Comparison of deconvolution techniques using a distribution …mignotte/Publications/JEI02.pdf ·...

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Comparison of deconvolution techniques using adistribution mixture parameter estimation:

Application in single photon emission computedtomography imagery

M. MignotteDIRO,�

Departement� d’Informatiqueet de RechercheOperationnelle�C.P�

. 6128Succ.�

Centre-villeMontreal� � Que

�´bec�

H3C 3J7CanadaE-mail:

[email protected]

J.�

MeunierINRIA, Institut Nationalde Rechercheen Informatiqueet Automatique

France�

J.-P�

. SoucyOttawa

Hospital—CivicCampus1053CarlingAvenue

Ottawa �

Ontario �

K1Y�

4E9 Canada

C.�

JanickiHo� ˆpital� Notre-Dame� CHUM

� �1560rue SherbrookeEst

Montre� ´al� � Que

� ´bec� �

H2L�

4M1 Canada

Abstract. Thanks to its ability to yield functionally rather thananatomically-based information, the single photon emission com-puted� tomography (SPECT) imagery technique has become a greathelp in the diagnostic of cerebrovascular diseases which are thethird most common cause of death in the USA and Europe. Never-theless, SPECT images are very blurred and consequently their in-terpretation is difficult. In order to improve the spatial resolution ofthese images and then to facilitate their interpretation by the clini-cian, we propose to implement and to compare the effectiveness ofdifferent existing ‘‘blind’’ or ‘‘supervised’’ deconvolution methods. Tothis end, we present an accurate distribution mixture parameter es-timation procedure which takes into account the diversity of the lawsin the distribution mixture of a SPECT image. In our application,parameters� of this distribution mixture are efficiently exploited in or-der to prevent overfitting of the noisy data for the iterative deconvo-lution techniques without regularization term, or to determine theexact support of the object to be restored when this one is needed.Recent blind deconvolution techniques such as the NAS–RIF algo-rithm, [D. Kundur and D. Hatzinakos, ‘‘Blind image restoration viarecursive filtering using deterministic constraints,’’ in Proc. Interna-tional Conf. On Acoustics, Speech, and Signal Processing, Vol. 4,pp.� 547–549 (1996).] combined with this estimation procedure, canbe�

efficiently applied in SPECT imagery and yield promising results.©�

2002 SPIE and IS&T. [DOI: 10.1117/1.1426082]

1 Introduction

Single�

photon emission computedtomography � SPECT� �

imagesareobtainedby the measureof radiations � gamma�rays� coming from radioactiveisotopesinjectedin the hu-man! body. Contrary to other medical imaging techniques,such" as x-ray, computertomography, magneticresonanceimaging,etc.,this imageryprocessis ableto give function-ally� ratherthananatomically-basedinformation,suchasthemetabolic! behaviorof organs # like

$the humanbrain% , b& y

measuring! and visualizing the level of blood flow. Thisstudy" of regionalcerebralblood flow can aid in the diag-nosticof cerebrovasculardiseasesandbraindisorders' e.g.,(Alzheimer)

’s disease,Parkinson’s disease,etc.* by�

indicat-ing+

lower, or abnormalhigher, metabolicactivity in somebrain�

regions.Due to the peculiar imaging process,SPECT suffers

from,

poorstatisticsandpoorspatialresolution.Poorstatis-tics-

result from the small numberof photonsthat can beacquired� for eachimage;principally owing to the low sen-sitivity" of the collimator and the low doseof the injectedradiopharmaceutical.� Factorsinfluencingthe spatialresolu-tion-

aremainly thescatteringof theemittedphotonsand,toa� lesserdegree,the intrinsic resolutionof thecamera.Con-sequently" , resultingcross-sectionalSPECTimagesareveryblurred�

andtheir interpretationby the nuclearphysicianisoften. difficult, labor-intensive,andsubjective.If the objectto-

be visualized is small compared to the source-to-collimator distance,this degradationphenomenonmay be

Paper/

99056 receivedOct. 12, 1999; revised manuscriptreceivedMay 3, 2001;accepted0 for publicationSept.5, 2001.1017-9909/2002/$15.00© 2002SPIEandIS&T.

Journal1

of Electronic Imaging 11(1), 1–0 (January 2002).

Journal1

of Electronic Imaging / January 2002 / Vol. 11(1) / 1



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considered to beapproximatelyshift-invariantand,neglect-ing+

noise,this one can be modeledby a convolutionpro-cess betweenthe true undistortedimage and the transferfunction of the imagingsystem.1 A body of theoreticalandexperimental( work has led to approximatethis transferfunction, 2

also� calledthe point spreadfunction 3 PSF4 576

by�

atwo-dimensional-

symmetricGaussianFunction.2,3 In8

orderto-

improve the spatial resolutionof SPECTimages,someauthors� havethusinvestigatedtheSPECTimagedeblurringproblem� with this classof Gaussiantransferfunction andby�

usingclassicalWienerfilter techniques1,2 or. supervisedmaximum entropy filter-baseddeconvolutiontechnique.4

9This restorationprocedure,alsocalleda deconvolutionpro-cedure, is an important considerationin SPECTmedicalimaging+

wheretheremaybe localizedsingularitiesor cold/hot spotsin the true image,associatedwith lesionsor tu-mors.Theselocalizedsingularitiesmaynot bevisible in theblurred�

image, owing to the diffusive effects associatedwith: the convolution process,which averagesout differ-ences( in neighboringvalues.A deconvolutionschemecouldthen-

be very useful in order to detectsuchsingularitiesbyimproving+

the spatialresolutionof SPECTimages.Under;

the assumptionthat the blur operationis exactlyknown,manyiterativemethodshavebeenproposedby theimageprocessingcommunity for tackling this deconvolu-tion-

procedureand for facing the usualdifficulties relatedto-

this ill-posed problem.Amongst the existing methods,some" of themarestructuredin thecontextof regularizationproblem� to makethe inversionwell behaved.5–

<8 Others

areunregularized= andrequirea terminationcriteria in order tostop" the iterative procedureat the point where there is abalance�

betweenthe fit to the imagedataand the amplifi-cation of the noise, inherent to this ill-posed inverseproblem.� 9–1

>3 Nevertheless,?

let usnotethat thesesuperviseddeconvolution@

methodsremainlimited andsensitiveto theassumption� madeon the natureof the blurring function.Theoretically, the PSFcan be measureddirectly from theSPECT�

cameraby visualizing the blurredresultof a pointsource" againsta uniform background,but suchexperimentis generallydifficult to obtainin practiceanddoesnot nec-essarily( yield a reliablePSF. In applicationssuchasmedi-cal imaging,whenlittle is knownaboutthePSF, it canturnout. often more relevantto estimatedirectly the PSFfromthe-

observedinput image.This problemof simultaneouslyestimating( the PSF A or. its inverseB and� restoring an un-knownC

imageis called ‘‘blind deconvolution’’ or ‘‘decon-volutionD with blur identification.’’ Recenttechniquesexistand� canbe usedin the SPECTimagerycontext.

In8

this paper, we proposea comparativestudyof exist-ing+

blind or superviseddeconvolutionmethods.We discussand� comparetheir respectiveeffectivenessfor improvingthe-

spatialresolutionof realbrainSPECTimages.First,webriefly�

review classicalsuperviseddeconvolutionmethodswhich: assumetheblur is exactlyknownaE priori and,� in thiscontext, we exploit the two-dimensionalGaussianassump-tion-

for thePSFproposedby someauthors.1–3 For theclassof. the superviseddeconvolutiontechniquewithout regular-ization+

term, we presentan accuratedistribution mixtureparameter� estimationwhich takesinto accountthediversityof. the laws in the distributionmixture of a SPECTimage.In our application,parametersof this distribution mixtureare� efficiently exploitedin orderto find a reliablestopping

rule for theseiterative methodsand then to prevent theamplification� of thenoise.Then,recentblind deconvolutiontechniques-

are briefly presentedand tested.We will showthat-

the joint estimationof the imageandPSFcanlead,forsome" of them,to betterrestorationresultsandalsothat theGaussianF

assumption,proposedby someauthors,is only arough� approximation.Finally, for the classof the blind de-convolution techniquein which the exact supportof theobject. to be recoveredis needed,we proposea novelsupport-finding" algorithmexploitingalsotheparametersofthe-

aforementioneddistribution mixture estimationproce-dure.@

This paper is organizedas follows. Section II brieflydescribes@

superviseddeconvolution methods and recentblind�

deconvolutiontechniquesthat we will compare.InSec.�

3, we detail thedistributionmixtureparameterestima-tion-

procedure. Deconvolution experimental results onphantoms,� synthetic and real brain SPECT images aregiven� in Sec.4. Finally, a conclusionandperspectivesaregiven� in Sec.5.

