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PubTeX output 2001.04.11:0922 - College of Engineering and ...

Caffaro, approvata la "cassa" in derogaTORVISCOSA - (PT) Buone notizie, dopo mesi d'attesa, per le circa 70 maestranze ex-Caffaro in cassaintegrazione che non hanno ancora ricevuto l'assegno dall'inizio dell'anno. La Femca-Cisl ha reso notaieri l'approvazione da parte del Ministero del lavoro del decreto per la cassa integrazione in deroganazionale, dal primo gennaio al 30 giugno 2013, quindi, di fatto, con effetto retroattivo. Si tratta di undocumento che consente al'Inps di erogare gli assegni che fino a oggi erano rimasti in sospeso, sommeche entreranno nelle disponibilità degli ex-lavoratori entro un paio di settimane. Femca-Cisl pensa giàal termine dell'ammortizzatore sociale, a inizio luglio. L'ex-assessore regionale Brandi, che avevaseguito tutta la vicenda fino al rinnovo del governo del FriuliVg, si era impegnata per ottenere, sempredal ministero, altri 6 mesi di cassa in deroga nazionale. Se ciò non fosse stato possibile, l'alternativa,per le 70 maestranze, altrimenti destinate alla mobilità, era quella dell'attivazione di una cassa in derogama di tipo regionale. Prossimo, quindi, l'incontro della Cisl con il nuovo assessore regionale al lavoroper capire se le intenzioni e gli obiettivi siano rimasti gli stessi. Continuano a lavorare nel polo chimicoCaffaro, intanto, circa 145 operai. E nella giornata di oggi, per quel che attiene al percorso del nuovoimpianto cloro-soda, sarà firmato il contratto di cessione della "macroarea 7" ad Halo Industry;sull'appezzamento sorgerà il nuovo impianto di ultima generazione.PORDENONEDomino, fiducia dai creditoriCristina Antonutti Ulteriore passo avanti per il salvataggio della Domino Spa di Spilimbergo, giàammessa al concordato preventivo in continuazione. Ieri si è tenuta l’adunanza dei creditori chirografi,convocata dal giudice Francesco Petrucco Toffolo. Il commissario giudiziale Paolo Fabris ha illustratola proposta di risanamento ottenendo il 20% dei voti a favore e il 2,5% a sfavore. Nei prossimi ventigiorni i chirografi avranno la possibilità di inviare ulteriori espressioni di voto. Il silenzio verràinterpretato come un assenso al concordato (per passare dovrà ottenere il 50% di consensi). A luglio,infine, il Tribunale deciderà sull’omologa del concordato.Per i 117 dipendenti e le loro famiglie ci buone possibilità. Il piano di risanamento industriale efinanziario, curato dall’avvocato milanese Giulia Santamaria, non prevede licenziamenti o la vendita dibeni per pagare i fornitori. L’organico sarà mantenuto grazie ai contratti di solidarietà: 90 sono quellisiglati. Per circa una trentina di dipendenti l’uscita sarà invece concordata con i sindacati.Finora si sta procedendo in linea con la proposta prevista dal concordato. Tra le novità vi è lacessazione dell’attività dello stabilimento di Castelraimondo, in provincia di Macerata, dove eraconcentrata una produzione di livello medio-basso che ora verrà inglobata nella sede di Spilimbergo(nel capannone era occupata una trentina di lavoratori).Ora si spera in una ripresa del mercato e si conta su alcune commesse arrivate dalla Russia, dalla Cina edall’India, mercati nuovi, che possono riservare grandi possibilità per l’azienda spilimberghese. LaDomino, infatti, radicata nel territorio sin dagli inizi degli anni ’80 e con un indotto affatto trascurabile,avrà delle chancee al fatto che la Certina ha internazionalizzato l’azienda facendola anche al di fuoridell’Europa.


