Image reconstruction in optical tomography in the presence ...€¦ · Image reconstruction in optical tomography in the presence of coupling errors Martin Schweiger, Ilkka Nissilä,

Image reconstruction in optical tomography in thepresence of coupling errors

Martin Schweiger, Ilkka Nissilä, David A. Boas, and Simon R. Arridge

Image reconstruction in optical tomography is a nonlinear and generally ill-posed inverse problem. Noisein the measured surface data can give rise to substantial artifacts in the recovered volume images ofoptical coefficients. Apart from random shot noise caused by the limited number of photons detected atthe measurement site, another class of systematic noise is associated with losses specific to individualsource and detector locations. A common cause for such losses in data acquisition systems based onfiber-optic light delivery is the imperfect coupling between the fiber tips and the skin of the patientbecause of air gaps or surface moisture. Thus the term coupling errors was coined for this type of datanoise. However, source and detector specific errors can also occur in noncontact measurement systemsnot using fiber-optic delivery, for example, owing to local skin pigmentation, hair and hair follicles, orinstrumentation calibration errors. Often it is not possible to quantify coupling effects in a way thatallows us to remove them from the data or incorporate them into the light transport model. We presentan alternative method of eliminating coupling errors by regarding the complex-valued coupling factors foreach source and detector as unknowns in the reconstruction process and recovering them simultaneouslywith the images of absorption and scattering. Our method takes into account the possibility that couplingeffects have an influence on both the amplitude and the phase shift of the measurements. Reconstructionsfrom simulated and experimental phantom data are presented, which show that including the cou-pling coefficients in the reconstruction greatly improves the recovery of absorption and scatteringimages. © 2007 Optical Society of America

OCIS codes: 100.3190, 170.3010, 110.4280.

1. Introduction

Diffuse optical tomography is a novel medical imagingmodality1–3 that seeks to reconstruct 3D images of tis-sue optical parameters and related functional imagesof tissue physiology, such as blood volume and oxygen-ation,4 particularly in peripheral muscle, breast tis-sue,5 and the brain.6–9 Data acquisition systemsdeliver light to the surface of the body and measure thelight transmitted through the tissue at multiple detec-tor locations on the surface. Due to high scattering of

IR light by biological tissue, the relationship betweenvolume parameters and measurements is nonlinear,and the measurement data carry little spatial in-formation, which generally requires an iterativeand regularized model-based approach to image re-construction. We have previously presented a recon-struction algorithm that uses a finite element modelfor simulating diffuse light propagation in scatteringmedia and a regularized Newton–Krylov approach toimage reconstruction of the absorption and scatteringdistribution inside a 3D volume from boundary mea-surements of modulation amplitude and phase shiftof radio-frequency-modulated input sources.10 Thismodel assumes constant coupling between each fiberand the tissue or phantom and perfect calibration ofthe instrument, without losses in the light deliverysetup or at the skin surface. In practice such losses dooccur and must be included in the model to avoid thepropagation of noise into the image. In this paper wewill extend the current image reconstruction schemeto account for a class of systematic noise known ascoupling errors. By this we understand errors in am-plitude and phase that are specific to individualsource and detector channels but do not vary tempo-

M. Schweiger ([email protected]) and S. R. Arridge arewith the Centre for Medical Image Computing, University CollegeLondon, London WC1E 6BT, UK. I. Nissilä is with D. A. Boas atthe Athinoula A. Martinos Center for Biomedical Imaging, Massa-chusetts General Hospital, 149 13th Street, Charlestown, Massa-chusetts 02129, USA, and is also at the Laboratory of BiomedicalEngineering, Helsinki University of Technology, P.O. Box 2200,02015 HUT, Helsinki, Finland.

Received 4 August 2006; revised 21 December 2006; accepted 6January 2007; posted 10 January 2007 (Doc. ID 73784); published23 April 2007.

