absorption and scattering images of heterogeneous scattering media can be simultaneously...

Absorption and scattering images of heterogeneousscattering media can be simultaneously reconstructedby use of dc data

Yong Xu, Xuejun Gu, Taufiquar Khan, and Huabei Jiang

We present a carefully designed phantom experimental study aimed to provide solid evidence that bothabsorption and scattering images of heterogeneous scattering media can be reconstructed independentlyfrom dc data. We also study the important absorption–scattering cross-talk issue. In this regard, wedevelop a simple normalizing scheme that is incorporated into our nonlinear finite-element-based recon-struction algorithm. Our results from the controlled phantom experiments show that the cross talk ofan absorption object appearing in scattering images can be eliminated and that the cross talk of ascattering object appearing in absorption images can be reduced considerably. In addition, these care-fully designed phantom experiments clearly suggest that both absorption and scattering images can besimultaneously recovered and quantitatively separated in highly scattering media by use of dc measure-ments. Finally, we discuss our results in light of recent theoretical findings on nonuniqueness for dcimage reconstruction. © 2002 Optical Society of America

OCIS codes: 170.3010, 170.3830, 170.6960, 170.3890.

1. Introduction

Diffuse optical tomography �DOT�, as a potentialmedical imaging modality, has been investigated formore than a decade.1 The idea of DOT is the re-construction of the spatial distribution of opticalproperties within tissue by use of tomographic mea-surements of near-infrared diffusive light along thetissue boundary. Light propagation within tissueis generally described by photon diffusion approxi-mation, which has proven to be capable of providingaccurate methods in many practical situations.1 Aunique advantage of DOT is its capability of ex-tracting both tissue structural maps and functionalinformation such as hemoglobin, water content, andlipid concentration. One important potential appli-cation of DOT is for breast cancer detection. Afterevaluation of various DOT methods with extensivetissuelike phantom experiments, several groups re-cently reported their initial studies of DOT on human

The authors are with Clemson University, Clemson, South Caro-lina 29634. Y. Xu, X. Gu, and H. Jiang �[email protected]� arewith the Department of Physics and Astronomy, and T. Khan iswith the Department of Mathematical Sciences.

Received 14 December 2001; revised manuscript received 9 May2002.

0003-6935�02�255427-11$15.00�0© 2002 Optical Society of America

breasts.2–7 This represents a significant progressionof this emerging technology from laboratory studiesto clinical examinations.

In DOT, three major methods based on time reso-lution, frequency domain, and continuous wave �cw�have been used to probe turbid media or tissues.Although time- and frequency-domain-basedmethods8–11 may provide more optical informationthan cw approaches, a number of researchers havebeen interested in cw image reconstruction for sev-eral years, in part because of the relative simplicity,high signal-to-noise ratio, and low cost of cwtechniques.12–17 A recent study showed that, whendeconvolved, cw data appear to provide significantlybetter spatial resolution than frequency-domain da-ta.18 To date, significant progress in cw DOT hasbeen made with both phantom and in vivo clinicaldata,2,6,7,12–17,19 which suggests that research of cwDOT should be performed parallel to that of otherDOT methods.

Although several cw- or dc-based reconstructionalgorithms have been developed, our own research inthis regard has been based on a powerful nonlinearfinite-element reconstruction algorithm.13,19 Inthese early studies, we showed that qualitative re-covery of embedded objects with both absorption andscattering contrast can be obtained from simulatedand experimental dc data. Recently Iftimia andJiang20 reported quantitative reconstruction of ob-

1 September 2002 � Vol. 41, No. 25 � APPLIED OPTICS 5427

jects with both absorption and scattering contrast withdc measurements when a novel data-preprocessing–optimization scheme is used before image reconstruc-tion. This data-preprocessing scheme provides amethod to optimize the critical parameters needed forreconstruction, including the boundary-conditions co-efficient and the initial estimates of optical proper-ties. The use of this scheme has also eliminated theneed for a reference or calibration measurement withhomogeneous phantoms.13,19 We have extended theimproved algorithm from two-dimensional �2D� tothree-dimensional �3D� cases and demonstrated full3D reconstruction of both absorption and scatteringimages by use of dc phantom data.21 More recentlywe achieved successful 2D and 3D image reconstruc-tions by use of in vivo breasts and bones and jointsfrom dc data.6,7,22,23

