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Fluorescence-enhanced optical imaging of large phantoms using singleand simultaneous dual point illumination geometries

A. GodavartyPhoton Migration Laboratory, Texas A&M University, College Station, Texas 77843-3573

C. Zhang and M. J. Eppsteina)

Department of Computer Science, University of Vermont, Burlington, Vermont 05405

E. M. Sevick-Muracab)

Photon Migration Laboratory, Texas A&M University, College Station, Texas 77843-3573

~Received 2 July 2003; revised 4 November 2003; accepted for publication 18 November 2003;published 8 January 2004!

Fluorescence-enhanced optical tomography is typically performed using single point illuminationand multiple point collection measurement geometry. Single point illumination is often insufficientto illuminate greater volumes of large phantoms and results in an inadequate fluorescent signal tonoise ratio~SNR! for the majority of measurements. In this work, the use of simultaneous multiplepoint illumination geometry is proposed for acquiring a large number of fluorescent measurementswith a sufficiently high SNR. As a feasibility study, dual point excitation sources, which arein-phase, were used in order to acquire surface measurements and perform three-dimensional re-constructions on phantoms of large volume and/or significant penetration depth. Measurementswere acquired in the frequency–domain using a modulated intensified CCD imaging system underdifferent experimental conditions of target depth~1.4–2.8 cm deep! with a perfect uptake opticalcontrast. Three-dimensional reconstructions of the fluorescence absorption from the dual pointillumination geometry compare well with the reconstructions from the single point illuminationgeometry. Targets located up to 2 cm deep were located successfully, establishing the feasibility ofreconstructions from simultaneous multiple point excitation sources. With improved excitation lightrejection, multiple point illumination geometry may prove useful in reconstructing more challeng-ing domains containing deeply embedded targets. Image quality assessment tools are required todetermine the optimal measurement geometry for the largest set off imaging tasks. ©2004 Ameri-can Association of Physicists in Medicine.@DOI: 10.1118/1.1639321#

I. INTRODUCTION

Fluorescence-enhanced optical imaging has been rapidly de-veloping toward three-dimensional~3-D! tomographic stud-ies in phantoms as well as inin-vivo animal studies.1 To date,tomographic studies have been performed using sequentialpoint illumination and sequential point collection schemes asthe measurement geometries. More recently, rapid tomogra-phic fluorescence measurements are acquired using a charge-coupled device~CCD! camera for simultaneous area collec-tion of light received from multiple boundary points in bothcontinuous wave~cw!2,3 and time-dependent4 measurementapproaches.

For small rodent studies or for phantom studies, which areeither small in volume or offer limited penetration depths,sufficient fluorescence generation, and collection across thedepth of the entire phantom is possible with sequential illu-mination at single boundary points. Consequently, three-dimensional reconstruction of the target location and size isfeasible with the single point illumination measurementgeometry.5–7 However, in studies involving large phantomsof greater volumes and penetration depths, a single pointillumination of the phantom surface may be insufficient togenerate a fluorescent signal that arises throughout the entirephantom with a sufficient SNR. Weak fluorescent signals areusually dominated by noise, thus impacting the measurement

precision and accuracy, and eventually hindering the accuratereconstruction of the target location and size.

In this work, we explore the feasibility of using simulta-neous multiple point illumination measurement geometry inorder to increase the volume of illumination, and the totalnumber of fluorescent measurements with robust SNR. As apreliminary study toward applying the multiple point illumi-nation geometry, initial studies are performed using dualpoint illumination geometry for different experimental con-ditions. In the past, researchers employed dual excitationsources that were 180° out-of-phase~destructive interferingphoton density waves! for 2-D spatial localization of an ab-sorbing~or fluorescing! target located in a 3-D phantom,8–12

without actually reconstructing the target in three dimen-sions. However, in the current work, surface fluorescencemeasurements obtained from dual excitation sources, whichare in-phase, are used to reconstruct the target location andsize in three dimensions. The current work represents thefirst time that fluorescence measurements acquired usingmore than a single point of excitation illumination have beenused to reconstruct 3-D targets in large tissue phantoms forcomparison to single points of excitation illumination.

Herein, time-dependent fluorescence measurements areacquired on large breast phantoms using a frequency–domain imaging system employing a gain-modulated inten-sified CCD~ICCD! camera for rapid data acquisitions. Image

183 183Med. Phys. 31 „2…, February 2004 0094-2405 Õ2004Õ31„2…Õ183Õ8Õ$22.00 © 2004 Am. Assoc. Phys. Med.

reconstructions using the dual point illumination as well assingle point illumination measurement geometry are pre-sented for different conditions of target depth~1.4–2.8 cmdeep! under perfect uptake conditions without fluorescentdye in the background.