0003-6935/07/142743-14$15.00/0© 2007 Optical Society of America

10 May 2007 � Vol. 46, No. 14 � APPLIED OPTICS 2743

rally. The most common cause for this type of error isthe imperfect delivery or collection of light on thesurface owing to losses at the interface between theskin and light delivery system. In many data acqui-sition systems, light is delivered and collected by op-tical fibers. In these systems, it is possible that an airgap between the fiber tip and the skin surface affectsthe characteristics of light delivery. Hair and localskin surface features, such as moles and follicles, willalso have an influence on the coupling where surfaceareas of light delivery and collection are small, as iscommonly the case for fiber-based systems. Otherchannel-specific effects, such as losses in individualoptical fibers or differences in detector gain, can alsobe included in the coupling parameters.

Since coupling effects are difficult to predict andquantify, they cannot normally be included directly inthe light transport model. Instead we propose to treatthem as unknowns and include them in the recon-struction process. A previous study by Boas et al.11 hasused a similar approach for amplitude calibration only,and promising results of absorption reconstructionfrom simulated data were presented. Stott et al.12 haveextended this method to include the correction for po-sitional errors in the source and detector locations.Schmitz et al.13 have presented a method of calculatingreal-valued source and detector coupling coefficientsfor a linear transfer function used for measuring dy-namic changes with a CCD detection system.

Oh et al.14 have expressed the problem of couplingcoefficient recovery in a Bayesian regularizationframework, where optical parameters and couplingcoefficients were updated sequentially in each iter-ation step of the nonlinear maximum a posteriorireconstruction scheme. Other authors15–17 showedthat errors induced by the measurement system canbe reduced by incorporating reference measurementsobtained from phantoms with known optical proper-ties. Recent work on calibrating measurement data toaccount for coupling effects includes Tarvainen et al.18

and Nissilä et al.19

In this paper we present a method that takes intoaccount coupling factors for both the amplitude andphase data obtained from frequency domain data ac-quisition systems. Phase coupling can occur, for exam-ple, from reflections in the optical system or dispersionof the signal in the fiber. We will show that the recon-struction algorithm can recover amplitude and phasecoupling coefficients in addition to the simultaneousreconstruction of absorption and scattering imagesand that the inclusion of the coupling coefficients canimprove convergence and image quality dramatically.The results from applying the method to both simu-lated and experimental data will be presented.

2. Forward Problem with Coupling Coefficients

A. Light Transport Model

Consider the problem of light propagation in a highlyscattering domain � bounded by ��, where absorp-tion coefficient �a�r� and scattering coefficient �s�r�satisfy �s�r� �� a�r� everywhere. Further let the dis-

tance between the boundary source and detector lo-cations be large compared to the mean free transportpath 1��s. Then the diffusion equation20,21 is an ad-equate model for light transport in �.22,23 In the fre-quency domain, the diffusion equation is given by

�� · ��r�� a�r� �i�c ��r, �� q�r, ��, r � �,

(1)

where � is the diffusion coefficient given by � 1��3��a � �s��, with �s� �1 � g��s and g being theanisotropy coefficient for the single-scattering phasefunction. q is the boundary source distribution mod-ulated at frequency �, and � is the photon densityfield in the medium. Where the diffusion equation isnot sufficient, a higher-order approximation to theBoltzmann equation must be used.24,25 The discus-sion of coupling coefficients in this paper can be ap-plied directly to these models. We employ a Robinboundary condition:

��, �� 2��n��, �� 0, � � ��, (2)

where � is a coefficient that incorporates the refrac-tive index mismatch at the tissue–air interface, andoperator �n denotes the outward normal to theboundary at point �. The complex boundary flux � isobtained by boundary operator

�� n��, �� 12�

��, ��, � � ��. (3)

Consider a data acquisition system consisting of Sboundary source distributions qi��, � � ��, i 1 . . . S, which are illuminated sequentially, and Mboundary detectors, with sensitivity profiles wj��,j 1 . . . M, along the boundary. Then the signalyij � � measured by detector j under illuminationfrom source i is given by

yij ��

wj�� i��d� ��

wj��2�

i��, ��d�, (4)

where i is the solution to Eq. (1) for source distribu-tion qi. The forward problem [Eqs. (1)–(4)] may beformally represented by the operator

y f�x�, (5)

where y � �SM is the vector of complex projectiondata, and x �i, �i � �2B is an array of basis coef-ficients for the optical parameters � and , expressedin a suitable spatial basis expansion bk�r� of dimen-sion B, such that a parameter distribution x�r� can beapproximated by

x�r� � �k1

B

xkbk�r�, (6)