EPPSTEIN et al.: THREE–DIMENSIONAL BAYESIAN OPTICAL IMAGE RECONSTRUCTION WITH DOMAIN DECOMPOSITION 153measurements ) as discussed in Section I. The recursive aspectof the AEKF can also be exploited for domain decomposition.With domain decomposition, different subsets of the parameters(modeling different physical regions of the domain) areconditioned incrementally, independent of whether or not datadecomposition is employed. There are many ways in which domaindecomposition can be implemented; here we describe a“sliding window” approach.In the sliding window approach to domain decomposition, theabsorption in the entire 3-D domain is initially modeled as homogeneous,using a single stochastic variable (zone). This zonecan be spectroscopically conditioned on measurements, usingthe AEKF, to obtain a better starting estimate (as in [24]), althoughin the experiments reported here we simply started withthe correct background value as the initial estimate everywhere.A 3-D “window” comprises some number of adjacent voxels,defining a spatially contiguous subvolume of the domain (i.e.,a “subdomain”). In these experiments, a window is defined as afixed number of adjacent – planes, and the number of planes(i.e., the number of voxels in the -dimension) is referred to asthe “window size.” When the window is “moved over” a portionof the domain, the formerly homogeneous estimate of absorptionwithin that subdomain is reparameterized into several small(in this case, single-voxel) zones modeled with 3-D stochasticvariables. The APPRIZE method is then used to condition andfurther segment the estimates of absorption in the subdomain[Fig. 3(a)]. Subdomains can be overlapping or nonoverlapping.In these experiments we used nonoverlapping subdomains, withthe exception that the final subdomain in a reconstruction wasallowed to overlap with its adjacent subdomain, if needed, inorder to maintain a uniform window size for every subdomain ina given experiment. Note that the forward simulator always usesthe current absorption estimate of the full 3-D model, regardlessof the conditioning window size or location. This contrasts withtwo alternate implementations of geophysical domain decompositionemployed in [25], [26]. Thus, even when the windowsize is one (i.e., each subdomain is a single – plane), we donot introduce the out-of-plane effects that would be caused byusing a 2-D forward simulator to predict measurements for 3-Dsystems.The optical absorption in each subdomain is conditionedusing a subset of those measurements whose sources anddetectors are located within the subdomain, thereby limitingthe maximum source-detector distance (presumably improvingthe signal-to-noise ratio), as well as keeping the size of matrixinversion small. In these experiments we continue conditioning,using data from each source in the subdomain incrementally,until either the estimate converges or until measurements fromall sources in the subdomain have been used. A subdomain isconsidered to have converged when no parameter value haschanged more than 1% during the most recent pass of the filter.We did not iterate on any data here (i.e., condition on the samemeasurements more than once within a given subdomain),although this is certainly an option (as in [19], [24]). Additionally,other measurements that traverse the subdomain canbe used, if desired, although here we restricted measurementsto those between sources and detectors fully contained withinthe current subdomain. When DDZ is applied in the course(a)(b)Fig. 3. (a) The domain is conditioned in smaller subdomains inside a “slidingwindow”; the number of adjacent layers (i.e., voxels in the z-dimension) in thesubdomain is referred to as the “window size.” When a subdomain lies withinthe sliding window, its parameterization is refined into small, voxel-based,zones that are subsequently conditioned and merged using the APPRIZEinverse method. (b) Initially, absorption in the domain is parameterized with asingle homogeneous zone (H); as the estimation proceeds, the sliding window(W) is moved from one end of the domain to the other, until the entire domainhas been estimated (E).of the APPRIZE algorithm, the number of parameters in thesubdomain, and hence the computation time required forsubsequent conditioning, typically is dramatically reduced.After a subdomain is estimated in this manner, the windowis conceptually slid over the next subdomain and the process isrepeated. This continues until the optical properties within theentire domain have been estimated, as depicted in Fig. 3(b). Theintegrated APPRIZE method using domain decomposition, datadecomposition by source and subdomain, AEKF, and DDZ isdescribed by the high-level pseudocode shown at the top of thenext page.III. COMPUTATIONAL EXPERIMENTAL DESIGNNote that the number of free parameters (zones) used tomodel the absorption in the domain fluctuates as the estimationproceeds, going up when the parameterization of a new subdomainis refined and going down as DDZ is applied. The aspectsof the APPRIZE method that act as a computational bottlenecktherefore change dynamically as the number of zones varies


