POPART - Performance Optimized Algebraic Reconstruction Technique

POPART Optimized Performance OPtimized Technique Algebraic Reconstruction Technique Algebraic Hanson* K. M. Hanson* P940 MS P940 Los Alamos Los Alamos National Laboratory, MS USA 87545 USA Los Los Alamos, Alamos,NM NM 87545

Abstract specified visual A method for for optimizing image-recovery image- recoveryalgorithms algorithmsisispresented presentedthat that is is based on how well a specified visual A by assessed by numerically assessed performance isis numerically task performance Visual task images. Visual reconstructed images. task can be performed using the reconstructed appropriate scenes of generation the including process imaging complete imaging process including the of scenes simulation of the complete Carlo simulation Monte Carlo a Monte recovery, and image recovery, to the desired and performance performanceofofthe the stated stated task subsequent data taking, image application, subsequent desired application, to Technique (ART), Algebraic Reconstruction Technique optimize the Algebraic to optimize is used to based on on the the final final image. image. This method is based whichreconstructs reconstructs images imagesfrom fromtheir theirprojections, projections,by byvarying varyingthe therelaxation relaxationfactor factor employed employedininthe theupdating updating which ART, constrained ART, of constrained optimization of the optimization procedure. In some of the the imaging imaging situations situations studied, studied, itit is is found found that the some of procedure. little is little There is which aa nonnegativity nonnegativity constraint constraint is invoked, invoked, can can vastly vastly increase increasethe the detectability detectability of objects. objects. There in which ART. improvement attained attained for unconstrained ART. improvement

Introduction scene being The overall overall purpose purpose of of an an imaging imaging system system isis to to provide providevisual visualinformation informationabout about the object or scene information of information imaged. orientedimaging imagingsystems, systems,the thetype type of of scenes scenesexpected expectedand and the the kind of mission-oriented For missionimaged. For basis the basis optimized on the system should be optimized imaging system an imaging case an such a case desired can can frequently frequentlybe be specified. specified. In In such desired of how howwell wellthe thespecified specifiedvisual visualtasks taskscan canbe beperformed performedusing usingthe theresulting resultingimages. images. Here Here this this approach approach to to of complete imaging optimization isis applied applied to to only only one one aspect aspect of the complete imaging system, system, that that of the image reconstruction optimization practical. distinctly practical. is distinctly optimization is an optimization such an shown that such algorithm. It It is shown algorithm. image-ofimage optimization of Several classes classesof ofmeasures measureshave havebeen beenemployed employedininthe thepast past on on which which to to base the optimization Several conventional recoveryalgorithms algorithms [1], [1]. Some Some are are based based on on the the fidelity fidelity of of the the reconstructed reconstructed images, such as the conventional recovery measure of of the the rms rms difference differencebetween betweenthe thereconstruction reconstruction and and the original image, image, simply called called the rms error. measure There images. There of images. usefulness the with correlated be to seem always not does this that us Experience teaches teaches us that does always correlated with the usefulness of Experience the measurement measurement reproduces reconstruction estimated the closely how closely on how measures based on are are alternative measures estimated reconstruction reproduces reconstruction constraints reconstruction Unfortunately, without without further constraints residual. Unfortunately, mean-square the mean example, the for example, data, for -square residual. [1]. worse,illill-posed evenworse, oreven knowntotobebeillill-conditioned residualisisknown mean-square -square residual - conditioned or -posed [1]. the mean based on minimizing the well how of basis the onthe basis of how well ratedon algorithmisis rated an algorithm here, an presented here, optimization presented algorithm optimization In the approach to algorithm performance [2], task performance Ref. [2], in Ref. shown in As shown images. As reconstructed images. the reconstructed using the one can one can perform perform stated stated visual tasks using well specified in a well specifiedimaging imagingsituation situation isis readily readily assessed assessednumerically numericallythrough throughaa Monte Monte Carlo Carlo technique technique that that maximizing task involves maximizing procedure involves optimization procedure The optimization process. The imaging process. complete imaging the complete simulate the used to simulate is used is performance by varying varying whatever whatever free freeparameters parameters exist exist in in the the reconstruction algorithm. performance by