2744 APPLIED OPTICS � Vol. 46, No. 14 � 10 May 2007

where xk is a 2B-dimensional vector of basis coeffi-cients. We have implemented the model Eq. (5) as afinite-element model (FEM), by subdividing domain �into nonoverlapping elements of simple shape, such astetrahedra or regular voxels, and defining a basis con-sisting of local polynomial basis functions.26,27

B. Coupling Coefficients

We now consider each source qi to be contaminatedlinearly with a coupling coefficient �i � �, producinga modified field

i�r, �� ii�r, ��, i 1 . . . S, (7)

and each measurement to be contaminated with acoupling coefficient �j � �, producing modified data

ij�� j

12�

i��, �� i�j ij��, i 1 . . . S,

j 1 . . . M, (8)

leading to a modified forward operator

y f�x�, (9)

where the augmented solution vector x now includesthe unknown coupling coefficients, x x, �, �, andthe forward model is modified by the multiplicativecoefficients

fij�x� �i�jfij�x�. (10)

Note that as each measurement is affected by aproduct �i�j of two coupling coefficients, the set ofcoefficients �, � is uniquely defined only up to anarbitrary scaling factor11 � � �: ��, ��1�. In the it-

erative reconstruction algorithm described below thisambiguity is implicitly taken into account by recov-ering coupling coefficients that minimally vary fromtheir initial estimates ��0�, ��0�, such that

� arg min�

��i

��i � �i�0��2 � �

j��1�j � �j

�0��2�.

(11)

3. Inverse Solver

We have previously presented a damped Gauss–Newton scheme for recovering the optical parametersfrom boundary measurements,10 which uses a Krylovsolver with implicit representation of the Hessianmatrix to solve the linearized problem at each Gauss–Newton step k:

xk�1 xk � �k�JT�xk�J�xk� � �� xk��1�JT�xk��y � f�xk�� xk�� (12)

where J is the Jacobian of the forward model, is aregularization term with �� and �� being its first andsecond derivatives with respect to the parameters xk,and � is a regularization hyperparameter. �k is a steplength obtained by a line search at each iteration.

To extend this solver to the simultaneous recon-struction of the optical and coupling coefficients, xmust be replaced by the augmented solution vector x,forward model f by the extended model f defined inEq. (10), and the Jacobian J by an augmented matrixJ that includes the derivatives of the projection datawith respect to the source and detector coupling co-efficients.

A. Calculation of the Jacobian

The nonlinear reconstruction scheme for recovery ofthe optical parameters x now must be adapted torecover the extended solution vector x including the

Fig. 1. Block structure of the augmented Jacobian including optical parameters, source and detector coupling blocks.


coupling coefficients. Our reconstruction approachuses a Gauss–Newton scheme that requires the cal-culation of the Jacobian of the forward model, Jij,k �fij�x��xk. After extending for the additional cou-pling coefficient parameters in the solution vector,this becomes

J � fij�x��xk

� fij�x��l

� fij�x��m

�. (13)

With Eq. (10), the components of the modified Jaco-bian are evaluated as

� fij�x��xk

�i�j

�fij�x��xk

, (14)

� fij�x��l

��jfij�x� if i l0 otherwise, (15)

� fij�x��m

��ifij�x� if j m0 otherwise. (16)

Commonly the measurements are expressed in loga-rithmic polar form, in terms of logarithmic amplitudeand phase:

yij Aij exp i�ij )�ln Aij Re�ln yij��ij Im�ln yij�

. (17)

Similarly, the coupling coefficients can be expressedin polar form:

�i Ai�� exp i�i

��, �j Aj�� exp i�i

��. (18)

Recasting Eq. (10) in logarithmic form leads to

f ij�x� : ln fij�x� ln�Aij� � i��ij�, (19)

with

Aij AijAi��Aj

��, �ij �ij � �i�� j

��. (20)

The Jacobian J for f is organized so that the ampli-tude and phase components are stored in separateblocks, leading to a real-valued matrix:

Fig. 2. (Color online) Target and reconstructed images of absorption (top) and scattering distributions (bottom) for the 2D test case. Imagecolumns from left to right: target, reconstruction from noise-free data, and reconstruction from data with 1% additive Gaussian noise.