2 Deconvolution

2.1G

Introduction

In8

our application,the degradationof a SPECTimagecanbe�

representedas the result of a convolution of the trueimagewith a blurring function H the

-PSFI plus� an additive

term-

to model the noise from the physical system.If theimaging+

systemis assumedto be linear andshift invariant,the-

degradationprocesscan then be expressedby the fol-lowing linear model:

gJ K xL ,& yM NPO fQ R

xL ,& yM S * hT U

xL ,& yM VPW nX Y xL ,& yM Z ,&where: gJ (

[xL ,& yM )

\is the degradedor blurred image, f

Q([xL ,& yM ) i

\s

the-

undistortedtrue image,hT

([xL ,& yM )

\is the PSFof the imag-

ing+

systemand nX ([xL ,& yM )

\is the additivecorruptingnoise.In

this-

notation, the coordinates(xL ,& yM )\

representthe discretepixel� locationsand* is the discretelinear convolutionop-erator( .

2.2 Supervised Deconvolution Methods

Assumingthat the blurring function hT

([xL ,& yM )

\is known, the

problem� is thento determinefQ

([xL ,& yM )

\given the observation

gJ ([xL ,& yM )

\. This one is generallyill-posed owing to the exis-

tence-

of the additive noise. This meansthat there is nounique= least square solution of minimal norm ] gJ (

[xL ,& yM )

\^ fQ

([xL ,& yM )

\* hT

([xL ,& yM )

\ _ 2. Besides,a small perturbationof thegiven� dataproduceslarge deviationsin the resultingsolu-tion.-

An appropriatesolutionmaybechosenthroughproperinitialization of the algorithm or by using deterministicprior� information about the original image ` viaD a regular-ization+

terma to-

makethe inversionwell behaved.In thisway: , iterativeapproacheshavebeenproposed.Their mainadvantages� arethat thereis no needto explicitly implementthe-

inverseof an operatorand the processmay be moni-tored-

asit progresses.Someof themarebriefly presentedinthis-

sectionandareoptimal; in theleastsquaresense,underconstraints 5

<or. not,9

>in the maximum likelihood b ML c

sense" 10 or. in the maximuma posteriori d MAPe sense." 6,1f

1

Mignotte et al.

2 / Journal of Electronic Imaging / January 2002 / Vol. 11(1)



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2.2.1 Van Cittert’s algorithm

Vg

an Cittert12 proposed� the following iterativealgorithm:

fQ ˆ

kh i

1 j xL ,& yM kPl fQ ˆ

kh m xL ,& yM nPoqpsr gJ t xL ,& yM uwv h

T xxL ,& yM y * f

Q ˆkh z xL ,& yM {}| ,&

where: ~ is a convergenceparametergenerallyset to 1. In

this-

iterativescheme,the estimatedimage fQ ˆ ([xL ,& yM )

\is modi-

fied at eachiteration by addinga term proportionalto theresidual� r� ([ xL ,& yM )

\ �gJ ([xL ,& yM )

\ �hT

([xL ,& yM )

\* fQ ˆ

kh ([ xL ,& yM ).

\2.2.2G

Landweber’s algorithm

Another)

iterative algorithm, proposed by Landweberet� al.,& 9> is provided by the minimization of the norm�gJ ([xL ,& yM )

\ �hT

([xL ,& yM )

\* fQ ˆ

kh ([ xL ,& yM )

\ � 2� and� leadsto the following it-eration,(fQ ˆ

kh �

1 � xL ,& yM �P� fQ ˆ

kh � xL ,& yM �P�q� h

T ��xL ,& � yM � * � gJ � xL ,& yM �

� hT �

xL ,& yM � * fQ ˆ

kh � xL ,& yM �}� .

This algorithm,also called the one stepgradient,leadstosimply" movetheestimatef

Q ˆ ([xL ,& yM )

\iteratively in thenegative

gradient� direction.

2.2.3G

Richardson–Lucy’s algorithm

The Richardson–Lucy’s � RL� algorithm� 10 is an iterativetechnique-

which attemptsto maximizethelikelihood of therestored� image by using the expectationmaximizationalgorithm� 14 when: the image is assumedto come from aPoissonprocess.This iterativealgorithmmaybesuccinctlyexpressed( as

fQ ˆ

kh �

1 xL ,& yM ¡P¢ fQ ˆ

kh £ xL ,& yM ¤ h

T ¥�¦xL ,& § yM ¨ * gJ © xL ,& yM ª

hT «

xL ,& yM ¬ * fQ ˆ

kh xL ,& yM ® .

In this form of notation,thedivision andthemultiplicationis donepoint-by-point.

2.2.4 Tichonov–Miller’s algorithm

This algorithm, also called the constrainedleast squaresrestoration,� consistsin choosingthe estimatef

Q ˆ ([xL ,& yM )

\that

minimizesthe following cost function:

fQ ˆ ¯ xL ,& yM °P± ar� gmin

f² ³µ´ gJ ¶ xL ,& yM ·P¸ h

T ¹xL ,& yM º * f

Q ˆ » xL ,& yM ¼¾½ 2�¿qÀÂÁ

cÃ Ä xL ,& yM Å * fQ ˆ Æ xL ,& yM Ç¾È 2� É ,&

where: the term cÃ ([xL ,& yM )

\* fQ ˆ ([xL ,& yM )

\generallyrepresentsa high

pass� filteredversionof theimagefQ ˆ ([xL ,& yM )

\. This is essentially

a� smoothnessconstraintwhich suggeststhat most imagesare� relatively flat with limited high-frequencyactivity, andthus-

it is appropriateto minimize the amountof high-passener( gy in therestoredimage.Onetypical choicefor cÃ (

[xL ,& yM )

\is the two-dimensionalÊ 2DË Laplacianoperator. The mini-mization! of the earlier equationleads,with the methodofsuccessive" approximation,proposedin Ref. 5, to the fol-lowing$

iterativeestimationschemefor fQ ˆ ([xL ,& yM ):

\

fQ ˆ

kh Ì

1 Í xL ,& yM ÎPÏ fQ ˆ

kh Ð xL ,& yM ÑPÒÔÓÖÕ gJ × xL ,& yM Ø * h

T ÙµÚxL ,& Û yM Ü

ÝßÞ Aà há â xL ,& yM ãPäqå A

àcæ ç xL ,& yM è}é * f

Q ˆkh ê xL ,& yM ëíì ,&

with: fQ ˆ

0î ï xL ,& yM ðPñÔòôó gJ õ xL ,& yM ö * h

T ÷µøxL ,& ù yM ú}û .

Aà

há ([ xL ,& yM )

\ ühT

([xL ,& yM )

\* hT

([ ý

xL ,& þ yM )\

andAà

cæ ([ xL ,& yM )\

aretheautocor-relation� functionsof h

T([xL ,& yM )

\andcÃ (

[xL ,& yM )

\, respectively. ÿ is

+called theregularizationparameterwhich mustbecarefullychosen for reliable restoration.This iteration converges if0� ��

(2[

/ � � max� )\ , where max� is+

the largesteigenvalueofthe-

matrix Ahá ([ xL ,& yM )

\ � �Acæ ([ xL ,& yM ).

