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Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization Keith D. Paulsen and Huabei Jiang Optical image reconstruction in heterogeneous turbid media is sensitive to noise, especially when the signal-to-noise ratio of a measurement system is low. A total-variation-minimization-based iterative algorithm is described in this paper that enhances the quality of reconstructed images with frequency- domain data over that obtained previously with a regularized least-squares approach. Simulation experiments in an 8.6-cm-diameter circular heterogeneous region with low- and high-contrast levels between the target and the background show that the quality of the reconstructed images can be improved considerably when total-variation minimization is included. These simulated results are further verified and confirmed by images reconstructed from experimental data by the use of the same geometry and optically tissue-equivalent phantoms. Measures of imaging performance, including the location, size, and shape of the reconstructed heterogeneity, along with absolute errors in the predicted optical-property values are used to quantify the enhancements afforded by this new approach to optical image reconstruction with diffuse light. The results show improvements of up to 5 mm in terms of geometric information and an order of magnitude or more decrease in the absolute errors in the reconstructed optical-property values for the test cases examined. r 1996 Optical Society of America 1. Introduction Interest is growing rapidly in biomedical applica- tions of optical imaging. For example, the potential advantages of optical breast-cancer detection and tissue-oxygenation mapping have been motivating researchers in several groups worldwide to develop optical image-reconstruction techniques that are ap- propriate for tissues. 1 A number of methods for probing centimeter-scale depths in tissue with scat- tered light at near-infrared wavelengths are now being investigated. 2–11 Within this context, the idea of the indirect use of measured optical data to form an image of the tissue optical-property distributions with a mathematical model of light propagation is gaining acceptance. 15–22 This type of reconstruction approach offers the possibility of the extraction of information contained in the measured optical data that would otherwise be obscured in images formed by direct methods. Several reconstruction algorithms have been devel- oped to date for imaging problems in heterogeneous media including the Newton iterative methods based on the diffusion equation, 20,21 perturbation meth- ods, 15 and the Born approximations. 16 Recently there has been considerable interest in frequency- domain schemes, and results from both simulated and experimental data in the frequency domain have begun to appear. 16–19 In our own work, 18,19 exten- sive simulations and experiments with frequency- domain measurements have been conducted, and simultaneous recovery of absolute absorption and scattering coefficients in multicentimeter phantom geometries has been achieved with a Newton itera- tive algorithm used in conjunction with finite- element methods. Although encouraging successes with model-based image reconstruction have been reported, noise can be a significant limitation for these techniques. For example, the early studies of Arridge et al. 23 indicate that 5% added noise to measured time-resolved data would lead to fairly poor reconstructed images. Pogue et al. 17 have shown seriously degraded image reconstructions with added noise levels of 1% in the ac intensity and 5° in phase using frequency-domain data. Our own experience in this regard has re- vealed that the quantitative quality of reconstructed The authors are with the Thayer School of Engineering, Dart- mouth College, Hanover, New Hampshire 03755. Received 2 October 1995; revised manuscript received 25 Janu- ary 1996. 0003-6935@96@193447-12$10.00@0 r 1996 Optical Society of America 1 July 1996 @ Vol. 35, No. 19 @ APPLIED OPTICS 3447

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Page 1: Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization

Enhanced frequency-domainoptical image reconstruction in tissuesthrough total-variation minimization

Keith D. Paulsen and Huabei Jiang

Optical image reconstruction in heterogeneous turbid media is sensitive to noise, especially when thesignal-to-noise ratio of a measurement system is low. A total-variation-minimization-based iterativealgorithm is described in this paper that enhances the quality of reconstructed images with frequency-domain data over that obtained previously with a regularized least-squares approach. Simulationexperiments in an 8.6-cm-diameter circular heterogeneous region with low- and high-contrast levelsbetween the target and the background show that the quality of the reconstructed images can beimproved considerably when total-variation minimization is included. These simulated results arefurther verified and confirmed by images reconstructed from experimental data by the use of the samegeometry and optically tissue-equivalent phantoms. Measures of imaging performance, including thelocation, size, and shape of the reconstructed heterogeneity, along with absolute errors in the predictedoptical-property values are used to quantify the enhancements afforded by this new approach to opticalimage reconstruction with diffuse light. The results show improvements of up to 5 mm in terms ofgeometric information and an order of magnitude or more decrease in the absolute errors in thereconstructed optical-property values for the test cases examined. r 1996 Optical Society of America

1. Introduction

Interest is growing rapidly in biomedical applica-tions of optical imaging. For example, the potentialadvantages of optical breast-cancer detection andtissue-oxygenation mapping have been motivatingresearchers in several groups worldwide to developoptical image-reconstruction techniques that are ap-propriate for tissues.1 A number of methods forprobing centimeter-scale depths in tissue with scat-tered light at near-infrared wavelengths are nowbeing investigated.2–11 Within this context, the ideaof the indirect use of measured optical data to forman image of the tissue optical-property distributionswith a mathematical model of light propagation isgaining acceptance.15–22 This type of reconstructionapproach offers the possibility of the extraction ofinformation contained in the measured optical datathat would otherwise be obscured in images formedby direct methods.