where the individual blocks are given by

� ln Aij

�xk

1Aij

�Aij

�xk

� ln Aij

�xk, (22)

��ij

�xk

��ij

�xk, (23)

� ln Aij

� ln Ak�� 1 if i k

0 otherwise, (24)

��ij

� ln Ak�� 0, (25)

� ln Aij

� ln Ak�� 1 if j k

0 otherwise, (26)

��ij

� ln Ak�� 0, (27)

� ln Aij

��k�� 0, (28)

��ij

��k�� 1 if i k

0 otherwise, (29)

� ln Aij

��k�� 0, (30)

��ij

��k�� 1 if j k

0 otherwise. (31)

The block structure of the resulting matrix is shownin Fig. 1. Whereas the blocks relating to the opticalcoefficients are dense, the coupling coefficient compo-nents are sparse.

B. Parameter Scaling

Due to the heterogeneity of both the data and param-eter spaces, a rescaling of both spaces is necessary tocompensate for ill conditioning of the linearized prob-lem.10 For the augmented solution vector x, scaling ofthe parameter space now must also incorporate thecoupling coefficients.

In this paper we use the solution space transfor-mation

J �� ln Aij

��a k

� ln Aij

��k

� ln Aij

� ln Ak��

� ln Aij

��k��

� ln Aij

� ln Ak��

� ln Aij

��k��

��ij

��a k

��ij

��k

��ij

� ln Ak��

��ij

��k��

��ij

� ln Ak��

��ij

��k��

�, (21)

Fig. 3. Log amplitude and phase data at all detectors for a singlesource. Data without coupling contamination (solid curve) andwith the three cases of coupling coefficients.

Table 1. Levels of Coupling and Random Error in the Simulated DataSets Used for 2D Reconstructions

Data Set

Coupling Error Relative Random Error

�(ln A( ,�)) �(�( ,�)) �R(ln A, �)

C0N0 0 0 0C1N0 0.2 0.01 0C2N0 1 0.05 0C3N0 2 0.1 0

C0N1 0 0 0.01C1N1 0.2 0.01 0.01C2N1 1 0.05 0.01C3N1 2 0.1 0.01


x → T�x� �ln�a

�� a�0�, ln

�

��0�, ln�

��0�, ln�

��0��,

(32)

where each of the components is rescaled by the av-erage of its initial estimate, indicated by superscript0, and then transformed to its logarithm.

The Jacobian J has to be rescaled accordingly. Thedata and solution space transformations are repre-sented by the premultiplication and postmultiplica-tion with a diagonal matrix, respectively. For moredetails on the application of data and solution spacetransformations see Ref. 10.

4. Results

A. Simulated Two-Dimensional Data

To assess the effectiveness of the coupling coeffi-cient recovery, reconstructions from simulated 2Ddata contaminated with artificial coupling coeffi-cients were performed. The target image is a circleof radius 25 mm with embedded circular and ellipti-cal absorption and scattering objects (Fig. 2). Thebackground parameters are �a 0.025 mm�1 and

�s 2 mm�1. The parameter ranges of the inclusionsare 0.01 mm�1 � �a � 0.06 mm�1 and 0.8 mm�1

� �s � 5 mm�1.Data are generated for 32 source and 32 detector

positions evenly spaced along the circumference of theobject. Each measurement consists of logarithmic am-plitude ln A and phase � at a modulation frequency of100 MHz, using a finite-element mesh discretizationconsisting of 32,971 nodes and 7261 ten-noded trian-gles, supporting a piecewise cubic polynomial basisexpansion.

The forward model employed by the reconstructionalgorithm uses a lower-resolution mesh with 1015nodes, 1950 three-noded triangles, and a piecewiselinear basis expansion. The FEM data are mappedinto a regular bilinear basis grid with dimensions of40 � 40. After removing the grid points outside thesupport of the domain, the basis is represented by1332 basis coefficients. The dimensions of the real-valued solution space is thus 1332 � 2 � �32 � 32�� 2 2772, including 2664 optical coefficients and128 coupling coefficients.

