Spatial deconvolution technique to improve theaccuracy of reconstructed three-dimensionaldiffuse optical tomographic images

Harry L. Graber, Yong Xu, Yaling Pei, and Randall L. Barbour

A straightforward spatial deconvolution operation is presented that seeks to invert the information-blurring property of first-order perturbation algorithms for diffuse optical tomography (DOT) imagereconstruction. The method that was developed to generate these deconvolving operators, or filters, wasconceptually based on the frequency-encoding process used in magnetic resonance imaging. The compu-tation of an image-correcting filter involves the solution of a large system of linear equations, in whichknown true distributions and the corresponding recovered distributions are compared. Conversely,application of a filter involves only a simple matrix multiplication. Simulation results show that appli-cation of this deconvolution operation to three-dimensional DOT images reconstructed by the solution ofa first-order perturbation equation (Born approximation) can yield marked enhancement of image qual-ity. In the examples considered, use of image-correcting filters produces obvious improvements in imagequality, in terms of both location and �a of the inclusions. The displacements between the true andrecovered locations of an inclusion’s centroid location are as small as 1 mm, in an 8�cm-diameter mediumwith 1.5�cm-diameter inclusions, and the peak value of the recovered �a for the inclusions deviates fromthe true value by as little as 5%. © 2005 Optical Society of America

OCIS codes: 170.3880, 170.3010, 100.1830, 100.6950, 100.6890.

1. Introduction

The subject of this paper is a novel application oflinear deconvolution techniques for the problem ofimproving the spatial resolution of three-dimensional(3D) reconstructed images in the field of diffuse opti-cal tomography (DOT). In brief, DOT involves theillumination of tissue structures with one or morewavelengths of visible or near-infrared radiation atmultiple positions on the tissue boundary, multiple-site detection of the light that is reemitted across theboundary, and reconstruction of images of the spatialdistributions of the tissue’s absorption or scatteringcoefficients. The clinical utility of DOT lies in therelations between these coefficients and physiologicalparameters such as blood oxygen saturation and tis-

sue blood volume; furthermore, it is recognized thatDOT offers specific practical advantages relative toestablished functional imaging modalities such aspositron emission tomography and single-photonemission computed tomography (it does not entailadministration of radionuclides) or functional mag-netic resonance imaging (higher temporal resolution,possibly less ambiguity in the interpretation of re-sults).

In a series of papers published during the past 4–5years, our group has demonstrated the advantages ofperforming rapid repeated DOT measurements, thenextracting dynamic information by applying varioustime-series analysis methods—many of which are al-ready widely used to process optical spectroscopydata—to the reconstructed images.1–7 As a practicalmatter, a requirement for use of these algorithms isthe availability of a reconstruction algorithm that canrecover many images in a short time. The basicframework we have adopted for this purpose is thenormalized difference method (NDM),8 which is ro-bust to many of the biases and types of noise that arecommonly encountered when laboratory or clinicalDOT measurements are performed and is also able toaccommodate many refinements. One of these, thenormalized constraint method,9 was shown to suc-

H. L. Graber (harry.graber@downstate.edu), Y. Xu, and R. L.Barbour are with the Department of Pathology, State University ofNew York Downstate Medical Center, Box 25, 450 Clarkson Ave-nue, Brooklyn, New York 11203. Y. Xu, Y. Pei, and R. L. Barbourare with NIRx Medical Technologies LLC, 15 Cherry Lane, GlenHead, New York 11545.

Received 18 December 2003; revised manuscript received 18July 2004; accepted 26 October 2004.

0003-6935/05/060941-13$15.00/0© 2005 Optical Society of America

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cessfully distinguish absorption and scattering per-turbations in a diffusing medium over a broad rangeof medium optical properties when applied tocontinuous-wave DOT measurement data. For casesin which the test medium for the normalized con-straint method had spatially coincident absorptionand scattering perturbations, we assessed the algo-rithm’s ability to separate them by assigning a qual-itatively different form of temporal fluctuation toeach optical parameter.10 These dynamics acted astags by which interparameter cross talk was pre-cisely quantified.

While the preceding study was in progress, it wasrecognized that the tagging of optical parameterswith temporal information immediately suggests ageneral mechanism to characterize the action of DOTimage reconstruction algorithms on their inputdata.11–14 Namely, let a distinct mode of temporalfluctuation be assigned to each optical parameter ofinterest in every volume element of a medium, and leta time series of forward- and corresponding inverse-problem solutions be computed. Then a map of mag-nitude versus location for a given mode of fluctuationwithin the image space reveals precisely how the re-construction algorithm distributes the correspondingoptical parameter throughout that space. A mappingfrom object space into image space obtained in thisway is conceptually analogous to a point-spread func-tion (PSF),15 which characterizes the physical accu-racy and resolution of an optical device. In view ofthis analogy, the term information spread function(ISF) has been adopted as our descriptor for the typeof object–space to image–space mapping describedhere.

The ISF formulation has an important implica-tion that provides the main topic of the presentpaper. Namely, once the behavior of a reconstruc-tion algorithm has been characterized under agiven set of initial conditions (e.g., the NDM for aparticular reference medium and arrangement ofsources and detectors), its ISFs can be used in de-convolution computations to improve the spatialresolution and accuracy of reconstructed images.This application is analogous to the establishedpractice in which the point-spread functions of anoptical measuring device are used as the basis forcalibration and correction procedures to reduce im-age blurring or aberrations.16–19 Results presentedin this paper demonstrate that highly significantimprovements in the quality of DOT image spatialinformation can be achieved in this way. It is worthstressing that the technique described here can beapplied to media of arbitrary shape and internalcomposition, and arbitrary source–detector geome-tries, in a computationally efficient manner.

It is well appreciated that solutions to the DOTforward problem are nonlinear functions of the me-dium’s optical parameter values. Previously, groupsthat engaged in DOT research, ourselves included,always assumed that these nonlinearities are a prin-cipal cause of the distortions and low spatial resolu-tion typically seen in images recovered by linear

reconstruction algorithms.20,21 Accordingly, the usualprescription used to improve image quality has beento adopt a nonlinear reconstruction strategy, recom-puting solutions to the forward and inverse problemsin an alternating pattern.22 The findings presentedhere imply, however, that linear spatial convolution,or mixing of information from many target mediumlocations in any given image pixel or voxel, is actuallya more important source of the errors in recon-structed DOT images. The significance and limita-tions of this premise are expanded on in Section 4.