II. MATERIALS AND METHODS

Tomographic studies were performed using a large breastphantom of;1087 cm3 volume, as shown in Fig. 1. Thehemispherical portion of the phantom structure representedthe breast tissue and the cylindrical portion of the phantomrepresented the extended chest wall regions and the tissuesbeneath the breast. A total of 128 collection fibers~multi-mode optical fibers of 1 mm diameter! were glued symmetri-cally in concentric rings along the hemispherical surface.The other ends of the collection fibers were coupled onto aninterfacing plate as 2-D arrays for area imaging using anICCD camera. Twenty-seven source fibers available for point

illumination were placed intermittently between the concen-tric rings of collection fibers, at different planes from thecollection fibers. The phantom structure was filled with 1%Liposyn ~Abbott Laboratories, North Chicago, IL! to aheight of 2.5 cm of the cylindrical portion, providing a totalvolume of;1087 cm3 for photon migration. Further detailsof the tissue phantom are provided elsewhere.4

A. ICCD imaging system

Sinusoidally modulated light source at 100 MHz was gen-erated using a high-power laser diode~783 nm and 375 mW,HPD1105-9 mm-D-78505 model, High Power Devices, Inc.,NJ! driven by a laser diode driver~Thorlabs, Inc., NJ! and amaster oscillator~Marconi Instruments model 2022 D, UK!.The modulated light source was split into two equal intensitylight sources~;78 mW optical power!, which were in-phase,using a 50:50 beamsplitter~20Q20BS.2, Newport Corp., CA!and launched as two simultaneous point sources onto thephantom surface. The schematic of the dual point illuminat-ing imaging system is illustrated in Fig. 2. A fluorescentsignal was acquired using the collection fibers located on thephantom surface. The signal propagated through the collec-tion fibers onto the interfacing plate before it was imaged bythe intensified CCD camera~TE/CCD-512-EFT PhotometricCH12, Roper Scientific, NJ!. Details of the current ICCDimaging system are provided elsewhere.4 In short, the imageintensifier~FS9910C Gen III model, ITT Night Vision, VA!,which was optically coupled to the CCD camera, was modu-lated at 100 MHz at its photocathode using a slave oscillator~Programmed Test Sources model 10M201GYX-53, Little-ton, MA!, in order to perform homodyne detection. Thesteady-state intensity images on the phosphor screen of theimage intensifier were captured by the CCD camera at dis-crete phase delays~between 0 and 2p!, which were intro-duced between the rf signals generated by the slave and mas-ter oscillators. The values of intensity plotted as a function ofphase delay were used to extract the amplitude and phase ateach collection fiber location using fast Fourier transforms~FFT!.13–15

The experimental setup for the single point illuminatingimaging system was similar to that shown in Fig. 2 exceptthat only one of the two split sources from the beamsplitterwas used to illuminate the phantom at a given time.

B. Experimental parameters

Measurements of AC and phase~u! were acquired at theemission wavelength~830 nm! under perfect uptake condi-tions of the fluorescing agent present in the 1 cm3 target (13131 cm3) with its centroid located 1.4–2.8 cm deep fromthe phantom surface, using both the single and dual pointillumination geometry. One micromolar concentration of in-docyanine green~ICG! in 1% Liposyn was used as the fluo-rescing agent in the target, with no fluorescence in the 1%Liposyn background. The optical properties of the back-ground and target are provided in Table I. Since the current

FIG. 1. ~a! Actual phantom set-up;~b! x-y location of all 27 point illumina-tion ~asterisks! and 128 collection locations~hollow circles! on the hemi-spherical portion of the breast phantom.

184 Godavarty et al. : Fluorescence-enhanced optical imaging 184

Medical Physics, Vol. 31, No. 2, February 2004

studies are focused in demonstrating the feasibility of theproposed imaging technique, a 1 cm3 target was used underdifferent experimental conditions.

Fluorescent measurements were acquired at single anddual point illumination configurations, where the averagefluorescent signal of each interfacing plate, quantified interms of the average modulation depth~AC/DC!, was greaterthan 0.1~a value chosen based on the modulation depth ofnoise floor observed during experimentation!. The total num-ber of measurements acquired and the number of sourcecombinations used for different experimental conditions isprovided in Table II.