\2.2.5G

Super resolution algorithm

AssumingPoissonphotondistributionin the image,thenaBayesian�

andMAP derivationhasbeenproposedby Huntet� al.11 This

�oneleadsto the following iterativescheme:

fQ ˆ

kh �

1 � xL ,& yM �� fQ ˆ

kh � xL ,& yM ��

exp( gJ � xL ,& yM �hT �

xL ,& yM � * fQ ˆ

kh � xL ,& yM �� 1.0 * h

T xL ,& yM ! .

2.2.6G

Molina’s algorithm

Following�

the Bayesianparadigm,Molina et� al. have"

pro-posed� to incorporateprior informationto theRL # maximumlikelihood$ restorationmethod.6

%In order to model the aE

priori& smoothness" of the imageto be recovered,this oneisdefined@

by the following conditionalautoregressivemodel:

PF' ( f)�* exp( +-, 1

2 . f t/ 0

I 1 2 N3 4

f5 .In this matrix-vectornotation,f is the true imageorderedlexicographically$

by stackingthe rows into a vector. 6 is+

the-

unknownregularizationparameter, matrix N3

is suchthatN3

i j 7 1 if cells i8

and� j9

are� spatialfour-neighbors: pixels� atdistance@

one; and� zerootherwise,andscalar < is just lessthan-

0.25.Theterm f= t/ ([ I> ? @

N3

)\f=

represents,� in matrix nota-tion,-

the sumof squaresof the valuesf=i minus! A times

-the

sum" of fifjB for neighboringpixels i

8and� j

9. Following theRL

method,which correspondsto MAP estimationwith a uni-form,

imageprior, Molina et� al. obtain. the following itera-tive-

scheme:

fQ ˆ

kh C

1 D xL ,& yM E�F�G kh H xL ,& yM I f

Q¯kh J xL ,& yM K�LNM 1 O�P k

h Q xL ,& yM RTS fQ ˆ

kh U xL ,& yM V

WhT XZY

xL ,& [ yM \ * gJ ] xL ,& yM ^hT _

xL ,& yM ` * fQ ˆ

kh a xL ,& yM b .

ckh ([ xL ,& yM )

\ d0�

correspondsto the classical RL restorationmethode we: recall that,in this form of notation,thedivisionand� themultiplicationaredonepoint-by-pointf . f

Q¯kh ([ xL ,& yM ) i

\s a

filtered versionof fQ

kh ([ xL ,& yM )

\in which eachpixel is the aver-

age� of its four-neighborspixels.

2.3 Blind Deconvolution Methods

Wheng

little is known aboutthe PSF, a solution for the de-blurring�

problemconsistsin achievinga blind deconvolu-

Comparison of deconvolution techniques . . .

Journalh




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tion-

technique.Blind imagedeconvolutionis the simulta-neousi estimationof the true imageand the PSFfrom theblurred�

observation.A commonly usedmethod for blinddeconvolution@

is by minimization of an error metric thatoptimizes. theform of therestoredimageandthePSF j or. itsinverse+ k

to-

fit the various constraintson the form of thesolution;" typically positivity andknown supportof the ob-jectl

to berecovered.Steepestdescentor conjugategradientmethodare generallyused to achieveoptimization.15,8 Asecond" method, usually called ‘‘grouped coordinatede-scent,’" ’ restoresthe image and the PSF separatelyin aniterative+

form. During eachcycle either the imageor thePSFis held staticwhile the other is updated,generallyus-ing one of the standarddeconvolutiontechnique.7,13

mIn

these-

methods,thatalternatebetweenrestorationof the im-age� and PSF, iterationsdo not necessarilyhaveto usethesame" restorationalgorithm. In this section, we describebriefly�

four recent blind deconvolutiontechniquesstem-ming! from thesetwo differentapproaches.

2.3.1 The iterative blind deconvolution method

The iterativeblind deconvolutionn IBD o method,proposedby�

AyersandDainty,7m

requires� that the imageandthe PSFbe�

non-negativewith known finite support p the-

supportisdefined@

as the smallestrectanglecontainingthe entire ob-jectl q

. After an initial guessis madefor the true image,thealgorithm� alternatesbetweenthe image and Fourier do-mains,! enforcing known constraints in each. The con-straints" arebaseduponinformationavailableaboutthe im-age� and the PSF. The image domain constraintscan beimposed+

by replacingnegativevaluedpixels within the re-gion� of supportwith zero and nonzeropixels outsidetheregion of support with the backgroundpixel value. TheFourierdomainconstraintinvolvesa Wiener-like filter forthe-

imageandthePSF. This filter allows to efficiently sup-press� noiseamplificationresultingfrom theill-posednatureof. the restorationproblem

Hr ˆ

kh s ut ,& uwv�x G

y zut ,& {}| Fk

h ~1* � ut ,& �w��

Fkh �

1 � ut ,& �w�� 2 � � /� �

Hkh �

1 � ut ,& �w�� 2 ,&

F� ˆ

kh � ut ,& �w�� G

y �ut ,& �w� H

r ˆkh �

1* � ut ,& �w��Hr ˆ

kh

1 ¡ ut ,& ¢w£�¤ 2 ¥ ¦ /� §

F� ˆ

kh ¨

1 © ut ,& ªw«�¬ 2 .

where: Hkh ([ ut ,& ),

\Gy

([ut ,& ® )

\, andF(

[ut ,& ¯ )

\representthe2D fast

Fourier�

transformof the PSF, the original image and thetrue-

image, respectively. Subscriptsdenote the iterationnumberof thealgorithmand(.)* is thecomplexconjugateo. f (.). The real constant ° representsthe energy of theadditive� noise and must be carefully chosenfor reliablerestoration.� Figure1 givesanoverviewof this scheme.Thealgorithm� is run for a specifiednumberof iteration,or untilthe-

estimatesbegin to converge. The major drawbackofthis-

method is its lack of reliability; the uniquenessandconver gencepropertiesare uncertainand the algorithm issensitive" to the initial imageestimateandcanexhibit insta-bility�

.

2.3.2G

The Biggs–Lucy’s algorithm

This method13 alternates� betweenrestoringthe imageandthe-

PSFusingthe RL algorithm ± by�

simply swappingvari-ables� h

T([xL ,& yM )

\and f

Q([xL ,& yM )

\in the RL iteration² . The image

and� the PSFestimatesaregiven by

hTˆ

kh ³

1 ´ xL ,& yM µ�¶ 1·

fQ ˆ

kh ¸ xL ,& yM ¹

hTˆ

kh º xL ,& yM »

¼fQ

kh ½¿¾ xL ,& À yM Á * gJ Â xL ,& yM Ã

hT

kh Ä xL ,& yM Å * f

Q ˆkh Æ xL ,& yM Ç ,&

fQ ˆ

kh È

1 É xL ,& yM Ê�Ë 1Ì

hTˆ

kh Í

1 Î xL ,& yM ÏfQ ˆ

kh Ð xL ,& yM Ñ

ÒhT

kh Ó

1 ÔZÕ xL ,& Ö yM × * gJ Ø xL ,& yM ÙhT

kh Ú

1 Û xL ,& yM Ü * fQ ˆ

kh Ý xL ,& yM Þ .

This methodrequiresa goodinitial guessfor thePSFandadif@

ferent numberof iterationsfor the imageand the PSF,expressed( by an asymmetricfactor which is necessarybe-cause imageandPSFestimatesconverge at differentrates.Dependingon the type of the imageand the natureof thePSF4

, this factor is generallydifferentandmustbecarefullychosen for reliablerestoration.

2.3.3 The non-negativity and support constraintsrecursive inverse filtering algorithm

The�

non-negativityand support constraintsrecursive in-verseD filtering ß NAS

?–RIFà technique

- 15 is applicableto situ-ations� in which anobjectof finite supportis imagedagainst

Fig. 1 IBD algorithm.

Mignotte et al.