The authors are with the Thayer School of Engineering, Dart-mouth College, Hanover, New Hampshire 03755.Received 2 October 1995; revised manuscript received 25 Janu-

ary 1996.0003-6935@96@193447-12$10.00@0r 1996 Optical Society of America

Several reconstruction algorithms have been devel-oped to date for imaging problems in heterogeneousmedia including the Newton iterativemethods basedon the diffusion equation,20,21 perturbation meth-ods,15 and the Born approximations.16 Recentlythere has been considerable interest in frequency-domain schemes, and results from both simulatedand experimental data in the frequency domain havebegun to appear.16–19 In our own work,18,19 exten-sive simulations and experiments with frequency-domain measurements have been conducted, andsimultaneous recovery of absolute absorption andscattering coefficients in multicentimeter phantomgeometries has been achieved with a Newton itera-tive algorithm used in conjunction with finite-element methods.Although encouraging successes withmodel-based

image reconstruction have been reported, noise canbe a significant limitation for these techniques. Forexample, the early studies of Arridge et al.23 indicatethat 5% added noise to measured time-resolved datawould lead to fairly poor reconstructed images.Pogue et al.17 have shown seriously degraded imagereconstructions with added noise levels of 1% in theac intensity and 5° in phase using frequency-domaindata. Our own experience in this regard has re-vealed that the quantitative quality of reconstructed

1 July 1996 @ Vol. 35, No. 19 @ APPLIED OPTICS 3447

Page 2: Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization

images is affected considerably by 5% noise added toboth the phase and the ac intensity.19In this paper we propose the incorporation of a

total-variation-minimization scheme in our fre-quency-domain reconstruction algorithm to reducethe effects of noise and to enhance the quality of thereconstructed images. The approach is evaluatedwith simulated data in which up to 10% randomnoise is added to the measured boundary data in acircular test geometry containing a localized hetero-geneity having either low-contrast 12:12 or high-contrast 110:12 ratios to the background. Quantita-tive measures of image quality, including the size,location, and shape of the heterogeneity, along witherrors in its recovered optical-property values areused to quantify the success of the new approach.Reconstructions from experimental data obtained ina laboratory environment are also examined andquantified with the same performance measures.The results show that the imaging enhancementachieved with total-variation minimization duringsimulation experiments can also be realized in alaboratory setting, with perhaps the most impres-sive improvements arising in the absolute values ofthe recovered optical properties of the target.

2. Total-Variation-Minimization Scheme

The concept of total variation was originally con-ceived as a way of restoring 1enhancing2 images 1i.e.,denoising, deblurring, etc.224 and has only very re-cently been applied to the image reconstructionproblem.25,26 Several existing reconstruction algo-rithms in optical imaging are based on nonlinearleast-squares criteria20,21 that stand on the statisti-cal argument that the least-squares estimation isthe best estimator over an entire ensemble of allpossible pictures. Total variation, on the other hand,measures the oscillations of a given function anddoes not unduly punish discontinuities.24,25 Hence,one can hypothesize that a hybrid of these twominimization schemes should be able to providehigher-quality image reconstructions. In fact, thestrategy of finding minimal total-variation recon-structions has recently proved considerably success-ful in applications such as electrical-impedancetomography,25 microwave imaging,26 image process-ing,24,27,28 and optimal design.29Two typical approaches exist for minimizing total

variation: a constrained minimization through thesolution of a nonlinear diffusion equation23,24 and anunconstrained minimization by the addition of thetotal variation as a penalty term to the least-squaresfunctional.26–28,30 Solutions obtained with these twomethods are essentially the same; however, from acomputational standpoint, unconstrained minimiza-tions aremuch easier to implement than constrainedproblems.27 Further, from the perspective of imagereconstruction, the addition of the total-variationterm to the least-squares functional requires modestmodifications to the existing algorithm. Hence, theunconstrained total-variationminimization has beenimplemented for the purpose of this work.

3448 APPLIED OPTICS @ Vol. 35, No. 19 @ 1 July 1996

To describe our hybrid least-squares–total-varia-tion-minimization method, we first briefly introducethe regularized least-squares reconstruction algo-rithm we have used previously.18–20 Our approachcasts the image-formation task as a nonlinear param-eter-estimation problem in which the optical-prop-erty distribution is adjusted to provide a best fit 1in aleast-squares sense2 between a discrete set of mea-sured and computed optical quantities. The under-lying mathematical model is the frequency-domaindiffusion equation

= · D=F 2 1µa 2iv

c 2F 5 2S, 112

whereD is the diffusion coefficient 1which is predomi-nantly related to the inverse of the scattering coeffi-cient in highly scattering media such as tissue, seeRef. 202, µa is the absorption coefficient, v is themodulation frequency, c is the light velocity in themedium, F is the photon density, and S is theharmonically modulated light source. The solutionof Eq. 112 for estimated optical properties is accom-plished with the finite-element method, for which wehave adopted a standard Galerkin weighted-re-sidual formulation that leads to the weak formstatement