For reference, Fig. 2 shows the result of a recon-struction of the optical parameters only from datathat are not contaminated by coupling coefficients,

Fig. 4. (Color online) Reconstructions from data in the presence of coupling errors. Shown are absorption images (left) and scatteringimages (right) for the three levels of coupling contamination, without additional random noise. In each image pair, the left image showsthe conventional reconstruction without accounting for coupling noise, while the right image is obtained when coupling coefficients areincluded in the reconstruction.


both without additional random data noise and withadded Gaussian noise at a level of � 1%. Theseresults have been presented previously and showgood localization and quantitative recovery of the in-clusions for the noiseless case, and some degradationof the recovered shape of the inclusions when datanoise is present.

The simulated projection data are then contami-nated by adding synthetic source coupling coeffi-cients ln A��, �� and detector coupling coefficientsln A��, �� to each of the source and detector sites. Thevalues of the coupling coefficients are drawn fromGaussian-distributed random samples. We considerthree cases of increasing severety of coupling noise.In addition to the coupling effects, random Gaussiannoise with a standard deviation of 1.0% of the datavalues, simulating measurement shot noise, wasadded to the data. As a best-case scenario, recon-structions from data including coupling noise, butexcluding the random noise, are also presented forcomparison. The noise parameters of the resulting sixdata sets are listed in Table 1.

Figure 3 shows an example of the boundary projec-tion data obtained from one source after the couplingcoefficients have been applied for each of the threecases.

The synthetic data are then applied to the Gauss–Newton reconstruction scheme defined in Eq. (12).To assess the effect of the coupling coefficient re-construction, we perform reconstructions both ex-cluding and including the recovery of the couplingcoefficients, with and without additional randomnoise. In all cases the reconstructions start froman initial estimate of the optical parameters of �a

0.025 mm�1 and �s 2 mm�1. In the case of cou-pling coefficient reconstruction the initial estimate ofall coupling coefficients ln A��,�� and ��,�� is 0. For theregularization functional we choose a first-orderTikhonov scheme. Hyperparameter � is set to 10�5 forcoupling cases 1 and 2 and to 10�4 for case 3.

When applying the data contaminated with cou-pling coefficients to a conventional reconstructionscheme that recovers the distribution of optical coef-ficients only, by employing the forward model f, thecoupling errors will manifest themselves as artifactsin the recovered �a and �s images. Inclusion of thecoupling coefficients in the reconstruction process,using the extended model f instead, is expected toreduce or eliminate the amount of image artifact.

Figure 4 shows reconstruction results for the threecases of coupling contamination of the measurement

Fig. 5. (Color online) Reconstructions from data in the presence of coupling and random errors. The arrangement of images is thesame as for Fig. 4.


data, when reconstructed both with and without re-covering the coupling coefficients. It can be seen thatthe presence of coupling coefficients severely de-grades the image quality of the conventional recon-structions with loss of localization of the inclusionsand the appearance of boundary artifacts. When thecoupling coefficients are reconstructed together withthe optical parameters, the results are significantlyimproved. For cases 1 and 2 the image quality iscomparable with that achieved from uncontaminateddata. In case 3 the images begin to deteriorate butstill provide significant improvement in localizationcompared to a reconstruction that ignores the cou-pling coefficients.

When additional random noise is added to the datasets, the resulting reconstructions of both the imagesand coupling coefficients are affected. The images aretherefore degraded not only by the statistical noisebut also by the cross talk from the incompletely re-covered coupling coefficients. Therefore we expect theeffect of the coupling reconstruction to decrease athigher levels of random noise. Figure 5 shows recon-structions from data degraded by coupling and ran-dom noise. The inclusion of the coupling coefficientsin the reconstructions still significantly improves thequality of the recovered images, even though the con-trast and edge localization of the inclusions is re-

duced compared with the case without random noise.The amount of boundary artifact is also increased,indicating that residual coupling errors are affectingthe images.

The reconstructed coupling coefficients are shownin Fig. 6. The results for case 1 are scaled by a factorof 10 and those for case 2 by a factor of 2 so that theycan be compared with the same target coefficients. Itcan be seen that the coefficients are recovered verywell in all cases.