Performance Task Performance Calculate Task Method to Calculate effects of For linear imaging imaging systems systems the the effects of image imagenoise noiseon ontask taskperformance performance can can be be predicted predicted for for a variety For linear effects of measurement masking effects The masking effects of of simple simpletasks tasks [3], [3]. The The same same cannot cannot be be said of the effects of artifacts. artifacts. The of being measurements being setofofmeasurements eachset ineach results in process results noiseprocess randomnoise Therandom nature. The noise are truly truly random random in nature. noise are manifest However, many different,even evenwhen whenthe the scene scene being being imaged imaged does doesnot not change. change. However, many types types of of artifacts manifest different, number contract under Energyunder Departmentofof StatesDepartment UnitedStates the United *This work was was supported supported by the Energy contract number W-W-7405-ENG-36. 7405- ENG -36. "This work

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not. are not. sense, they are themselves as seemingly seeminglyunpredictable unpredictableirregularities irregularitiesthat that look look like likenoise, noise,but but in in aa strict sense, themselves as to completely inadequate to scene is completely simple scene of a simple single realization of scene, aa single Since Sincethese theseartifacts artifacts depend depend on the scene, response the response ofthe average of meaningfulaverage statisticallymeaningful obtainaastatistically to obtain necessary to judge judge a reconstruction algorithm. ItIt isis necessary cope. scenes with which of scenes ensemble of the ensemble of the realizations of an algorithm algorithm to many realizations which it must cope. of an pseudo-random employs pseudoA Monte Monte Carlo Carlo technique, technique, one one that employs randomnumbers numberstotogenerate generateits its results, results, isis used used to A can because it can performance, because final task performance, imaging process simulate the simulate the entire entire imaging process from from scene scene generation generationto to the the final ensemble. readily readily provide provide the the above above variations within the ensemble. following manner: thefollowing in the problem in entire problem specification of the entire complete specification The The method requires first aa complete a) Define Define the the class class of of scenes scenes to to be be imaged imaged with with as as much much complexity complexity as as exists exists in in the the intended application. uncerblur, uncersuch as blur, deficiencies in The deficiencies Define the b) Define the geometry geometry of the measurements. The in the measurements such specified. be specified. should be (noise) should measurements (noise) the measurements in the geometry, and uncertainties in tainties tainties in the geometry, signal and the concerning what Details concerning performed. Details c) Define Defineclearly clearlythe the task task to to be performed. what isis known known about about the signal c) explicitly. stated explicitly. background must must be be stated background appliintended applithe intended consistent with the should be consistent performance. This method should Define the d) Define d) the method method of task performance. information. known information. priori known cation cation and and the a priori following: thefollowing: doingthe bydoing performedby then performed is then The simulation procedure is Carlo Monte Carlo of aa Monte means of by means measurement data by corresponding measurement the corresponding scene and the representative scene e) e) Create a representative simulation simulation technique. tested. being tested. algorithm being scene with the algorithm Reconstruct the scene f) Reconstruct image. reconstructed the using task specified g) g) Perform the specified reconstructed image. the on the statistics on necessary statistics the necessary obtain the times to obtain number of times sufficient number g) aasufficient through g) e) through steps e) Repeat steps h) h) Repeat accuracy of of the the task performance. accuracy performed: Finally, determine how how well well the the task has been performed: Finally, determine of performance. measure of relevant measure the relevant using the performance using task performance i) Evaluate the task nonstasituations, nonstaimaging situations, complex imaging handles complex readily handles The The advantage advantage of this this numerical approach is is that itit readily tionary imaging imaging characteristics, characteristics,and and nonlinear nonlinearreconstruction reconstructionalgorithms. algorithms.Its Its major major disadvantage disadvantage isis that that it tionary investigated. specific imaging situation investigated. the specific for the only for valid only is valid provides provides an an evaluation evaluation that that is

