optical image reconstruction using frequency-domain data: simulations and experiments

Jiang et al. Vol. 13, No. 2 /February 1996 /J. Opt. Soc. Am. A 253

Optical image reconstruction using frequency-domain data: simulations and experiments

Huabei Jiang, Keith D. Paulsen, and Ulf L. Osterberg

Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire 03755

Brian W. Pogue and Michael S. Patterson

Hamilton Regional Cancer Center, 699 Concession Street, Hamilton, Ontario L8V 5C2, Canada

Received June 5, 1995; revised manuscript received September 13, 1995; accepted September 18, 1995

Optical image reconstruction in a heterogeneous turbid medium with the use of frequency-domain measure-ments is investigated in detail. A finite-element reconstruction algorithm for optical data based on a diffusionequation approximation is presented and confirmed by a series of simulations and experiments using phan-toms having optical properties in the range of those expected for tissues. Simultaneous reconstruction ofabsorption and scattering coefficients is achieved both theoretically and experimentally. Images with differ-ent target locations and contrast levels between target and background are also successfully recovered. Allreconstructed images from both simulated and experimental data are derived directly from absolute opticaldata in which no differential measurement scheme is used. Results from the use of simulated and measureddata suggest that quantitative images can be produced in terms of absorption and scattering coefficient valuesand location, size, and shape of heterogeneities within a circular background region over a range of contrastlevels. Further, the effects of modulation frequency are found to be relatively modest, although boundaryconditions appear to be important factors. 1996 Optical Society of America

1. INTRODUCTION

Recently there has been considerable interest in bio-medical applications of optical imaging.1 – 6 For example,studies have shown that optical imaging may holdpromise in either a complementary or competitive rolewith respect to conventional x-ray mammography forbreast cancer detection.7 – 10 Although breast transillu-mination has been described as early as 1929,11 consider-able progress in optical imaging of breast tissue has beenmade in the last few years.1,2,6 Several advanced tech-niques for light transmission and/or reflection throughtissue over distances relevant to breast imaging are nowbeing investigated, including continuous-wave (cw), time-domain, and frequency-domain approaches.5,7,12 – 14 Atthe moment no consensus exists as to the method ofchoice for breast tissue imaging, which is not surprisingsince these techniques are largely in preclinical stagesof development; hence considerable interest remains ineach approach, and a number of tradeoffs appear to exist.For example, although time-domain schemes can providemore information than is available through cw excitation,the cost of system components is high, whereas a low-cost alternative can be realized in the frequency domain,which provides essentially the same information that isavailable in the time domain.15,16 On the other hand, inthe time domain direct image formation from received sig-nals is intuitive and natural, since first-arriving photonsapproximate the direct input/output distance in the tissuegeometry, whereas for cw or frequency-domain methodsfurther processing of measured data is warranted. Al-though interesting images based on direct imaging withfrequency-domain data have been reported,4,14 a number

0740-3232/96/020253-14$06.00

of investigators are studying reconstruction algorithms inorder to extract all of the information contained in suchdata17 – 19(this kind of computational approach is being in-vestigated in the time domain as well20,21 and is showingsome potential advantages22).

Although studies from simulated cw and frequency-domain data have been encouraging,19,23 promising ex-perimental images have been obtained only recently.Graber et al.24 have shown reconstructed images withgood quality from experimental DC data when the tar-get region consists of only an absorption heterogeneity.More recently, O’Leary et al.17 have presented successfulreconstructed images using frequency-domain measure-ments and a differential imaging technique when thetarget region consists of only an absorption or only a scat-tering heterogeneity. Using frequency-domain measure-ments and a multigrid finite-difference method, Pogueet al.18 have obtained images for simultaneous recon-struction of both absorption and scattering coefficientswhen a differential imaging technique is applied. Jianget al. have also demonstrated experimental images foronly the scattering coefficient10 using cw data and forsimultaneous reconstruction of both absorption and scat-tering profiles using DC25 and AC26 data from frequency-domain measurements, respectively. In those works theimages were obtained from absolute measured data, andno differential imaging technique was invoked.

As shown in Ref. 27 with simulated images and dem-onstrated with experimental images in Refs. 10 and 25,quantitative, simultaneous reconstruction of both absorp-tion and scattering profiles using cw or DC data appearsto be nearly impossible in multicentimeter tissue geome-tries. In this paper we demonstrate that quantitative,

1996 Optical Society of America

254 J. Opt. Soc. Am. A/Vol. 13, No. 2 /February 1996 Jiang et al.

simultaneous reconstruction of both absorption and scat-tering coefficients is theoretically and experimentally pos-sible using phase information as an additional observableduring the image formation. In terms of computationaltechnique we utilize a finite-element-based image recon-struction algorithm for tomographically collected opticaldata around the perimeter of an object. Although muchof the detail concerning the algorithm can be found inRef. 27, features relevant to the use of frequency-domaindata will be highlighted here, especially when we dealwith the inverse property-recovery procedure. The per-formance of the reconstruction algorithm is documentedand confirmed through a series of simulations and ex-periments using phantoms having optical properties inthe range of those expected for tissues, where simulta-neous reconstruction of both absorption and scatteringprofiles is achieved theoretically and experimentally.Images with different target locations and different con-trast levels between target and background are also suc-cessfully demonstrated. It is important to note that allreconstructed images from both simulated and experi-mental data are derived directly from absolute opticaldata in which no differential measurement scheme isused. The algorithm is found to be computationallypractical and can be implemented without difficultiesin a workstation computing environment. All recon-structed images have been analyzed both qualitativelyand quantitatively. Results from the use of simulateddata suggest that quantitative images can be producedin terms of absorption and scattering coefficient values,location, size, and shape of heterogeneities within a cir-cular background region 8.6 cm in diameter over a rangeof contrast levels. These simulated results are furtherconfirmed by experiments using phantoms with the samegeometry and optical properties. Images are obtainedboth theoretically and experimentally for different modu-lation frequencies, and no significant differences in imagequality are found. Boundary conditions are also dis-cussed, as they appear to be an important considerationin frequency-domain optical imaging.