7D=F · =ci8 1 71µa 2iv

c 2Fci8 5 7Sci8 1 r D=F · n̂cids,

122

where ci is the ith member of a complete set ofweighting functions, 7· · ·8 indicates integration overthe problem domain, and r denotes integration overthe enclosing boundary surface. Following the pro-cedures outlined in Refs. 19 and 20, the discretiza-tion of Eq. 122 is completed by the expansion of F, D,and µa as the sum of coefficients multiplied by a set oflocally spatially varying Lagrange-polynomial basisfunctions. A final matrix equation,

3A45F6 5 5b6, 132

where the elements of matrix 3A4 are

ai j 5 7ok51

K

Dkck=cj · =ci 1 ol51

L

1µlcl 2iv

c 2cjci8 ,i 5 1, 2, . . . , N, j 5 1, 2, . . . , N,

and the column vectors 5b6 and 5F6 are

bi 5 7Sci8 2 a oj51

B

Fj r cjcids,

F 5 5F1, F2, . . . , FN6T,

is generated by the selection of a finite set of thesesameLagrange polynomials as both basis andweight-

Page 3: Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization

ing functions. In this case we have invoked the typeIII boundary condition 2D=F · n̂ 5 aF, where a is acomplex-valued coefficient whose size depends onboundary reflections 1as discussed by Haskell et al.31andAronson322 and whose value has been empiricallydetermined in Ref. 19. In Eq. 132, Fi is the photondensity at node i, N is the number of nodes in thefinite-element mesh, and B is the number of bound-ary nodes. Note that the expansions used to repre-sent the diffusion and absorption profiles in Eq. 112are K and L terms long when K fi L fi N, in general;however, in the work reported here K 5 L 5 N.To form an image from a presumably uniform

initial guess of the optical-property distribution weneed a method of updating D and µa from theirstarting values. This update is accomplishedthrough the least-squares minimization of

F1F, D, µa2 5 oj51

M

1Fjo 2 Fj

c22, 142

where Fjo and Fj

c are the observed and computedphoton densities, respectively, at the boundary forj 5 1, 2, . . . ,M. One can easily show that theminimization of the above functional is identical tothe approximated equations through the Taylor ex-pansion of the photon density, as reported in Refs.18–20. Using a regularized Newton method, weobtain the following equation for updating D and µa:

1JTJ 1 lI2Dx 5 JT1Fo 2 Fc2, 152

defining a new functional:

F̃1F, D, µa2 5 F1F, D, µa2

1 71wD2 0=D 02 1 wµ

2 0=µa 02 1 d221@28, 162

where wD, wµ, and d are typically small positiveparameters that need to be determined numerically.The introduction of the small number d is needed toovercome any nondifferentiability in the total varia-tions of D and µa.27,28 The minimization of Eq. 162proceeds in standard fashion by the differentiationof F̃ with respect to each nodal parameter thatconstitutes the D and µa distributions; simulta-neously all these relations are set to zero. Thesesteps lead to the nonlinear system of equations

≠F̃

≠x15 2o

j51

M

1Fjo 2 Fj

c2≠Fj

c

≠x11 V1 5 0,

≠F̃

≠x25 2o

j51

M

1Fjo 2 Fj

c2≠Fj

c

≠x21 V2 5 0,

···

≠F̃

≠x2N5 2o

j51

M

1Fjo 2 Fj

c2≠Fj

c

≠x2N1 V2N 5 0, 172

where x 5 5D1, D2, . . . , DN, µa1, µa2, . . . , µaN 6T,

for i 5 1, 2, . . . ,N, and

Vi 5 ewD

21≠ci

≠x ok51

N

Dk

≠ck

≠x1

≠ci

≠y ok51

N

Dk

≠ck

≠y 25wD

231ok51

N

Dk

≠ck

≠x 22

1 1ok51

N

Dk

≠ck

≠y 22

4 1 wµ231o

l51

N

µl≠cl

≠x 22

1 1ol51

N

µl≠cl

≠y 22

4 1 d261@2

dxdy,

Vi 5 ewµ

21≠ci

≠x ol51

N

µl≠cl

≠x1

≠ci

≠y ol51

N

µl≠cl

≠y 25wD

231ok51

N

Dk

≠ck

≠x 22

1 1ok51

N

Dk

≠ck

≠y 22

4 1 wµ231o

l51

N

µl≠cl

≠x 22

1 1ol51

N

µl≠cl

≠y 22

4 1 d261@2

dxdy,

where J is a Jacobian matrix consisting of deriva-tives of F 1at each boundary-observation point2 withrespect to the discrete set of parameters that de-scribe theD and µa distributions,18–20 Dx is the vectorthat contains the discrete set of perturbations for Dand µa, I is the identity matrix, and l is a regulariza-tion parameter.33We now incorporate the total variations of D and

µa as penalty terms, as reported in Refs. 25–27, by

for i 5 N 1 1,N 1 2, . . . , 2N, which can be solved bythe Newton method. Defining G as

