introduced during theimaging-based reconstruction of a heart-torso model, can some-times be quite large. Both sorts of errors have been studied be-fore [23], [25], [40]–[42]. However, most of the methods forerror modeling, such as Tikhonov and TSVD, shared a commonassumption, i.e., only the right-hand side of the (2) is contami-nated by measurement errors, and less attention has been paid tothe system errors appearing in matrix . In this paper, we applythe truncated total least squares (TTLS) method [43]–[45] to theECG inverse problem. The TTLS method can process the mea-surement and geometric errors at the same time and has beensuccessfully applied in other fields, such as ultrasound inversescattering [46]. The proposed algorithm is tested using a realisti-cally shaped heart–lung–torso model with inhomogeneous con-ductivities, and the performance is compared with other avail-able regularization approaches.

II. METHOD

A. Forward Problem

To study the ECG inverse problem using a model-basedmethod, the first step is to construct both a forward model andconstruct the transfer matrix . Here, we adapt our previouslydeveloped model obtained from CT scans of the human body[36], [47] for the current investigation. Fig. 1 shows our originalheart, lung, and torso models, in which there are about 165nodes and 315 elements in the heart, 297 nodes and 586 ele-ments in the lung, and 210 nodes and 416 elements in the torso,respectively. Details of the model parameters can be found in[47] and references therein. In this paper, we keep the heartand lung model unmodified and employ the h-adaptive BEMtechnique to refine the torso model (see Fig. 2) [26]. In our newtorso model, there are about 421 nodes and 838 elements, andin particular, dense meshes are used in the frontal chest area.

Details of the procedure of h-adaptive BEM can be foundin our previous paper [26]. Here, only the main steps are de-scribed. The h-adaptive BEM starts from the calculation of elec-trical potentials from the initial triangular BE meshes. Then thediscretization errors of the boundary element solutions are es-

timated by the whole meshes’ error estimation and the errorindicator of each mesh . The former parameter determines

SHOU et al.: TRUNCATED TOTAL LEAST SQUARES 1329

Fig. 1. Original geometry model of heart, lung, and torso [36], [47]. Node andelement information: heart (165, 315), lung (297, 586), torso (210, 416).

Fig. 2. Refined geometry model using h-adaptive BEM. Node and element in-formation: heart (165, 315), lung (297, 586), torso (421, 838). The red nodesshow the electrodes applied to the ECG inverse problem.

the stopping criteria for refinement/iteration, and the latter oneis used to find those elements to be refined by adding new nodesin the center of the refined triangles. The mesh refinement pro-cedure is repeated and triangular meshes are reconstructed until

convergence. The final BE meshes are then obtained. In the pro-cedure of h-adaptive BEM, error estimation is important, and wetake the residual-type error indicator as shown in Fig. 1 in Bäch-told’s paper [28] in which the error is estimated by the residualof boundary integral equation [26]. The triangular meshes arerearranged using joint quality factors of triangular pairs [48].The refinement degree of the element is controlled by a param-eter (k) in Fig. 3, which is the flowchart for the h-adaptive BEMmesh generation.

B. Inverse Problem

1) The TTLS Method: The total least squares (TLS) methodis a generalized version of the original least squares method, andit is motivated by linear models in which both and

have errors [41]. Instead of using standard least squares for-mulation [29], we now state the problem with TLS formulation[41] as follows:

subject to (4)

where denotes the Frobenius norm, and are the errorversions of and , respectively. Corresponding to TSVDand Tikhonov regularizations (see the Appendix) in LS, the reg-ularization in TLS includes the truncated TLS (TTLS) method[44] and the regularized TLS (R-TLS) [45]. In this work, weapply the TTLS method to consider both sides of the matrix withnoise at the same time. The TTLS algorithm can be summarizedas follows [44].

TTLS Algorithm:1. Compute the SVD of the augmented matrix

(5)

2. Select the regularization parameter .3. Block-partition as

(6)

4. Compute the TTLS solution as

(7)

In (7), is the pseudoinverse of and. The small singular values of are neglected

through selection of the regularization parameter . We note thatthe determination of the regularization parameter is not an easytask and after a series of investigations, we find that the general-ized cross-validation (GCV) is the best approach and can lead tooptimal solution for the ECG inverse problem [36]. Therefore,GCV has been our default method for the determination of theregularization parameters.