2. RECONSTRUCTION ALGORITHMAlthough most of the details of the reconstruction tech-niques used in this study are identical to those that wedescribed in Ref. 27, we focus some attention on the fewdifferences. The previous evaluation of our reconstruc-tion algorithm in Ref. 27 was limited to the use of DC(or steady-state) data, although a generalized frequency-domain image formation process was presented. Basedon our current experience with simulated and measuredfrequency-domain data, we have modified our approachin terms of construction of the Jacobian matrices neededfor the inverse procedure. To place these modificationsin context, we briefly review our image reconstructionframework.

Since we will utilize time-harmonic light illumina-tion, our image reconstruction is based on the frequency-domain diffusion equation, which can be stated as13,18

= ? Dsrd=Fsr, vd 2

"masrd 2

iv

c

#Fsr, vd 2Ssr, vd ,

(1)

where Fsr, vd is the radiance, Dsrd is the diffusion coef-ficient, masrd is the absorption coefficient, c is the lightspeed in the medium, and v is the modulation frequency.The diffusion coefficient can be written as

Dsrd 1

3fmasrd 1 m0ssrdg

, (2)

where m0ssrd is the reduced scattering coefficient (since

m0s .. ma in a turbid medium, we work directly with dif-

fusion coefficient D and absorption coefficient ma duringreconstruction). Ssr, vd is the source term in Eq. (1),which for a point source can be written as S S0dsr 2 r0d,where S0 is the source strength and dsr 2 r0d is the Dirac-delta function for a source at r0.

For a known D and ma distribution Eq. (1) becomes astandard boundary-value problem for the spatially vary-ing radiance subject to appropriate boundary conditions(BC’s). In this paper we identify two typical BC’s of in-terest that are common with the diffusion equation28 – 30:(1) type I, F 0, and (2) type III, 2D=F ? n sa 1 ibdF,where n is the unit normal vector for the boundary surfaceand a, b are coefficients that are related to the internalreflection at the boundary. The imaginary part is intro-duced here to allow the phase of reflected photon densitywaves to be changed at the boundary, typically delayedas we will see in the next section. Although the imple-mentation of our reconstruction algorithm27 incorporateseither of these two BC situations, we will use only thetype III BC’s in this study because of the large errors thatoccur in the diffusion model relative to experimental mea-surements when type I BC’s are applied, as will be shownin detail in the next section.

Making use of finite-element discretization of Eq. (1),we can obtain a matrix equation in terms of a discreteset of spatially distributed optical properties and radiancevalues:

fAghFj hbj . (3)

Following the procedures outlined in Ref. 27, one findsthat the elements of matrix fAg are aij k2

PKk1 Dkck

=cj ? =ci 2PL

l1 sm1c1 2 ivycdcj cil, where k l indicatesintegration over the problem domain, and F, D, and ma

have been expanded as the sum of coefficients multipliedby a set of locally spatially varying Lagrangian basisfunctions cj , ck, and cl, and the entries in column vectorshbj and hFj are

bi 2kScil 1 sa 1 ibdMX

j1Fj

Icj ci ds ,

F hF1, F2, . . . , FN jT , (4)

whereH

expresses integration over the boundary surfacewhere type III BC’s have been applied. Fi is the radi-ance at node i, N is the number of nodes in the finite-element mesh, and M is the number of boundary nodes.Note that the expansions used to represent the diffusionand absorption profiles in Eq. (3) are K and L terms long,where K fi L fi N in general; however, in the work re-ported here K L N .


In order to form an image from a presumably uniforminitial guess of the optical property distribution, we needa way of updating D and ma from their starting values.To accomplish this, we Taylor-expand F about an as-sumed sD, mad distribution that is a perturbation awayfrom some other distribution, sD, mad, such that a discreteset of radiance values can be expressed as

FsD, mad FsD, mad 1≠F

≠DDD 1

≠F

≠maDma 1 · · · , (5)

where DD D 2 D and Dma ma 2 ma. If the assumedoptical property distribution is close to the true profile,the left-hand side of Eq. (5) can be considered as true data(either imposed or observed) and the relationship can betruncated to produce

IDx Fo 2 Fc, (6)

where I is the Jacobian matrix consisting of derivativesof F with respect to D or ma at each boundary observationnode,27 Dx is the vector that expresses perturbations of Dand ma, and Fo and Fc are the observed and the computedradiance at the boundary.