The L2 errors of the recovered coupling coefficientswith respect to the target values are shown in Fig. 7as functions of iteration count. For moderate couplingtargets (case C1N0), the coefficients are well recov-ered after approximately ten iterations, while conver-gence is slower for case C2N0. For the most severecoupling coefficients (case C3N0) the residual error ofthe recovered coupling coefficients remains signifi-cantly higher. The addition of Gaussian random noiseon the data results in an increase of the residual errorof the recovered coupling coefficients in all threecases (bottom graph of Fig. 7).

Figure 8 shows the convergence of the reconstruc-tion cost functions for all three cases as a function ofiteration count for the data without additional ran-dom noise (top graph) and with random noise on thedata (bottom graph). We find again that cases C1 and

Fig. 6. (Color online) Reconstructed source and detector coupling coefficients for ln A and �. The results for cases 1, 2, and 3 are scaledso that they can be compared to common target coefficients (solid curve).


C2 converge to similar residuals, although at dif-ferent rates, while in case C3 the residual remainssignificantly higher. Similarly, the residuals for thenoisy data are higher than for the noiseless data inall three cases.

To assess the effect of the coupling reconstructionon superficial perturbations, a further simulationwas performed on a homogeneous circular mesh witha single semicircular absorption perturbation of ra-dius 3 mm located directly under a source site [Fig.9(a)]. Data without coupling or random noise contam-ination for 32 source and 32 detector sites were thengenerated and used for reconstructions both exclud-ing and including coupling coefficients. Cross sectionsthrough the target absorption image and both recon-structed absorption images are shown in Fig. 9(b).The reconstructed amplitude and phase coupling co-efficients are shown in Fig. 10. As can be seen in thecase of reconstructing for the coupling coefficients,the surface perturbation is partly assimilated intothe coupling coefficients, leading to an underestima-tion of the reconstructed object contrast and spurious

coupling coefficients on the sources and detectorsclose to the object. Sometimes this effect of suppress-ing superficial features in the reconstructed imageswill be desirable to suppress boundary artifacts, e.g.,from local variation in skin pigmentation. However,in situations where superficial objects must be de-tected, for example, in optical mammography, whereit is frequently the case that the tumor is located nearthe surface, this method is not applicable.

B. Experimental Phantom Data

To test the effectiveness of the coupling parameterreconstruction with realistic data, reconstructionswere performed with data obtained with an experi-mental frequency domain instrument19 from a cylin-drical phantom with embedded inclusions. The effectof coupling errors was simulated by placing hair be-tween the phantom surface and the tips of the fiberoptics delivering light from the sources and collect-ing light at the detectors. Hair is one of the mostlikely sources of coupling variations when optodesare placed on the head for data acquisition in brainimaging.

Fig. 7. L2 error of reconstructed coupling coefficients as a functionof iteration count for the three cases considered. Top, results fromdata without additional random noise. Bottom, results from datawith additional 0.5% Gaussian-distributed random noise.

Fig. 8. Objective functions for the three coupling cases as a func-tion of iteration count. Top, no additional random data noise (N0).Bottom, 0.5% Gaussian-distributed random data noise.


The phantom had a diameter of 70 mm and aheight of 110 mm. It consisted of a homogeneousbackground material with �a � 0.01 mm�1 and �s�� 1 mm�1. One of the two small cylindrical inclusionshad a contrast of �2 in �a, the other a contrast of �2in �s�. A second, homogeneous cylinder with identicalbackground parameters was used to obtain baselinemeasurement used for difference image reconstruc-tions.

Sixteen source and sixteen detector fibers werearranged with equal angular spacing in two ringsaround the cylinder mantle above and below thecentral plane, as shown in Fig. 11. Only 15 of thedetectors were used because of a technical problemwith one of the detector channels. For each sourcethe measurements from the four closest detectorswere discarded to reduce boundary artifacts thatcan arise as a result of mismatches between modeland measurement data projected into spacially lo-calized sensitivity regions. This leads to a total of 180measurements, where each measurement consisted ofthe amplitude and phase of the transilluminated sig-nal, arising from source input amplitude modulated at100 MHz.