ART function reconstructs aa function The Algebraic Algebraic Reconstruction Reconstruction Technique Technique (ART) (ART) [4] [4]isisan aniterative iterativealgorithm algorithm that that reconstructs function estimating aa function for estimating successful algorithm, particularly for verysuccessful be aavery to be proven to has proven from from its its projections. projections. It has when there when there isis aa limited limited amount amount of of data data available. available. Assume Assumethat that N projection measurements are made of the the which will unknown function/,f, which will be be considered consideredaavector. vector. As As these these measurements measurementsare arelinearly linearlyrelated relatedtoto /,f, unknown function they may may be be written as (1) 9i = ..., N, gi = Hif, Hit, i == 1,l,...,JV, (1) ART The ART corresponding row the corresponding is the measurement and where where gigiisis the the ith ith measurement and Hi is row of of the the measurement measurement matrix. matrix. The updated is updated Q. Then algorithm proceeds proceedsas asfollows. follows.An Aninitial initialguess guessisismade, made,for forexample, example,f°f° == O. Then the estimate is algorithm by iterating on the individual measurements measurements taken in turn:

fk +l= fk +AkHTt

gi

-Hifk HTHi

(2)

are constraints are applicable constraints Anyapplicable update. Any kth update. factor for the kth relaxation factor is aa relaxation Xk is and Ak mod(JV)+l and where ii = k mod(N)+1 where enforced, invoked invokedafter aftereach eachupdate. update. For example, for constrained ART ART in which which aa nonnegativity nonnegativity constraint is enforced, when that when such that is such (2) is of (2) normalization of the normalization constraints, the absenceofofconstraints, theabsence set f fkk+l 0, set k+l < 0, when ffk+1 when +l =—0.0.InInthe Ak = one nomenclature one standardnomenclature the standard measurement equation satisfy the measurement guaranteed to l is fk+ 1, fk — 1, Xk +1 is guaranteed to satisfy equation (1). (1). In the to measurements has iteration isis completed iteration completedafter after the the full full set set of N measurements has been been processed. processed. We We use use the the index index K to here to used here are used factors are damping) factors (or damping) relaxation (or Variable relaxation mt(k/N)). indicate the indicate the iteration iteration number number (K == int(k /N)). Variable as factor as express the relaxation factor will express We will successive updates attenuate successive updates during during the reconstruction. We AK A*

rÀ)K -1 = X0 (r,)K-\

(3)

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Figure randomly generated generated scene scene consisting consisting of of 10 10 highhigh-contrast 10 low low-contrast Figure 1: 1: The first randomly contrast and 10 - contrast discs. discs. The evaluation of average over over ten similar similar scenes. scenes. evaluation of task task performance performance is is based on an average The proper There is is very very little little guidance guidance on on this The proper choice choiceof ofthe the relaxation relaxationfactor factorisisthe theissue issueatat hand. hand. There choice known [5] [5] that ifif aa solution solution to to the themeasurement measurement equations equations exists, exists, the the ART ART choiceininthe theliterature. literature. It isis known algorithm willconverge convergetotoititinin the the limit limit of of an an infinite infinite number numberof ofiterations iterationsprovided providedthat that 22 >> ak algorithm will A& > 0. A > O. value of [6] have unconstrained ART ART ultimately ultimatelyconverges converges value ofunity unity isis often often suggested. suggested. Censor Censor et al. [6] have shown shown that that unconstrained minimum-norm zero slowly slowly enough. However, However, to a minimum -norm least-squares least -squaressolution solutionififthe the relaxation relaxation factor approaches zero \K will asymptotically approach zero for any value of r\ < 1. The value appropriate to a finite number of AK will asymptotically zero for any value of ra < 1. The value appropriate to a finite number of iterations previous work work the author has has assumed assumed for Ao AQ and the nominal nominal values values of iterations remains remains uncertain. uncertain. In previous and r\ ra the 1.0 and 0.8 0.8 for involving aa limited number of projections, and and 0.2 0.2 and and0.8 0.8for forproblems problemsinvolving involving 1.0 for problems problems involving many views [2]. [2]. This Thischoice choice for for ra r\ makes makes the the final final AK A^ at ten ten iterations iterations about aboutseven seven times times smaller smaller many (~100) (100) views initial one one A0. A0 . Next Next we we discuss discuss a way way to find the best best choice choice for the relaxation relaxation parameters parameters for for aa than the initial given problem.