2. Methods

A. Spatial Deconvolution Algorithm

The strategy described here was conceptually moti-vated by a consideration of the physical basis of im-age formation in magnetic resonance imaging (MRI).There, spatial discrimination is possible because theimposition of a magnetic field gradient creates arange of position-dependent resonance frequencies.This same concept is applied here to the image for-mation problem of optical tomography, although itseems likely (see Section 4) that it will be applicableto other inverse problems as well. The basic idea isthat first-order solutions to linear perturbation prob-lems often produce less than ideal solutions, withimage blurring evident. Now suppose there is someway to specifically encode the pixel information in theobject space and to track where it is recovered in theimage space. As in MRI, this can be accomplishedwhen the absorption coefficient ��a� is tagged in everyobject pixel with a unique time-varying function. Thereduced scattering coefficient ��s�� can be simulta-neously tagged in the same way to assess the degreeof interparameter cross talk in the recovered images.However, to simplify the presentation in this initialdemonstration, only �a was modulated in the exam-ples presented in this paper.

In the implementation that was used to producethe results presented below, the particular taggingfunctions used were simple sine waves with incom-mensurable frequencies. We achieved incommensu-rability by using the square roots of successive primenumbers as the tag frequencies (i.e., f1, 2, 3, 4, . . .

� 1, 21�2, 31�2, 51�2, . . . Hz) so that the ratio of anytwo is an irrational number; by adopting this se-quence, any nonlinear phenomena that might occurcannot produce harmonics or sum–difference fre-quencies that are equal to any of the tag frequencies.The tagging functions were evaluated at a large num-ber Nt �Nt � 16, 384 � 214� of successive time points,with a constant time interval �t��t � 0.005 s� thatwas short relative to the difference between the os-cillation periods of the two highest-frequency taggingfunctions. This choice of Nt and �t assures that noneof the tagging functions are undersampled, whichwould produce aliasing artifacts, and that they allcan be resolved in the frequency domain.

At each time point a forward problem for a tomo-graphic measurement (i.e., multisource, multidetec-tor) was solved, as described in Subsection 2.B. We

942 APPLIED OPTICS � Vol. 44, No. 6 � 20 February 2005

subsequently solved the inverse problem, as de-scribed in Subsection 2.C, by using the NDM to pro-duce a system of linear equations whose solution wascomputed by LU decomposition.23 In this way a timeseries of reconstructed images was built up. Inspec-tion of the power spectra of the images (i.e., spatialmaps of power versus position for selected frequen-cies) directly identifies where and to what extent in-formation from each pixel in the object domain ismapped to the image domain. As described in Section1, a spatial map of this sort is referred to as an ISF.The details of the information spread, a two-dimensional (2D) example of which is shown in Fig. 1,are specific to the properties of the object and theillumination geometry used and to the algorithm em-ployed for image recovery. Accordingly, inspection ofthese maps can provide valuable insight into the ac-tion of a reconstruction algorithm. The ISF in Fig. 1,for example, demonstrates that the reconstructionalgorithm used yields images containing two com-monly observed types of error. First, the optical pa-rameter (absorption coefficient, in this example)value present at a specific location in the medium[Fig. 1(a)], as identified by the amplitude of the sinu-soidal fluctuation used to tag its location, is attrib-uted to a much larger area in the image space [Fig.1(b)]. Second, the location of the high-amplitude peakin the image is displaced from its correct locationtoward, in this case, the center of the image space.That is, the reconstruction algorithm used here pro-duces a blurred and distorted image.

The potentially most important application of theknowledge obtained when we compute ISFs would beto use them as the basis of an image enhancementalgorithm. That is, knowing how optical parameterinformation is mapped from object space to imagespace should enable one to devise a suitable correc-tion that would in effect remove the effects of linearspatial convolution from images. In practice, in thepreferred strategy for doing this, we do not use thepower spectra of the recovered temporal fluctuations(or, more generally, the temporal covariance be-tween the assigned and recovered tagging func-

tions) as the input data for derivation of imagedeconvolution operators. The reason is that compu-tation of the relevant deconvolution operator typi-cally is an ill-conditioned problem. Consequently,for media comprising Ns voxels, our attempts to gen-erate such operators by inverting Ns � Ns matrices ofISFs have mostly yielded unsatisfactory results.12

(The issue of ill-conditioning is also important to un-derstand why in practice the best corrections are ob-tained when Nt �� Ns, as discussed in Subsection4.B.) Instead, the assigned and recovered time-varying �a in all pixels should be compared in thetime domain directly. That is, the input is collectedinto an Ns � Nt matrix Y and the reconstructed �a’sare collected into an Ns � Nt matrix X, where theelement yij is the assigned (tag) value of �a in voxel iat time j and the element xij is the recovered value of�a in voxel i at time j. The objective is to determine anNs � Ns filter matrix F that will transform X into Y.That is, a solution to the equation Y � FX is sought.This F will subsequently be applied to reconstructedexperimental images in the hope of improving them.Thus if u is a particular reconstructed image of a testmedium (i.e., a medium different from all the onesused to generate F), then v � Fu is the correspond-ing corrected, or deconvolved, or filtered image.

In general X and Y are not square matrices, asNt �� Ns in the situations of interest. For example, inthe results to follow there were 982 or 984 voxels [i.e.,finite-element method (FEM) mesh nodes] in the 3Dmedia that were studied, which is more than an orderof magnitude smaller than the number of time steps.Then X and Y cannot in general be inverted, and theequation Y � FX cannot hold. This situation is com-mon in statistical problems; the matrix equation rep-resents Nt linear equations in Ns unknowns; andunless the coefficients are linearly dependent in someway, there is no exact solution. A commonly usedstrategy in these cases is to try to make the differencematrix Y � FX as small as possible, in the least-squares sense. That is, the elements fji of F are chosenso that the quantity

I��k�1

Nt

�j�1

Ns �yjk��i�1

Ns

fjixik�2

, (1)

which is the sum of the squares of the errors in theindividual terms when Y is approximated by FX isminimized. Setting the derivative of I with respect toeach element of F equal to zero yields

�I�fji

��2�k�1

Nt �yjk��l�1

Ns

fjlxlk�xik�0, (2)

or

�k�1

Nt

yjkxik��k�1

Nt

�l�1

Ns

fjlxlkxik, ∀ i, j. (3)

Fig. 1. (a) Spatial map of the amplitude of one selected temporalmodel function assigned to the absorption coefficient ��a� of a FEMmesh node located at the intersection of the horizontal and verticallines. In this example, the mesh is an 8�cm-diameter 2D disk withhomogeneous �s� and homogeneous time-averaged �a. (b) Spatialmap of the amplitude in the time series of reconstructed �a imagesof the same temporal frequency as that assigned to the indicatedFEM mesh node in (a).