1330 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 4, APRIL 2008

Fig. 3. Flowchart of the construction of transfer matrix A using h-adaptive BEM approach: A: Flowchart of adaptive triangular mesh generator. B: Process ofrefinement of an element. Here, �̂: body surface potentials; R̂ : error indicator of the element � ; R̂ : error indicator of the overall element.

Fig. 4. Epicardial and body surface potentials distribution induced by a dipole setting in the heart center; the left column is the anterior side, while the right is theposterior side. (For detailed setting, see text.)

C. Evaluation of the Inverse Solution

The inverse solution is evaluated using relative error (RE)and/or correlation coefficient (CC), and RE is defined as

RE (8)

and CC is given by

CC (9)

where is the mean value of the EPs, is the assumed“exact” solution from simulation.

SHOU et al.: TRUNCATED TOTAL LEAST SQUARES 1331

III. NUMERICAL SIMULATIONS

In this paper, the performance of the TTLS method wastested by the reconstruction of EPs from BSPs in a realisticheart–lung–torso model. The EPs are generated from a dipoleinside the heart and are normalized in the range of 10 mV[39]. Fig. 4 shows the corresponding epicardial and torso sur-face potentials distributions. It is noted that the visualization ofthe EP distributions is implemented with an extra interpolationprocedure [49] on the triangulated 3-D surface of the heartmodel.

In the inverse calculation of epicardial potentials, we chooseonly part of the nodes over the torso surface (248 nodes, markedwith red nodes in Fig. 2), which are those measurable nodes forclinical practice. Therefore, the dimension of the transfer matrix

is 248 165. The maximum and minimum singular values ofare , respectively. It can be

seen that the linear system is severely ill-posed, with a conditionnumber of .

To comprehensively simulate the noise involved in the inverseproblem, we modify the linear system equation as

(10)

where the body surface potential is added with simulatedvarious Gaussian white noise, and the transfer matrix is addedwith geometry error matrix , which is a random error gener-ated by a Matlab function randn. This noise is normally dis-tributed with zeroth mean value and unit standard variation,and normalized such that . Different error levelsare considered by adjusting the parameter . This error setupsomehow different from other approaches [3], [25], [37] suchas moving the heart model inward or outward about 1 cm in thetorso model, or considering the cardiac motion. In this paper,we characterized all these sorts of errors using random noisein the transfer matrix . The regularization methods, Tikhonov,TSVD, and TTLS are carried out in the Regularization Toolspackage under the MATLAB environment [50].

In the first simulation, only measurement noise is considered,and the noise level is about 50- and 30-dB signal-to-noise ratio(SNR) [9], [25], [37]. The matrix for Tikhonov, TSVD, andTTLS is the identity matrix, the first, and the second deriva-tive operators, respectively. The simulation results are summa-rized in Table I. From this table, we can see that if there is nomeasurement noise in BSPs, the three regularization methodsare all able to achieve good inverse solutions. With increases innoise levels, the inverse solutions are less accurate. Interestingly,the second-order regularization performs much poorer than theother two. Therefore, in the following studies, we only considerthe zeroth-order regularization, which is also the most popularapproach for ECG inverse problem solution [3], [8], [35]. It isalso found that the three regularization approaches can lead tosimilar results, with Tikhonov and TTLS performing slightlybetter than TSVD.

In the second simulation study, geometric noise is considered,and the control parameter is adjusted in a range of 0.01–0.1.Table II summarizes the inverse solution results from three ze-roth-order regularization methods with both measurement and

TABLE ISUMMARY OF THE INVERSE SOLUTIONS FOR THE CASE WITH MEASUREMENT

NOISE IN BSPS. NOISE PROPERTY: GAUSSIAN WHITE NOISE, 50-dBAND 30-dB SNR. REGULARIZATION APPROACHES: TIKHONOV,

TSVD, AND TTLS METHODS

TABLE IISUMMARY OF THE INVERSE SOLUTIONS FOR THE CASE WITH MEASUREMENT

NOISE IN BSPS AND GEOMETRY ERRORS IN TRANSFER MATRIX A. NOISE

PROPERTY: MEASUREMENT NOISE: GAUSSIAN WHITE NOISE, 50-dB AND

30-dB SNR; GEOMETRY NOISE: RANDOM ERROR. REGULARIZATION

APPROACHES: TIKHONOV, TSVD, AND TTLS METHODS (L: IDENTITY MATRIX)

geometric noise. In this case, TTLS method performs muchbetter than Tikhonov and TSVD, especially for the case withgeometric errors only.