PROO

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a� uniform or noisy backgroundwhich is our case.It com-prises� a 2D variablefinite impulseresponsefilter ut (

[xL ,& yM ) o

\f

dimension@

Ná

xuâ ã Ná

yuä with: the blurredimagepixels gJ ([xL ,& yM )

\as� input. The outputof this filter representsan estimateofthe-

true image fQ ˆ ([xL ,& yM )

\. This estimateis passedthrougha

nonlineari filter which uses a nonexpansivemapping toproject� the estimatedimageinto the spacerepresentingtheknowncharacteristicsof the true image.Thedifferencebe-tween-

this projectedimage fQ ˆ

NLå ([xL ,& yM )

\and f

Q ˆ ([xL ,& yM )

\is usedas

the-

error signal to updatethe variablefilter ut ([xL ,& yM )

\. Figure

2æ

givesan overviewof this scheme.The imageis assumedto-

be non-negativewith known support.The cost functionused= in this restorationprocedureis definedas

Jç è

ué ê�ë ì(íxâ ,yä )î ïñð

supò fQ ˆ 2� ó

xL ,& yM ô 1 õ sgn" ö fQ ˆ ÷ xL ,& yM ø¿ø2æ

ù ú(íxâ ,yä )î ûñü¯

supòýfQ ˆ þ xL ,& yM ÿ�� LB

� � 2 �� (íxâ ,yä )î ut xL ,& yM �� 1

2

,&

where: fQ ˆ ([xL ,& yM )

\ �gJ ([xL ,& yM )

\* ut ([xL ,& yM )

\, and sgn(f

Q)\ ��

1 if fQ �

0�

and� sgn(fQ

)\ �

1 if fQ �

0.� �

sup is thesetof all pixelsinsidethe

region� of support,and �¯ sup is+

the set of all pixels outsidethe-

region of support.The variable � in the third term isnonzeroonly whenLB is zero,i.e., thebackgroundcolor isblack.�

The third term is used to constrainthe parameteraway� from thetrivial all-zeroglobalminimumfor this situ-ation.� Authorshaveshownthat the earlierequationis con-vexD with respectto ut (

[xL ,& yM )

\, so that convergenceof the al-

gorithm� to the global minimum is ensured using theconjugate gradientminimizationroutine.15

2.3.4G

The You–Kaveh’s algorithm

This�

method8 attempts� to minimizea costfunctionconsist-ing of a restorationerror measureand two regularizationterms,-

onefor the imageandthe other for the blur

C � fQ ˆ ,& hTˆ �� ar� gminf²

,há � 1

2� � gJ � xL ,& yM �� h

T xL ,& yM ! * f

Q ˆ " xL ,& yM #%$ 2�& 1

2 ')( cÃ * xL ,& yM + * fQ ˆ , xL ,& yM -�. 2� / 1

2 021 aE 3 xL ,& yM 4 * hTˆ 5 xL ,& yM 6�7 2� 8 ,&

where: aE ([xL ,& yM )

\andcÃ (

[xL ,& yM )

\areregularizationoperator9 e.g.,(

a� high-passfilter suchas the Laplacian: . ; and� < are� theregularization� parametersthat control the tradeoff between

fidelity to the observationandsmoothnessof the estimatedimage+

andthe estimatedPSF. In orderto takeinto accountthe-

scaleproblem,inherentto this cost function, an alter-nating minimization using steepestdescentor conjugategradient� methodis proposed.Note that, usingsteepestde-scent" method,resultingiterativeproceduresarecloseto theiteration+

schemeproposedby Landweber= with: a regular-ization termfor theblur> for thealternaterestorationof theimageandthe PSF.

3?

Distribution Mixture Parameter Estimation

3.1?

Introduction

In this section,we presentan estimationprocedureallow-ing+

to estimatethe gray level statisticaldistributionassoci-ated� to eachclass@ also� calledthenoisemodelA of. a SPECTimage.We will showalsohow this informationcanbe ex-ploited� in theaforementionedsupervisedor blind deconvo-lution$

methods.To this end, we considera couple of randomfields ZB ([XC

,& Gy )\, whereG

y D([Gy

sE ,& sF G SH

)\

representsthe field of ob-servations" locatedon a lattice S

Hof. Ná

sites" sF I associated� tothe-

Ná

pixels� of the SPECTimageJ ,& andXC K

([XC

sE ,& sF L SH

)\

thelabel field M relatedto the N

áclass labelsXsE of. a segmented

SPECT�

imageN . Eachaforementionedlabel is associatedtoa� specificbrain anatomicaltissue;the ‘‘CSF’’ areadesig-natesthe region that is normally due to the lack of radia-tions.-

In this distributionmixtureparameterestimation,thisregion� designatesthe brain regionfilled with cerebrospinalfluidO P

without: blood flow and thus without radiationQ and�also� the areaoutsidethe brain region.The ‘‘white matter’’and� ‘‘gray matter’’ R brightest

�regionS are� associatedto a low

and� a higher level of blood flow, respectively.16 Each

Gy

sE takes-

its value in (0, . . .& ,255)& (256 gray levelsT ,& andeach( X

CsE in+

(e� 1 U ‘‘CSF’’, e� 2� V ‘‘white matter’’ , e� 3

W X ‘‘graymatter’’ Y .

In thefollowing, theparametersin uppercaseletterdes-ignate+

the randomvariableswhereasthe lower caselettersrepresent� the realizationsof theseconcernedrandomvari-ables.� In this estimationstep,the distribution of (X,& Gy ) i

\s

defined,@

first, by prior distribution PZ

X([xL )\, supposedto be

Markovianandsecondly, by the site-wiseconditionallike-lihoods$

PZ

G[

s\ /]X^

s\ ([ gJ sE /�xL sE )\ whose shapeand parameter_ (

íxâ s\ )î

depends@

on the concernedclasslabel xL sE ([gJ sE designates

@the

gray� level intensity associatedto site sF )\ . Finally, we as-sume" independencebetween each random variable G

ysE

given� XsE . The observableGy

is called the ‘‘incompletedata,’@

’ andZ`

the-

‘‘completedata.’’

3.2?

Estimation of the Distribution MixtureParameters for the Complete Data

Assuming)

the segmentationresult xL is+

known, the param-eters( of the gray level statisticaldistribution associatedtoeach( class,can thenbe easilycomputedwith the ML esti-matorof the completedata.

Experimentations

haveshownthatwe canrightly modelthe-

statisticalgraylevel distributionin thebackgroundor inthe-

CSFareaby a exponentiallaw a see" Ref. 3 andalsotheleft partof thehistogramreportedin Fig. 3b . This led us tothink-

that the noisein this regionis approximatelyPoisso-niani with the following statisticalgray level distribution:

Fig. 2 NAS–RIF algorithm.


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dG[ e gJ ; f2g�h 1i

exp( j gJ k,&

with: gJ l 0.�

Let now Gy m

([Gy

1 ,& ..., Gy

Mn ) b\

e M randomvari-ables,� independentand identically distributedaccordingtoa� ‘‘single’’ exponential law o G

[ ([gJ ; p )

\, and gJq (

[gJ 1 ,& ..., gJ M)

\a realizationof G

y. The ML estimatorofr

MLs for thecompletedatais simply themeanof thesample

gJ .17

In8

orderto describethe luminancewithin thewhite mat-ter-

and the gray matterregions,we model the conditionaldensity@

function for theseregionsby two Gaussianlaws.This�

assumptionof normality is a reasonableapproxima-tion-

due to the reconstructionphysical processused inSPECT�

imageryin which the gray level of a given pixel,hereinconsideredasa randomvariable,aresumsof manyvariablesD andthe ‘‘central limit theorem’’ canbe applied.17

The�

correspondingML estimatorof thecompletedata,for asample" gJ distributed

@accordingto a normal law, is defined

simply" by the empiricalmeanandthe empiricalvariance.

3.3?