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Equation (3) is just the matrix equation YXT

� FXXT, where XT is the transpose of X. Since XXT isa square matrix and the columns of X are linearlyindependent, XXT can be inverted to give F � �YXT��XXT��1.

Finally, it should be noted that if the process isinverted, to produce the optimal approximation X� GY in the least-squares sense, it is found that G� �XYT��YYT��1 and, in general, F and G are notinverses of each other.

B. Solution of the Forward Problem

Forward-problem solutions were computed for thetwo FEM meshes shown in Fig. 2. The hemisphericmesh shown in Fig. 2(a), which approximates themeasurement geometry for DOT mammographicstudies, contains 4309 tetrahedral elements with 982nodes; the curved-slab mesh [Fig. 2(b)], which ap-proximates the measurement geometry for DOTbrain imaging, contains 4274 tetrahedral elementswith 984 nodes. The diameter of the hemisphere is8.0 cm. The distance between the convex and concavesurfaces of the curved slab is 3 cm, and it can becircumscribed by a parallelepided of dimensions7.2 cm �length� � 5.2 cm �width� � 4.33 cm (thick-ness). The starting point in its generation was a7.2 cm � 5.2 cm � 3.0 cm rectangular slab, whichwas next subjected to two thickness-preserving cir-cular bends, first in the length dimension and then inthe width dimension.

Defined sets of points on the convex mesh surfaceswere used as the locations for isotropic sources ordetectors. For the hemispheric mesh, 29 detector lo-cations were designated [only 14 are visible in Fig.2(a)], and 25 of these were also used as sources, for atotal of 725 source–detector channels. For thecurved-slab mesh, 24 source and detector locationswere designated, as shown in Fig. 2(b), for a total of576 source–detector channels.

We acquired tomographic data for the simulatedtissue models by using the FEM to solve the diffusionequation with Robin boundary conditions and a dcsource term. For a spatial domain � with boundary��, this is represented by the expression

· �D(r) (r)� � �a(r)(r) � ��(r � rs), r � � (4)

where �r� is the photon intensity at position r, rs isthe position of a dc point source, and D�r� and �a�r�are the position-dependent diffusion and absorptioncoefficients, respectively. Here the definition used forthe diffusion coefficient was D�r� � 1�3��a�r�� �s��r��, where �s��r� is the position-dependent re-duced scattering coefficient. For all computationsconsidered in this paper, both media shown in Fig. 2had spatially homogeneous and temporally invariantscattering with �s� � 10 cm�1.

For the forward-problem computations that gener-ated the object versus image comparisons that wereused to compute filter (F) matrices, the spatial dis-tributions and temporal fluctuations of the dynamic,heterogeneous media’s absorption coefficients are de-scribed in Subsection 2.A. The spatiotemporal meanvalue was ��a� � 0.06 cm�1, and the amplitude of thesinusoidal temporal variation at each FEM meshnode was 0.0048 cm�1 (8%) for the hemisphere and0.009 cm�1 (15%) for the curved slab. For the compu-tations that generated the data used in the subse-quent tests of the filters’ image-correcting power, thestatic, spatially heterogeneous medium structuresshown in Fig. 3 were employed. In this step the FEMmeshes used for computation of detector readingsintentionally were made finer than those used forgeneration of the F matrices and for subsequent im-age reconstruction (hemisphere, 2212 nodes and

Fig. 2. Three-dimensional FEM meshes and source–detector ge-ometries used to compute and test the image deconvolution oper-ators. (a) Hemispheric mesh containing 982 nodes and 4309elements. (b) Curved-slab mesh containing 984 nodes and 4274elements. Source–detector locations are marked with small whitecircles.

Fig. 3. Heterogeneous test media used in demonstrations of theefficiency of deconvolution to improve reconstructed image accu-racy. (a)–(c) Hemisphere with one inclusion; (d)–(f) hemispherewith three inclusions; (g)–(i) curved slab with three inclusions.First column [(a), (d), (g)] shows the x–y projections; the secondcolumn [(b), (e), (h)] shows the x–z projections; and the third col-umn [(c), (f), (i)] shows the y–z projections of the 3D test media.Numbers along the gray scales give the quantitative value of thespatially varying �a, whereas �s is homogeneous.

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10,305 elements; curved slab, 2062 nodes and 8698elements) so that we would not be committing a sim-ple inverse crime. In these test media the inclusionsare approximately spherical (limited by the coarse-ness of the FEM meshes), with average diameters of1.5 cm for the hemispheres and 1.2 cm for the curvedslabs. The background regions of these media haduniform absorption, with �a � 0.06 cm�1, whereasthe inclusions were more strongly absorbing, with�a � 0.12 cm�1 for the hemisphere and either0.12 cm�1 (Fig. 9) or 0.3 cm�1 (Figs. 6 and 8) for thecurved slab. The latter perturbation was made largerstrictly for reasons of computational convenience, i.e.,because backscattering measurements were beingsimulated in this case and the inclusions’ volumeswere less by almost a factor of 2 ��1.2 cm�1.5 cm�3

� 0.512�, a higher value of �a in the inclusions wasneeded to produce detector reading perturbations ofthe same magnitude as those obtained for the hemi-spheric medium. However, it should be stressed—and the results presented in Fig. 9 prove—that theimage reconstruction and image enhancement algo-rithms do not require this degree of contrast betweenstructures in the medium to perform successfully.

Imaging operators (see Subsection 2.C) were com-puted, in the manner described in Ref. 24, for eachsource–detector channel. In brief, each row of thematrix Wr [Eq. (5)] is a function of two forward-problem solutions: a product of forward and adjointintensities for perturbations of �a and a dot product offorward and adjoint intensity gradients for perturba-tions of D. For each simulated medium (i.e., targetmedium) shown in Fig. 2, a single set of imagingoperators was used for all inverse-problem computa-tions. These were computed for a homogeneous ref-erence medium having the same shape, size, andmeasurement geometry as the (heterogeneous) targetand with optical parameters equal to the mean values��a � 0.06 cm�1, �s� � 10 cm�1� of those in the tar-get.