The present study has been motivated by two fac-tors. First, there is a recent interest in studying theabsorption–scattering cross-talk problem �e.g., local-ized variations in absorption appear as localized vari-ations in scattering in the reconstructed image or viceversa�. This issue has been studied theoretically bySchweiger and Arridge24 and discussed experimen-tally by McBride et al.4 Pei et al.25 developed anormalized-constraint algorithm, which has shown tobe capable of reducing the interparameter cross talkconsiderably with dc measured data. Second, thereis a theoretical issue brought up by Arridge andLionheart26: They claimed that simultaneousunique recovery of absorption and scattering coeffi-cients cannot be achieved from dc data. To addressthe nonuniqueness issue, we present solid experi-mental evidence here. In their paper, Pei et al.25

attempted to provide a number of rather thoughtfulexplanations to support why simultaneous recoveryof absorption and scattering coefficients is possible byuse of dc data. In this paper we discuss severalcritical differences between dc DOT and the theoret-ical derivations and offer an initial mathematicalrationale for a possible unique solution with dcdata. To attack the interparameter cross-talk prob-lem, we propose a simple normalizing scheme andincorporate it into our finite-element reconstructionalgorithm. Using carefully designed phantom ex-periments, we provide solid evidence that we cansimultaneously reconstruct and quantitatively sep-arate absorption and scattering coefficients of het-erogeneous scattering media using dc data.

The organization of this paper is as follows. Ourfinite-element-based reconstruction algorithm for dcmeasurements is briefly described, and a novel, to ourknowledge, normalizing method-based algorithm isdetailed in Section 2. Experimental materials andmethods are presented in Section 3. Imaging re-sults are given in Section 4 and discussed in Section5. Conclusions are also presented in Section 5.

2. Reconstruction Algorithm

Our reconstruction algorithm uses a regularized New-ton’s method to update an initial optical property dis-tribution iteratively to minimize an object function

composed of a weighted sum of the squared differencebetween computed and measured optical data at themedium’s surface. The computed optical data �i.e.,photon density� is obtained by solution of the photondiffusion equation with the finite-element method.The mathematical details of our algorithm and itsevaluation are described elsewhere.20,27 The core pro-cedure in our reconstruction algorithm is the iterativesolution of the following regularized matrix equation:

��T� � �I��q � �T��m� � ��c��, (1)

where � is the photon density; �a is the absorptioncoefficient; D is the diffusion coefficient, which can bewritten as D � 1�3��s where ��s is the reduced scat-tering coefficient. I is the identity matrix, and � canbe a scalar or a diagonal matrix. �q � ��D1,�D2, . . . , �DN, ��a,1, ��a,2, . . . , ��a,N�T is the up-date vector for the optical property profiles, where Nis the total number of nodes in the finite-elementmesh used; ��m� � ��1

�m�, �2�m�, . . . , �M

�m�� and ��c� ��1

�c�, �2�c�, . . . , �M

�c��, where �i�m� and �i

�c�, respectively,are measured and calculated data for i � 1, 2, . . . , Mboundary locations. � is the Jacobian matrix that isformed by ��D and ��a at the boundary mea-surement sites as follows:

�

� ��1

D1

�1

D2· · ·

�1

DN

�1

�a,1

�1

�a,2· · ·

�1

�a,N

�2

D1

�2

D2· · ·

�2

DN

�2

�a,1

�2

�a,2· · ·

�2

�a,N

······

· · ····

······

· · ····

�M

D1

�M

D2· · ·

�M

DN

�M

�a,1

�M

�a,2· · ·

�M

�a,N

� .

(2)

In DOT, the goal is to update the �a and D or ��sdistributions through the iterative solution of Eq. �1� sothat a weighted sum of the squared difference betweencomputed and measured data can be minimized.

The idea of the normalizing scheme is to balancethe variations of ��D and ��a in the Jacobianmatrix by normalization of each of these derivativeswith their respective maximums. Assuming thatmD and m� are the maximums of ��D and ��ain an iterative computation, respectively, that is,

mD � Max��1

D1,

�1

D2, · · ·

�1

DN,

�2

D1,

�2

D2,

· · ·�2

DN, · · ·

�M

D1,

�M

D2, · · ·

�M

DN� , (3)

m� � Max� �1

�a,1,

�1

�a,2, · · ·

�1

�a,N,

�2

�a,1,

�2

�a,2,

· · ·�N

�a,N,�M

�a,1, · · ·

�M

�a,2, · · ·

�M

�a,N� , (4)

5428 APPLIED OPTICS � Vol. 41, No. 25 � 1 September 2002

we then obtain the normalized ��D and ��a as

�

D�

1mD

�

D, (5)