G 5 1≠F̃≠x1

, ≠F̃

≠x2

, . . . ,≠F̃

≠x2N2T

5 1 f1, f2, . . . , f2N2T

leads to a regularized Newton iteration requiring the

1 July 1996 @ Vol. 35, No. 19 @ APPLIED OPTICS 3449

Page 4: Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization

solution of the matrix system

1G 1 lI2Dx 5 2G, 182

where Dx 5 x1n2 2 x1n212 and

G 5 3≠f1≠x1

≠f1≠x2

· · ·≠f1

≠x2N

≠f2≠x1

≠f2≠x2

· · ·≠f2

≠x2N

······

· · ····

≠f2N≠x1

≠f2N≠x2

· · ·≠f2N≠x2N

4 .Rewriting Eq. 182 in a form that can be related to Eq.152, one sees that the minimizers of Eq. 162 areiteratively found by solving the matrix system

1JTJ 1 R 1 lI2Dx 5 JT1Fo 2 Fc2 2 V, 192

where

R 5 3≠V1

≠D1· · ·

≠V1

≠DN

≠V1

≠µ1· · ·

≠V1

≠µN

···· · ·

······

· · ····

≠VN

≠D1· · ·

≠VN

≠DN

≠VN

≠µ1· · ·

≠VN

≠µN≠VN11

≠D1· · ·

≠VN11

≠DN

≠VN11

≠µ1· · ·

≠VN11

≠µN

···· · ·

······

· · ····

≠V2N

≠D1· · ·

≠V2N

≠DN

≠V2N

≠µ1· · ·

≠V2N

≠µN

4 ,V 5 1V1, V2, . . . , VN, VN11, VN12, . . . , V2N2T as de-fined in Eqs. 172, and Dk 1for k 5 1, 2, . . . , N2 and µ11for l 5 1, 2, . . . , N2 are the reconstruction param-eters for the optical-property profile. Given thestructure of Eq. 192 relative to Eq. 152, it is tempting toview the addition of total variation as another formof regularization. Although this may be perfectlyvalid, we believe it is more powerful to consider thisnew algorithm as resulting from the minimization ofa different functional 3i.e., Eq. 162 versus Eq. 1324,which has a different objective. Specifically, theaddition of the total-variation term serves to counter-balance the desire of the least-squares objectivefunction to produce smooth solutions not only be-cause it allows sharp discontinuities to exist, butalso because it reinforces their existence during theminimization process. After Eq. 192 has beenreached, our solution procedure 1standard Gaussianelimination2 and regularization parameter selection1Marquardtmethod2 are identical to those used previ-

3450 APPLIED OPTICS @ Vol. 35, No. 19 @ 1 July 1996

ously in Refs. 19 and 20; hence it becomes clear thatthe only additions to our new algorithm result fromthe assembly of matrix R and the construction ofcolumn vector V.

3. Results

In this section the enhanced reconstruction algo-rithmwith total-variation minimization is evaluatedfirst with a few simulation examples and then withsome experimental data. Both the simulated andexperimental test cases to be presented are identicalto those we have utilized elsewhere18,19; hence onlyan abbreviated discussion of their important fea-tures is provided here. For comparative purposes,reconstructions without the total-variation-minimi-zation enhancement 1i.e., those which use ourpreviously described18,19 regularized least-squares-minimization algorithm2 are also presented. In thesimulations image reconstruction is performed un-der conditions of no measurement noise and anadded 10% random noise to both the real andimaginary components of the complex-valued photondensity 1which results in up to a 10% variation in theac amplitude and a 5° variation in phase shift2.Reconstructions using both low-contrast 12:12 andhigh-contrast 110:12 levels between the target and thebackground are also considered. With respect tocontrast, both the absorption and scattering coeffi-cients were increased in the target regions by thestated amounts; however, it is important to recog-nize that the results that follow are displayed interms of the diffusion coefficient, which actuallydecreases proportionally to the increase in scattering.In the reconstruction examples involving experimen-tal data, all geometric and optical parameters corre-spond to those used in the simulated data, andimaging performance is verified by the use of thesame qualitative and quantitative metrics.The test geometry is a circular, black plastic

phantom 86 mm in diameter that has a 25-mm-diameter small region 110-mm offset along the hori-

Fig. 1. Diagram of the phantom geometry for the off-center-target case under study: R1 5 43 mm, R2 5 12.5 mm.

Page 5: Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization

mental setup, phantom design, and materials have

1a2 1b2 1c2

1d2 1e2 1f2

Fig. 2. Simulated simultaneous reconstructions of both thediffusion and absorption coefficients with a 2:1 contrast level foran eccentrically located target without added random noise: 1a2exact image of D, 1b2 reconstruction of D with no total-variation1TV2 minimization, 1c2 reconstruction of D with TV minimization,1d2 exact image of µa, 1e2 reconstruction of µa with no TV minimiza-tion, and 1f 2 reconstruction of µawith TV minimization.

zontal axis at 3 o’clock2 representing a tumor ortarget area 1Fig. 12. For both the simulated andexperimental data we have used the same opticalproperties for the background and the target; specifi-cally, the background medium has values of µs8 5 0.6mm21 1reduced scattering coefficient2 and µa 5 0.006mm21, against which the low-contrast 12:12 and high-contrast 110:12 levels 1target to background2 havebeen studied. The modulation frequency was se-lected to be 150 MHz, and multiple excitation andmeasurement positions were used to collect theboundary information needed in the reconstructions.Specifically, measurements were recorded at 16equally spaced 1circumferentially 22.5° apart2 detec-tors for a single fixed-source position 1halfway be-tween two detectors2, then the target was rotated22.5° and the measurements repeated in successionuntil data from 16 equivalent source positions wereobtained, which yielded a total of 256 separateamplitude- and phase-shift observations for eachimage reconstruction. Further details of the experi-