C. Solution of the Inverse Problem

The reconstruction algorithm that was used to gen-erate the results presented below seeks to solve amodified perturbation equation whose form is

Wr · �x � �Ir, (5)

where �x is the vector of differences between theoptical properties [e.g., absorption and scattering(diffusion) coefficients] of a target and a defined ref-erence medium; Wr, the imaging operator or weightmatrix, is the weight matrix describing the influencethat each voxel has on the surface detectors for theselected reference medium; and �Ir is proportional tothe difference between detector readings obtainedfrom the target in two distinct states (e.g., the differ-ence between data collected at two different instantsor the difference between instantaneous and time-averaged data). Also, to improve the condition num-ber of the linear systems being solved, zero-order

Tikhonov regularization (ridge regression) wasused in all image reconstructions. Thus we recon-structed the images not by computing �x� Wr

� · �Ir � WrT�WrWr

T��1 �Ir in accordance with Eq.(5), but by computing the regularized quantity

�x � WrT(WrWr

T � I)�1�Ir. (6)

For all inverse-problem calculations carried out forthis paper, the numerical value of the regularizationparameter was � 0.01. As is shown subsequently,the presence of regularization term is important, butthe parameter value used has, over a 5 order-of-magnitude range from � 10�5 to � 100, littleeffect on image quality.

The distinction between Eqs. (5) and (6) and a stan-dard linear perturbation equation lies in the struc-ture of the right-hand side. Here we used thepreviously described NDM,8 in which the quantity �Ir

on the right-hand sides of Eqs. (5) and (6) is definedby

(�Ir)i �(I � I0)i

(I0)i(Ir)i. (7)

In Eq. (7), Ir is the computed detector readings cor-responding to a selected reference medium. For thefilter-generating computations, I and I0 represent theintensity at a specific time point and the time-averaged intensity, respectively. For the filter-testingcomputations, I and I0 are the intensities (i.e., detec-tor readings) computed for the heterogeneous targetmedium and the homogeneous reference medium, re-spectively.

A Levenberg–Marquardt (LM) algorithm was usedto compute the numerical solutions to Eq. (6).22 Atime period of less than 1 min was required for com-putation of the weight matrix and the subsequent LUdecomposition. [Note that the L (lower triangular)and U (upper triangular) factors could be read from alibrary of precomputed matrices and that the decom-position is a one time only event in any case.] Subse-quently, the time required to reconstruct each image(i.e., backsubstitution step) was approximately 0.3 son a PC with a 2.4�GHz Pentium IV processor and1 Gbyte of RAM. The weak dependence of imagequality on the degree of regularization used, for � 10�5, is shown in Fig. 4, where the root-mean-squared error (RMSE) of the reconstructed image rel-ative to the original target medium is plotted versus� for the three-inclusion hemispherical medium de-picted in Figs. 3(d)–3(f). For the uncorrected imagesthat we reconstructed by solving Eq. (6) (dashedcurve), the RMSE increases rapidly with decreasing �for � 10�6. However, the RMSE is only 10% largerat � 100 than at the absolute minimum of thecurve, which occurs at � 5 � 10�5. The depen-dence of RMSE on � is even weaker for the correctedimages (solid curve) and actually decreases with in-creasing � over the same interval in which the un-

20 February 2005 � Vol. 44, No. 6 � APPLIED OPTICS 945

corrected image RMSE increases. The specific value, � 10�2, that we used in generating the resultspresented in Section 3 was chosen as a compromisevalue. Also worthy of note is the absence of crossingsbetween the curves in Fig. 4, which indicates thatthere is no degree of regularization for which decon-volution fails to enhance the reconstructed image.Consistent with this, inspection of the uncorrectedimages (not shown) reveals that no value of � yieldsreconstructed images substantially better than thosepresented below in Figs. 6(a)–6(c).

Initial tests of the spatial deconvolution strategyfor image enhancement were made on noise-freedata. Subsequently, to assess the robustness of themethod to noise in detector data, the same imagereconstruction and deconvolution steps were applieda second time to the media in Figs. 3(b) and 3(c), afterthe detector readings were corrupted with multipli-cative Gaussian noise, at a uniform level of 1%. Thusthe noise vector element corresponding to the ithsource–detector channel was sampled from theGaussian distribution N�0, 0.01�I0�i�. Then I� � I� n, where n is the noise vector, was substituted forI in Eq. (7).

3. Results

Three orthogonal 2D projections of the hemispherictarget medium used in the first test of the spatialdeconvolution approach are shown in Figs. 3(a)–3(c).The corresponding views of the image reconstructedobtained by the solution of Eq. (6) are shown in Figs.5(a)–5(c). It can be seen that the presence, location,and direction (i.e., inclusion more strongly absorbingthan background) of the inclusion are recovered rea-sonably accurately, whereas more quantitative fea-tures such as its shape and volume and themagnitude of the �a contrast are not. Noteworthy isthe frequently seen tendency of some DOT image

reconstruction algorithms to push the recovered per-turbation into regions of the medium with the lowestweight, which in this case lies along the planarboundary of the hemisphere. The same views of thecorrected image obtained by application of the filter-ing scheme described here is shown in Figs. 5(d)–5(f).The improvement in image quality and the similaritybetween the image and the original target are strik-ing. Comparison of the gray scales of the uncorrectedand corrected images with those of the original re-veals that the improvement is quantitative and dem-onstrates the possibility of accurately recovering theabsolute value of optical parameter variations with areconstruction algorithm that operates on relativechanges in detector readings. Because of space limi-tations, extensive numerical analyses of qualitativeand quantitative accuracy10,25 are not presented here,but will be included in a more in-depth report cur-rently in preparation. We do note here, however, thatin this example the location of the inclusion’s centroidin the corrected image lies within 1 mm of its trueposition, and the peak recovered �a lies within 5% ofthe correct quantitative value.

Although the preceding single-inclusion studymight lead some to conclude that the quality of theuncorrected image is adequate, and the computa-tional effort that went into generating the filter ma-trix is therefore gratuitous, it does not address theimportant issue of the reconstruction algorithm’s re-solving power. To examine the effect of our image-correcting procedure on this feature, the next testmedium that was considered contained three inclu-sions, as shown in Figs. 3(d)–3(f). Here the averagedistance between the centroids of pairs of inclusionswas 2.68 cm. The image that we reconstructed byusing the method of Subsection 2.C, shown in Figs.6(a)–6(c), exhibits the same types of quantitative in-accuracy as was seen in the one-inclusion case, buthere the qualitative accuracy also suffers as a conse-

Fig. 4. Root-mean-squared difference between the three-inclusion hemispherical medium shown in Figs. 3(d)–3(f) and theuncorrected (dashed curve) and corrected (solid curve) recon-structed images of the same target, versus the magnitude of theTikhonov regularization parameter �.