154 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 3, MARCH 2001relative to the number of measurements used during any givenpass of the AEKF.We designed three sets of computational experiments to testthe performance of the domain decomposition extension to AP-PRIZE. The purpose of the first set of experiments was to establishobjective criteria for discriminating between detected heterogeneitiesand artifacts. In the second set of experiments, weused the established criteria to aid in assessing the performanceof the domain decomposition method using a variety of windowsizes. Finally, in the third set of experiments, we tested the robustnessof the reconstruction method to the presence of unmodeledbackground variations in scattering and absorption due tononfluorescing chromophores (parameters that were assumedhomogeneous and known while reconstructing absorption dueto a fluorescent contrast agent such as ICG).In all sets of computational experiments, a series of simulateddomain models were created for testing the method. Each modelrepresents a physical domain 8 4 8cm that has been discretizedonto a 17 9 17 finite difference mesh with uniform0.5 cm spacing between nodes for both forward and inverse calculations.For simulated data, this grid spacing is adequate sincethere is no system noise. When inverting actual data, the gridspacing for the forward model should be selected to minimizesystem noise within computational limitations. In preliminaryresults, not otherwise reported here, we have found grid spacings≤ 0.25 cm to give a reasonable match between predicted andmeasured values. This result is consistent with results reportedby Pogue et al. [53] in which the finite difference solution wascompared to the analytic solution of the diffusion equation. Thegrid spacing for the inverse model is chosen to maximize resolutionof the reconstructed image while maintaining numericalstability, and keeping computational costs practical. On the peripheryof each of the 17 - planes there are four simulated NIRsources in the center of each side and 24 simulated fiber opticdetectors, spaced 1 cm apart (Fig. 4), for a total of 68 sourcesand 408 detectors (27 744 potential source-detector pairs) in theentire 3-D domain.The absorption in each domain was simulated to includerandomly located heterogeneities in an otherwise homogeneousFig. 4. The open circles depict the locations of the 24 optical detectors and theasterisks denote the locations of the four sources on the periphery of each x-yplane. Axis numbering denotes nodal discretization for the forward simulator.background. Each heterogeneity was one voxel (0.5 0.50.5cm ) in size and had a true absorption value of0.13 cm ; this represents a 10 : 1 contrast above the backgroundvalue of 0.013 cm . We tested our algorithmwith the smallest possible heterogeneities that could be representedin our models (single voxels), since our ultimate aim isto be able to detect small tissue abnormalities. Each domain wassynthesized to include either one or three distinct single-voxelsas randomly located heterogeneities. All other optical propertieswere considered homogeneous and known, as shown in Table I.The validity of this assumption was partially tested by the thirdset of experiments, discussed below. Each of the six boundaryfaces was assumed to have a reflectance coefficient of 0.0222(where zero represents no reflection and one is complete reflection).This low reflectance value was determined by the methoddescribed in Haskell et al. [54] to model the water/acrylic interfaceof a tissue-mimicking phantom (with the size and shape ofthe domain models employed here) that we are currently usingto experimentally validate the method.In the experiments presented here, synthetic frequency-domainphase measurements at the emission wavelength ( )were inverted to estimate spatially heterogeneous fluorescenceabsorption of light at the excitation wavelength ( ) in the3-D domain model. Each inversion started with an initiallyhomogeneous absorption estimate and was accomplishedusing the previously described APPRIZE inverse method withdomain decomposition.The first set of experiments was performed using 41 differentdomains with a sliding window size of six (i.e., each subdomainwas 17 9 6 voxels in size), in order to determine criteria fordiscriminating between detected heterogeneities and artifacts.Of these 41 domains, 21 contained a single randomly locatedheterogeneity and 20 contained three randomly located heterogeneities.In the second set of experiments, each of 17 additional domainswas reconstructed using sliding window sizes that variedfrom one to nine voxels thick. At window sizes greater thannine, we could not complete the reconstructions due to computer


EPPSTEIN et al.: THREE–DIMENSIONAL BAYESIAN OPTICAL IMAGE RECONSTRUCTION WITH DOMAIN DECOMPOSITION 155TABLE IASSUMED AND INITIAL VALUES FOR OPTICAL PROPERTIES IN SYNTHETIC DOMAINSmemory limitations. Of the 17 domains, ten contained a singlerandomly located heterogeneity and seven contained three randomlylocated heterogeneities.In the third set of experiments, each of the ten single-heterogeneitydomains used in the second set of experiments were reconstructedusing a window size of six voxels thick. However,in these experiments, we added randomly generated beta-distributedbackground variations to the “true” synthetic absorptionand isotropic scattering , even though these propertieswere still considered homogenous and known during image reconstruction.Beta-distributed background noise was created bytransforming Gaussian random noise with a pseudo-beta transform[see, (7)], using the desired lower