The acquired fluorescence measurements of AC and phaseon each interfacing plate and for each excitation source com-bination were referenced with respect to the collection fiberlocation that exhibits maximum amplitude value. Referenc-ing was performed in order to account for the instrumenteffects arising from the quantum efficiency of the image in-tensifier and the CCD camera, transmission efficiencies ofoptical filters used, and the loss of signal along the collectionfibers. It was assumed that the collection fibers were of equal

length and that the two split point sources in the dual pointillumination system were of equal intensity and in-phase.

There are a number of possible dual and multiple excita-tion source configurations that can be employed in order toobtain surface fluorescence measurements for improved to-mographic reconstruction. Once feasibility of improved to-mographic reconstructions are demonstrated, the optimalmeasurement configuration must be determined through im-age quality assessment tools and observer studies for a largeimaging set of varying target size, number, contrast, anddepth.

III. IMAGE RECONSTRUCTIONS

The fluorescent light propagation in the scatteringLiposyn medium was modeled using the coupled diffusionequations employing the Galerkin approximation of the finiteelement method.16,17A total of 6956 nodes and 34 413 tetra-hedral elements constituted the finite element mesh of thebreast phantom. For the dual point illumination system, theexcitation source term in the partial current boundary condi-tion was modeled as two point sources of equal amplitudeand in-phase. For the single point illumination system, theexcitation source term was modeled with amplitude of unityand phase delay of 0°.

The model match between referenced measurements andsimulations from the forward model of the coupled diffusionequation were determined for the dual and single point illu-mination systems. Referenced measurements of ln~AC! andphase regularized by empirically determined measurementerror variance~error between measurement repetitions!, astatic estimate of model error variance~estimated to be 1/4 ofthe mean of measurement error variance!, and a dynamicrecursive estimate of the parameter error variance, to recon-struct the target location using a computationally efficientvariant4,18 of the approximate extended Kalman filter

FIG. 2. Dual point illumination imag-ing system using an ICCD camera.

TABLE I. Optical properties of background and target in the perfect uptakecase for all the target depth studies.

Optical properties~cm21!

Perfect uptake~1:0!

Target Background

Excitation maxfa 0.30 0.00

msxb 10.88 10.88

Emission mamfc 0.05 0.00

msmd 9.82 9.82

aAbsorption coefficient due to fluorophore at the excitation wavelength.bIsotropic scattering coefficient at the excitation wavelength.cAbsorption coefficient due to fluorophore at the emission wavelength.dIsotropic scattering coefficient at the excitation wavelength.



~AEKF! algorithm.5 Given the estimates of the measurementerror, model error, and initial parameter estimation errors, thealgorithm minimizes the parameter error variance, as op-posed to simply minimizing the parameter error. Details ofthe AEKF algorithm used in the current study are brieflydescribed in the Appendix.4,5,18,19

Comparing measurements to simulations from the for-ward model for several known phantom domains, it was ob-served that the model mismatch was markedly elevated forsome measurements whose modulation depth was less than0.025. Hence, only measurements with a modulation depthgreater than 0.025 were included in the reconstruction algo-rithm ~see column 5 of Table II!. The advantages of prefil-tering the measurements for performing reconstructions in-clude ~i! improvement in the convergence of thereconstructions and generation of minimal artifacts, sincemeasurements with significant errors were excluded; and~ii !improvement in the computational efficiency, in terms ofmemory requirements~small matrices! and computationalspeed~fewer calculations for the Jacobian sensitivities andsmaller matrices to invert!.

IV. RESULTS AND DISCUSSION

A. Effect on volume illuminated and signal strength

The total volume illuminated within the phantom in-creased when dual points were used instead of single pointsfor excitation illumination. The signal strength at collectionfibers located around both the points of excitation in the dual

point illumination system was higher, thus generating astrong fluorescent signal in comparison to the signal aroundthe point of excitation in the single point illumination sys-tem. Also, more measurements above the set noise floor wereacquired in the dual point illumination imaging system~Table II, column 3!.

B. Model match between experiments and forwardmodel

A reasonably good match was observed between refer-enced measurements and simulated predictions of the loga-rithmic AC ratio@ln~ACR!# and relative phase shift~RPS! fortarget depths of 1.4 and 2.0 cm in both the illumination ge-ometries. However, the bias and variance in the error be-tween experiments and simulations was much lower for thedual point illumination system than the single point illumi-nation system~Table III!.