Fig. 5. Reconstructed image of one-inclusion hemispheric testmedium [see Figs. 3(a)–3(c)]. (a)–(c) Uncorrected image, which isthe solution to Eq. (6); (d)–(f) corrected image obtained by applyingthe spatial deconvolution to the result in (a)–(c). First column [(a)and (d)] shows the x–y projections; the second column [(b) and (e)]shows the x–z projections; and the third column [(c)–(f)] shows they–z projections of the 3D images. Numbers along the gray scalesgive the quantitative value of the spatially varying �a.

946 APPLIED OPTICS � Vol. 44, No. 6 � 20 February 2005

quence of the inability of the reconstruction to resolvethe perturbation into three distinct objects. In con-trast, the corrected image displayed in Figs. 6(d)–6(f)exhibits the same degree of qualitative and quanti-tative accuracy improvement as obtained in the one-inclusion case. Significantly, there is no appreciabledegradation in the performance of the filtering strat-egy, even though the strongly absorbing volume islarger and the potential for nonlinear interactionsamong the inclusions exists in this case.20,21 This ob-servation suggests that the primary cause of inaccu-racies in the uncorrected images produced with thesolution of Eq. (6) are a consequence of linear spatialconvolution arising from the algorithm’s ISF. It isreasonable to suppose that a similar result would beobtained with a different choice of reconstruction al-gorithm, in which case the conceptual basis of non-linear iterative reconstruction schemes would becalled into question (see Section 4).

In the preceding two tests, the uncorrected imageswere reconstructed from fully tomographic sets ofmeasurement data. Two important questions not yetaddressed are the dependence of the filtering strate-gy’s performance on the geometry of the target me-dium and on the disposition of sources and detectors(i.e., measurement geometry). To begin to examinethese issues, the image-correction procedure was ap-plied in the case of the three-inclusion curved-slabtest medium shown in Figs. 3(g)–3(i). Here the dis-tance between centroids of adjacent inclusions was2.23 cm, and all three lay at the same depth. Conse-quently, nonlinear phenomena ought to have asmaller effect on the quality of the reconstructed im-age here than in the preceding hemispheric mediumcase. However, the quality of the image we recoveredby using the method of Subsection 2.C is, as can beseen in Figs. 7(a)–7(c), so poor as to make it unusable.Application of the image-correcting filter appropriatefor this case produces the result in Figs. 7(d)–7(f). Thecorrected image has the same levels of qualitativeand quantitative accuracy as that obtained in thepreceding tests, which means that an appreciablyhigher degree of improvement was obtained. The re-sults in Figs. 7(d)–7(f) imply that information regard-

ing the correct spatial distribution of �a in the targetmedium is present in the uncorrected image, but in ahighly spatially convolved form. This strengthens thehypothesis that nonlinear image-correction ap-proaches, when applied to DOT problems, are basedon a false premise regarding the origin of the inaccu-racies in the reconstructed images. It further sug-gests (see Section 4) that many apparent DOT imagereconstruction failures actually have been unrecog-nized successes, requiring only an appropriate post-processing step to put them into a visuallyrecognizable form.

Because the preceding results were obtained underideal conditions, it is important to examine the effectof factors that impair the quality of real experimentaldata on the effectiveness of the spatial deconvolutionprocedure. Chief among these is the inevitability ofsome level of random error in the detector readingsused by the image reconstruction algorithm. In othermedical imaging fields, it has been shown that de-blurring operations can amplify image degradationresulting from noise.26,27 Therefore it is important toinclude test cases in which noise was added to thedetector data and to compare the output of the image-correction scheme to the corresponding noise-free re-sult. The uncorrected and corrected images obtainedof the three-inclusion hemispheric [Figs. 3(d)–3(f)]and curved-slab [Figs. 3(g)–3(i)] test media when thedetector readings were corrupted with multiplicativenoise as described in Subsection 2.C are shown inFigs. 8 and 9, respectively.

Comparing Figs. 8(d)–8(f) with Figs. 6(d)–6(f), itcan be seen that the noise does not appreciably affectthe accuracy of either qualitative or quantitative fea-tures of the recovered inclusions. Rather, the onlyapparent effect of noise is to introduce a number ofirregularly shaped blobs of spurious �a contrast intothe background region. That is, although noise placeslimits on spatial resolution and sensitivity, just as itdoes for any other imaging modality (i.e., inclusionswith physical dimensions or contrast smaller than

Fig. 6. Reconstructed image of a three-inclusion hemispheric testmedium [see Figs. 3(d)–3(f)]. See caption to Fig. 5 for explanationof gray scales and of row and column assignments.

Fig. 7. Reconstructed image of a three-inclusion curved-slab testmedium [see Figs. 3(g)–3(i)]. See caption to Fig. 5 for explanationof gray scales and of row and column assignments.

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those of the noise blobs cannot be distinguished fromthe effects of noise), it does not pose any extra burdenon the image-correcting filter approach outlined here.

For the curved-slab case, a comparison of Figs. 9(d)and 7(d) could suggest that noise has a larger effecton the ability of the filtering scheme to resolve inclu-sions and to accurately recover their qualitativeshape features. However, the views of the imagesthat show the depth dependence of the recovered �a

spatial distributions [Figs. 7(e) and 7(f) and 9(e) and9(f)] reveal that the inclusions actually are well re-solved. Also interesting is that the spurious blobsproduced by the data noise have a marked depthdependence, and in this example most of them lie inthe region more superficial than the inclusions. It isimportant to determine, in future research, whetherthis observation continues to hold when the relativenoise level increases with increasing source–detectorseparation,28 as one would expect to find in experi-mental or clinical data. If results of the type seen in

Fig. 9 are obtained consistently, then in subsequentefforts to incorporate regularization into the decon-volution step (in addition to the Tikhonov regulariza-tion that is used, as described in Subsection 2.C, toreconstruct the uncorrected images), the idea of em-ploying a depth-dependent regularization functionalshould be explored. The regularization magnitudewould increase with decreasing depth, as some pre-vious empirical studies already have recommended.29