Estimation of the Distribution MixtureParameters for the Incomplete Data

Wheng

the segmentationresult is unknown t i.e., the classlabel of eachpixel is not supposedto be knownu ,& the con-sidered" problem is more complex. In order to determinevxwzy|{

(íe} 1)î ,& ~ (

íe} 2� )î ,& � (

íe} 3� )î � ,& we use the iterative conditional

estimation( � ICE8 �

algorithm.� This procedure,describedindetail@

in Ref. 18 relieson an estimator� ˆ ([X,& Gy )

\with good

asymptotic� properties, like the ML estimator, for com-pletely� observeddata case.When X

�is+

unobservable,thisprocedure� startsfrom an initial parametervector � (0)

í �not

too-

far from the optimal one� and� generatesa sequenceofparameter� vectorsleadingto the optimal parameters,in theleastsquaresense,with the following iterativescheme:

� (íp� � 1) � 1

nX �� ˆ � xL (1)í ,& gJ �%� �� ˆ � xL (

ín� )î ,& gJ �� ,&

where: xL (íi)î ,& with i

8 �1,2, . . . ,nX ,& , are realizationsof X

�drawn@

according to the posterior distributionP�

X�

/�G ¡ xL /

�gJ ,& ¢ (

íp� )î £

. In order to decreasethe computationalload, we can take nX ¤ 1 without altering the quality of theestimation.( 19 Finally, we can use the Gibbs sampleralgorithm� 20 to

-simulaterealizationsof X

�according� to the

posterior� distribution. For the local aE priori model! of theGibbsF

sampler, we adoptan isotropic Pottsmodel with afirst orderneighborhood.21

�In this model,therearetwo pa-

rameters,� called ‘‘the clique parameters’’ denoted ¥ 1 ,& ¦ 2and� associatedto thehorizontalandverticalbinarycliques,respectively. § Cliques

�aresubsetsof siteswhich aremutual

neighbors.i 21 GivenF

this aE priori model,! the prior distribu-tion-

PX([xL )\

canbe written as

P�

X� © xL ª�« exp( ¬®¯

s° ,t± ²2³ st° ´ 1 µ·¶¹¸ xL s° ,& xL t± º¼» ,&

where: summationis taken over all pairs of neighboringsites" and ½ (

[.) is the Kroneckerdelta function. In order to

favor,

homogeneousregionswith no privileged orientationin the Gibbs samplersimulation process,we choose¾ st°¿ÁÀ

1 ÂÁÃ 2 Ä 1. Finally, Å (íp� Æ 1) is computedfrom Ç (

íp� )î

inthe-

following way:

• StochasticÈ

step: usingtheGibbssampler, onerealiza-tion-

xL is simulatedaccordingto the posteriordistribu-tion-

P�

X�

/�G ([xL /�gJ )\, with parametervector É (

íp� )î.

• Estimation step: the parametervector Ê (íp� Ë 1) is esti-

mated! with the ML estimatorof the completedatacorresponding to eachclass.Ì If N

á1 Í #(sÎ Ï S

Ð:xL s° Ñ e� 1)

\is thenumberof pixelsof

the-

CSF area, the ML estimator Ò ˆ (íe} 1)î of. Ó is

given� by Ref. 17: Ôˆ ([xL ,& gJ )

\ Õ(1[

/Ná

1)\ Ö

s° × SØ

:xâ sÙ Ú e} 1gJ s° .Û If

8Ná

2� Ü #(

ÝsÎ Þ S

Ð:xL s° ß e� 2

� )\ andNá

3W à #(

ÝsÎ á S

Ð:xL s° â e� 3

W )\pixels� are located in the white matter and graymatter regions, respectively, the correspondingML estimatorof eachclassis given by the em-pirical� meanand the empirical variance.For in-stance," for the white matter class,we have forã ˆ

(íe} 2� )î :ä ˆ å xL ,& gJ æèç 1

Ná

2

és° ê SØ:xâ sÙ ë e} 2 gJ s° ,&

ì ˆ 2í xL ,& gJ îèï 1ðNá

2� ñ 1ò ó

s° ô SØ

:xâ sÙ õ e} 2� ö gJ s° ÷ùø ˆ ú 2.• Repeatuntil convergenceis achieved;i.e., if û ˆ (

íp� ü 1)ý /� þ ˆ (

íp� )î,& we returnto stochasticstep.

Figure�

3 representstheestimateddistributionmixtureofthe-

SPECTimageshownin Fig. 4ÿ b� � . The threesite-wiselikelihoods$

P�

G

sÙ /�X�

sÙ ([ gJ s° /�e� kh ),\ k

� �1,2,3, � weighted: by theesti-

matedproportion � kh of. eachclasse� k

h )\ aresuperimposedtothe-

imagehistogram.Correspondingestimatesobtainedbythe-

estimationprocedure,requiringaboutten iterations,aregiven� in Table1.

Fig. 3 Image histogram of the picture reported in Fig. 4(b) (solidcurve) and estimated probability density mixture obtained with theICE procedure (dotted and dashed curves).

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3.4?

Determination of the Support and Stopping Rule

In thecaseof superviseddeconvolutiontechniqueswithoutregularization� term, suchas the Van–Cittert, the Landwe-ber�

, the RL, andthe superresolutionalgorithms,the itera-tive-

deconvolutionprocedureis generallymonitoredas itprogresses� and stoppedafter someiterations,generallybyvisualD inspection.This iterationnumbermaybevery differ-ent( for eachSPECTimageand is generallyrelatedto thebehavior�

of eachiterativemethodnearthe convergence.Infact, at somepoint of the iteration procedure,the solutionfit�

more to the noise than the image data.Therefore,forthese-

methods,the processhasto be stoppedat the pointwhere: thereis a balancebetweenthe fit to the imagedataand� theamplificationof noise.To this end,in orderto stopautomatically� thesealgorithmsbeforethe amplificationofthe-

noise, we proposeto computethe parametersof thedistribution@

mixture of fQ ˆ

kh regularly� , namelyevery k

�itera-+

tions-

(k�

depends@

on the speedof convergenceof the con-sidered" deconvolutionmethod� . If theparameterassociatedto-

thebackgroundnoise � i.e.,+ �

)\

is abovea fixed threshold,we: decideto stop the procedure.Of course,this thresholdhasto be fixed empirically like the iterationnumber. Nev-ertheless,( contrary to the iteration number, this thresholddoes@

not dependof the adoptedunregularizedmethodorthe-

speedof convergenceof eachmethodas well as theused= SPECTimages.Besides,it doesnot requirea visualinspection,+

for eachiteration of the deconvolutionproce-dure,@

that can be cumbersomeand unreliablefor an auto-matic deconvolutionof a setof SPECTimages.

For thesuperviseddeconvolutionmethodsusinga regu-larization$

term e.g.,( the Tichonov–Miller ’s algorithm , o& rprior� information � e.g.,( the Molina’s algorithm� ,& the termi-nation criteria consistssimply in stopping the algorithmwhen: thesolutionis stable.Neverthelessthesemethodsre-quire a regularizationparameterwhich must be chosen

carefully for reliablerestoration.This parametercanbealsoderived@

efficiently from the proposednoisemodel estima-tion-

procedure.In8

the caseof blind deconvolutiontechniques,in whichthe-

rectangularsupport of the object to be restored isneededand unknown,we can also efficiently exploit theparameters� of the distributionmixture of the input imagegJby�

adoptingthefollowing strategy;we assumethat therow�i � SÐ

contains the object � to-

be restoredif we canfindtwo-

consecutivesites �� i for which

P� �

gJ i j /� �

CSF� ��

P� �

gJ i j /� �

white: matter�� ,&where: the subscriptsi

8,& j refer� to the pixel locatedat the i

8th-

row� and the j th-

columnandgJ to-

the luminance.We adoptan� identical reasoningfor the column and the object sup-port� is then accuratelydeterminedby the set of pixels gJ i j

which: belongto a row ! i and� a column " j# containing the

object. $ . Figure 4 displaysexamplesof rectangularsup-port� determinationfor somecross-sectionalbrain SPECTimages.+

A moreaccuratesupportcould be given by an un-supervised" Markovian segmentationbasedon parametersgiven� by the ICE procedure.Finally, let us recall also thatfor theseblind deconvolutiontechniques,thereis no needto-

implementa stoppingrule and convergenceis reachedwhen: the estimatedPSFandimagearestable.