1a2 1b2

1c2 1d2

Fig. 3. Comparison of exact and simulated reconstructions along transects AB and CD 1Fig. 12 for the images appearing in Fig.2. Profiles of 1a2 D along transect AB, 1b2 µa along transect AB, 1c2 D along transect CD, and 1d2 µa along transect CD. The vertical axesindicate values of either D or µa in millimeters and inverse millimeters, respectively, and the horizontal axes indicate the distance alongeither transect AB or CD in millimeters.

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Page 6: Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization

1a2 1b2 1c2

1d2 1e2 1f2

Fig. 4. Simulated simultaneous reconstructions of both thediffusion and absorption coefficients with a 2:1 contrast level foran eccentrically located target with an added 10% random noise:1a2 exact image of D, 1b2 reconstruction of D with no total-variation1TV2 minimization, 1c2 reconstruction of D with TV minimization,1d2 exact image of µa, 1e2 reconstruction of µa with no TV minimiza-tion, and 1f 2 reconstruction of µawith TV minimization.

been described in Ref. 19, and in fact the experimen-tal data used here are identical to those utilized inRefs. 18 and 19. The radial location of each sourcewas positioned inside the physical boundary by adistance of d 5 1@µs8 1µs8 is the reduced scatteringcoefficient of the background medium2 for the point-source excitation implemented in the reconstructionalgorithm.20,21 Mixed boundary conditions 1Type III2were applied for both the simulated and experimen-tal data. The finite-element mesh used in the ex-amples shown herein consisted of 492 nodes and 918triangle elements.As mentioned above, the parameters wD, wµ, and d

were determined through numerical experimenta-tion. We have found that the small number d is notas critical as the weights wD and wµ and that d has avalue of d 5 0.001 for both the simulations andexperiments presented here, whereas some numeri-cal experimentation was needed to find suitablevalues for the weights, the sizes of which have beensuggested to be related to the signal-to-noise ratio ofthe data.28 To limit the potential parameter space,

1a2 1b2

1c2 1d2

Fig. 5. Comparison of exact and simulated reconstructions along transectsABandCD 1Fig. 12 for the images appearing inFig. 4. Profiles of 1a2Dalong transectAB, 1b2µa along transectAB, 1c2Dalong transectCD, and 1d2µa along transectCD. Thevertical axes indicate values of eitherD orµainmillimeters and inversemillimeters, respectively, and the horizontal axes indicate the distance along either transectAB orCD inmillimeters.

3452 APPLIED OPTICS @ Vol. 35, No. 19 @ 1 July 1996

Page 7: Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization

we have considered only the case in whichwD 5wµ atthis point, and we have not explored the sensitivityof the weights to other experimental parameters,such as the modulation frequency, measurementgeometry, etc. We have also not explored any typeof strategy for iteratively altering the weightingduring image reconstruction, as is done with theregularization parameter l. For the cases pre-sented, values of wD 5 wµ 5 1.05 3 10210 for thesimulations without random noise added,wD 5 wµ 51.05 3 1026 for the simulations with 10% randomnoise added, and wD 5 wµ 5 1.05 3 1028 for theexperimental data appear to provide excellent results.All the final images reported are each the result of 20iterations of the algorithm at a cost of 1 min periteration on an IBM RISC 6000 workstation for thefinite-element mesh used.

A. Reconstructions from Simulated Data

In each simulation, the so-called measured data isgenerated by the use of a forward-diffusion model

1a2 1b2 1c2

1d2 1e2 1f2

Fig. 6. Simulated simultaneous reconstructions of both thediffusion and absorption coefficients with a 10:1 contrast level foran eccentrically located target without added random noise: 1a2exact image of D, 1b2 reconstruction of D with no total-variation1TV2 minimization, 1c2 reconstruction of D with TV minimization,1d2 exact image of µa, 1e2 reconstruction of µa with no TV minimiza-tion, and 1f 2 reconstruction of µawith TV minimization.

with the exact optical properties in place. Figures21a2 and 21d2 show the basic character of the exactoptical-property distribution for the 2:1 contrast-level case, the reconstruction of which is sought fromboundary-only observations. The other images inFig. 2 are representative of reconstructions for the2:1 contrast level without 3Figs. 21b2 and 21e24 and with3Figs. 21c2 and 21f24 the total-variation minimizationinvoked under conditions of no noise added to themeasured data. As can be seen, enhancement ofthe reconstruction by the incorporation of the total-variation minimization is obvious over that achievedwith the regularized least-squares minimization.To provide a more quantitative assessment of theseimages, Fig. 3 is included, in which the reconstructedoptical-property distributions are displayed alongtwo transects through the domain—one through thecenters of both the target and the background re-gions 1transect AB in Fig. 12 and the other throughthe center of the target but perpendicular to the firsttransect 1transect CD in Fig. 12—without and withthe total-variation minimization compared with theexact values.Figure 4 presents image reconstructions having