Finally, because a principal conclusion drawn fromthis study is that image-correcting procedures of thetype presented here should either replace or comple-ment the strategy of iteratively updating theforward- and inverse-problem computations, it is im-portant to directly compare the performance of thesetwo approaches when they are applied to the same setof detector data. The curved-slab target medium usedfor this comparison has the same anatomical struc-ture as that shown in Fig. 3(b), but here theinclusion/background �a contrast ratio is 2:1, insteadof the 5:1 used for the computations whose results areshown in Figs. 7 and 9. The first-order reconstructedimage (not shown) is qualitatively similar to that inFigs. 7(a)–7(c), and the image recovered after teniterations of a previously described22 LM algorithm isshown in Figs. 10(a)–10(c). The total computationtime was approximately 90 min on the same comput-ing platform used for the filter generation and spatialdeconvolution computations. Although evidence of athree-inclusion structure is present here, only twoinclusions are well defined in terms of shape andcentroid location, and only one has a well-definedboundary. It also can be seen that the error in theapparent depth of the inclusions is not corrected afterthis number of iterations. The corresponding resultthat we obtained by deconvolving the first-order im-age is shown in Figs. 10(d)–10(f). In this result allthree inclusions are well resolved with well-definedboundaries, and their depth is correctly identified.

Fig. 8. Reconstructed image of three-inclusion hemispheric testmedium [see Figs. 3(d)–3(f)] when detector data are corrupted withnoise in the manner described in Subsection 2.C. See caption toFig. 5 for explanation of gray scales and of row and column assign-ments.

Fig. 9. Reconstructed image of three-inclusion curved-slab testmedium [see Figs. 3(g)–3(i)] when detector data are corrupted withnoise in the manner described in Subsection 2.C. See caption toFig. 5 for explanation of gray scales and of row and column assign-ments.

Fig. 10. Comparison of the performance of the nonlinear iterativeimage reconstruction algorithm and the spatial deconvolution ap-proach. The test medium is a curved slab with three inclusions [seeFigs. 3(g)–3(i)]; detector data are noise free. See caption to Fig. 5 forexplanation of gray scales and of row and column assignments.

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It is recognized that the particular iterative recon-struction algorithm used in this example has notbeen optimized with respect to the convergence ratefor the imaging problem that was considered. Un-doubtedly the quality of the image in Figs. 10(a)–10(c) would improve with more iterations, and it isprobable that algorithms can be found that requirefewer iterations for convergence. In any case, thetime required to recover the final answer would bemany times greater than the 0.01 s needed to applythe filter correction. In marked contrast to iterativestrategies, the time required for filter generation canprecede the collection of any given data set and sodoes not affect the time interval between data collec-tion and production of the final image.

4. Discussion

In this paper a straightforward spatial deconvolutionoperation is presented that seeks to invert theinformation-blurring property of first-order perturba-tion algorithms for DOT image reconstruction. Thecomputation of such a deconvolving operator, or filter,involves the solution of a large system of linear equa-tions, in which known true distributions and the cor-responding recovered distributions are compared.The application of a filter is by simple matrix multi-plication. Thus, although their generation entails sig-nificant computational effort, for any given type ofDOT measurement they can be precalculated, andonly the application time ( 10�3 s or 6–7 orders ofmagnitude smaller than the generation time for theillustrative examples presented here) contributes tothe postmeasurement computational burden.

The method that was developed to generate theimage-correcting filters was conceptually based onthe frequency-encoding process used in MRI. It islargely for this reason that, for the examples pre-sented in this paper, we tagged each medium’s voxelsby assigning incommensurable sinusoidal fluctua-tions to their optical coefficients. Because of this par-ticular choice of tagging functions, the technique wedeveloped for generating the filters was referred to inpreliminary reports as frequency encoding of spatialinformation.11,13 In general, however, many otherfunctional forms should be usable for this purpose.Consequently, the generic term we apply to proce-dures of the type described above in Subsection 2.A istemporal encoding of spatial information.

A seemingly cogent objection to the strategy pre-sented here can be raised, namely, that the inaccu-racies and artifacts that the described spatialdeconvolution corrects for are only those that thereconstruction algorithm itself introduces into theimage. Then the appropriate remedy should be tosubstitute a better one, or at least to make a betterchoice of control variables (e.g., the optical parame-ters assigned to the reference medium or initialguess, regularization parameters, truncation point ina truncated singular value decomposition algorithm,whether and how to scale rows or columns of Wr),rather than to attempt a posteriori repair on the out-

put of one that is bad or suboptimally tuned. There isa three-part answer to this objection: First, nothingmore is being claimed for our image enhancementprocedure than that it can substantially improve re-sults obtained when linear reconstruction techniquesare used, but that the image quality attainable bythese methods may not have been fully appreciatedby some in the DOT community; second, that thereconstruction algorithm used may have properties(e.g., speed, robustness) that compel its use and arelacking in other algorithms that yield more accurateanswers under ideal conditions8–10; third, that inpractice it may not be possible to know in advancewhat the optimal values for all control variables are,so that an efficient data-driven technique for a pos-teriori correction has considerable practical value.

The specific application area of primary interest toour group is diffuse optical tomography. However, thestrategy developed and presented here can be ex-pected to have applicability to a much larger set ofproblem areas. This follows from the fact that thereare many fields in which a linear transformation isused to convert sets of observations or measurementsinto interpretable results. A correction operationanalogous to that derived here for the specific case ofDOT should be possible in many of these areas. Elec-trical impedance tomography, microwave tomogra-phy, magnetoencephalographic imaging, andpositron emission tomography and single-photonemission computed tomography are medical imagingmodalities that could be expected to benefit fromthese operations, and they might also be usefullyapplied to inverse problems in electroencephalogra-phy. The potential for application to problem areas ingeology, oceanography, atmospherics and meterol-ogy, and astronomy, wherein interpretation of obser-vations also ultimately entails a solution to aninverse problem derived from a transport equation,also seems clear. Outside of the natural sciences,however, it also would seem that there is potentialutility for the general correction strategy outlinedhere in economics and other social sciences, where,interestingly, inverse diffusion problems also are con-sidered.30,31

A. Summary and Implications of Results

Here it has been shown that application of the de-convolution operation to DOT images reconstructedby a solution to a first-order perturbation equation(Born approximation) can yield marked enhance-ment of image quality. These corrected (filtered)images are quantitatively accurate in terms of tar-get location, size, and shape. In the 3D examplesthat are considered here, use of image-correctingfilters produced obvious improvement in imagequality in terms of both location and �a of the inclu-sions. The displacements between the true and therecovered locations of an inclusion’s centroid locationwere as small as 1 mm, in an 8�cm-diameter mediumwith 1.5�cm-diameter inclusions, and the peak valueof the recovered �a for the inclusions deviated fromthe true value by as little as 5% (see Figs. 5, 6, and 8).