Left-multiplying Eq. (6) by the transpose of I, we ob-tained an equation for updating D and ma as described inRef. 27. From there we evaluated our reconstruction al-gorithm using DC data. When we adopted this approachfor reconstructions using AC data, we found that the per-turbations for D and ma in each iterative update becamecomplex valued as a result of unavoidable rounding effectsresulting from the poorly conditioned matrix system rep-resented in Eq. (6), which was undesirable in this prob-lem. To overcome this difficulty, we now decouple Eq. (5)into its real and imaginary parts as follows:

FR sD, mad FR sD, mad 1≠FR

≠DDD 1

≠FR

≠maDma 1 · · · ,

(7a)

FI sD, mad FI sD, mad 1≠FI

≠DDD 1

≠FI

≠maDma 1 · · · ,

(7b)

where FR and FI are the real and imaginary parts ofF, respectively. This maneuver produces an equationsimilar to Eq. (6):

IDx Fo 2 Fc, (8)

where

I

26666666666666666664

≠FR ,1

≠D1

≠FR,1

≠D2· · ·

≠FR,1

≠DK

≠FR,1

≠m1

≠FR,1

≠m2· · ·

≠FR,1

≠mL

≠FI ,1

≠D1

≠FI ,1

≠D2· · ·

≠FI ,1

≠DK

≠FI ,1

≠m1

≠FI ,1

≠m2· · ·

≠FI ,1

≠mL...

.... . .

......

.... . .

...≠FR,M

≠D1

≠FR,M

≠D2· · ·

≠FR,M

≠DK

≠FR,M

≠m1

≠FR,M

≠m2· · ·

≠FR,M

≠mL

≠FI ,M

≠D1

≠FI ,M

≠D2· · ·

≠FI ,M

≠DK

≠FI ,M

≠m1

≠FI ,M

≠m2· · ·

≠FI ,M

≠mL

37777777777777777775

, s9ad

Dx sDD1, DD2, . . . , DDK , Dm1, Dm2, . . . , DmLdT , (9b)

Fo sFoR,1, Fo

I ,1, FoR,2, Fo

I ,2, . . . , FoR,M , Fo

I ,M , dT , (9c)

Fc sFcR,1, Fc

I ,1, FcR,2, Fc

I ,2, . . . , FcR,M , Fc

I ,M , dT , (9d)

and FoRsI d,i and F

cRsI d,i are observed and calculated real (or

imaginary) parts of the radiance [based on the estimatedsD, mad distribution] for i 1, 2, . . . , M boundary loca-tions. Dk, k 1, 2, . . . , K, and ml, l 1, 2, . . . , L, are thereconstruction parameters for the optical property profile.At this point the procedure again becomes identical to thatdiscussed in Ref. 27 and involves left-multiplying Eq. (8)by the transpose of I and invoking regularization meth-ods to stabilize the decomposition of the square system ofequations.

In the results (both simulations and experiments) re-ported herein we have used a low-pass filter to smooththe reconstructed parameters at the end of the iterativeprocess. We found that the use of this low-pass filter notonly enhanced the visual quality of the reconstructed im-ages but also improved the images quantitatively. It actsto average the D and ma values of a given node with val-ues of the surrounding nodes in a weighted manner suchthat the influence of the surrounding nodal values can besystematically controlled. This filtering is realized pointby point with the following equation:

xnew i s1 2 udxoldi 1u

Np

NpXj1

xoldj , (10)

where u is a factor between 0 and 1 and the summationis over the values of the Np nodes directly connected tonode i. We have found that u 0.35 appears to give anoptimal result for the cases studied to date.

3. EXPERIMENTAL MATERIALSAND METHODSIn this paper the reconstruction algorithm describedabove will be tested and verified by a series of simula-tions and experiments using tissue-equivalent phantomsfor a cylindrical optically heterogeneous medium. Ourexperimental setup is standard and shown schematicallyin Fig. 1. Light from a 751-nm, 4-mW diode laser is di-rectly modulated by a signal generator (Marconi 2202A)at a selected frequency (50–300 MHz) and is deliveredthrough a fiber bundle (3 mm in diameter) into the phan-tom medium. A second fiber bundle is used to deliver areference signal (5% light), and a third fiber bundle is ap-plied to detect photon density waves along the boundarysurface of the phantom. Both detected light and refer-


Fig. 1. Experimental setup used for the frequency-domain mea-surements. Ref., reference; Sig., signal.

Fig. 2. (a) Phantom geometry for the off-centered target case.The centered target case is identical except that the center of theinternal heterogeneity is concentric with the background region.(b) Photograph of the phantom system used in this study. Onthe top of the phantom a target suspension system has beenincorporated into a rotatable stage (scaled precisely with lessthan 0.5± error), which provided accurate manipulations duringthe data collection procedures.

ence light are mixed with a signal from a second generatorat a nominally higher frequency s1100 Hzd through photo-multiplier tubes (Hamamatsu R928) to form a heterodynesignal. Intensity, phase shift, and modulation can thenbe recorded by a computer. The signal-to-noise ratio fora typical measurement with this system is approximately100.18,31

A circular phantom (black plastic) geometry (86-mm di-ameter) was used in this study as shown in Fig. 2(a). Allexperiments were performed with a scattering mediumcomposed of a fat emulsion suspension (Intralipid) withIndia ink added as an absorber. The absorption of thesuspension is essentially due to water 31 – 33; hence the inkprovided a controlled way of achieving a higher level of ab-

Fig. 3. Comparison between measured and computed data us-ing type I and type III BC’s at the 16 detector positions aroundthe phantom for one source excitation location (0± at the sur-face) in a homogeneous medium having m0

s 0.6 mm21 andma 0.006 mm21: (a) normalized logarithmic AC amplitude,(b) phase shift between the detector and the source position.The horizontal axes express angle along the boundary surface(in degrees).