4 Experimental Results

The�

effectivenessof eachdeconvolutionmethodwastestedon. severalcross-sectionalphantoms,syntheticandreal hu-

T%

able 1 Estimated parameters for the picture reported in Fig. 4(b).& stands for the proportion of the three classes within the SPECTimage. ' are the exponential law parameter. ( and ) 2

*are the

Gaussian law parameters.

ICE procedure

+(,e1)-final

.0.52(

, /)- 11(, 0

)-1

(,e2)-final 0.26(

, 2)- 100(

, 3)- 648(

, 4 2)-5

(,e3)-final

.0.22(

, 6)- 172(

, 7)- 383(

, 8 2)-

Fig. 4 Examples of support determination for some cross-sectional brain SPECT slices.

Fig. 5 Original PSF defined as a two-dimensional Gaussian distri-bution with variance 9 2 : 1.5 in a 7 ; 7 support.


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manbrain SPECTimagesof 64 = 64>

pixels sizewith 256gray? levels.Thosepresentedin this sectionareonly a fewexamples.@

Except for the Tichonov–Miller ’s algorithm, the initialestimated@ imageof theseiterative schemesis the originalinput image A i.e., f

B ˆ0C (D xE ,F yG )

H IgJ (DxE ,F yG )

H] . Besides,exceptfor the

NASK

–RIF blind deconvolutiontechnique,the original PSFestimate,@ aL priori fixed

Mfor the superviseddeconvolution

methods,N is approximatedfor the real SPECTimagesby atwo-dimensionalO

Gaussian distribution P i.e., hQˆ

0C (D xE ,F yG )

HRTS

xâ ,yä (D U 2V)H] with varianceW 2

V X1.5 Y i.e., about3 pixels of

widthZ at half maximumasshownin Fig. 5[ . This variancevalue\ hasbeenchosenempirically, for eachset of decon-volution\ experimentspresentedin this section,in order toobtain] the best supervisedrestorationresults.The initialinversefinite impulseresponseFIR_ filter requiredby theNASK

–RIF algorithm is simply the Kronecker deltafunction15 and` we haveusedacb 0

dbecausethebackground

of] the SPECTimagesis not completelyblack. Finally, pa-rameterse and` f ,F usedin theYou andKaveh’s algorithm,

are` givenby theestimationmethodproposedby theauthorsin Ref. 8. In order to objectivelycomparethe spatialreso-lutiong

improvementsandthecontrastenhancementbetweentheO

original andestimatedimagesaswell asthe resolutionimprovementof thesedifferent restorationapproaches,wehavestretchedthehistogramof theestimatedimageat con-ver\ gence h i.e.,

ifB ˆ

final(DxE ,F yG )

H] in order to get the samemean

value\ asthe original input imagegJ (DxE ,F yG ).

HForj

the unregularizedsuperviseddeconvolutionmeth-ods,] the terminationcriteria is given by the stoppingstrat-egy@ presentedin Sec.3.4 k seel Table 2m . For the blind de-convolutionn methodsrequiring the exact support of theobject] to berestored,deconvolutionresultsarebasedon thesupport-findingl algorithmpresentedin this samesection.

Thecomputationalcostfor a SPECTimageandfor eachsupervisedl or blind deconvolutionprocedureis indicatedinToable3.

Table 2 Iteration number for each supervised deconvolutionmethod as chosen by the proposed stopping rule. Respectively, theVan Cittert (VC), the Landweber (LW), the Richardson–Lucy (RL),thep

super resolution (SR), and the Molina’s (MO) algorithms.

Iteration number

VC LW RL TM SR MO

10 4 200 50 100 10

Table 3 Computational cost for each deconvolution method. Re-spectively, the Van Cittert (VC), the Landweber (LW), theRichardson–Lucy (RL), the super resolution (SR), the Molina (MO),thep

IBD, the Biggs–Lucy (BL), the NAS–RIF and finally, the You–Kaveh’s (YK) algorithm. Results are obtained on a standard Sun-Sparc 2 workstation and are expressed in seconds.

Computational cost

Supervised methods Blind methods

VC LW RL TM SR MO IBD BL NAS–RIF YK

3 1 18 8 18 2 30 120 129 345

Fig. 6 Examples of brain SPECT image deconvolutions. (a) Original image. (b)–(g) Supervised de-convolution methods, respectively; (b) Van Cittert, (c) Landweber, (d) RL, (e) Tichonov–Miller, (f) superresolution, (g) Molina’s algorithm. (h)–(k) Blind deconvolution methods, respectively; (h) IBD, (i)Biggs–Lucy, (j) You–Kaveh, (k) NAS–RIF algorithm.

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Figures6 presentsexamplesof brain SPECTimagede-convolutionsn obtainedby thesedifferentmethods.Figure7displaysq

the PSFestimatedby theYou–Kaveh’s algorithm.Amongstthesuperviseddeconvolutionschemes,theVan

Cittert’r

s methodseemsto improveslightly theresolutionoftheO

original SPECT image. The Landweber’s algorithmseemsl to give quitegoodresultsrelatively to its implemen-tationO

simplicity andits low computationalcomplexity. TheTichonov–Miller and the Molina’s algorithms,which im-poses aL priori smoothnessl of the true imagein an effort tocontroln noise,seemto fail to detectall detailsand singu-laritiesof the trueundistortedimage.In fact, theusedpriormodelseemsto bemuchtoo simpleto modelaccuratelyalltheO

propertyof the true unblurredimage.The RL and thesuperl resolutionalgorithmsgivesimilar resultsandallow toimprovei

slightly the spatialresolutionof theseSPECTim-ages.`

Amongsttheblind deconvolutiontechniques,the IBD isunablet to converge for 200 iterationsandmore.The algo-rithm fails to producea reliableestimateof the true imageforu

all thepresentedSPECTimages.Deconvolutionexperi-mentsN with theexactrectangularsupportof theobjectto berestored,various initial conditionsand different noisepa-rameterv values w produceds poor results as well. TheBiggsx

–Lucy andtheYou–Kaveh’s algorithmsseemto givequitey goodcontrastenhancementresultsbut alsoshowun-desirableq

artifactsall around z and` maybeinside{ theO

objecttoO

be restored.In addition, thesetechniquesremainsensi-tiveO

to theinitial PSFgivento thedeconvolutionprocedure.

A randominitial guessfor the PSFor an initial Kroneckerdeltaq

function lead to poor results.Let us note that thesemethodsN arenot ensuredto converge to the global minima|and` remain highly sensitiveto the initial conditions.Fi-nally, theNAS–RIF techniqueseemsto convergeto a goodestimate@ of the solution without aL priori information

ior

good? initial guessaboutthe PSF. Figure8 givesexamplesof] five cross-sectionalSPECTimagedeconvolutionsof hu-manbrain given by the NAS–RIF algorithm.