Optimization of ART The numerically calculated task task performance can be be used to search The numerically calculated performance can search for for the the optimum optimumchoice choice of of Ao A0 and r\ for the ART algorithm. algorithm. For For the the present present purpose, purpose, the thescene scene isis assumed assumed to toconsist consist of ofaanumber numberof ofnon non-ra for overlapping example, each each scene scene contains contains10 10 high high-contrast discs overlapping discs discsplaced placed on on aa zero zero background. background. For this example, -contrast discs of 10 low-contrast discs are randomly placed placed within a of amplitude amplitude 1.0 1.0 and and 10 low- contrastdiscs discswith withamplitude amplitude0.1. 0.1. The The discs circle diameter of circleof ofreconstruction, reconstruction,which whichhas hasaadiameter diameterof of 128 128pixels pixelsininthe thereconstructed reconstructedimage. image. The The diameter each The first first of of the the series series of of images images generated generated for for these tests tests is is shown shown in in Fig. Fig. 1. In each disc discisis 88 pixels. pixels. The this computed computed tomographic measurements are are assumed assumed to to consist consist of ofaaspecified specified number number tomographic (CT) problem, the measurements of of parallel parallel projections, projections,each each containing containing128 128samples. samples.Ten Teniterations iterationsof ofART ARTare are used used in in all all of the present examples. the visual visual task task to to be beperformed performed is is the the detection detectionof ofthe thelowlow-contrast discs. To examples. ItIt is is assumed assumed that that the contrast discs. produce noise is added to to the the projection projectionmeasurements measurements using using aaGaussian Gaussian-distributed produce noisy noisy data, data, random noise -distributed number generator. random number The 12 noiseless noiseless views shown in Fig. Fig. 2. The The result result of reconstructing Fig. 11 from from 12 views spanning spanning 180° 180° isis shown seemingly random limited number of of seemingly randomfluctuations fluctuationsininthe the background backgroundare are actually actually artifacts artifacts produced produced by by the limited projections and arise mainly from the high-contrast discs. As the artifacts depend on the positions of the projections and arise mainly from the high- contrast discs. As the artifacts depend on the positions of the discs, itit is allow for for random discs, is important important to allow random placement placement of of the the discs discs to to allow allowfor for the the full full range range of of artifacts. artifacts. It 320 //SP SPIE Vol. 7007 /E Vol. 1001 Visual Communications and Image Image Processing Processing'88 '88

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Figure 2: Reconstructions of Fig. 11 from from12 12 noiseless noiseless parallel projections projections subtending subtending 180° 180° obtained Figure 2: Reconstructions obtained with 10 iterations (left) without without the nonnegativity nonnegativity constraint. These 10 iterations of of the the ART ART algorithm algorithm (right) (right) with with and and (left) constraint. These reconstructions were 1.0 and rr^, = 0.8. 0.8. These reconstructions wereobtained obtained with with A0 ao == 1.0 These images images are are displayed displayed at at high high contrast contrast to