Fig. 4. Simulated simultaneous reconstruction of both diffusionand absorption coefficients with 2:1 contrast for an eccentricallylocated target: (a) exact D image, (b) D reconstruction with nonoise added, (c) D reconstruction with 1% random noise added,(d) D reconstruction with 5% random noise added, (e) exact maimage, (f ) ma reconstruction with no noise added, (g) ma recon-struction with 1% random noise added, (h) ma reconstructionwith 5% random noise added.


sorption. It is also known that the added ink should nothave a significant impact on the scattering coefficient.34

A 25-mm-diameter transparent tube (50-mm thickness)containing different concentrations of Intralipid and In-dia ink was used to simulate heterogeneities. Thesephantom materials have been used in several other in-vestigations, where their optical properties were char-acterized by cw and frequency-domain techniques.31,35

The background medium had m0s 0.6 mm21 and ma

0.006 mm21, and different contrast levels (2:1, 5:1, and10:1 for both m0

s and ma) between the heterogeneity andthe background have been studied. These values aresimilar to those for human soft tissues, especially sincethey are scaled relative to the geometric size of the phan-tom, which is large compared with typical breast dimen-sions. Measurements for both centered and off-centered

Fig. 5. Comparison of exact and simulated reconstructions along transects AB and CD shown in Fig. 2(a) for an eccentrically locatedtarget with different noise levels: (a) D profiles along transect AB, (b) ma profiles along transect AB, (c) D profiles along transect CD,(d) ma profiles along transect CD. The horizontal axes indicate either transect AB or CD with millimeter units.

Table 1. Image Errors (Absolute Difference between the True and the Reconstructed Values)for Images from Simulated Data with Different Noise Levelsa

Diffusion Coefficient (mm) Absorption Coefficient (mm21)

Noise Levels Maximum Average Maximum Average

No noise 0.1338 0.01926 0.004896 0.00063931% noise 0.2041 0.02633 0.007780 0.00084665% noise 0.2269 0.04984 0.01111 0.002215

aThe true values for the background are ma 0.006 mm21 and m0s 0.6 mm21 (i.e., D 0.56 mm); for the target region they are ma 0.012 mm21

and m0s 1.2 mm21 (i.e., D 0.28 mm). The target is located at 3 o’clock (off-center 10 mm).


(10-mm offset along the horizontal axis at 3’oclock) het-erogeneities have been conducted.

Photon density waves were transmitted and detectedat 16 circumferential sites around the boundary surface.For each source excitation all detector locations wererecorded, then the phantom was rotated 22.5±, and theprocess was repeated. A total of 16 3 16 measurementswere obtained for each phantom configuration. A photo-graph of the phantom system that provided these pre-cise manipulations in the laboratory is shown in Fig. 2(b),where the source and detector fiber bundle and detectionwindows are readily visible. On the top of the phantom atarget suspension system has been incorporated into a ro-tatable stage (scaled precisely with less than 0.5± error),which provided accurate manipulations during the datacollection procedures.

Homogeneous medium (i.e., without the heterogeneity)measurements were used to calibrate the source term andthe boundary conditions in the reconstruction algorithm.Although it is straightforward to do this, a trial-and-errorprocess is needed to complete the calibration procedure.We found that the amplitude of the source largely deter-mined the computed AC amplitude values and that thephase of the source directly impacted the phase values ofthe computed photon density waves similarly, as expectedin a linear system. We also found that BC’s could af-fect the computed overall light distribution dramatically.Figure 3 shows results from the calibration procedure.Interestingly, we see that type III BC’s produce almostthe same match as that for type I BC’s for the intensitydistribution when computed and measured data are com-pared around the boundary surface [Fig. 3(a)], whereastype I BC’s generate large errors in the computed phasedata relative to the measured phase values [Fig. 3(b)].Both computed intensity and phase under type III BC’sshow excellent agreement with the experimental resultswhen the background optical properties are used in thediffusion model. This clearly suggests that type I BC’swill be unable to provide accurate reconstructions, at leastin terms of absolute measured data, in practical situ-ations such as the ones studied here and that type IIIBC’s should be used. This result is also consistent withthat obtained in Haskell et al.,28 where a similar conclu-sion was reached for photon density wave measurementsobtained for a semi-infinite medium. It should be notedthat since both types of BC’s (type I and type III) givecorrect results for intensity computations, there is no

essential difference between using type I and type IIIBC’s when only intensity data are involved in imagereconstructions. This finding has been demonstratedelsewhere, where images were formed with only cw orDC data.25