and upper bounds andno skew. In this case, the bounds on the beta distributions weredetermined by taking the desired means of the optical propertyvalues and adding or subtracting a specified percentage, asshown in Fig. 5.In previous studies we have added random noise to simulatedoptical measurements [19], [24]. However, since the purposehere was to see how the fundamental performance of the reconstructionalgorithm varied with different levels of domaindecomposition, no system or measurement noise was added topredicted or “observed” values of . Nonetheless, for applicationof the APPRIZE inversion algorithm to this synthetic data,system noise was assumed to be independent with a variance of0.02, and measurement noise was assumed to be independentwith a variance of 0.01 (i.e., and are diagonal matriceswhose dimension matches the number of measurements usedin a given pass through the AEKF), in order to ensure adequateregularization of the inversions. The initial variance of the transformedabsorption was set uniformly at 40 with an initialcorrelation length of 0.IV. RESULTSA. Criteria for Discriminating Between DetectedHeterogeneities and ArtifactsThe results of the first set of experiments using a window sizeof six layers showed that the method was relatively robust overmost of the domains tested. A typical result is shown in Fig. 6(a),where two of the single-voxel heterogeneities are very accuratelyreconstructed and the third (located on -level 14) is properlylocated but the size and value are over- and under-estimated,respectively. The root mean square (rms) error in the phase predictions( ) over all 27 744 source-detector pairs is reducednonmonotonically as the estimation progresses, converging onpartial estimates as each of the three subdomains is conditioned[Fig. 6(b)]. Note that the number of sources used varies betweensubdomains, as convergence may be achieved before allsources in a given subdomain have been used in the conditioningFig. 5. Probability distributions of different levels of random beta-distributedspatial heterogeneity in “true” synthetic optical properties due to nonfluorescingchromophores. Absorption values are shown on the lower horizontal axisand isotropic scattering values are shown on the upper horizontal axis.process. The number of free parameters used to discretize absorptionin the entire domain can be seen to rise sharply as thesliding window moves to each new subdomain (when the newsubdomain is reparameterized from a single homogeneous zoneinto voxel-based stochastic zones), then fall rapidly as the AP-PRIZE algorithm conditions and merges zones in the subdomain[Fig. 6(c)]. Note that, using APPRIZE without domain decomposition,the maximum number of parameters used to model absorptionwould be 2601. In this example, using domain decompositionwith a window size of 6, the number of free zones neverexceeded 922 at any given time.Data for all zones in the 41 reconstructed domains that hadestimated values ≥ 0.018 cm (i.e., at least 0.005 cmhigher than background absorption of 0.013 cm ) are plottedin Fig. 7. It is evident that there is a nonlinear inverse relationshipbetween the size and value of the estimated heterogeneities( 0.91). The reconstructions identified seven single-voxelheterogeneities with values between 0.018 cm and 0.032cm that did not correspond to actual heterogeneities. Theseartifacts were never larger than one voxel, presumably dueto the large inaccuracies in model predictions that would becaused by large artifacts. We also note that all single-voxelheterogeneities that were correctly identified as such had estimatedvalues > 0.04 cm . These observations were usedto establish the following criteria; any multivoxel zone with anestimated value ≥ 0.018 cm , or any single-voxel zonewith an estimated value ≥ 0.04 cm , was considered tobe a heterogeneity, as depicted by the shaded region of Fig. 7.


156 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 3, MARCH 2001Fig. 6. (a) A sample reconstruction of a domain containing three single-voxel heterogeneities using a window size of six layers. The exact locations of the threesingle-voxel heterogeneities are as shown by the arrows. Note that scales vary for the three axes. (b) The rms phase error at emission wavelengths is reduced as thesliding window moves over the three subdomains and each heterogeneity is identified. (c) Dynamic changes in the number of free parameters. This domain wasselected for presentation because it happened to have one heterogeneity in each subdomain, thereby making the results easier to interpret by the reader.Fig. 7. Heterogeneities were usually correctly located (asterisks and stars) and were frequently correctly identified as single-voxels with estimated absorptionvalues close to the actual value of 0.13 cm ; however, it was not uncommon for the size of some heterogeneities to be overestimated, with concomitantunderestimations in absorption value. Performance was similar for domains including one or three heterogeneities. Four estimated heterogeneities weremislocated by a single-voxel from their true locations (open triangles). A few single-voxel zones had elevated absorption estimates in locations that did notcorrespond to actual heterogeneities (open squares and diamonds). The shaded region depicts the criterion for discrimination of “heterogeneities” from artifactsand background values.