show the the low - contrast discs discs of of interest. interest. show low-contrast appears the nonnegativity nonnegativity constraint constraint improves improves the reconstruction reconstruction considerably has reduced reduced appears that that the considerablyin in that that it has the confusion confusion caused in the the background. background. However, However, some some of of the the low low-contrast discs have caused by by the fluctuations in -contrast discs have been reproduced. reproduced. Also, there still remain many fluctuations in the background that not been that may may mislead mislead one one to to suspect the presence of discs in places where none exist in reality. Thus, on the basis of this single example, suspect presence discs places where none exist reality. Thus, on the basis of this single example, one cannot say say with with certainty whether -contrast discs one cannot whether or or not not the thedetection detectionofofthe thelow low-contrast discs is is improved improved by the the nonnegativity constraint. constraint. A between reconstructions reconstructions with with and and without without nonnegativity A statistically statistically significant significant comparison comparison between the constraint must be made made to to assess assess its its value. value. The - contrast discs. The task to to be beperformed performedisisthe thedetection detectionofofthe thelow low-contrast discs. It is is assumed assumed that the the position position of a possible disc isis known known beforehand, beforehand,asas isis the the background. background. To perform possible disc perform the stated stated task taskofofdetection, detection, it is variable. This is assumed assumed that the the sum sum over over the the area area of ofthe thedisc disc provides provides an an appropriate appropriate decision decision variable. This is an approximation of the the optimum decision variable when when the the image image isis corrupted corrupted by by additive uncorrelated an approximation of decision variable uncorrelated Gaussian noise [7]. [7], After reconstruction, reconstruction, the sums sums over over each each region region where where the the lowlow-contrast objects are contrast objects known to calculated, as well as those over over each These two two data sets sets known to exist exist are are calculated, as well as those each region region where wherenone noneexist. exist. These may be represented as histograms may be represented as histograms in this decision decision variable. variable. The The degree degree of of separation separation between between these these two two distributions distributions isis often often characterized characterized by by the the detectability index d', d', given given by the the difference difference between between the means means of these square root of of the the average average of of their their variances. variances. This This isis sometimes sometimes called called these distributions distributions divided divided by by the square the signal signal-to-noise area -to -noiseratio ratio(SNR) (SNR)for fordetection. detection.As Asnoted notedininRef. Ref. [2], [2],the the detectability detectability index based on the area under curve d& d1 under the the receiver receiver operating operating characteristic curve dAisismore moreappropriate appropriatefor forthe the binary binary decision decisiontask. task. But d' statistical accuracy accuracy than thandA d& and and is is more more likely likely to continuous function has better statistical to be be a continuous function of of the the parameters parameters that can makes dd'1 preferable preferable for for the purpose purpose of of optimization. optimization. can be be varied varied in in the the reconstruction procedure. procedure. This makes When When ten randomly randomly generated generated scenes scenes similar similar to Fig. Fig. 1 1 are reconstructed reconstructed and analyzed analyzed to obtain obtain an an averagevalue valuefor ford', d', itit is average is found found that that d'd is f is0.871 0.871for forunconstrained unconstrained reconstructions reconstructions and 2.054 2.054 for for those those employing Indeed, the nonnegativity constraint constraint has has improved improved detectability. detectability. employingthe thenonnegativity nonnegativity constraint. constraint. Indeed, Fig. 33 shows for optimization functions depend shows how various choices choices for depend on on Ao A0 and and rA r\ for constrained constrained ART. ART. L1 difference differenceisisbased basedon onthe the LI L1norm normof ofthe the difference differencebetween betweenthe thereconstruction reconstruction and the original image. The LI XZ. There There is a definite The rms residual is is essentially essentially V^?definite minimum minimum in these functions functions indicating optimum The operating points points for for these these two two parameters. parameters. However, the minima minima are are at different operating However, the different positions. Which Which one one do do

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ORIGINAL) 100 RMS DIFF(RECON. DIFF(RECON. - ORIGINAL) 100 xx RMS

100 /d' 1.00 1.00 -

1.00 -

0.98 -

0.98 -

0.96 -

0.96 -

0.94 -

0.94 -

0.92 0.92 -

0.92 -

0.90 0.90 -

0.90 -

ra

2.6

2.8

3.0

3.2

3.4 À0

AO

ORIGINAL) 100 xx LI 100 Li DIFF(RECON. -- ORIGINAL)

100 xx RMS 100 RMS RESIDUAL

.—i—i—|—.—i—i—|—i—i—i—|—i—i—i—|—i—r

1.00 1.00 -

1.00 1.00

0.98

0.98 -

0.96

0.96 -

r

ra

0.94

0.94 -

0.92 0.92

0.920.92

0.90 -

0.90 2.6

2.8 2.8

3.0

3.2

3.4 3.4

Ao AO

2.6

2.8

3.0

3.2

3.4

A0

parameters relaxation parameters the relaxation of the function of functions plotted Figure 3: Contour plots of plotted as a function optimization functions four optimization of four Figure 3: noiseless, 12 of consist measurement The algorithm. Al) and rA used in the constrained ART reconstruction algorithm. The measurement consist of 12 noiseless, reconstruction A0 and r\ used in the constrained ART by necessitated functions, these of points) 10 x (10 sampling coarse sampling (10 x 10 points) of these functions, necessitated by 180°.°. The The coarse parallel projections spanning 180 effects. scalloping the for accounts evaluation, accounts for the scalloping effects. function evaluation, each function for each the lengthy computation required for time required computation time