Figure 12 shows a subset of the difference datavectors between inhomogeneous and homogeneousphantoms for both log amplitude (top graph) andphase measurements (bottom graph). It compares thecoupling-free case (no hair under the optodes) withthe coupling case (hair was placed between all fibertips and the phantom surface). It can be seen that theeffect of coupling errors is higher in log amplitudethan in phase. For the log amplitude data the effect ofthe coupling errors is significantly higher than theeffect of the inclusion. For the phase data the cou-pling effect is comparable in magnitude to the per-turbations caused by the inclusions, except for somespikes on the coupling data for a few measurements.

To construct a forward model, the cylinder wasdiscretized by a finite-element mesh consisting of181,278 tetrahedral elements and 35,228 nodes, withthe node density highest at the surface of the mesh.

The initial parameter distributions for the re-constructions were set to homogeneous values of

Fig. 9. (Color online) Effect of coupling coefficient reconstructionon the recovery of a superficial perturbation. (a) Target image withboundary absorption inclusion. (b) Cross sections of the target andreconstructed absorption images along the line indicated in (a).

Fig. 10. (Color online) Reconstructed coupling coefficients of logamplitude (top) and phase (bottom) for the superficial inclusion inFig. 9.


�a 0.0103 mm�1 and �s� 0.807 mm�1. These ini-tial values were obtained by a global homogeneousparameter reconstruction from the measurements.

1. Difference ReconstructionsWe first consider the case of difference imaging,where two sets of data, representing two differentstates of the probed medium, are acquired, and thereconstruction recovers the parameter differences. Inthis phantom experiment, a baseline measurementwas performed on a homogeneous cylinder that didnot contain the inclusions. The baseline measure-ment was acquired without coupling contamination,while the measurements of the inhomogeneous cyl-inder were obtained both with and without hair un-der the optodes.

Reconstruction in dynamic imaging is performedby calculating projection data f�x0� for the initial pa-rameter estimate x0 and adding the measured differ-ence data y�inhomog� � y�homog� to obtain a syntheticabsolute measurement y f�x0� � y�inhomog� �y�homog�. Note that in situations where y�inhomog� andy�homog� are contaminated with the same coupling co-efficients, these effects cancel in the difference data,and the reconstruction of the coupling coefficients isless critical.

Figure 13 shows cross sections of dynamic recon-structions of absorption (top row) and scatteringdistribution (bottom row) in the central plane of thecylinder. The left column is a reconstruction fromthe inhomogeneous cylinder acquired without hair.The middle column shows reconstructions from datawith hair but without recovery of the coupling co-efficients. The right column used the same data forthe reconstruction but also recovered the couplingcoefficients. We find that the effect of coupling con-tamination of the data leads to a reduction of con-trast (in the scattering image) and localization (inthe absorption image) if the coupling coefficientsare not reconstructed. The inclusion of the coupling

coefficients in the reconstruction recovers contrastand localization well although some residualboundary artifacts in absorption and scattering re-main.

2. Absolute ReconstructionsAbsolute reconstructions from measurement data arerequired where no baseline data are available and ifabsolute values of absorption and scattering must berecovered. The same data from the inhomogeneouscylinder as in the previous section are used, both withand without coupling contamination with hair.

Cross sections of the reconstructed absolute param-eter distributions are shown in Fig. 14 for absorption(top row) and scattering coefficients (bottom row). Be-cause of the large differences in dynamic range be-tween the reconstructions, the images were scaledindividually. The images in the left column show thereconstruction from data without coupling contamina-tion. Both inclusions are recovered although at lowercontrast and with more artifacts in the backgroundmedium than for the corresponding difference recon-

Fig. 11. (Color online) Phantom setup for experimental data ac-quisition: location of embedded inclusions and arrangement ofsources and detectors on the surface.

Fig. 12. Subset of 100 measurements of the experimental datavectors of log amplitude (top) and phase (bottom). Shown are thedata differences between inhomogeneous and homogeneous phan-tom for the two cases with and without hair under the optodes.


struction. In particular, the scattering image suffersfrom boundary artifacts similar in magnitude to therecovered target object. The central column of imagesshows the reconstruction results from data with cou-pling coefficients induced by hair under the optodes,

where the coupling coefficients were not reconstructed.In this case the reconstruction fails entirely withcomplete loss of the inclusions, and the appearanceof high-contrast boundary artifacts under individ-ual optode positions. When the coupling coefficients

Fig. 13. (Color online) Cross sections of reconstruction of absorption (top) and scattering distributions (bottom) from difference phantomdata. Columns from left to right, reconstruction of uncontaminated data, reconstruction of hair data without recovery of couplingcoefficients, reconstruction of hair data with coupling coefficients. Target locations are marked with a black circle.