10% added random noise to the measured datawithout 3Figs. 41b2 and 41e24 and with 3Figs. 41c2 and 41f24the total-variation minimization. Again, consider-able improvement can be observed in the recon-structed images when the total-variation minimiza-tion is invoked compared with only the regularizedleast-squares minimization. Similarly to Fig. 3,Fig. 5 shows quantitative information for the imagesdisplayed in Fig. 4 along the two transects shown inFig. 1. As an additional evaluation of our recon-struction algorithm, we present Fig. 6, in whichimages having a 10:1 contrast level between thetarget and the background are obtained from noise-free data without and with the total-variation mini-mization. Clearly an improved image quality isagain apparent.To obtain further quantitative information about

the reconstructed images in these figures, the imageerrors 1the maximum and average differences in thereconstructed optical-property profiles comparedwiththe exact distribution2 were determined and the

Table 1. Comparison of Image Errors for Simulated-Data Images with Differing Noise and Contrast Levels, both with and without the Total-VariationATVB Minimization a

Noise State andContrast Level

Absolute Error

D Image (mm) µa Image (1022 mm21)

Without TV With TV Without TV With TV

Maximum Average Maximum Average Maximum Average Maximum Average

No noise2:1 Contrast 0.20 0.021 0.096 0.005 1.64 0.085 0.40 0.00310:1 Contrast 0.21 0.045 0.092 0.009 1.83 0.12 0.51 0.005

10%Added noise 0.23 0.18 0.16 0.019 3.65 1.22 0.61 0.008

aImage errors constitute the absolute difference between the true and the reconstructed values. Background values: µa 5 0.006mm21, µs8 5 0.6 mm21, i.e.,D 5 0.56 mm. Target values for the 2:1 contrast case: µa 5 0.012 mm21, µs8 5 1.2 mm21, i.e.,D 5 0.28 mm.Target values for the 10:1 contrast case: µa 5 0.06 mm21, µs8 5 6.0 mm21, i.e.,D 5 0.056 mm.

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Page 8: Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization

Table 2. Geometric Information for Reconstructed Simulated-Data Images with Differing Noise and Contrast Levels, both with and without theTotal-Variation ATVB Minimization a

Noise, Contrast Level,and TV State

Target Characteristics

Location Size Shape

D-ImageCoordinates

µa-ImageCoordinates

D-ImageTransect Length

µa-ImageTransect Length

D-ImageTransect Ratio

µa-ImageTransect Ratio

x y x y EF GH EF GH EF/GH EF/GH

Exactb 10.4 0.0 10.4 0.0 27.0 28.0 27.0 28.0 0.96 0.96No noise, 2:1 contrastlevel, without TV

15.6 0.6 7.8 0.8 24.5 29.6 29.3 31.0 0.83 0.95

No noise, 2:1 contrastlevel, with TV

10.6 20.5 11.2 0.2 26.7 28.6 28.4 29.3 0.93 0.97

10%Added noise with TV 11.5 1.4 7.9 22.4 23.8 24.1 25.8 26.7 0.99 0.97No noise, 10:1 contrastlevel, without TV

15.8 0.3 8.5 0.5 26.4 29.1 28.9 30.2 0.91 0.95

No noise, 10:1 contrastlevel, with TV

10.5 20.3 10.8 0.3 27.3 28.5 26.5 27.4 0.96 0.97

aThe x, y coordinates 1in millimeters2 are those of the target center; EF and GH are the transect lengths 1in millimeters2 of the targetregion along the x and y directions, respectively 1see Fig. 12. The case of 10% added noise without TV is not included because noidentifiable target region was recovered 3see Figs. 41b2 and 41e24.

bNote that EF fi GH for the exact case because the property profile is modeled as a linear interpolation across the jump discontinuityassumed in the optical properties 3see Figs. 21a2 and 21d24.

location, size, and shape, of the reconstructed targetwere calculated through the estimation of the FWHMof the reconstructed optical-property profiles alongthe two transects EF and GH.19 Tables 1 and 2present the results from these calculations for allimages shown in Figs. 2, 4, and 6.

B. Reconstruction from Experimental Data

In this section experimental data are used to verifyand confirm our findings from the simulations.Figure 7 shows reconstructed images obtained forthe eccentrically located target having a 2:1 contrast

1a2 1b2 1c2

1d2 1e2 1f2

Fig. 7. Simultaneous reconstructions of both the diffusion andabsorption coefficients from the experimental data with a 2:1contrast level for an eccentrically located target: 1a2 exact imageof D, 1b2 reconstruction of D with no total-variation 1TV2minimiza-tion, 1c2 reconstruction of D with TV minimization, 1d2 exact imageof µa, 1e2 reconstruction of µa with no TV minimization, and 1f 2reconstruction of µawith TV minimization.