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Figure Figure 4: 4: Optimized reconstructions of Fig. 11 from 12 noiseless noiselessparallel parallelprojections projections subtending subtending 180° 180° obtained obtained with left is is obtained with constrained constrained ART. ART. The The reconstruction on the left obtained with with Ao A0 = 2.96 2.96 and rA rA = = 0.975, 0.975, which which is the optimum optimum for is obtained obtained with with Ao A0 — 3.25 and rA for detectability. detectability. The reconstruction on the right is r\ = — 0.975, 0.975, = 3.25 which difference between which produces producesthe the smallest smallest rms rms difference betweenthe the reconstruction reconstruction and and the original image. image. Although Although the rms error in the the reconstruction reconstruction is the rms is aa common common measure measure for the quality quality of of reconstruction, reconstruction, itit yields yields more more visible from its its optimum optimum of of 23.5 23.5 to to 12.6. 12.6. visibleartifacts artifacts and and reduces reduces dd'9 from we we choose? choose?Fig. Fig. 44 shows showsthe the reconstructions reconstructions obtained obtained using using the the relaxation relaxation parameters parameters for optimization with respect respect to to 100/d' 100 /d' and and the rms error in the reconstruction. reconstruction. There There is is an an enormous enormous improvement improvement in the quality quality of both reconstructions reconstructions over those shown shown in Fig. Fig. 2. 2. Optimization Optimization with 100/d' with respect respect to to 100 /d' appears appears to to be preferable twice as optimization with preferablebecause becauseitityields yieldsaadd'f that that is twice as large large as as the the optimization with respect respect to to rins rms error. error. The The latter also leads annoying streak artifacts, which which are quite visible visible in leads to annoying in a good display display of of the the reconstruction. reconstruction. same kind The same kind of contour contour plots plots for for unconstrained ART are relatively flat and uninteresting. uninteresting. Fig. shows reconstructions Because of the views, the Fig. 5 shows reconstructions obtained obtained from from noisy noisy data. data. Because the large number of views, the data are complete. complete. For the unconstrained unconstrained and and constrained constrained reconstructions, reconstructions, d' d 1 isis found found to tobe be1.995 1.995and and1.825, 1.825, are respectively. case the nonnegativity constraint constraint has worsened worsened detectability, respectively. In In this this case the nonnegativity detectability, contrary contrary to to what what might might be concluded from calculate these detectabilities be concluded fromaa first first glance. glance. The The CPU time required to calculate detectabilities took about about one one 8700, which whichisis about about four four times faster than hour on a VAX VAX 8700, than aaVAX VAX 785. 785. The optimum values for Ao and r\ ra were AQ and were found for various various conditions conditions of data data collection collection using aa function function minimizer called E04JB3. finds the parameters parameters for for the the minimum minimum of of aa minimizerfrom fromthe the NAG NAGlibrary library'1 called E04JB3. This routine finds function after function. From From 20 20 to to 100 100 function function after many many evaluations evaluations of of the the function. function evaluations evaluations are are required required for for the the cases cases studied studiedhere. here. Table Table 11 tabulates tabulates the results obtained with with unconstrained ART. ART. In In most most cases cases relatively relatively little improvement improvement in in detectability detectability is is achieved achieved by by optimization optimization compared comparedto to that that obtained with the the nominal nominal relaxation the noiseless noiseless cases, cases, aa value value of of unity unity for for AK A^ yields relaxation factors. factors. In the yields essentially essentially the same results as as the the optimized values, agreement with common common practice. However, for optimized values,aa choice choicethat that is is in agreement practice. However, for noisy noisy data itit seems seems desirable desirablefor for r\ rAtotobe be less less than than unity and, and, when when there there are aremany manyviews, views, Ao AQ should should be be small. small. These Thesechoices choices are reasonable significant averaging are reasonable as as they they promote promote significant averagingover overall allthe theviews. views. As As aa rule rule of thumb, thumb, for noisy noisy but complete completedata, data, the the relaxation factor should should be approximately approximately equal to the the reciprocal reciprocal of of the the number number of ofviews views for few iterations. for the last few The results The results of of optimizing optimizing constrained constrained ART ARTare are presented presentedinin Table Table 2.2. The nonnegativity nonnegativity constraint constraint is Numerical 'Numerical Algorithm Algorithm Group, Group, 7 Banbury Road, UK Road, Oxford Oxford OX2 OX2 6NN, 6NN, UK