To confirm TTLS’s performance, we consider the geometryerror in a relatively larger range, where the parameter iswithin the range of 0.001–1. The corresponding RE and CC

1332 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 4, APRIL 2008

Fig. 5. Inverse solution for the case geometry error involved only. Regularization solver: Tikhonov, TSVD, and TTLS.

values are shown in Fig. 5(A) and (B). From this test, we ob-serve that the TTLS method produces the better inverse solu-tion. When is equal to 0.01 and 0.1, the RE value of TTLSmethod is decreased by about 10%, and CC value is increasedby 5% compared with those of the Tikhonov and the TSVDmethods. Fig. 6 demonstrates the EP distribution reconstructedfrom the case where . It is clear that the EP distri-bution generated by TTLS is closer to the forward model result(see Fig. 4), particularly for the maximum and minimum valuelocations. In addition, the TTLS reported epicardial potentialprofile is smoother than those of Tikhonov and TSVD methods.

In this paper, the TTLS method is implemented in Matlab,and it only takes about 2 s for a solution on a Dell computer withIntel Core2 CPU 1.80 GHz, and 2.00-GB RAM. The algorithmis therefore efficient and can be used for real-time applications.

IV. DISCUSSION

In the ECG inverse problem, in addition to the measurementerrors in the BSPs, the transfer matrix is also contaminated bynoise which is contributed to by: geometric inaccuracies in themodel anatomy, the beating of the heart, and the representationof conductivities of the volume conductor. These system errorsmake the estimation problem very difficult, and it is challengingto find a good, stable inverse solution. In this paper, we use theTTLS regularization method as introduced by Fierro et al. [44]to solve this discrete ill-posed inverse problem of ECG.

The TTLS method was implemented and compared withthe most commonly used regularization methods, includingTikhonov and TSVD. These three regularization approachesachieve similar solutions when there is only measurement error.However, TTLS is able to produce much better results thanTikhonov and TSVD when there are geometric errors involved.The results from the TTLS method, when compared to thoseof Tikhonov and TSVD methods can be readily understood byexamining the geometric meaning of the TLS [43]: Considera curve that minimizes the distances from the curve to a givenset of points, then it is the vertical distance used in the LSapproach, while the perpendicular distance is used in the TLSapproach. When there is only measurement error, the curve issubstantially flat, and the vertical and perpendicular distancesare equivalent, leading to inverse solutions at the same level ofaccuracy. If there is error in , however, the curve’s slope isbigger, and the perpendicular distance is shorter than vertical

distance, which makes the TLS solution better than the LS.The results also support that the TTLS method suits stronglyill-posed problems, which has been pointed out by Fierro et al.[44]. The TTLS performance is evident by the RE, CC valuesand EP profiles obtained in the simulations in this ECG inverseproblem study. The TTLS algorithm is capable of handling bothgeometry errors and measurement noise, which is importantfor practical applications. To achieve more accurate inversesolutions in the presence of noise, careful solution constraints,such as additional time and spatial constraints, need furtherinvestigation.

Another feature of this study is that we have applied theh-adaptive BEM for the ECG forward modeling. The h-adap-tive BEM uses an iteration procedure to find the optimal BEmeshes for torso representation. The algorithm has been de-tailed in our previous paper [26], in which both a concentricspheres model and a realistic heart–lung–torso model areinvestigated. Through forward and inverse ECG modeling, theh-adaptive BEM has been validated and is demonstrated tobe able to improve the numerical accuracy of the BEM withminimal computational cost. Therefore, the h-adaptive BEMis a promising technique to construct the forward model forinverse studies.

The current torso–lung–heart model is reconstructed from theCT scans of the human body; MRI-based solution may be betterfor the reconstruction of the model, especially for the considera-tion of cardiac motion. The TTLS seems more robust than otherregularization schemes when the model geometry becomes animportant issue in potential clinical applications. In addition,our adaptive BEM scheme can represent the model geometrywell and therefore further improve the system accuracy. Furtherinvestigations are underway toward practical usage.