The observed mismatch for the 1.4 and 2.0 cm targetdepth cases could possibly be due to~i! the assumption ofequal source strengths in the case of dual point excitationsources;~ii ! the assumption of equal fiber lengths for all thecollection fibers in dual and single point illumination system;~iii ! the degree of discretization in the finite element mesh;~iv! experimental error in the target location; and/or~v! pre-cision error in the normalized location of the source andcollection fibers on the phantom surface.

In the case of the 2.8 cm deep target, the model mismatchwas significantly greater than for shallower targets for bothof the illumination geometries. Apart from the reasons statedabove, the major reason for the loss in accuracy is due toexcitation light leakage through the optical filters in the pres-ence of a weak fluorescence signal from deeply embeddedtargets, thus contaminating the measured emission signal,and hence increasing the model mismatch. In general, duringthe fluorescence-enhanced imaging process the strong exci-tation signal, which is at least three to four orders of magni-tude greater than the fluorescent signal, cannot be completelyrejected using a stack of optical filters without further attenu-ating the transmission of the weak fluorescent signal. Cur-rently, work is in progress to employ customized filters foroptimal excitation light rejection in order to detect deeplyembedded targets.

TABLE III. Model mismatch in ln~ACR! and RPS for different experimental conditions.

Targetdepth~cm!

Pointillumination

geometry

ln~AC ratio!Relative phase shift

~RPS!

Mean Variance Mean Variance

1.4 Dual 20.327 0.931 20.269 0.278

Single 20.779 1.641 20.565 0.479

2.0 Dual 20.463 0.938 20.127 0.472

Single 21.045 2.131 20.513 0.843

2.8 Dual 21.409 0.719 20.204 1.329

Single 21.508 0.851 20.145 0.969

TABLE II. The number of measurements and sources for different experi-mental conditions.

Targetdepth~cm!

Pointillumination

geometry

# ofmeasurements

acquired

# of pointilluminationcombinations

# of measurementsused in

reconstructions

1.4 Dual 832 13 pairs 439

1.4 Single 320 5 single 126

2.0 Dual 1408 22 pairs 773


2.8 Dual 1088 17 pairs 515




C. Measurement error

Spatially variant measurement error variance was empiri-cally determined from five repetitions of each observation ofAC and phase at all the experimental source–detector com-binations, for use in the reconstruction algorithm. The aver-age measurement error is reduced when dual point illumina-tion was employed over single point illumination at targetdepths of 1.4 and 2.0 cm~Table IV!. However, the measure-ment error was high, particularly in phase, for the 2.8 cmdeep target case, in either of the illumination geometries.Excitation light leakage limits both the measurement geom-etries from obtaining precise measurements of a weak fluo-rescent signal, which is reflected in the measurement errordata when the target is 2.8 cm deep.

D. Image reconstructions

Reconstructions were performed for the experimentalcases of 1.4 and 2.0 cm deep targets using the dual and singlepoint illumination systems individually. Reconstructionswere not performed on the 2.8 cm deep target case, since thesystem was limited by excitation leakage~or noise!, as re-flected from the model mismatch and measurement errordata shown in Tables III and IV, respectively.

Three-dimensional reconstructions were performed usingthe computationally efficient variant of the AEKF algorithm.The referenced fluorescence measurements were recon-structed as blind data, without prior knowledge of the targetlocation, size, and optical contrast with respect to the back-ground. As an initial guess, the unknown parameter (maxf) tobe reconstructed was assumed to be homogeneous with a lowvalue of 0.003 cm21 for the entire phantom. The reconstruc-tion process was carried out recursively until the root meansquare output error~RMSE! decreased by less than 1%. Asthe number of measurements increased, the computationaltime also increased from a few minutes to a few hours for thesame number of unknowns, due to the increased dimensionsof the matrices involved in the computations~see the Appen-dix!. Consequently, although the inverse code is identical forsingle source and dual source datasets, the reconstructionsfrom these dual source datasets took longer only becausethey comprised a larger number of measurements above thenoise floor. Alternate image reconstruction algorithms arecurrently under development in order to improve the compu-tational costs for solving 3-D tomographic problems.