�

�a�

1m�

�

�a(6)

and rewrite the regularized Eq. �1� as

�� T�� I��q� � �� T��m� � ��c��, (7)

where

3. Experimental Materials and Methods

The experimental setup used is schematically shownin Fig. 1, which is an automated multichannelfrequency-domain system �we needed just dc mea-surements to reconstruct the absorption and scatter-ing images reported here�. This system wasdescribed in detail elsewhere.7,20 Briefly, anintensity-modulated light at 150 MHz from a 785-nm50-mW diode laser is sequentially sent to the phan-tom by sixteen 3-mm fiber-optic bundles. For eachsource position, the diffused light is received at 16detector positions along the surface of the phantomand sequentially delivered to a photomultiplier tube�PMT�. A second PMT is used to record the refer-ence signal. The multiplexing of the source–detector fibers is accomplished by two automaticmoving stages. dc, ac, and phase-shift signals are

obtained by use of the standard heterodyne techniquecontrolled by fast Fourier transform LABVIEW routines.The total data-collection time for 256 measurementsis approximately 8 min. The measured dc data arethen input into our reconstruction software to gener-ate a 2D cross-sectional image of the phantom at thesource–detector plane.

The phantom materials used consisted of Intralipidas scatterer and India ink as absorber. We used1–2% agar powder to solidify the Intralipid and Indiaink solutions.28 The background phantom was a 50-mm-diameter solid cylinder with �a � 0.007 mm1

and ��s � 1.0 mm1. One or two 14-mm-diametercylindrical holes were drilled in the homogeneousbackground phantom for inclusions of targets withvarious optical contrasts. Figure 2 depicts the geo-metrical configurations for the test cases understudy. To investigate the interparameter cross-talkissue, we tested the following seven cases:

Case 1 �one target, absorption contrast only�:�a � 0.014 mm1, ��s � 1.0 mm1.

Case 2 �one target, absorption contrast only�:�a � 0.028 mm1, ��s � 1.0 mm1.

Case 3 �one target, scattering contrast only�: �a �0.007 mm1, ��s � 1.7 mm1.

Case 4 �one target, scattering contrast only�: �a �0.007 mm1, ��s � 2.5 mm1.

Case 5 �one target, absorption and scattering con-trast�: �a � 0.014 mm1, ��s � 1.7 mm1.

Case 6 �one target, absorption and scattering con-trast�: �a � 0.028 mm1, ��s � 2.5 mm1.

Case 7 �two targets, absorption or scattering con-trast only�: Target A �at 3 o’clock�—�a � 0.014mm1, ��s � 1.0 mm1; Target B �at 10 o’clock�—�a �0.007 mm1, ��s � 1.7 mm1.

4. Results

Experiments with the above seven configurationswere performed. A 2D finite-element mesh with 717nodes and 1368 triangular elements was used in thereconstructions for both forward and inverse solu-tions. All the reconstructions were the results of 20iterations, after which no noticeable improvementwas observed. The computations were conducted ina 600 Pentium III personal computer. Both gray-scale images reconstructed and quantitative plots of

Fig. 1. Schematic of the DOT system.

�q� � �mD�D1, mD�D2, · · · mD�DN, m��a,1, m��a,2, · · · m��a,N�T, (8)

�� 1

mD

�1

D1

1mD

�1

D2· · ·

1mD

�1

DN

1m�

�1

�a,1

1m�

�1

�a,2· · ·

1m�

�1

�a,N

1mD

�2

D1

1mD

�2

D2· · ·

1mD

�2

DN

1m�

�2

�a,1

1m�

�a,2

�a,2· · ·

1m�

�2

�a,N

······

· · ····

······

· · ····

1mD

�M

D1

1mD

�M

D2· · ·

1mD

�M

DN

1m�

�M

�a,1

1m�

�M

�a,2· · ·

1m�

�M

�a,N

� . (9)


one-dimensional �1D� profiles of the exact and recov-ered property distributions contained in these imagesare presented in this section.

To present the cross-talk problem, we show in Fig.

3 images from cases 2 �Figs. 3�a� and 3�b�� and 4 �Figs.3�c� and 3�d�� in which the images were reconstructedby use of our previous algorithm without the normal-izing scheme. We note that an unexpected scatter-ing object appears in Fig. 3�b� and that an unexpectedabsorption heterogeneity exists in Fig. 3�c�, whereasthe expected absorption �Fig. 3�a�� and scattering�Fig. 3�d�� objects were recovered with good quality.These results clearly indicate that there is cross talkbetween absorption and scattering coefficients in op-tical image reconstruction.