EPPSTEIN et al.: THREE–DIMENSIONAL BAYESIAN OPTICAL IMAGE RECONSTRUCTION WITH DOMAIN DECOMPOSITION 157Fig. 8. Estimates of an “easy” domain at window sizes of (a) 1, (b) 5, and (c) 9. The true locations of the single-voxel heterogeneities are as shown by the arrowsin (a). Final root mean square errors (rmse) for phase error at emission wavelengths are shown beneath the reconstructions in each panel.Using these criteria, only two artifacts were misclassified asfalse positives. As these were both in the same domain andwere located only two voxels away from a pair of slightlymislocated heterogeneities, they might more accurately bedescribed as mislocated heterogeneities. Heterogeneities areconsidered correctly located if the identified zone includes thevoxel at which the true heterogeneity is located.Summarizing the data for all 81 individual heterogeneitiesincluded in the 41 domains, 40% of the heterogeneities werenearly perfectly reconstructed as correctly located, single-voxelheterogeneities with estimated values between 0.11 cm and0.13 cm ; 88% of the heterogeneities were correctly located;and 94% of the heterogeneities were located either correctly orwithin one voxel of the correct location. Only 6% of the heterogeneitieswere missed completely (false negatives), and onlyone domain (2%) had any false positives.B. Optimization of Subdomain SizeThe second set of experiments tested the effects of window(subdomain) size on 17 distinct domains containing a total ofindividual 31 heterogeneities. Some domains were relativelyeasy to reconstruct, and performance was good at all windowsizes tested (e.g., Fig. 8). However, most domains had markedlyworse reconstructions at very small window sizes (e.g., Fig. 9),and artifacts (or false positives) were common with a windowsize of 1. A couple of domains proved difficult to reconstruct regardlessof the window size (e.g., Fig. 10), in which both falsenegatives and false positives occur, even in the best reconstructions.Presumably, these domains were difficult to reconstructdue to the locations of heterogeneities relative to sources anddetectors, although we have not yet ascertained any consistentpatterns.Since it is not practical to produce images of all 153 reconstructionsfrom the window size experiments here, we have summarizedthe data in Fig. 11. Trends were similar for the singleheterogeneityand triple-heterogeneity domains, so these dataare lumped in Fig. 11(a)–(d). The percent reduction in root meansquare (rms) phase error and the final rms absorption error areshown in Fig. 11(a) and (b), respectively. Although these arecrude measures of the quality of a reconstruction for diagnosticpurposes, they do indicate that intermediate window sizes (3–6voxels thick) yield the best reconstructions. The sensitivity ofthe method for identifying true positives [Fig. 10(c), solid line]is high at all window sizes, but tends to decrease slightly aswindow size increases. On the other hand, the percentage of domainsthat contain false positives [Fig. 11(c), dashed line], aswell as the number of false positives per domain [Fig. 11(d)],is highest with a window size of 1. The percentage of domainsthat had essentially perfect reconstructions is also highest forintermediate window-sizes [Fig. 11(c), dash-dot line]. In addition,the computation time for our research code is seen to belowest for the intermediate window sizes on the single-heterogeneitydomains [Fig. 11(e), solid line], although the CPU timesfor the more complicated triple-heterogeneity domains are relativelyindependent of window size. Total CPU times for imagereconstruction of these 2601 voxel domain models ranged from15 to 43 minutes (note that each application of the forward simulatorfor one optical source required approximately 1 s). Thesetimes are intended to be compared in a relative sense and not toindicate production level computing times.C. Sensitivity to Unmodeled Background Variations in OpticalPropertiesThe third set of experiments examined the question ofwhether it is valid to treat optical absorption and isotropic


158 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 3, MARCH 2001Fig. 9. Estimates of the same domain shown in Fig. 6(a) at window sizes of (a) 1, (b) 5, and (c) 9, showing the presence of artifacts at window size 1. True locationsof single voxel heterogeneities are as indicated by the arrows in (a). Final rmse for phase error at emission wavelengths are shown beneath the reconstructions ineach panel.Fig. 10. (a) The heterogeneities labeled h1 and h3 in this random domain proved to be difficult to reconstruct at any window size, although h2 was always correctlyidentified. (b) The best reconstruction was at window size four, where h1 and h3 were faintly detected, as was an artifact on z-level 17. (c) At window size five,neither h1 nor h3 were detected at all, and four artifacts were present on z-levels 16 and 17. Final rmse for phase error at emission wavelengths are shown beneaththe reconstructions in panels (b) and (c).scattering as homogeneous, for the purpose of reconstructingabsorption due to fluorophores from phase shiftat emission wavelengths, even though these properties willundoubtedly contain some level of heterogeneity within biologicaltissues. For the ten single-heterogeneity domains tested,sensitivity was 100% (i.e., no false negatives). Remarkably, thishigh sensitivity was not diminished, even with large amountsof unmodeled background variations in nonfluorescence ab-