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Figure 5: Reconstructions of Fig. from 100 100 noisy noisy parallel Figure 5: Reconstructions Fig. 11 from parallel projections projections subtending 180° 180° obtained with the the ART algorithm ART algorithm (right) (right) with with and and (left) (left) without without the the nonnegativity constraint. The rms noise nonnegativity constraint. noise added added to the the projection measurements measurements is is 8, 8, which which isis ten ten times times the peak projection value for one of the lowlow-contrast contrast discs. These XQ == 0.2 These reconstructions reconstructions were were obtained obtained with Ao 0.2 and and r^ ra = 0.8. 0.8. seen seento tobe be generally generallyuseful usefulwith withthe thenominal nominalrelaxation relaxationfactors, factors,particularly particularlywhen whenthe thedata data are are limited limited by by the the measurement optimization, huge measurement geometry. geometry. But But with optimization, huge improvements improvementsinin detectability detectability are are obtained obtained in in these these cases. cases. Very Very large large relaxation relaxation factors factors are are preferred, preferred, in in fact fact much much larger larger than might might be be expected. expected. However, However, when is realized realized that the the nonnegativity nonnegativity constraint constraint has has the the effect when itit is effect of of undoing undoing the the agreement agreement with with each each measurement seems reasonable that measurement that that should result result from an update, itit seems that overrelaxation overrelaxation is needed. needed. Neither Neither use of nonnegativity the use nonnegativity nor nor the the optimization has has much much benefit benefit when when the the data are complete complete but noisy. noisy.

Discussion some of imaging situations studied, the the use In some of the the imaging use of of the the nonnegativity nonnegativity constraint constraintininART ARTsignificantly significantly increases the objects, especially especially when increases the detectability detectability of objects, when the the data consist consist of of aalimited limitednumber numberofofnoiseless noiseless

Table Table 1:1: Summary Summary of ofthe the effect effectof ofoptimization optimizationwith withrespect respecttotothe the detectability detectability index index d' d' on reconstructions reconstructions obtained using using 10 10 iterations iterations of unconstrained ART. ART. The optimum operating point point was found by varying the parameters that control control the therelaxation relaxationfactor factorused usedininthe theART ARTalgorithm, algorithm,Ao parameters that A0 and and?À, TA, as as discussed discussed in the text. text. There There isis generally generally little little improvement observed in detectability. detectability.

number proj. 100 100 88 12 12 16 16 16 16 16 16

AO A0 (deg.) 180 L 180 180 180 180 180 180 180 90 90 180 180

rms noise 88 00 00 00 00 22

nominal

optimized

AQ Ao

Ta TX

d'

Ao AO

rT X

d' d>

0.2 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8

1.995 0.464 0.871 1.960 1.122 1.653

0.107 0.915 0.427 1.047 1.714 2.247

0.820 0.463 0.729 0.998 0.993 0.635

2.013 0.485 0.932 1.969 1.202 1.662

324 //SP SPIE Vol. 1001 1001 Visual /E Vol. Image Processing Processing '88 '88 Visual Communications and Image

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reconstruc­ on ART reconstrucTable 2:2: Summary Summary of ofthe the effect effectof ofoptimization optimizationwith withrespect respecttotothe the detectability detectability index index d' on Table reconstruction geometry limits the reconstruction measurement geometry the measurement When the tions tions incorporating incorporating the the nonnegativity nonnegativity constraint. When possible. be possible. to be seen to is seen detectability is in detectability improvement in dramaticimprovement rather than noise data,dramatic noise in the data number proj. 100 100 88 12 12 16 16 16 16 16 16

AO A0 (deg.) (deg.) 180 180 180 180 180 180 180 180 90 90 180 180

rms noise noise 88 00 00 00 00 22

A0 A0

0.2 0.2 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

nominal nominal d' rA TA 0.8 0.8 1.825 1.825 0.653 0.8 0.653 0.8 2.054 0.8 2.054 0.8 0.8 4.782 0.8 4.782 0.8 2.050 0.8 2.050 0.8 2.372 2.372 0.8

optimized ao A0 0.052 0.052 3.450 3.450 2.959 2.959 2.794 2.794 2.782 2.782 3.012 3.012

ra T-A 0.859 0.859 0.959 0.959 0.975 0.975 0.951 0.951 0.967 0.967 0.712 0.712

d' 1.908 1.908 4.91 4.91 23.46 23.46 40.13 40.13 6.30 6.30 2.747 2.747

initial projections. Optimization Optimization isis accomplished accomplishedbybyvarying varyingthe therelaxation relaxationfactor, factor,both bothinin terms terms of of its its initial