Theo

effectivenessof thesedeconvolutiontechniquesisalso` testedon a realSPECTphantom} i.e.,

ia physicalplexi-

glas? headphantomfilled with radioactivematerialandmea-suredl by a SPECTsystem~ for which the groundtruth ofthisO

segmentedphantom is exactly known and thus forwhichZ the performanceof eachdeconvolutionmethodcanthenO

beobjectivelyjudged.Figure9 presentsanexampleofdeconvolutionq

results,on this SPECTphantom,obtainedbytheO

different aforementioneddeconvolutionmethods.Wecann easilynoticethat this SPECTvolumeis lessnoisy andlessblurred than the real humanbrain SPECTslice previ-ously] presentedandprocessed� due

qto severalfactorssuch

as` a differentdoseof radioactiveisotopescontainedin eachuniformt region of this SPECTphantom,a longer acquisi-tionO

time, the stillnessof this simulatedbrain during theSPECT�

process,a reducedattenuation,etc.� . In order tofullyu

assessthe successof this restorationprocedure,weuset the specific evaluation criteria proposedin Ref. 4,based�

on the estimationof the threefollowing measures:

i.i

First, the averagecontrastof the image,definedby�

C � (1D �

m| 2 /�m| 3� )H , where m| 2 and` m| 3

� are` themeanof the pixel value in the white matterandgray? matterarea,respectively.

ii. � ii � Second,�

the image mottle M 2 in the whitematterN region,characterizedby taking the ratio oftheO

standarddeviation � 2 of] pixel valuesin thisarea` to the meanm| 2

V .

iii. � iii � Third, theimagemottleM 3� in thegraymatter

area.`Thesetwo last parametersallow to measurethe ampli-

fication of the noiseand/ormeasurethe presenceof unde-sirablel artifacts that can be createdby the deconvolutionprocedures in a uniform regionof the real SPECTphantom�thusO

with ideally uniform radioactiveactivity� . Due to thedifq

ferenceof proportionof pixels belongingto eachbrain

Fig. 7 Estimated PSF by the You–Kaveh’s algorithm in a 7 � 7 sup-port.

Fig. 8 Examples of human brain cross-sectional SPECT image deconvolutions given by the NAS–RIFalgorithm. Top: five consecutive real cross-sectional SPECT slices. Bottom: deconvolution results.


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anatomical` tissue, we consider the total mottle measuregiven? by M

� ��2M�

2 �� 3� M� 3

� ,F with � 2 and` �3� designatesq

theproportions of pixel belongingto the white matterandgraymatterarea,respectively. A reliableSPECTimagerestora-tionO

techniquewill thenallow to enhancethecontrastof theimagei

with little increasein mottle,i.e.,without amplifyingtooO

much the noiseand/orwithout creatingfalse artificialfeaturesu �

technicallyO

, an increaseby a factor of 10%–15%of] theoriginal mottleof theimageremainsacceptableif thecontrastn enhancementis significantly increased� .4� Due totheO

difference of thickness between the cross-sectionalslicesl of therealandsegmentedphantom,theseabovemen-tionedO

measures are estimated on the whole three-dimensionalq �

3D� �

phantoms after this one has beenregisteredv 22

Von] the groundtruth of the segmentedphantom

volume\ � seel Fig. 10 wheresomeconsecutiveslicesof thesegmentedl phantomareshown� . Table4 givesthe contrastand` imagemottle for eachdeconvolutiontechniqueappliedon] this SPECTphantom.

Amongst

the superviseddeconvolution schemes,theLandweber¡

’s algorithmallows to increasesignificantly thecontrastn of the image but at cost of an unacceptablein-

creasen of the mottle of the image ( ¢ 33.0%�

of mottle£ .Deconvolution¤

results,obtainedon this SPECTphantom,by�

the Van Cittert, the RL, the Tichonov–Miller and thesuperl resolutionalgorithmarenearlysimilar; theyallow toobtain] a goodcontrastenhancementbut alsopresentsomeartifacts,` visible all aroundthe object to be restored.Moli-na’s algorithm gives the best resultsfor this SPECTvol-ume;t i.e., a good contrastenhancementwith only a littleincreasei

of the mottle. Experimentshave shown that thismethodN is well suitedfor cross-sectionalSPECTimagesnottooO

blurred.Amongst the blind deconvolutiontechniques,the IBD

algorithm` fails to producea reliable estimateof the trueimage.i

TheYou–Kaveh’s algorithmallows oneto increasetheO

contrastof the image but this techniquealso createsundesirablet artifacts and/or an unacceptableamplificationof] the noise ( ¥ 30.2%

�of mottle¦ . Deconvolution result

given? by the Biggs–Lucy’s algorithmis very poor. Finally,theO

NAS–RIF blind deconvolutiontechniqueproducecon-trastO

enhancementresultasgoodasthebestsupervisedde-convolutionn technique§ i.e.,

ithe Molina’s algorithm along`

withZ the slightestincreaseof the mottle amongstthe con-

Fig. 9 Examples of phantom SPECT image deconvolutions. (a) Original image. (b)–(g) Superviseddeconvolution methods, respectively; (b) Van Cittert, (c) Landweber, (d) RL, (e) Tichonov–Miller, (f)super resolution, (g) Molina’s algorithm. (h)–(k) Blind deconvolution methods, respectively; (h) IBD, (i)Biggs–Lucy, (j) You–Kaveh, (k) NAS–RIF algorithm.

Fig. 10 Examples of some consecutive cross-sectional slices of the segmented phantom (groundtruth).p

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sideredl supervisedandblind deconvolutiontechniques.Ex-perimentss haveshownthat this methodis well suited forboth�

very blurredSPECTslicesandalsoin thecaseof lessblurred�

SPECTimages.W©

e havealsotestedthe effectivenessof thesedeconvo-lutiong

techniqueson a cross-sectionalslice of a syntheticSPECT�

volume. In order to simulate at best the typicalcharacteristicsn of realhumanbrainSPECTimages,we haverecreatedthreehomogeneousregionsandaddedthe corre-spondingl noisefor eachones,accordingto the gray levelstatisticall distribution alreadyestimatedon a real human

brain�

SPECTslice ª seel the distribution mixture presentedini

Fig. 3 and parametersgiven in Table 1« . We havealsoadded` a 3D Gaussianblur in orderto simulatethe3D scat-teringO

of the emittedphotons.Figure11 showsthe groundtruthO

of a segmentedsyntheticslice, the syntheticSPECTslice,l andfinally the deconvolutionresultsobtainedby ourdifq

ferentrestorationmethods.Amongstthesuperviseddeconvolutionschemes,theVan

Cittertr

and the Landweber’s algorithmgive quite good re-sultsl althoughat costof a slight amplificationof the noiseini

eachuniform region of the syntheticSPECTslice. The

Table 4 Contrast and image mottle enhancement from the original input image and for each decon-volution method (enhancement expressed in percentage). Respectively; the Van Cittert (VC), theLandweber (LW), the Richardson–Lucy (RL), the super resolution (SR), the Molina (MO), the IBD, theBiggs–Lucy (BL), the NAS–RIF and finally, the You–Kaveh’s (YK) algorithm.

Supervised Blind

VC LW RL TM SR MO IBD BL NAS–RIF YK

¬C 34.5% 62.0% 35.1% 28.7% 35.1% 28.0% 6.1% 12.2% 24.7% 29.0%®M 15.3% 33.0% 14.9% 13.5% 14.9% 12.6% 3.8% 13.8% 11.5% 30.2%

Fig. 11 Examples of synthetic SPECT image deconvolutions. (a) Top: ground truth of the segmentedsynthetic slice. Bottom: synthetic SPECT slice. (b)–(g) Supervised deconvolution methods, respec-tively;p

(b) Van Cittert, (c) Landweber, (d) RL, (e) Tichonov–Miller, (f) super resolution, (g) Molina’salgorithm. (h)–(k) Blind deconvolution methods, respectively; (h) IBD, (i) Biggs–Lucy, (j) You–Kaveh,(k) NAS–RIF algorithm.