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Multiple inclusions were present in all but one case,and in all of these the spatial deconvolution strategysuccessfully resolved the inclusions while also accu-rately locating each one.

Corruption of the simulated detector data withmultiplicative Gaussian noise (Figs. 8 and 9) did notbring about a reduction of spatial resolution or in thequalitative (centroid location) or quantitative (peak�a value) accuracy of the recovered inclusions. Thus itis reasonable to progress from simulation studies totests involving laboratory phantoms to determinewhether the approach presented here is likewise ro-bust to all the forms of noise that are present inexperimental data. The principal effect of the noisewas the appearance of small, irregularly shaped re-gions of spurious absorption contrast. Notably, for thecase in which a backreflection measurement was sim-ulated (Fig. 9), the latter regions showed a pro-nounced depth dependence, decreasing in size andmagnitude with increasing depth. This pattern sug-gests that it could be profitable to implement depth-dependent regularization when Eq. (6) is solved or tocombine the deconvolution step with a long-passspatial-frequency filtering operation having a depth-dependent threshold.

The relatively low spatial resolution of recoveredimages in DOT applications has been observed andcommented on many times. Also well known is thetendency of image reconstruction algorithms to pro-duce appreciable error in the depth component of aninclusion’s location, although the direction of the biasis a function of the specific algorithm that is em-ployed. An important implication of the results inFigs. 5–9 is that much previous research on develop-ing and improving image reconstruction algorithmshas been based on a mistaken premise regarding theorigin of these phenomena. We believe this point isone of the more significant conclusions of the presentstudy and that some elaboration is called for here.

It is certainly true that if a perturbation in theoptical coefficients of a medium is large in terms ofmagnitude or volume, then the first derivatives thatconstitute the imaging operator do not correctlymodel the effect of changes in the medium’s opticalcoefficients on the detected near-infrared signals.Nonlinear reconstruction strategies proceed from theassumption that the inaccuracies in the images re-covered by solution of the first-order perturbationequations are caused by these deviations from linearrelationships between perturbation and signal. But ifthat assumption were correct, then the linear spatialdeconvolution approach developed here would not beexpected to enhance the accuracy of the test mediumimages, and in fact should make them worse. Thequality of the results actually obtained, particularlyin the direct comparison of spatial deconvolution anda nonlinear LM reconstruction algorithm (Fig. 10),implies that the information-spreading properties ofthe reconstruction algorithm are the true cause of thelow spatial resolution and depth bias commonly seenin DOT images. It is recognized that ten is almostcertainly not the optimum number of LM iterations;

employing the optimal number of iterations wouldimprove the quality of the result in Figs. 10(a)–10(c),and the algorithm itself could probably be adjusted toyield a higher rate of convergence. In any case, thepostmeasurement computational time and costwould invariably be many times greater for any ap-proach of this type than for application of the image-correcting filter, which requires only a matrixmultiplication.

The preceding is not an assertion that nonlinearrelationships between optical parameter perturba-tions and detector readings are not present, or thatthey cannot in certain cases place important limits onthe performance of linear reconstruction algorithms.In fact, we have explored the relative importance oflinear information spread and nonlinear effects andhave found (results not shown) that with sufficienteffort the filtering strategy can be made to fail. In thetest media for which this occurred, the inclusionswere located at more than one depth, with superficialones obstructing every source–detector channel’sview of the deep ones. In at least some cases it wasfound that the nonlinear reconstruction algorithmeventually recovered all the inclusions, and the fil-tering approach recovered only the superficial ones. Itis noteworthy, however, that in every such case elim-inating the deep-lying inclusions from the mediumhad only a very small effect on the detector data (i.e.,at best in the third significant digit). Then the addi-tion of even a low level of noise or any sort of system-atic error (i.e., the tests were all carried out on ideal,noise-free data) would bring us back to the situationin which the linear and nonlinear algorithms producecomparable answers.

An important implication of these considerations,as already alluded to, is that many earlier assess-ments of DOT reconstructed images, if based princi-pally or exclusively on visual inspection of theimages, may have been unduly pessimistic. An espe-cially vivid example of this phenomenon can be seenin Fig. 7 (others almost as striking can be seen inFigs. 4 and 6 of Ref. 12). Here the first-order imageobtained from a multiple-inclusion test medium hasalmost no hint of the actual structure of the medium,but the corrected image demonstrates that the cor-rect answer is actually present in the former result,only in a visually unrecognizable form. It is our ex-pectation that application of the appropriate spatialdeconvolution operator to apparently unsatisfactoryimage reconstruction results obtained by many of thegroups engaged in DOT research would have a com-parably gratifying outcome. (In the future we hope toinvite other groups to allow us to apply spatial de-convolution to some examples of their image data asa test of the preceding prediction.)

B. Theoretical and Numerical Considerations

There are several useful ways to conceptualize themethod used to determine F that is outlined in Sub-section 2.A. For one, it can be thought of as a partic-ular type of general linear model calculation, inwhich the columns of X are the models that are pos-

950 APPLIED OPTICS � Vol. 44, No. 6 � 20 February 2005

tulated to account for most of the variability in eachcolumn of Y (i.e., X is the general linear model designmatrix). Limiting our consideration to linear combi-nations of the models (i.e., main effects) is equivalentto assuming that interactions among two or more ofthem are unimportant. In principle, we could obtaina mapping from X into Y that is accurate over a largerrange of medium optical coefficient values by solvinghigher-order equations,

yi � ai ��j�1

Ns

bijxj ��j�1

Ns

�k�j

Ns

cijk�xj1xk1 xj2xk2 . . . xjNtxkNt�T

� . . . , (8)

for the unknown coefficients ai, bij, cijk, . . .. But to doso is a practical impossibility because of the prohibi-tively large number of additional terms, even if onlytwo-way interactions are included in the model. Al-ternatively, the generation of F can be thought of asa system calibration problem, more computation in-tensive but no different in principle from commonlaboratory practices such as use of standard com-pounds with well-established melting points to pro-duce a correction curve for a thermometer. Anotheralternative is to regard X and Y as the equivalent ofa training set in a neural network computation and Fas the equivalent of the connectivity pattern thatresults from the training process.