4. RESULTSIn this section we will use our reconstruction algorithmdescribed in Section 2 to conduct a series of simulationtests and experimental confirmations and evaluations.In the simulation studies we first show simulated resultsthat demonstrate a working implementation of the imageformation process under conditions of no measurementnoise and with added noise up to 5% for both AC ampli-tude and phase shift; second, we evaluate image recon-structions from simulated data with the use of differentcontrast levels between the target and the background;third, we present reconstructed images with differentmodulation frequencies (from 50 MHz to 300 MHz) inwhich the effect of modulation frequency on the imagequality can be investigated. We use an image error inthe optical property values, a contrast level comparisonbetween the target and the background, an optical prop-erty ratio of the target between different contrasts, andthe location, the size, and the shape of the target to quan-tify the reconstructed images. In the experimental work

Fig. 6. Reconstructed images from simulated data (no addednoise) obtained from an eccentrically located target having differ-ent contrasts with the background medium: (a) D image with2:1 contrast level, (b) D image with 5:1 contrast level, (c) Dimage with 10:1 contrast level, (d) ma image with 2:1 contrastlevel, (e) ma image with 5:1 contrast level, (f ) ma image with10:1 contrast level.

Table 2. Geometric Information for Reconstructed Images from Simulated Data withDifferent Noise Levelsa

Target Location Target Size Target Shape

D Image ma Image D Image ma Image D Image ma Image

X Y X Y EF GH EF GH EF/GH EF/GH

Exact 10.4 0.0 10.4 0.0 27.0 28.0 27.0 28.0 0.96 0.96No noise 10.5 0.0 10.5 1.0 23.5 28.0 31.0 29.0 0.84 1.071% noise 10.6 1.0 6.3 21.5 18.5 24.0 15.0 25.0 0.77 0.605% noise 10.4 1.6 8.2 0.8 18.0 19.7 22.4 22.4 0.91 1.0

aX and Y refer to the x and y coordinates (in millimeters) of the target center, respectively. EF and GH are the transect length (in millimeters) ofthe target region along the x and y directions, respectively [see Fig. 1(a)]. This table is for the off-centered target case with 2:1 contrast level betweenthe target and the background medium. Note that EF fi GH in the exact case, because the property profile is modeled as being linearly interpolatedacross the jump discontinuity assumed in the optical properties [see Figs. 4(a) and 4(e)].


all of these situations are also verified and evaluated withmeasured data and the same qualitative and quantitativemetrics of imaging performance.

For both simulations and experiments we have usedthe same optical properties for the background and thetarget as those described in Section 3 and the same back-ground and target dimensions as those shown in Fig. 2(a).In addition, both centered and off-centered target loca-

tions were considered, and we used multiple excitationand measurement positions to collect (for experiments) orproduce (for simulations) the boundary information usedin the reconstructions. The radial location of each sourcewas positioned inside the physical boundary by a dis-tance d 1ym0

s (m0s is the reduced scattering coefficient

of the background medium) for the point-source excitationused in the computational algorithm.3,21,27 Type III BC’s

Fig. 7. Comparison of exact and simulated reconstruction profiles along transects AB and CD shown in Fig. 2(a) for an eccentricallylocated target with different contrast levels: (a) D profiles along transect AB, (b) ma profiles along transect AB, (c) D profiles alongtransect CD, (d) ma profiles along transect CD. The horizontal axes indicate either transect AB or CD with millimeter units.

Table 3. Reconstructed Contrast Levels between the Target and the Background andOptical Property Ratios of the Target between the

Different Contrast Levels for Images from Simulated Data, Where the Target Is Located Off-center a

ma ma ma D D D ma(5:2) ma(10:5) D(5:2) D(10:5)

Exact 2:1 5:1 10:1 1:2 1:5 1:10 2.5:1 2:1 1:2.5 1:2Reconstructed 1.99:1 3.7:1 4.8:1 1:1.5 1:1.7 1:1.7 2.1:1 1.6:1 1:1.2 1:1.1

aAverage values for optical properties in the target and background regions have been used. Note that (5:2) and (10:5) in the 8th–11th columnsexpress target comparisons of contrast pairs 5:1 and 2:1 and contrast pairs 10:1 and 5:1, respectively.


Fig. 8. Simulated reconstructions (no added noise) for a cen-trally located target having 5:1 contrast excited at differentmodulation frequencies: (a) exact D image, (b) D image atf 50 MHz, (c) D image at f 200 MHz, (d) D image atf 300 MHz, (e) exact ma image, (f ) ma image at f 50 MHz,(g) ma image at f 200 MHz, (h) ma image at f 300 MHz.

were applied for simulations as well as for experiments.The finite-element mesh used in this study consisted of492 nodes and 918 triangle elements. The final imagesreported for both simulations and experiments are theresult of iteration until the initial sum of squared errorsbetween measured and computed intensity and phase val-ues at the measurement site locations is reduced 5 ordersof magnitude. Reaching this level of reduction in the ini-tial sum of squared errors typically required 50 iterationsat a cost of 1 min per iteration for the finite-element meshused herein.