Figures 4–11 give the results from the seven testcases with the normalizing scheme–based algorithm.Figure 4 presents simultaneous reconstruction ofboth absorption and scattering images from cases 1and 2 for a pure absorbing target with increasinglevels of contrast �2:1 and 4:1�. The recovered ab-sorption and scattering images from cases 3 and 4 fora pure scattering target with different levels of con-trast �1.7:1 and 2.5:1� are shown in Fig. 5. Figure 6displays the reconstructed images from cases 5 and 6for a target with both absorption and scattering con-trast. The recovered absorption and scattering im-ages from case 7 for two targets with a pure absorbingor a pure scattering object are given in Fig. 7. The1D profiles of the exact and recovered property dis-tributions �along transects CD and EF or C*D* andE*F* in Fig. 2� contained in these images are pre-sented in Figs. 8–11.

5. Discussion and Conclusion

Interesting observations can be made on the basis ofthe reconstructed images presented in Section 4.From Fig. 4, we note that no measurable cross talk ofthe absorbing object appears in the scattering images

Fig. 2. Phantom geometry for �a� cases 1–6 and �b� case 7.

Fig. 3. Reconstructed absorption and scattering images by use of the existing algorithm �without the normalizing scheme�: �a� absorp-tion image for case 2, �b� scattering image for case 2, �c� absorption image for case 4, and �d� scattering image for case 4.


�Figs. 4�b� and 4�d��. Similarly from Fig. 5, we cansee that no measurable cross talk of the scatteringobject exists in the absorption images �Figs. 5�a� and5�c��. This indicates that, when the target is a pureabsorber or a pure scatterer relative to the back-ground, the cross talk between the �a and the ��simages can be minimized with our normalizingscheme–based algorithm. Figure 6 shows the quan-titative recovery of �a and ��s images when the target

has both absorption and scattering contrast. Forthe two-target case, we immediately note from Fig.7�a� that there is a small amount of cross talk con-tributed by the scattering object, which appears as asecond absorbing object in the absorption image.

To obtain further quantitative information aboutthe reconstructed images, we have calculated the re-covered optical property distribution along twotransects through the center of the target �transects

Fig. 4. Reconstructed absorption and scattering images by use of the normalizing scheme–based algorithm: �a� absorption image forcase 1, �b� scattering image for case 1, �c� absorption image for case 2, and �d� scattering image for case 2.



CD and EF or C*D* and E*F*; see Fig. 2�. All thecalculated 1D �a and ��s distributions are shown inFigs. 8–11, in which we have plotted the exact opticalproperty profiles in each figure for comparisons.From these 1D �a and ��s profiles, we note that therecovered images are quantitative in terms of thetarget location, size, and shape. We found that theoptical properties of the object can be recovered

within 23% and 18% of the expected peak values for�a and ��s, respectively.

Taking a close examination of the �a profiles shownin Figs. 10�a� and 10�c� and Fig. 8, we observe that the�a peak values seen in Figs. 10�a� and 10�c� are over-estimated by as much as 8% more than that seen inFig. 8. Meanwhile, no noticeable difference in the ��sprofiles is observed between Figs. 10�b� and 10�d� andFig. 9. These comparisons suggest that, when thetarget has both absorption and scattering contrast,there may exist a small amount of cross talk contrib-uted by scattering contrast and that absorption con-trast generally does not contribute cross talk toscattering images. Interestingly, a similar observa-tion was made on the �a and ��s images obtained withfrequency-domain data.4 Finally, from Figs. 11�a�and 11�b�, we found for the two-target case that thecross-talk contribution of the scattering object is shownas a 1.28:1 absorption contrast in the �a image.

The results presented here clearly suggest that si-multaneous reconstruction of both �a and ��s imagesof heterogeneous scattering media can be obtainedwith dc data. In fact, several papers published re-cently by us and others have come to the sameconclusion.13,19–21,25,29 These experimental results,however, are in striking contrast to the theoreticalfindings reported by Arridge and Lionheart,26 whopresented a simple theoretical proof that simulta-neous recovery of absorption and scattering coeffi-cients cannot be achieved with dc measurements.To understand these seemingly contradictory views,we do not intend here to provide rigorous mathemat-ical derivations for reconciling these different conclu-sions; rather, we discuss several critical differencesbetween the practical reconstruction in dc DOT and


Fig. 7. Reconstructed absorption and scattering images by use ofthe normalizing scheme–based algorithm: �a� absorption imagefor case 7 and �b� scattering image for case 7.


the theoretical derivation of Arridge and Lionheartand offer an initial mathematical rationale for a pos-sible unique solution with dc data.