EPPSTEIN et al.: THREE–DIMENSIONAL BAYESIAN OPTICAL IMAGE RECONSTRUCTION WITH DOMAIN DECOMPOSITION 159Fig. 11. (a) Mean percent reduction in rms phase error due to reconstruction using various window sizes in 17 random domains. Vertical bars denote standarddeviation. (b) Mean rms absorption errors of the final reconstructed estimates of the 17 domains using various window sizes. Vertical bars denote standard deviation.(c) Percent sensitivity of the method (S: solid line), percent of domains that were essentially perfectly reconstructed (PR: dash-dot line), and percent of domainsthat included false positives (FP: dashed line) as a function of window size. (d) Mean number of false positives in the 17 domains reconstructed using variouswindow sizes. Vertical bars denote minimum and maximum values. (e) Mean CPU time for the ten single-heterogeneity domain reconstructions (solid line) andthe seven triple-heterogeneity domain reconstructions (dashed line) as a function of window size. Vertical bars denote standard deviation.sorption and isotropic scattering ,upto 90% aboveand below mean for absorption and up to 50% aboveand below the mean for isotropic scattering . This rangeof optical properties is considerably larger than reported forhuman breast tissues [55]. In addition, specificity remainedhigh over the entire range of absorption variations tested;even at 90%, the average number of artifacts per domain wasonly 0.2 (Fig. 12, solid line). With unmodeled variations inisotropic scattering within 10% of the mean there werestill less than 0.4 artifacts per domain but, as the unmodeledvariation in isotropic scattering increased, an increasingnumber of artifacts were reconstructed (Fig. 12, dashed line).The responses to unmodeled variations in isotropic scatteringand absorption appear to be additive. For example,we ran a series of tests where isotropic scattering includedvariations of 10% of the mean and absorption includedvarying amounts (from 0% to 90% of the mean) of unmodeledvariation in absorption . The average number ofartifacts remains less than one per domain (Fig. 12, dotted line),and is the trend is nearly parallel to, but slightly higher than,the average number of artifacts with no unmodeled variation inisotropic scattering (Fig. 12, solid line).V. DISCUSSIONWe have shown previously that the APPRIZE method can beused to accurately reconstruct 2-D [19] and small 3-D [24] opticaldomains in a few minutes on a general purpose microcom-


160 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 3, MARCH 2001Fig. 12. Changes in specificity of fluorescence absorption imagereconstruction as a function of the amount of unmodeled beta-distributedheterogeneity in “true” nonfluorescence optical properties (see Fig. 5),as reflected by the average number of artifacts present in reconstructeddomains where these properties were considered homogeneous and known.Error bars indicate the minimum and maximum numbers of artifacts inany reconstructions. The solid line refers to reconstructions where theisotropic scattering was homogeneous, but varying degrees of unmodeledheterogeneity were present in absorption . The dashed line refers toreconstructions where there were varying amounts of unmodeled heterogeneitypresent in isotropic scattering , but absorption was homogeneous.The dotted line refers to reconstructions where unmodeled heterogeneity inisotropic scattering was beta-distributed between 610% of the mean andvarying degrees of unmodeled heterogeneity were present in absorption .puter, even in the presence of noisy measurements. The inclusionof DDZ makes sequential recursive passes of the algorithmsuccessively faster as the number of parameters is progressivelyreduced, and renders the method much faster than alternativeinversion methods (e.g., see [19]). However, the number of parametersin the initially fully-discretized domain still governsthe computational memory demands of the algorithm, and thereforelimits the size of the domain to which the method can bedirectly applied.By employing a domain decomposition approach, in whichoverlapping and/or nonoverlapping 