A. SimulationsIn each simulation the measured data were generatedwith a forward diffusion model with the exact opticalproperties in place. Figures 4(a) and 4(e) show the basiccharacter of the exact optical property distribution forthe 2:1 contrast level case with an off-centered target,the reconstruction of which is sought from boundary-onlyobservations. The other images in Fig. 4 are represen-tatives of reconstructions for the 2:1 contrast level casewith an off-centered target under conditions of no noiseand with added noise up to 5% for both intensity andphase shift. A 150-MHz modulation frequency has beenused. As can be seen, the images formed are qualita-tively correct, even for those with a 5% noise level. Fig-ure 5 and Table 1 provide a more quantitative assessmentof these images. Figure 5 displays the reconstructed op-tical property distribution along two transects through thedomain—one being through the centers of both the targetand the background cylinders [transect AB in Fig. 2(a)]and the other being through the center of the targetbut perpendicular to the first transect [transect CD in

Table 4. Image Errors (Absolute Difference between the True and Reconstructed Values) forImages from Simulated Data with a Centered Target and Different Modulation Frequency a


Modulation Frequency Maximum Average Maximum Average

50 MHz 0.1717 0.02521 0.03416 0.002561200 MHz 0.1479 0.02501 0.03572 0.002646300 MHz 0.1714 0.02819 0.02914 0.002319

aThe contrast level used for this table is 5:1 between the target and the background medium. The true values for the background are ma 0.006 mm21

and m0s 0.6 mm21 (i.e., D 0.56 mm); for the target region they are ma 0.03 mm21 and m0

s 3.0 mm21 (i.e., D 0.112 mm).

Fig. 2(a)]—for the no-noise and 1% and 5% noise condi-tions compared with the exact values. We note that theimages for no noise and with 1% added noise do appearto be quantitatively recovered. Table 1 shows the maxi-mum and average differences in the reconstructed opticalproperty profiles compared with the exact distribution forall the images shown in Fig. 4. To obtain further quan-titative information about the reconstructed images, wecalculated the location, the size, and the shape of the tar-get. We estimated these parameters by calculating thefull width at half-maximum of the reconstructed opticalproperty profiles along the two transects in order to ob-tain a measure of reconstructed target size, shape, andlocation. Table 2 presents the results from these calcu-lations for all the images displayed in Fig. 4.

Figure 6 displays image reconstructions having an off-centered target without added noise as a function ofcontrast level between the target and the background.

Fig. 9. Simultaneous reconstruction of both diffusion and ab-sorption profiles based on experimental data obtained from acentrally located target having 2:1 contrast with the background:(a) exact D image, (b) reconstructed D image, (c) exact ma image,(d) reconstructed ma image.

Fig. 10. Same as Fig. 9, except that the eccentrically locatedtarget configuration is used.


Again, the reconstructed images qualitatively capture theoptical property distribution regardless of the contrastlevels used. Similarly to Fig. 5, Fig. 7 shows quantita-

tive information for the images presented in Fig. 6 alongthe two transects shown in Fig. 2(a). Further quanti-tative evaluations of image contrast recovery are pre-

Fig. 11. Comparison of exact and reconstructed profiles along transects AB and CD shown in Fig. 2(a) based on experimental dataobtained from eccentrically and centrally located targets with 2:1 contrast. (a) and (e): D profiles along transect AB for off-centeredand centered, respectively; (b) and (f ): ma profiles along transect AB for off-centered and centered, respectively; (c) and (g): D profilesalong transect CD for off-centered and centered, respectively; (d) and (h): ma profiles along transect CD for off-centered and centered,respectively. The horizontal axes indicate either transect AB or CD with millimeter units.


sented in Table 3, where the reconstructed contrast levelsbetween the target and the background and the recon-structed optical property ratios of the target between thedifferent contrast cases are shown. In these calculationsaveraged values for the target and background regionshave been used. Reconstructions with a centered targethave shown behaviors similar to those presented here.

As a final evaluation of our reconstruction algorithmusing simulated data, we present Fig. 8, where imageshaving a 5:1 contrast level between the target and thebackground are obtained for different modulations rang-ing from 50 MHz to 300 MHz. All these images arequalitatively similar, and quantitatively they also appearto be largely unaltered with increasing modulation fre-quency, as illustrated in Table 4. Cases with other con-trast levels have been found to have results similar tothose displayed in Fig. 8 and Table 4.

B. ExperimentsA series of experiments corresponding to the above simu-lations have been performed, which can be used to verifyand confirm our simulated findings in a systematic way.Figures 9 and 10 show experimentally reconstructed im-ages for a centrally located and a noncentrally locatedtarget having a 2:1 contrast level between the targetand the background, respectively, where again the exactimages are included for comparisons. As can be seen,the images formed are clearly shown to be qualitativelycorrect in visual content for both target locations. Fig-ure 11 provides a more quantitative assessment of theseimages, where the reconstructed optical property distri-butions along the two transects shown in Fig. 2 (a) arereported. Table 5 shows the maximum and average dif-ferences in the reconstructed optical properties comparedwith the exact distribution for the two target location con-figurations with a 2:1 contrast. The geometric informa-

Table 5. Image Errors (Absolute Difference between the True and Reconstructed Values)for Images from Experimental Data with Centered and Off-Centered Target Locationsa


Target Location Maximum Average Maximum Average

Centered 0.1939 0.03456 0.02121 0.003671Off-Centered 0.2278 0.03202 0.02365 0.003645

aThe true values for the background are ma 0.006 mm21 and m0s 0.6 mm21 (i.e., D 0.56 mm); for the target region they are ma 0.012 mm21 and

m0s 1.2 mm21 (i.e., D 0.28 mm). The off-centered target is located at 3 o’clock (approximately 10 mm away from the center of the phantom).