The reconstructions for both the single and dual pointillumination geometries gave comparable image quality foreither of the two target depth cases~1.4 and 2.0 cm!. Theconvergence curves for the dual and single point illuminationgeometries for both the target depths are shown in Fig. 3.The actual image of the phantom with the 1.4 cm deep targetis shown in Fig. 4~a! and the reconstructed images are shownin Figs. 4~b! and 4~c! for the single and dual point illumina-tion geometries, respectively. Similar reconstruction imagesare provided for the 2.0 cm deep target case in Fig. 5.

The target was identified from the reconstructed spatiallydistributed fluorescence absorption coefficientmaxf . A cutoffvalue of the fluorescence absorption coefficient was selectedto distinguish between the background and the target basedon the break between modes in the bimodal histogram of thereconstructedmaxf . Details of the reconstructed target loca-tion, volume, and integrated fluorescence absorption for boththe target depths are tabulated in Tables V and VI. The vol-

TABLE IV. Measurement error~in terms of mean of variance! in ln~AC! andphase shift for different experimental conditions.

Target depth~cm!

Point illuminationgeometry ln~AC!

Phase shift~radians!

1.4 Dual 0.001 54 1.99Single 0.002 17 2.51

2.0 Dual 0.002 21 3.54Single 0.003 02 4.69

2.8 Dual 0.004 47 23.44Single 0.004 66 18.64

FIG. 3. Convergence curves of reconstruction for the dual and single pointillumination systems with~a! 1.4 cm target depth; and~b! 2.0 cm targetdepth.



ume of the reconstructed target was evaluated as the sum ofvolumes associated with each node whose estimatedmaxf

was greater than the set cutoff value. This volume informa-tion was used to estimate the volume-integrated fluorescence

absorption estimate of the reconstructed target. The Euclid-ean distance between the value-weighted centroids of thepredicted and actual target location was also determined andis referred to asdistance-off.

FIG. 4. Actual and reconstructed targets inx-yandx-zplanes:~a! actual target,~b! the simulated target for a single point illumination system with a 1.4 cm targetdepth, and~c! the simulated target for a dual point illumination system with 1.4 cm target depth.

FIG. 5. The actual and reconstructed targets inx-y andx-z planes:~a! actual target,~b! the simulated target for single point illumination system with 2.0 cmtarget depth, and~c! the simulated target for dual point illumination system with 2.0 cm target depth.



V. CONCLUSIONS

Fluorescence-enhanced optical tomography was per-formed on large breast phantoms~;1087 cm3! using anICCD camera imaging system for rapid data acquisition,with single and dual point illumination measurement geom-etry.

From the perfect uptake studies performed under two tar-get depth conditions~1.4 and 2.0 cm deep! using both thepoint illumination measurement geometries, it was observedthat the dual point illumination system provided higher sig-nal strength, increased the total phantom volume illuminated,increased the total number of fluorescence measurements ob-tainable above the noise floor, and also reduced the modelmismatch and measurement errors in comparison to thesingle point illumination system. Despite the smaller datasetsobtained when using the single sources, in the two experi-mental cases tested there was still an adequate signal in thesingle source datasets to permit reconstruction. Three-dimensional reconstructions for the dual point illuminationmeasurement geometry, performed for the first time, success-fully reconstructed the target with results comparable to thereconstructions using the single point illumination geometry.Thus, we have shown that 3-D image reconstruction usingthe simultaneous, in-phase, dual point illumination system isalso feasible, thereby increasing the kinds of measurementconfigurations that can and should be considered in the de-sign of the next generation of instrumentation for fluores-cence tomography. A future evaluation of the optimal mea-surement geometries~i.e., the placement and number ofdetectors as well as simultaneous point sources of excitationillumination! must be comprehensively performed over alarge imaging set~i.e., of varying depth, size, contrast, andnumber of targets! using image quality assessment tools,such as the ideal observer study tool.20 Once the optimummeasurement variables are determined from the entire vari-

able space, select experimental validation can affirm the op-timal measurement configuration.

Theoretically, the potential advantages of~optimum! si-multaneous multiple point illumination geometry include~i!greater volume illumination,~ii ! higher signal strength,~iii !an increase in the total number of measurements in largevolume phantoms, and~iv! an improvement in data acquisi-tion rates in a practical application involving clinical studieson human subjects. The current studies are focused in dem-onstrating the feasibility of the technique to employ simulta-neous~herein, dual! illumination geometries for improved3-D tomographic reconstructions. A few of the theoreticalpotential advantages mentioned above were observed fromthe current studies, and the remaining advantages to be ad-dressed in our ongoing efforts toward improvement and op-timization of the measurement configuration.