3-D subdomains are sequentiallyconditioned, this limitation can be overcome. Thesliding window approach to domain decomposition presentedhere provides a memory-efficient means of applying thestatistically powerful APPRIZE methodology to very large 3-Dinversions. As the window moves over a new subdomain, thediscretization of absorption (and/or other parameters of interest)is refined, thereby increasing the number of parameters to beestimated. As the subdomain is conditioned with APPRIZE,the number of parameters is again reduced through DDZ. Themaximum size of the parameter covariance matrix is thusgenerally not much higher than the number of parameters in thelargest subdomain. In other words, we can achieve voxel-levelresolution without ever requiring the storage and manipulationof a voxel-based parameterization of the entire domain modelat any given step in the estimation process. Recently, Holbokeand Yodh [17] report optical image reconstructions of 1368unknowns requiring 5.6 hours on 20 parallel processors (20.8hours on a single processor). Our image reconstructions of 2601voxel domain models were performed using an implementationnonoptimized for speed, written in Matlab, and executed on asingle processor 350 MHz Pentium II workstation, but typicallyrequired only 20–30 minutes of CPU time. While these programsand processors are not directly comparable, these timesnonetheless illustrate the relative computational efficiencyof the APPRIZE inverse method, especially considering theadded burden of the covariance manipulations in our Bayesianmethod.The quality of reconstruction via APPRIZE can also beenhanced by domain decomposition. When a large 3-D domain(or subdomain) is alternately conditioned using data from asingle source and parameterized with DDZ, there may not beenough information accrued in estimates far from the sourcebefore zones are clustered and merged in the DDZ process. Webelieve this is why sensitivity of the method begins to declineat large window sizes. In reconstructions using actual measurements,if system noise of the forward simulator increaseswith distance from the source, this effect could be much morepronounced. Using smaller subdomains limits the maximumsource-detector separation and ensures that DDZ is not appliedto zones that are very far from the applied source.On the other hand, these computational experiments clearlydemonstrate that the quality of reconstruction is reduced atvery small window sizes (1–2 voxels thick). Although using awindow size of one is still better than a 2-D inversion (becausethe forward simulator is still run in 3-D over the entire domain),these results highlight the importance of 3-D reconstruction,since we observed increased numbers of false positives in thinsubdomains.Our data indicate that, for the domains tested, the use of moderatelysized but still fully 3-D subdomains (three to six voxelsthick) yields the most accurate and rapid reconstructions. Wehave demonstrated only one of many possible architectures fordomain decomposition. Properly applied, there is essentially nolimit to the size of the domain that can be tackled using domaindecomposition without exceeding realistic computer memorylimitations. Domain decomposition will also facilitate a parallelimplementation, in which subdomains are independently and simultaneouslyestimated on separate processors. In this case, itmay prove desirable to utilize overlapping subdomains to minimizeerrors at subdomain boundaries.When reconstructing fluorescence absorption , there areseveral other optical properties that must also be considered.Joint estimation of multiple optical properties compoundsthe difficulty of the problem, both in a theoretical sense (theproblem becomes even more illposed) and in a computationalsense (both computer time and memory requirements increasedramatically). Thus, there is incentive to assume that one ormore of these “secondary” optical properties is homogeneous.In the computational experiments reported here, we demonstratedthat, for the simple domains tested, the reconstructionof fluorescence absorption from phase shifts at the emissionwavelength alone is relatively insensitive to unmodeledvariation in nonfluorescence absorption (at least up to90% of the mean) and isotropic scattering (up to 10%of the mean), but that higher levels of unmodeled variations in