Table 6. Geometric Information for Reconstructed Images from Experimental Datawith Centered and Off-Centered Target Locationsa

Target Location Target Size Target Shape

D Image ma Image D Image ma Image D Image ma Image

X Y X Y EF GH EF GH EF/GH EF/GH

CenteredE 0.0 0.0 0.0 0.0 25.0 25.0 25.0 25.0 1.00 1.00CenteredR 0.0 0.0 0.0 0.0 34.2 35.0 17.8 19.0 0.98 0.94

Off-centeredE 10.4 0.0 10.4 0.0 25.0 25.0 25.0 25.0 1.00 1.00

Off-centeredR 10.2 0.0 8.0 0.25 22.6 27.2 21.2 21.6 0.83 0.98

aX and Y refer to the x and y coordinates (in millimeters) of the target center, respectively. EF and GH are the transect length (in millimeters)of the target region along the x and y directions, respectively [see Fig. 1(a)]. This table is for the case with 2:1 contrast level between the target andthe background medium. CenteredE and CenteredR indicate the centered target for exact and reconstructed data, respectively, and Off-centeredE andOff-centeredR indicate the off-centered target for exact and reconstructed data, respectively.

Fig. 12. Reconstructed images based on experimental data ob-tained from a centrally located target having different contrastlevels relative to the background medium: (a) D image with 2:1contrast level, (b) D image with 5:1 contrast level, (c) D imagewith 10:1 contrast level, (d) ma image with 2:1 contrast level,(e) ma image with 5:1 contrast level, (f ) ma image with 10:1contrast level.

Fig. 13. Same as Fig. 12, except that the eccentrically locatedtarget configuration is used.


tion for these images in terms of location, size, and shapeof the target is provided in Table 6.

Our reconstruction algorithm was also evaluated withincreasing contrast levels from 2:1 to 10:1 between thetarget and the background for both the centered andoff-centered target location configurations. Figures 12and 13 show reconstructed images that again qualita-tively capture the optical property profiles for each con-trast level. Figure 14 further quantifies these images byshowing the reconstructed optical properties along the two

Fig. 14. Same as Fig. 7, except that experimental data are used.

transects shown in Fig. 2(a). Table 7 provides quantita-tive information about the contrast level comparisons be-tween target and background and within each target itselffor the three different cases.

The effects of modulation frequency on the recon-structed images are demonstrated in Fig. 15, where fre-quencies from 50 MHz to 300 MHz for the centered targetlocation configuration were used. Clearly they confirmedthe results in the simulations. Table 8 provides a quan-titative verification of this observation.

Table 7. Reconstructed Contrast Levels between the Target and the Background andOptical Property Ratios of the Target between the Different Contrast Levels for

Images Obtained from Experimental Data, Where the Target Is Located Off-Centera

ma ma ma D D D ma(5:2) ma(10:5) D(5:2) D(10:5)

Exact 2:1 5:1 10:1 1:2 1:5 1:10 2.5:1 2:1 1:2.5 1:2Reconstructed 2.5:1 3.8:1 4.2:1 1:1.1 1:1.2 1:1.2 1.6:1 1.2:1 1:1.02 1:1.02

aAverage values for optical properties in the target and background regions have been used. Note that (5:2) and (10:5) in the 8th–11th columnsexpress target comparisons of contrast pairs 5:1 and 2:1 and contrast pairs 10:1 and 5:1, respectively.


Fig. 15. Same as Fig. 8, except that experimental data are used.

Finally, we show Fig. 16, where images were recon-structed from data that were measured when exactly thesame phantom solution as the background medium wasplaced into the centrally located target tube. Clearly nooptical heterogeneity is observed. The purpose of this re-construction is to verify that the thin target tube itself isnot directly responsible for, or does not cause significantperturbations in, the recovered optical target.

5. DISCUSSION/CONCLUSIONSThe results presented in the previous section indicate thata significant amount of useful information, both qualita-tive and quantitative, can be obtained from reconstruc-tions based on frequency-domain diffuse optical data.The simulations and the experiments have shown that themethodology outlined in Section 2 leads to a reconstruc-tion algorithm that can be implemented at a reasonablecomputational cost in a workstation computing environ-ment. Importantly, it has been demonstrated throughboth simulations and experiments that absolute opticalreconstructions can be obtained quantitatively with thisapproach in terms of not only the location, the size, andthe shape of the heterogeneity but also, for the most part,the optical property values themselves. Figures 4 and 5for simulated data and Figs. 9–11 for experimental dataclearly support these conclusions. A closer examinationof the simulated images in Fig. 4 and their experimentalcounterparts in Figs. 9 and 10 reveals that the recon-structed images for ma present a systematically betteroverall recovery of the shape of the heterogeneity thanthe reconstructed D images, and both have approximatelythe same target location accuracies. Tables 2 and 6 ver-ify this observation quantitatively and also indicate thatthe target shape recovery from experimental data for bothD and ma images is quite accurate compared with the ex-

act case. In terms of target size both D and ma imagesfrom simulated and experimental data exhibit approxi-mately the same accuracy when compared with the exactimages, with the exception of the 5% noise case for simu-lated data, for which the D image had a less accuratetarget size recovery than the ma image. The simulateddata presented in Figs. 4 and 5 have also shown that ourreconstruction algorithm is largely resistant to randomnoise. These random errors are larger than those usuallypresent in experimental systems, but the uncertaintiesand the errors from other sources such as measure-ment of the optical properties involved and positioningof the source/detector sites and the target may be moreimportant.