From any of the preceding interpretations for thefilter generation process, one can infer the principaltrade-off that must be borne in mind when specifyingY: The set of modeled target media must constitute arepresentative sample of the mathematical space ofall media to which the filter could possibly be applied;but at the same time, the magnitudes of the assignedperturbations cannot be so large that the relationshipbetween the modeled and the recovered optical coef-ficients becomes manifestly nonlinear. (Of course, itis understood that this trade-off places a limit on therange of target medium optical coefficients for whicha computed filter can have its image-correcting ef-fect.) In Fig. 11 we illustrate what is meant by rep-resentative sample; points whose coordinates are theabsorption coefficient values of two particular FEMnodes in the first 1000 time points of the curved-slabtraining set are plotted in Fig. 11(a), and the corre-sponding plot for the entire 214-point time series isplotted in Fig. 11(b). These plots are 2D projections ofthe 984-dimensional mathematical space (�a space)whose points correspond to all possible distributionsof �a in the curved slab. The plot in Fig. 11(b) makesit clear that, as the size of the training and calibrationset increases, it will contain at least one point thatfalls within any specified finite subregion of �a space.Equivalently, a sufficiently large training set guar-antees the existence of a training set point lying ar-bitrarily close, in �a space, to any given test medium.

The preceding considerations provide an importantpart of the answer to the question of why it wasnecessary that Nt be almost 17 times larger than Ns

in the filter-generating process, when intuitively it

would seem that Nt � Ns should suffice. The empiri-cal fact that Nt � Ns does not suffice is shown explic-itly in Fig. 12, where the results obtained byapplication of deconvolution operators computedfrom the first 1 � 103, 6 � 103, 1.2 � 104, and1.6 � 104 rows of X and Y to the uncorrected image ofthe three-inclusion hemispheric test medium [Fig.6(a)–(c)] are presented.

These results demonstrate that the deconvolutionoperator is not effective unless the number of timepoints taken into account is more than ten timesgreater than the number of FEM mesh nodes. Forsmaller values of Nt, the corresponding points in �a

space do not constitute a representative sample. Ad-ditional insight into the question of why it is neces-sary that Nt��Ns is provided by examination of thedependence of the condition number and effectiverank of X on Nt. Plots of the former, for both hemi-spheric and curved-slab media, are shown in Fig.13(a), and the latter, for both media, are plotted inFig. 13(b). Here, the condition number is defined asthe ratio of the smallest nonzero singular value to the

Fig. 11. Plots of assigned �a�n1, t� versus �a�n2, t� �cm�1�, wheren1 and n2 denote two specific curved-slab FEM mesh nodes. Inparticular, n1 and n2 are those nodes for which the modulationfrequencies are 21�2 and 51�2 Hz. (a) �a�n1, t� versus �a�n2, t� plotfor Nt � 1000; enlarged round and square dots indicate first andlast points in the training and calibration time series. (b) �a�n1, t�versus �a�n2, t� plot for Nt � 16, 384. As Nt increases, sampledpoints in �a space constitute more complete sampling.

Fig. 12. Effect of the deconvolution operator applied to the recon-structed image [Figs. 6(a)–6(c)] of a three-inclusion hemispherictest medium, as a function of Nt. From left to right: results forNt � 103, 6 � 103, 1.2 � 104, and 1.6 � 104. The top row shows thex–y projections; the middle row shows the x–z projections; and thebottom row shows the y–z projections of the 3D images. Numbersalong the gray scale give the quantitative value of the spatiallyvarying �a.

20 February 2005 � Vol. 44, No. 6 � APPLIED OPTICS 951

largest singular value (this is the reciprocal of theusual definition, but has the advantage that the re-sulting quantity in constrained to lie in the interval0.0–1.0); the closer this ratio is to zero, the morebadly conditioned the matrix. Effective rank is heredefined as the number of singular values larger thanε, a small positive number [in particular, �� 103�8 � 10�8 � 2.4�10�8 was used to compute thecurves in Fig. 13(b)]; this is a measure of how manycolumns of the matrix are linearly independent. FromFig. 13(a) it is apparent that the case of Nt � Ns

yields the most badly conditioned X, from which itfollows that computation of a useful F requires thatNt be significantly larger than Ns. At the same time itcan be seen that the condition of X does not alwaysimprove with increasing Nt, so another factor must beinvolved in producing the calibration-set-length de-pendence that can be seen in Fig. 12. The effectiverank curves in Fig. 13(b) supply the remaining pieceof the puzzle. They signify that until Nt exceeds Ns bya factor of 10, the effective rank of X is smaller thanthe number of nodes in the FEM mesh. That is, as apractical matter not all columns of Y can be simulta-neously equated to linear combinations of the col-umns of X.

C. Future Directions

The preceding characterizations of deconvolution op-erators and their mechanism of action suggest sev-

eral directions for further development andimprovement of the image enhancement strategypresented in this report. For one, it appears highlyprobable that a simultaneously fast and accurate hy-brid reconstruction algorithm can be synthesized bythe application of the spatial deconvolution and anonlinear updating scheme in an alternating fashion.The resulting procedure should converge more rap-idly than, say, the LM algorithm used to generate theresults shown in Fig. 10, while also permitting recov-ery of media in which nonlinear effects of the opticalcoefficient perturbations are significant. A secondmodification that is of equal importance, and whoseimplementation is more straightforward, is to tagboth the absorption and the scattering coefficients ofa medium simultaneously. The principal benefit ofthis is that the filter matrices thereby derived can beapplied to the output of algorithms that provide si-multaneous reconstruction of �a and �s. At the sametime, it would constitute a mechanism for quantify-ing the extent of interparameter cross talk associatedwith a given reconstruction algorithm10 and for re-ducing its effect where it does occur. An importantconceptual point is that such cross talk can be re-garded as another sort of information spread, onethat occurs in optical parameter space rather than inphysical space. As such, the same general strategythat is used to correct for the effects of a reconstruc-tion algorithm’s ISFs should also be able to reducecross-talk artifacts. A third objective will be to searchfor nonsinusoidal forms for the tagging functions thatare optimal in the sense of minimizing the Nt that isneeded to achieve a representative sampling of �a

space. In contrast to the first two tasks, this one isgeared more toward reducing total computationaloverhead than to directly increasing the range of ap-plicability of the method. However, it is reasonable toexpect that a reduction of the overall size of the linearsystems that must be solved [Eq. (3)] will assumeconsiderable practical importance when implementa-tion of the first and second objectives is attempted.

This research was supported by the National Insti-tutes of Health under grants R21-HL67387, R21-DK63692, R41-CA96102, and R43-NS49734 and bythe U.S. Army under grant DAMD017-03-C-0018.

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