V. CONCLUSION

In this paper, the TTLS method is proposed to study inverseproblems in ECG, with the inclusion of noise on both sides ofthe linear system equation. The simulations are implemented ina realistic heart–lung–torso model constructed using a h-adap-tive BEM technique. The results demonstrate that the proposedmethod can produce better solutions than conventional regular-ization methods and that the zeroth-order regularization is op-timal for the ECG inverse problem. In this paper, the reconstruc-tion of EPs from BSPs only has been considered; in future work,

SHOU et al.: TRUNCATED TOTAL LEAST SQUARES 1333

Fig. 6. Epicardial potential distributions reconstructed from body surface potentials. Noise property: geometry error only, � = 0:01. Left panel: anterior view;right panel: posterior view. (A) Zeroth-order Tikhonov method; (B) TSVD method; and (C) TTLS method.

we will employ this promising scheme to other cardiac sourcereconstruction and imaging problems.

APPENDIX

According to matrix theory, the singular value decomposition(SVD) of is given by

(11)

where and have or-thonormal columns, with

.

The ordinary LS solution can be written as

(12)

Due to the division by small singular values , the solutionmay be dominated by components associated with the

errors in . Therefore, regularization is necessary to stabilizethe solution.

The common and well-known regularization approach isTikhonov regularization [36], which approximately solves thelinear system by minimizing

(13)

where is a positive regularization parameter chosen to con-trol the size of the solution vector and is a matrix that defines

1334 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 4, APRIL 2008

a (semi)norm on the solution through which the “size” is mea-sured. In general, represents the first or second derivative op-erator (first-order or second-order). The equivalent problem of(13) is

(14)

As increases, the (semi)norm of the solu-tion vector decreases monotonically while the residual

increases monotonically. It is easy to prove thatif , then the solution of (14) is given by

(15)

Therefore, Tikhonov regularization suppresses (filters) thecomponents of the solution corresponding to the small singularvalues of .

The regularization in TSVD method is achieved by truncatingthe singular value for , where is the equivalent regular-ization parameter

(16)

ACKNOWLEDGMENT

The authors would like to thank the three reviewers forhelpful comments on the paper.

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Guofa Shou was born in Zhejiang Province, China,in April 1982. He received the B.S. degree in biomed-ical engineering from Northwestern PolytechnicalUniversity, China. He is currently working towardthe Ph.D. degree in the Department of biomedicalengineering at Zhejiang University.

His research interest includes the forward and in-verse problem of ECG, and the numeric algorithmsin bio-electromagnetism.

Ling Xia was born in Zhejiang Province, China,in October 1965. He received the B.S. degree inelectrical automation control and the Ph.D. degreein biomedical engineering from Zhejiang University,China, in 1987 and 1996, respectively.

He is currently a Professor and the Vice Director ofthe Institute of Biomedical Engineering of ZhejiangUniversity. His research interest includes the multi-scale heart modeling and simulation, the ECG inverseproblem, MRI key technology, and the biological ef-fect of electromagnetic field. He has published more

than 40 papers in journals and obtained several highest-level grants from Chi-nese government.

Mingfeng Jiang received the B.S. degree andthe M.S. degree in biomedical engineering fromChongqing University, China, in 2000 and 2003, re-spectively. He is currently working toward the Ph.D.degree in the Department of biomedical engineeringin Zhejiang University, China.

His current research interests are in the forwardand inverse problem of ECG and in the biomedicalsignal processing.

Qing Wei received the M.S. degree from Universityof Queensland (UQ), Brisbane, Australia, in 2002.

She is currently working toward the Ph.D. de-gree in the school of information technology andelectrical engineering, UQ. Her current researchinterests include numerical tool design in biomedicalengineering and scientific visualization.

Feng Liu received the Ph.D. degree in biomedicalengineering from Zhejiang University, Hangzhou,China, in 2000.

He is currently a Senior Research Fellow atthe School of Information Technology and Elec-trical Engineering, The University of Queensland,Brisbane, Australia. His current research interestsinclude magnetic resonance engineering, bioelectro-magnetism, and power electronics.

Stuart Crozier (M’93) received the Ph.D. degreein electrical engineering and the D.Eng. degreein biomedical engineering from the University ofQueensland (UQ), Brisbane, Australia, in 1991 and2002, respectively.

He is currently a Research Director in theSchool of Information Technology and ElectricalEngineering, UQ. He is the author or coauthor ofmore than 100 journal articles and is the holder ofnumerous patents. His current research interestsinclude magnetic resonance engineering, bioelectro-

magnetics, and the methodological development of magnetic resonance.Prof. Crozier is an Associate Editor of the IEEE TRANSACTIONS ON

BIOMEDICAL ENGINEERING.