EPPSTEIN et al.: THREE–DIMENSIONAL BAYESIAN OPTICAL IMAGE RECONSTRUCTION WITH DOMAIN DECOMPOSITION 161isotropic scattering resulted in more artifacts reconstructedin the images. While not conclusive, these experiments areencouraging in that they suggest that it may be reasonable totreat nonfluorescent optical properties as homogeneous whenreconstructing fluorescence properties from data at emissionwavelengths. Our results are consistent with experimental measurementswhich show that the changes in emission phase-shiftowing to the presence of a fluorescently tagged heterogeneityare greater than the corresponding phase-shifts measured atthe excitation wavelength. Indeed, the change in phase-delaymeasured at the excitation wavelength in response to thepresence of a perfect absorber in a lossless media is smallerthan the phase-delay measured at the emission wavelength ofa fluorescently tagged heterogeneity, even when backgroundfluorophore is present [31]. Thus, low levels of endogenous opticalcontrast may actually be used to advantage in fluorescenceimaging, if they allow background nonfluorescence absorptionand scattering to be treated as homogeneous.Image reconstructions from synthetic domains play a valuablerole in determining the potential applicability of an inversemethod for solving the problem under ideal conditions. Syntheticstudies also offer an easily controlled way of testing thesensitivity of the method to the relaxation of various assumptionswhich would be difficult, if not impossible, to assess reliablyusing experimental data. Historically, development of solutionsto inverse problems in medical imaging proceed fromcomputational experiment, to experimental phantom, and if successful,to imaging of biological tissues. In this paper, we havedemonstrated that the APPRIZE inverse method with domaindecomposition can accurately reconstruct 3-D fluorescence absorptionmaps in synthetic domains containing small heterogeneitiesusing noise-free frequency-domain measurements ofphase shift at emission wavelengths. The computation time requiredfor image reconstruction using APPRIZE is a fractionof the time reported for other approaches. However, many challengesremain as we attempt to validate the method with experimentaldata from tissue-mimicking phantoms and from invivo tissues. Notably, these include 1) determination of optimalways to characterize and minimize the system noise, 2) waysto minimize the measurement noise, 3) how to handle bias andspatial correlation that may exist in the system and/or measurementnoise, 4) how to initialize the a priori estimate ofthe parameter error covariance. Overcoming these challengeswill be critical for ensuring appropriate regularization of theseill-posed inverse problems. In addition, many questions remainregarding 5) optimal ways of decomposing both the data and thedomain, 6) the types of measurement data that should be used(e.g., combinations of phase-delay, amplitude, dc, “referenced”measurements), and finally 7) the combinations of optical propertiesthat might be jointly estimated, in order to detect clinicallymeaningful structure. Nonetheless, the statistically powerfuland computationally efficient APPRIZE algorithm providesa framework for physically based regularization of large3-D optical imaging problems within clinically practical computationalresources.The clinical applicability for accurate 3-D fluorescenceimaging is growing. For example, in a recent study of spontaneousmammary cancer in canines, systemic injection of ICGshowed localization in mammary masses as well as in regional,reactive lymph nodes [56]. The ability to detect fluorescencesignals originating from regional lymph nodes suggest thatFDPM fluorescence imaging, coupled with improved fluorescentdyes, may provide a valuable diagnostic method forassessing regional lymph node status in breast cancer as wellas melanoma patients. In cancer patients, lymph node statuscan be a powerful predictor of recurrence and survival, andthe number of lymph nodes with metastases provides crucialprognostic information regarding the choice of adjuvant therapy[57]. In the recent past, axillary lymph node involvement isassessed by resection and subsequent biopsy. More recently,gamma ray imaging of technetium-99 sulfur colloid injecteddirectly into the breast mass is used to track lymph flowand identify the sentinel nodes which can be located withinthe breast. Using a second agent, a blue dye to visually aidsurgeons in the resection of the sentinel lymph node, biopsy isperformed possibly sparing the axillary lymph nodes [58], [59].The successful development of fluorescent contrast enhancedimaging in three-dimensions could result in the replacement ofradiolabeled dyes with fluorescence dyes such as the cyanineand its derivatives as recently developed by Achilefu et al., [43]and Becker, et al., [60]. We plan to direct future developmentof the APPRIZE technology toward such clinical applicationsof 3-D fluorescence imaging.While the APPRIZE method has been demonstrated hereusing an application in fluorescence absorption imaging,variants of the method have been successfully applied to otherinverse problems [19], [25], [26], [47]. 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