Fig. 14. (Color online) Cross sections of reconstructions of absorption (top) and scattering distributions (bottom) from absolute phantomdata. Columns from left to right, reconstruction of uncontaminated data, reconstruction of hair data without recovery of couplingcoefficients, reconstruction of hair data with coupling coefficients. Target locations are marked with a black circle.


are reconstructed (right column), both inclusionscan be recovered well again, although at some lossof contrast. The absorption object exhibits slightbleeding toward the boundary, similar to the differ-ence reconstruction case. Interestingly, the recov-ery of the scattering inclusion appears improvedcompared with the reconstruction from uncontam-inated data. This may be attributable to the pres-ence of coupling effects in the experimental systemother than the deliberate effects caused by insertinghair.

The recovered coupling coefficients are shown inFig. 15. It can be seen that the predicted magnitudeof the coupling coefficients is significantly larger thanthe influence of the inclusions on the boundary data(cf. Fig. 12), which explains the large effect of thecoupling reconstructions on the recovered images. Toprovide an estimate of the quality of the recoveredcoupling coefficients, we compare in Fig. 16 the prod-uct of the reconstructed source and detector couplingcoefficients with the effect of the hair on the mea-sured data. The spatial variability of the amplitude

coefficients is reconstructed well. The correspondinggraph for phase coefficients shows a less perfectmatch. This is because of greater noise in the phasemeasurement and the lack of an explicit cause oflarge phase coupling effects in the data. The recon-struction of the phase coupling coefficients in thisexperiment also corrected some residual calibrationerrors owing to the low optical power of the calibra-tion measurements.

5. Conclusions

We have presented a method to address the problem ofreconstruction artifacts in optical tomography arisingfrom the effects of coupling losses in the complex-valued measurement data collected by data acquisitionsystems. In this context, the term coupling loss com-bines all types of measurement error that are specificto individual source and detector sites. This type ofmeasurement error is well known to be a problem inoptical tomography, in particular where fiber opticsare used for light delivery and where hair under theoptode tips, moisture, or air gaps can lead to atten-

Fig. 15. Source and detector coupling coefficients ln A��, �� (top)and ��, �� (bottom) recovered during absolute reconstruction of ex-perimental phantom data.

Fig. 16. (Color online) Effect of hair on the difference data for (a)log A, (b) phase. Dashed curve is the product of source and detectorcoupling coefficients obtained in the reconstruction; solid curve isthe measured difference between clean measurement of the inho-mogeneous phantom and a measurement where the hair is inplace.


uation of the detected signal. Other sources of cou-pling errors not specific to fiber-optic acquisitionsystems include differences in power or sensitivityof individual light sources and detectors. Couplingerrors are difficult to quantify and therefore cannotgenerally be incorporated into the forward model.Instead, the coupling coefficients are treated as un-knowns and appended to the solution space.

The method presented here considers coupling er-rors in both amplitude and phase data, which arereconstructed simultaneously with the volume distri-butions of absorption and diffusion coefficients.

We have shown reconstructions from simulated 2Ddata, which were contaminated with synthetic cou-pling errors, and demonstrated that the reconstruc-tions can be improved significantly even in severecases where the level of coupling errors causes a con-ventional reconstruction to fail entirely.

The algorithm was then applied to experimentalphantom data where coupling errors were simulatedby placing hair between the phantom surface and thefiber tips of source and detector fiber optics. We haveperformed reconstructions from both difference dataand absolute reconstructions and demonstrated thatin both cases the inclusion of the coupling coefficientsin the reconstruction can lead to improvements thatreduce artifacts and restore the ability to recover theinhomogeneities in optical coefficients.

The authors acknowledge grant support from theEngineering and Physical Sciences Research Council,grants GR�N148�01 and GR�S48837�02, and fromthe Medical Research Council.

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Image reconstruction in optical tomography in the presence ...€¦ · Image reconstruction in optical tomography in the presence of coupling errors Martin Schweiger, Ilkka Nissilä,