3454 APPLIED OPTICS @ Vol. 35, No. 19 @ 1 July 1996

level relative to the background, both with andwithout the total-variation minimization invoked;again, the exact images are included for comparisonpurposes. As can be seen, the images formed byincorporation of the total-variation minimization areclearly enhanced qualitatively in visual content rela-tive to images that received only the regularizedleast-squares minimization. Figure 8 provides amore quantitative assessment of these images andreports the reconstructed optical-property profilesalong the two transects shown in Fig. 1.Reconstructions involving a high-contrast 110:12

level between the target and the background, withand without the total-variation minimization, aredemonstrated in Fig. 9, and again the improvedimage quality resulting from the incorporation of thetotal-variation minimization as compared with theregularized least-squares minimization is readilyapparent. Further quantitative information aboutthe image errors and the location, size, and shape ofthe reconstructed target for the images shown inFigs. 7 and 9 are provided in Tables 3 and 4.

4. Discussion and Conclusions

The results presented in Section 3 indicate that asignificant amount of useful information, both quali-tative and quantitative, can be obtained from recon-structions based on frequency-domain diffuse opticaldata. Importantly, it has been demonstratedthrough both simulations and experiments that abso-lute optical reconstructions can be quantitativelyenhanced by the use of the total-variation-minimiza-tion scheme relative to those reconstructions ob-tained with only the regularized least-squares mini-mization, not only in terms of the location, size, and

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1a2 1b2

1c2 1d2

Fig. 8. Comparison of the exact and reconstructed profiles along transectsABandCD 1Fig. 12 for the images appearing inFig. 7. Profiles of 1a2Dalong transectAB, 1b2µa along transectAB, 1c2Dalong transectCD, and 1d2µa along transectCD. Thevertical axes indicate values of eitherD orµainmillimeters and inversemillimeters, respectively, and the horizontal axes indicate the distance along either transectAB orCD inmillimeters.

shape of the target, but also of the optical-propertyvalues themselves. Figures 2–6 and Tables 1 and 2for the simulated data and Figs. 7–9 and Tables 3and 4 for the experimental data clearly support theseconclusions.A further examination of the simulated images in

Fig. 2 and their experimental counterparts in Fig. 7suggests that the reconstructed images with thetotal-variation minimization for D result in a moredramatic improvement in terms of the recovery ofthe target location, size, and shape than do thereconstructed µa images when compared with thosewithout the total-variationminimization. The quan-titative measures in Tables 2 and 4 bear out thisvisual perception for the most part. The D imagesshow a consistent 3–5-mm improvement in location,a 1–2-mm improvement in size, and a 5–20% im-provement in the shape factor as compared with theµa images, which improve 1–2 mm in location, 1–4mm in size, and 1–3% in shape. The exact reasonsas to why theD images appear to improve more thantheir µa counterparts are unclear at the present time

1a2 1b2 1c2

1d2 1e2 1f2

Fig. 9. Simultaneous reconstructions of both the diffusion andabsorption coefficients from the experimental data with a 10:1contrast level for an eccentrically located target: 1a2 exact imageof D, 1b2 reconstruction of D with no total-variation 1TV2minimiza-tion, 1c2 reconstruction of D with TV minimization, 1d2 exact imageof µa, 1e2 reconstruction of µa with no TV minimization, and 1f 2reconstruction of µawith TV minimization.

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Table 3. Image Errors a for Experimental-Data Images with Differing Contrast Levels between the Target and the Background, both with and withoutthe Total-Variation ATVB Minimization

Contrast Level

Absolute Error

D Image (mm) µa Image (1022 mm21)

Without TV With TV Without TV With TV

Maximum Average Maximum Average Maximum Average Maximum Average

2:1 Contrast 0.24 0.035 0.12 0.016 1.85 0.10 0.62 0.00810:1 Contrast 0.32 0.19 0.18 0.019 3.96 1.22 0.68 0.011

aImage errors constitute the absolute difference between the true and the reconstructed values. Background values: µa 5 0.006mm21, µs8 5 0.6 mm21, i.e.,D 5 0.56 mm. Target values for the 2:1 contrast case: µa 5 0.012 mm21, µs8 5 1.2 mm21, i.e.,D 5 0.28 mm.Target values for the 10:1 contrast case: µa 5 0.06 mm21, µs8 5 6.0 mm21, i.e., D 5 0.056 mm. The off-center target is located at 3o’clock, approximately 10 mm away from the center of the phantom.

but may be due to the fact that the D images wereconsistently worse to start with 1i.e., without totalvariation2, or this observation may suggest that therelative weights between the D and µa total-varia-tion contributions to the functional to be minimized3i.e., Eq. 1624 should not be equal. This latter possibil-ity should be explored in the future. It is interest-ing to note that the improvements in the size of theµa target recovery from the experimental data areconsistently 1–2 mmmore than that occurring in thesimulations. With the total-variation minimizationof the experimental data, the overprediction in the µatarget size is reduced by several inverse millimeters.From Figs. 3 and 8, one can observe that there is

considerable enhancement in the recovery of theoptical-property values 1for both D and µa2 when thetotal-variation minimization is incorporated. Interms of optical properties, the µa images may actu-ally show more improvement than the D images,which experienced more improvement in the geo-metrical factors of recovered target size, shape, andlocation, as discussed above. Tables 1 and 3 indi-cate that the absolute differences between the exactand the recovered optical-property values on averageare reduced anywhere from a minimum of a factor of2 to a maximum of several orders of magnitude.Most impressive in this regard are the gains in thequantitative nature of the recovered absorption-