The detection of weak fluorescent signals against a strongexcitation signal in the background is a potential problem indeeply embedded or weakly fluorescing targets. The use ofcustomized rejection filters of greater blocking OD~opticaldensity.10) at the excitation wavelength is required. Al-though our current imaging system is limited for imagingweak fluorescent signals, upon adapting better excitationlight rejection techniques the use of simultaneous multiplepoint illumination geometry may enable reconstruction oflarger and more complicated domains. Thus, the modifiedICCD imaging system will have greater advantage in largephantoms of greater volumes and penetration depths as wellas in clinical applications for diagnostic imaging of humanbreast tissue with an improvement in the image acquisitionrates.

ACKNOWLEDGMENT

The research was supported by the National Institutes ofHealth ~NIH R01EB002763! and ~NIH R01 CA67176!.

TABLE V. Image quality of the actual and reconstructed targets for the 1.4 cm target depth case.

Pointillumination

geometry

maxf cut-offvalue

~cm21!

CentroidIntegratedmaxf ~cm2!

Volume~cm3!

Distanceoff ~cm!X (cm) Y (cm) Z (cm)

Actual casea ¯ 0.5 22.5 2.5 0.3 1.0 ¯

Dual 0.2 0.8 22.3 2.6 0.4 0.8 0.3

Single 0.2 0.6 22.3 2.5 0.3 0.6 0.2

aActual case: Details of the actual target location and its optical properties during experiments.

TABLE VI. Image quality of the actual and reconstructed targets for the 2.0 cm target depth case.

Pointillumination

geometry

maxf cut-offvalue

~cm21!

CentroidIntegratedmaxf ~cm2!

Volume~cm3!

Distanceoff ~cm!X (cm) Y (cm) Z (cm)

Actual casea ¯ 0.5 21.5 2.5 0.3 1.0 ¯

Dual 0.4 0.8 21.7 2.8 0.4 0.7 0.5Single 0.2 0.8 21.8 3.0 0.5 0.9 0.6

aActual case: Details of the actual target location and its optical properties during experiments.



APPENDIX: AEFK ALGORITHM

Three-dimensional reconstructions were performed usinga modified version4,18 of the approximate extended Kalmanfilter ~AEKF! algorithm.5 The pseudocode for the hybridAEKF algorithm is as shown below:

iterate

1:x5 f ~y!

for i 51 to # source illuminations

2:Ji5]xi

]y

3:K i5P•JiT•„Ri1Qi1Ji•P•Ji

T…

À1

4:Dyi5K i•„ziÀxi…

5:P]PÀK i•Ji•P

end

6: y]y1(i

Dyi

until convergence

Here, x represents the distributed predictions of the refer-enced measurements in terms of logarithmic AC and phaseshift, obtained using the forward simulatorf. The parametery represents the vector of current estimates of the unknownoptical property, which is an inverse pseudo-b transform19 ofthe absorption coefficient due to the fluorophore (maxf).Measurement error covarianceR was assumed uncorrelated,with variance experimentally determined for each source–detector pair from repeated observations at each detector lo-cation~five repetitions in these experiments!. It is difficult toestimate the model error covarianceQ for unknown do-mains. Here, the model error variance used in the AEKFalgorithm was empirically chosen to be one-fourth the meanof the measurement error variance. The measurement errorcovariance,R, and the model error covariance,Q, were usedalong with the parameter error covariance,P, and the Jaco-bian matrix,J, in order to determine the gain matrixK foreach point illumination,i. Subscripti means that the vectoror matrix contains only data associated with source illumina-tion i. The gain matrix is in turn used in updating the un-known optical parametery as well as the parameter errorcovarianceP. In the current study, the parameter error wasassumed uncorrelated with and initial variance estimate setto 0.001 for all the reconstruction cases. Although in thepseudocode we showP as a covariance matrix, only the vari-ances~diagonal elements ofP! were actually stored and ma-nipulated in this implementation. The inversion algorithmused here differs slightly from the AEKF, as reportedpreviously.5 This variant of the AEKF is more computation-ally efficient and preliminary studies indicate that its resultsare at least as accurate as the original AEKF algorithm.18

a!For details on the reconstruction algorithm, contact Dr. M. J. Eppstein [email protected] or~802!-656-1918~phone!.

b!For details on instrumentation and experiments contact Dr. E. M.Sevick-Muraca at [email protected] or~979!-458-3206~phone!.

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