obtained reconstructions obtained the reconstructions in the detectability in The detectability number. The value and and the the rate rate of its decline with iteration number. decline with value found It with constrained constrained ART ART isis .dramatically dramatically enhanced enhancedby by the the optimization optimization procedure procedureinin some some cases. cases. It is found with rms quality, such as rms reconstruction quality, measures of reconstruction conventional measures toconventional respect to with respect optimization of ART with that optimization differencefrom fromthe theoriginal originalimage, image,results resultsininreconstructions reconstructionswith withmore moreartifacts artifacts and and lower lower detectability. detectability. For difference achieved through optimization. unconstrained unconstrained ART, ART, little improvement was achieved basis of what image-reconstruction optimizeimageimportanttotooptimize It is reconstruction algorithms algorithms on on the basis concluded that itit isisimportant is concluded using the performed using be performed is most most important, important, which which can can often often be be defined definedinin terms terms of ofaa visual visualtask task that that is to be is process imaging process complete imaging the complete of the simulation of Carlo simulation Monte Carlo on aa Monte based on is based final image. image. The The approach taken here is final This from the the composition compositionofofthe the original originalscene scenetotothe the final final interpretation interpretation of of the the reconstructed reconstructed image. image. This from can only method isis consistent consistentwith with the the assertion assertion that that an algorithm can only be be properly properly evaluated evaluated by by testing testing itit on on aa method varied. uncontrollable variables statistically meaningful sample sample of of trials trials in in which which all all the the uncontrollable variables in in the problem are varied. statistically meaningful effect of net effect the net This numerical simulation simulation technique technique has has several severalgreat great advantages. advantages. It It can be used to evaluate the This numerical themselves lendthemselves not lend do not that do complex scenes complex sceneson onthe thereconstructed reconstructed images. images. ItIt is is particularly particularly useful useful in in situations situations that performance of optimize the performance algorithms like analysis, as analytic analysis, to analytic as in nonlinear algorithms like constrained constrained ART. ART. ItIt can optimize of by directly by addressed directly be addressed cannot be issues cannot These issues iterations. These number of iterations. iterative iterative algorithms algorithmsfor for an an arbitrary arbitrary number numerical Carlo numerical Monte Carlo the Monte on the relying on of relying disadvantage of major disadvantage The major theoretical approaches theoretical approaches to to optimization. optimization. The generalizations are and generalizations imaging situation specific imaging the specific only to the applies only technique isis that technique that each result applies situation tested and seldom possible. seldom possible.

References 1977. Cliffs,1977. EnglewoodCliffs, Prentice-Hall, Restoration.Prentice ImageRestoration. DigitalImage Hunt.Digital R. Hunt. [1] [1] H. H. C. C. Andrews Andrews and B. R. -Hall, Englewood image-recovery evaluate image[2] K. [2] K. M. M. Hanson. Hanson. Method to evaluate recoveryalgorithms algorithmsbased basedonontask taskperformance. performance. Proc. Proc. SPIE SPIE 1988. 914,336-343, 914, 336 -343, 1988.

1983. 60-67, 419, 60 SPIE 419, observer. Proc. SPIE [3] [3] K. K. M. M. Hanson. Hanson. Variations Variations in in task task and the ideal observer. -67, 1983. elec­ three-dimensional forthree techniquesfor reconstructiontechniques Algebraicreconstruction Herman.Algebraic G. Herman. and G. Gordon, R. Bender, and [4] [4] R. R. Gordon, - dimensional elec1970. 471-481, 29, Biol. Theor. J. photography. microscopy and xx-ray tron microscopy -ray photography. J. Theor. Biol. 29, 471 -481, 1970. Assoc. Commun.Assoc. reconstruction. Commun. image reconstruction. for image methods for Relaxationmethods [5] G. [5] G. T. T. Herman, Herman, A. A. Lent, Lent, and and P. H. Lutz. Relaxation 1978. 152-158, 21, 152 Comput. Comput. Mach. Mach. 21, -158, 1978. for method for Kaczmarz's method underrelaxation in Kaczmarz's Strong underrelaxation Gordon. Strong Eggermont, and [6] [6] Y. Y. Censor, Censor,P.P. P. P. B. B. Eggermont, and D. D. Gordon. 1983. 83-92, 41,83 Math. 41, Numer. Math. inconsistent inconsistent systems. systems. Numer. -92, 1983. 1971. York, 1971. New York, Academic, New Signals. Academic, of Signals. Detection of [7] [7] A. A. D. D. Whalen. Whalen. Detection

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