coefficient values when the total-variationminimiza-tion is included.Figures 4 and 5 show that the addition of the

total-variation minimization can significantly en-hance image reconstruction in the presence of ran-dom noise. With a large amount 110%2 of randomnoise added to themeasured data, it is not possible todistinguish between the target and the backgroundwhen no total-variation minimization 1i.e., regular-ized least-squares minimization2 is used 3Figures 41b2and 41e24, yet good reconstructed images for both Dand µa 3Figs. 41c2 and 41f 24 are obtained when thetotal-variation minimization is incorporated. Fur-ther quantification of this finding is provided inTables 1 and 2.Interesting observations are also possible from the

studies of the reconstructed images having a 10:1contrast level between the target and the back-ground, presented in Fig. 6 for simulations and Fig. 9for experiments. Again, considerably enhancedsimulated images are achieved with the total-variationminimization, and their experimental coun-terparts further confirm this result. It is especiallyintriguing to note that the D image for a 10:1contrast level without the total-variation minimiza-tion exhibits more pronounced artifacts around thetarget boundary, whereas a much cleaner D image isobtained with the total-variation minimization.

Table 4. Geometric Information for Reconstructed Experimental-Data Images with Differing Contrast Levels, both with and without the Total-VariationATVB Minimization a

Contrast Level,and TV State

Target Characteristics

Location Size Shape

D-ImageCoordinates

µa-ImageCoordinates

D-ImageTransect Length

µa-ImageTransect Length

D-ImageTransect Ratio

µa-ImageTransect Ratio

x y x y EF GH EF GH EF/GH EF/GH

Exact 10.4 0.0 10.4 0.0 25.0 25.0 25.0 25.0 1.0 1.02:1 Contrast without TV 15.4 0.7 7.4 0.6 23.5 30.6 29.8 31.6 0.77 0.942:1 Contrast with TV 10.2 20.2 11.8 0.4 25.5 26.8 27.3 28.1 0.95 0.9710:1 Contrast without TV 14.1 0.6 8.2 0.7 27.2 28.1 29.2 30.5 0.97 0.9610:1 Contrast with TV 10.8 20.4 11.2 0.5 24.8 26.7 25.9 26.0 0.93 0.99

aThe x, y coordinates 1in millimeters2 are those of the target center; EF and GH are the transect lengths 1in millimeters2 of the targetregion along the x and y directions, respectively 1see Fig. 12.

3456 APPLIED OPTICS @ Vol. 35, No. 19 @ 1 July 1996

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This result implies that the total-variation-minimiza-tion reconstruction algorithm can effectively reducethe artifacts caused by strong instabilities at theinterface between drastically distinct optical mediain the ill-conditioned inverse problem, making total-variation minimization an excellent choice for thereconstruction of images with a wide range of con-trast levels between the target and the background.These enhancements are further quantitatively dem-onstrated in Tables 1–4.We have shown that a total-variation criterion is

easily introduced in unconstrained minimizationproblems such as those arising in diffuse opticalimaging in tissue and that it’s introduction results inmodest changes to our particular finite-element-based Newton-method implementation. We alsofound that the increased computational costs associ-ated with total-variation minimization are nominal,especially when they are considered relative to thesignificant gains in recovered-image quality. How-ever, the one caveat to all this improvement is thatthe weighting of the total-variation term must befound empirically, which requires some a prioriknowledge of the target image. Further research isneeded to determine the ultimate utility of thisapproach in the image-reconstruction mode. It isclearly a powerful method, but finding a criterion forselecting an optimal weighting without any knowl-edge of what the final image is supposed to look likewill be critical for the success of the method inimage-reconstruction problems. As noted in Sec-tion 2, the addition of the total-variation term is akinto regularization 3compare Eqs. 152 and 1924; hence, wemay gain insight by viewing the determination of thetotal-variation weighting as similar to choosing aregularization parameter, the selection of which hasbeen discussed in the optical-imaging context 1e.g.,Ref. 212.In summary, we have demonstrated frequency-

domain optical image reconstructions using an algo-rithm that incorporates total-variationminimization.Both quantitative and qualitative enhancements ofthe reconstructed images have been achieved theo-retically and experimentally when the new algo-rithm is compared with the regularized least-squaresminimization. The results have shown thatthe inclusion of total-variation minimization in ourreconstruction algorithm is highly effective in thepresence of noisy data. It is also important to notethat this new algorithm can be used to overcome theartifacts caused by the regular instabilities in ill-conditioned inverse problems and is ideal for imagerecovery over a wide range of contrast levels betweenthe target and the background.

The authors would like to thank Ulf Osterberg,Michael Patterson, and Brian Pogue for many stimu-lated discussions and for the use of facilities atthe Hamilton Regional Cancer Center, Hamilton,Ontario, where the experiments were performed.This work was sponsored in part by the Alma HassMilham Fellowship in Biomedical Engineering.

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