From Figs. 5 and 11 it is interesting to note that thesimulations with no noise and 1% noise achieve an excel-lent recovery of the optical property values (both D andma), whereas the simulation with 5% noise and experi-ments for both centrally and noncentrally located targetsexhibit larger ma and smaller D values than are presentin the exact data—a result that generally indicates con-sistency between the simulations and experiments. Theimage errors shown in Tables 1 and 5 for both simulatedand experimental data further suggest that overall goodquantitative accuracies in the recovery of the optical prop-erty values have been achieved in the 2:1 contrast case.This is a promising result, since eventual clinical appli-cations of this type of optical imaging (e.g., breast cancerdetection) will likely rely on the ability to distinguish dif-ferences between normal tissues and benign and malig-nant tumors.

Interesting observations are also possible from thestudies of the reconstructed image data as a functionof contrast between the target and the background pre-sented in Figs. 6 and 7 for simulations and Figs. 12–14for experiments. Again, the simulated ma images showbetter recovery of the location, the size, and the shape ofthe target than the D images as the contrast increases,and their corresponding experimental images further con-firm this result. From Fig. 7 we find that the ma images

Fig. 16. Reconstruction in which the centrally located targettube was filled with the same medium as that of the background:(a) D image, (b) ma image.

Table 8. Image Errors (Absolute Difference between the True and Reconstructed Values) forImages from Experimental Data with Centered Target and Different Modulation Frequency a


Modulation Frequency Maximum Average Maximum Average

50 MHz 0.3406 0.05257 0.04273 0.005152200 MHz 0.3441 0.05543 0.03748 0.004831300 MHz 0.3807 0.05236 0.04546 0.004703

aThe contrast level used for this table is 5:1 between the target and the background medium. The true values for the background are ma 0.006 mm21

and m0s 0.6 mm21 (i.e., D 0.56 mm); for the target region they are ma 0.03 mm21 and m0

s 3.0 mm21 (i.e., D 0.112 mm).


have excellent recovery of their profile values for all threecontrast levels, whereas the D images show good value re-covery for only the lowest contrast (2:1) and do not appearto be sensitive to contrast changes. The correspondingplots for the experimental images displayed in Fig. 14generally verified this finding. Other useful quantita-tive contrast information for both simulations and experi-ments with noncentrally located targets is demonstratedin Tables 3 and 7, respectively. Excellent recoveriesof reconstructed contrast levels between the target andthe background and reconstructed optical property ratiosof the target itself between the different contrast caseswere achieved in the ma images, whereas these recoveriesfor the D images were generally not satisfactory. FromFigs. 6, 12, and 13 it is also interesting to note that theD images for the higher contrast levels (5:1 and 10:1)do exhibit more pronounced artifacts around the targetboundary, whereas the D images for the lower contrastlevel (2:1) show better quality. This can also be readilyobserved from the more quantitative data displayed inFigs. 7 and 14 and Tables 3 and 7 for both the simulatedand measured data, respectively. This has importantimplications because realistic contrast levels between tu-mor and normal tissue are believed to be in the range ofthe lower contrasts studied here.36 The large artifacts inthe higher contrast level cases could be a result of break-down of the perturbation expansion in the reconstructionalgorithm or breakdown in first-order diffusion theory atthe interface between drastically distinct optical media.

Evaluations of our reconstruction algorithm in termsof the effects of modulation frequency have been madewith both simulated and experimental data. The im-ages for different modulation frequencies (from 50 MHzto 300 MHz) shown in Figs. 8 and 15 demonstrate almostno difference in visual content. Tables 4 and 8 quantita-tively substantiate this observation for both the simulatedand experimental data.

Although the quantitative information from both simu-lations and experiments shown in this paper are the first(to our knowledge) reported in the context of simultaneousabsorption and scattering optical image reconstructionsusing absolute measured frequency-domain data, someof the qualitative results reported here do appear to begenerally consistent with the limited number of previousstudies. For example, Graber et al.24 showed excellentimages from experimental DC data when the target re-gion consists of only an absorption heterogeneity, whichis in agreement with the results of this study, since over-all better absorption coefficient images have clearly beendemonstrated. Another example is the investigation ofeffects of modulation frequency, which is consistent witha similar simulation study conducted by O’Leary et al.17

A detailed comparison between our reconstruction algo-rithm and that developed by Arridge et al.21 using DCdata for simulations has also been discussed in Ref. 27.

In summary, we have demonstrated frequency-domainoptical image reconstructions using extensive simulationsand experiments. Simultaneous reconstruction of bothabsorption and scattering profiles from simulated and ex-perimental data using different contrast levels betweenthe target and the background has been achieved. It isparticularly important to note that the quantitative re-constructed images have been realized from absolute im-

age reconstruction procedures that do not use differencingtechniques to enhance target definition by first subtract-ing the reconstruction of a homogeneous region.

ACKNOWLEDGMENTSThe Dartmouth authors would like to thank Brian C.Wilson for many informative discussions and for provid-ing the initial impetus to foster this collaboration withour Hamilton colleagues. This work was sponsored inpart by the Alma Hass Milham Fellowship in Biomedi-cal Engineering and by the Ontario Laser and LightwaveResearch Center.

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optical image reconstruction using frequency-domain data: simulations and experiments

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