The first difference between the DOT reconstruc-tion and the theoretical derivation of Arridge andLionheart is the use of regularization techniques inDOT. Owing to the ill-posedness of the inverse

problem involved in DOT, one has to use regulariza-tion �e.g., see Eqs. �1� and �7�� that minimizes thefollowing cost functional:

F��m�, ��c�; q� �

j�1

M

��j�m� � �j

�c�� q�2, (10)

Fig. 8. Comparison of exact �dashed line� and reconstructed �solid curve� optical property distribution along transects CD and EF shownin Fig. 2�a� for the images appearing in Fig. 4: �a� absorption profiles along transect CD for case 1, �b� absorption profiles along transectEF for case 1, �c� absorption profiles along transect CD for case 2, and �d� absorption profiles along transect EF for case 2.

Fig. 9. Comparison of exact �dashed line� and reconstructed �solid curve� optical property distribution along transects CD and EF shownin Fig. 2�a� for the images appearing in Fig. 5: �a� scattering profiles along transect CD for case 3, �b� scattering profiles along transectEF for case 3, �c� scattering profiles along transect CD for case 4, and �d� scattering profiles along transect EF for case 4.


where � is a regularization parameter and q �opticalproperties� is in a parameter set Q. The idea of reg-ularization is to make F �when � � 0� mathematicallyconvex for ill-posed problems. This means that oneis solving a different nearby optimization problemand finding a minimum by use of F� instead of F. Inother words, regularization allows one to transform ahard-to-find minimum to an easy-to-find minimum.For example, in Fig. 12, we plot F� by use of the 1D

diffusion approximation with Rubin or type IIIboundary conditions for a homogeneous backgroundmedium with �a � 0.012 mm1 and D � 0.33 mm.From Fig. 12, it is clear that without regularization�� 0� the function F is rather insensitive to param-eter q � ��a�D �i.e., any numerical method startingwith an overestimate of the true parameter is boundto fail�. But with regularization �� 106�, F� ismore convex. We note here that the regularization

Fig. 10. Comparison of exact �dashed line� and reconstructed �solid curve� optical property distribution along transects CD and EF shownin Fig. 2�a� for the images appearing in Fig. 6: �a� absorption profiles along transect CD for case 5, �b� absorption profiles along transectEF for case 5, �c� scattering profiles along transect CD for case 5, �d� scattering profiles along transect EF for case 5, �e� absorption profilesalong transect CD for case 6, �f � absorption profiles along transect EF for case 6, �g� scattering profiles along transect CD for case 6, and�h� scattering profiles along transect EF for case 6.


has changed the problem so that we are solving for aminimum q� that is no longer the same as the prob-lem without regularization, mainly q. Thus the cau-tion is that the use of appropriate regularizationtechniques are critical, which should warrant that q�

approaches the true solution q as � approaches 0.Second, we note that the minimization of F��m�,

��c�; q� involves a parameter set Q. This brings in anadded theoretical complexity in terms of uniquenessbecause uniqueness or nonuniqueness may dependon the parameter set Q over which the optimization ofF is to be performed. Suppose the solution is notunique in Q, but if we restrict our parameter set to Qc� Q, then the inverse problem may have a uniquesolution. To illustrate this point, we discuss an ex-ample given by Banks and Kunisch �Ref. 30, p. 94� onparameter uniqueness for the equation �d�dx��q�d��dx�� f on the interval �0, 1� with the Dir-ichlet boundary conditions ��0� � ��1� � 0, where f �200�9 on �0, 3�10�, f � 0 on �3�10, 2�5� and f � 50�9on �2�5, 1�. It is easily seen that if we let ��m� �

�100�9� x2 � �20�3�x on �0, 3�10�, ��m� � 1 on �3�10,2�5�, and ��m� � �25�9�x2 � �20�9�x � 5�9 on �2�5,1�, then ��m� satisfies �d�dx��q�d��dx�� f for theparameter q � 1. To answer the question of unique-ness of q � 1, one has to verify if q � 1 is the onlysolution for the given ��m�. The answer depends onthe choice of the parameter space. If we assume q �Q, consisting of all continuously differentiable func-tions on �0, 1� such that q�x� � � � 0 on �0, 1�, where� is strictly positive, then the solution is not unique.This can be seen if we fix q � 1 on both intervals �0,3�10� and �2�5, 1�; we can choose q arbitrarily on theinterval �3�10, 2�5� without affecting ��m� � 1 on�3�10, 2�5�. This is because since ��m� � 1 and f �0 on �3�10, 2�5�, ��m� satisfies �d�dx��q�d��dx�� ffor any arbitrary choice of q on �3�10, 2�5�. There-fore, in the parameter set Q, there are infinitely manysolutions. But if we restrict q � Qc � Q with theadded constraint that q � Qc if and only if q � Q andq � 1 on the interval �3�10, 2�5�, then q � 1 is theunique solution in the parameter subspace Qc � Q;

Fig. 11. Comparison of exact �dashed line� and reconstructed �solid curve� optical property distribution along transects CD and EF orC*D* and E*F* shown in Fig. 2�b� for the images appearing in Fig. 7: �a� absorption profiles along transect C*D* for target B, �b�absorption profiles along transect E*F* for target B, �c� absorption profiles along transect CD for target A, �d� absorption profiles alongtransect EF for target A, �e� scattering profiles along transect C*D* for target B, and �f � scattering profiles along transect E*F* for targetB.


i.e., we have forced our choices of q on the interval�3�10, 2�5� so that q is restricted to 1 there and qcannot be chosen arbitrarily. Therefore we getuniqueness in Qc. In our reconstructions we haveused an optimization procedure that allows us tosearch for the best parameter space �initial �a and��s�20 for iteratively solving Eq. �7�. On the basis of apriori information �e.g., mathematically, physically,or biologically feasible ranges of these initial param-eters�, the optimization procedure seeks the best pa-rameter space in the vicinity of the exact one and hasallowed the minimization of the objection function tobe confined in a ��a, ��s� parameter subspace that isclose or similar to the exact solution. Thus, al-though theoretically infinite ��a, ��s� solutions exist,practically all other solutions that significantly differfrom the exact solution have been excluded. A sim-ilar explanation was also discussed in a recent re-port.31

Finally, in addition to the regularization in Eq. �1�,we have also normalized or scaled the Jacobian ma-trix � for well conditioning of �T�. For an N � Nmatrix A that defines a linear system Ax � b, scalingthe rows of this system is equivalent to premultiply-ing A by a diagonal matrix D1 � diag�a1, a2, . . . , aN�,and scaling the columns is equivalent to postmulti-plying A by the diagonal matrix D2 � diag�b1, b2, . . . ,bN�. The precision of computed numbers is betterwhen the condition number ��A� � �A� �A1� issmall.32,33 If ��A� is large, then a small relative per-turbation in A and b will produce a large perturbationin x, and the problem of solving Ax � b is ill condi-tioned. The idea behind row or column scaling is toscale matrix A to A� � D1AD2 such that ��A�� A�. Therefore a row or column scaling aims to

minimize the condition number of the scaled matrix.In this paper we have used only a column scaling of �,which scales both the rows and columns of �T� in thespecial case of D1 � D2

T. This represents a scalingof the variables according to their importance andessentially changes the norm in which the error ismeasured. The scaling used in this paper improvesthe condition number of �T� that is to be invertedand thus numerically improves the accuracy. It is,however, necessary to point out the difference betweenregularization and scaling. Scaling does not changethe system itself but merely makes the matrix betterconditioned to avoid round-off errors in the solution,whereas regularization changes the problem to a dif-ferent one so that minimization is possible.

In summary, we have presented solid experimentalevidence that quantitative absorption and scatteringimages of heterogeneous scattering media can berecovered simultaneously by use of dc data. In ad-dition, the results from controlled phantom exper-iments have shown that the interparameter crosstalk in DOT can be reduced considerably with thenovel normalizing scheme–based reconstruction al-gorithm. We have also pointed out several impor-tant differences between the recent theoreticalderivations and practical DOT experiments and at-tempted to present an initial mathematical expla-nation regarding the uniqueness in DOT. Weencourage active involvement of mathematicians instudying theoretical issues in DOT and believe thatwith such an effort a real reconciliation betweentheoretical and experimental findings will bereached in the near future.

The authors thank Bruce Tromberg of Universityof California, Irvine for helpful discussion. This re-search was supported in part by grants from the Na-tional Institutes of Health, R01 CA 90533, and theDepartment of Defense, BC 980050.

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