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RL, the Tichonov–Miller and the super resolution algo-rithm showclearlysomeartifactsall aroundtheobjectto berestored.v Deconvolutionresultgiven by the Molina’s algo-rithmv is very poor for this synthetic image; experimentshaveshownthat this methodis not well suitedfor highlyblurred�

image.Amongst

the blind deconvolutiontechniques,the IBDdoesq

not converge.TheYou–Kavehandthe Biggs–Lucy’salgorithm` showclearly falseandundesirableartificial fea-turesO

createdby the iterative blind deconvolutionproce-dure.q

Onceagain,theNAS–RIF techniqueproducesa rela-tivelyO

goodrestorationresult.Finallyj

, in orderto attesttheeffectivenessof our param-eter@ estimationprocedure,we havealso comparedthe de-convolutionn results, on SPECT images,obtainedby ex-ploitings or not the optimal parametersgiven by ourdistributionq

mixture estimationmethod.Figure12 showsthe resultsof the NAS–RIF algorithm,

on] a cross-sectionalsliceof a phantomSPECTvolume,forrespectivelyv , the rectangularsupportestimatedby our pro-ceduren ° a` rectangleof 35 ± 43 pixels sizein this case,seeFig.j

4² b� ³µ´ ,F an overestimatedsupportsize (37 ¶ 45·

pixelssizel ¸ ,F andfinally an underestimatedsupportsize(33 ¹ 41

·pixelss sizeº . Theseexperimentsclearlyshowthat the resto-ration, even for an overestimationor underestimationof10%, produceslessgood deconvolutionresultscomparedtoO

the one obtainedby exploiting the rectangularsupportestimated@ by our estimationprocedure.In the caseof anoverestimated] » and` incorrect¼ supportl size,theblind decon-volution\ techniquedoesnot improve sufficiently the reso-lutiong

of the SPECTimage ½ cf.n Fig. 12¾ cn ¿µÀ . Moreover, intheO

caseof an underestimatedsupportsize,someexternalregionsof the brain aremissing Á cf.n Fig. 12Â dq ÃÅÄ . The otherblind�

deconvolutionalgorithms producepoor deconvolu-tionO

resultsfor underestimationor overestimationof sup-

ports sizeaswell. This leadsus to think thata goodestima-tionO

of the supportsize is given by our algorithmandthisaccurate` estimationis essentialin orderto rightly constraintheO

ill-posednatureof the blind deconvolutionalgorithms.As for unregularizedsuperviseddeconvolutionmethods,

Fig.j

13 showsthe resultsof NÆ Ç

4·

iterationsof the Land-weberZ ’s algorithm È as` chosenby theproposedstoppingruleand` by setting ÉËÊÍÌ 0

C whereZ Î0C is theparameterassociated

toO

the backgroundnoisefor the original input imageÏ and`theO

deconvolutionresult given by NÆ Ð

2 and NÆ Ñ

6>

itera-tionsO

of this iterativealgorithm.For NÆ Ò

2,Ó

the deconvolu-tionO

result is poor Ô theO

brain remainsblurredÕ and` for NÆ

Ö 6>

the amplificationof noisebeginsto be too important.Therefore,o

NÆ ×

4· Ø

chosenn by our algorithmÙ seemsl to be agood? compromisesolution. In fact, the contrastenhance-mentN increaseswith thenumberof iterationuntil N

Æ Ú4·

andkeeps nearly constant after N

Æ Û4 Ü at` around Ý CÞTß 60%

>). For N

Æ à6>

and over, the mottle of the imagebegins�

to significantly increaseto unacceptablelevels.

5á

Conclusion

In this paperwe haveshownthat a deconvolutionproce-dureq

noticeablyimprovesthe spatial resolutionof humanbrain�

SPECTimagesand can be a greathelp to facilitatetheirO

interpretationby the nuclearphysician.The proposeddistributionq

mixtureestimationprocedureallowsefficientlytoO

give a reliable terminationcriteria for the unregularizediterativei

deconvolutiontechniquesor to accuratelydeter-mineN the rectangularsupportof the object to be restoredwhenZ this oneis neededby someblind deconvolutiontech-niques.This estimationprocedureis quite generalandcanbe�

usedfor otherapplicationssuchasanunsupervisedMar-kovianâ

segmentationof brain SPECTimagesinto differentanatomical` tissues, to create realistic synthetic brain

Fig. 12 (a) Cross-sectional slice of a phantom SPECT volume, (b) deconvolution with the optimalparameters given by our distribution mixture estimation method [a rectangle of 35 ã 43 pixels size inthisp

case, see Fig. 4(b)], (c) overestimated support size (37 ä 45 pixels size), (d) underestimatedsupport size (33 å 41 pixels size).

Fig. 13 (a) Cross-sectional slice of a phantom SPECT volume. (b) N æ 4 iterations of the Landweber’salgorithm (as chosen by the proposed stopping rule). (c) N ç 2 iterations of the Landweber’s algorithm.(d) N è 6 iterations of the Landweber’s algorithm.

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SPECT�

imagesor to give relevantinformation in order toclassifyn thesebrain imagesinto differentpathologyclasses.Amongst

existingdeconvolutiontechniques,theNAS–RIFalgorithm` performs better than other deconvolutionschemesl for SPECTimagerestoration.This techniquecanbe�

efficiently combinedwith our estimationproceduretofindM

the supportof the object to be restoredandyield verypromisings resultswithout aL priori assumption` on thenatureof] the blurring function or for all type of SPECTimagesémore or less blurredê . Finally, let us note also that this

methodN can efficiently be extendedin order to take intoaccount` theinter-sliceblur inherentto this 3D imagingpro-cess.n This can be doneby consideringa 3D variableFIRfilter with a blurredSPECTvolumepixels as input.23

V

Acknowledgment

Theo

authorsthankINRIA ë Institutì

Nationaldela Rechercheen@ Informatiqueet Automatique,Franceí for

ufinancialsup-

ports of this work î postdoctorals grantï .

References

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volution of multiframeimages,’’ Pr�

oc. SPIE 3461�

, 33 � 1998� .14. A. P. Dempster, N. M. Laird, andD. B. Rubin,‘‘Maximum likelihood

from incompletedatavia theEM algorithm,’’ Royal Statistical Society� �, 1–38 ! 1976" .

15. D. KundurandD. Hatzinakos,‘‘Blind imagerestorationvia recursivefiltering usingdeterministicconstraints,’’ in Pr

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On Acoustics, Speech, and Signal Processing, Vol. 4, pp. 547–549#1996$ .

16. D. C. Costaand P. J. Ell, Brain Blood Flow in Neurology and Psy-chiatry, P. J. Ell, Ed. % 1991& .

17. S. Banks,Signal�

Processing, Image Processing and Pattern Recogni-tion, PrenticeHall, EnglewoodClif fs, New Jersey' 1990( .

18. F. SalzensteinandW. Pieczinsky, ‘‘UnsupervisedBayesiansegmenta-tion)

usinghiddenmarkovianfields,’’ in Proc. Int. Conf. On Acoustics,Speech,�

and Signal Processing, pp. 2411–2414 * May 1995+ .

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Max Mignotte received his DEA (post-graduate degree) in digital signal, imageand speech processing from the INPG Uni-versity@ , France (Grenoble), in 1993 and hisPhD degree in electronics and computerengineering from the University ofBretagne Occidentale (UBO) and from thedigital signal laboratory (GTS) of theFrench Naval Academy, France, in 1998.He was an INRIA postdoctoral fellow at theUniversity of Montreal (DIRO), Canada

(Quebec), from 1998 to 1999. He is currently with DIRO at the Com-puter Vision and Geometric Modeling Lab as an Assistant Professorat the University of Montreal. His current research interests includestatistical methods and Bayesian inference for image segmentation(with hierarchical Markovian, statistical templates, or active contourmodels), parameters estimation, tracking, classification, deconvolu-tion,A

and restoration issues in medical or sonar imagery.

Jean Meunier received his BSc degree inphysics from the University of Montreal in1981, his MScA degree in applied math-ematics in 1983, and his PhD in biomedicalengineering in 1989 from Ecole Polytech-nique de Montreal. In 1989, after postdoc-toralA

studies at the Montreal Heart Institute,he joined the department of computer sci-ence at the University of Montreal, wherehe is currently a full professor. He is also aregular member of the Biomedical Engi-

neering Institute at the same institution. His research interests are incomputer vision and its applications to medical imaging. His currentresearch focuses on motion assessment and analysis in biomedicalimages.

J.-P. Soucy went to the College Stanislasde Montreal, and received his ‘‘baccalau-reat francais’’ (with honors) from the Acad-emie de Caen in 1975. He then went on totheA

Universite de Montreal Faculty of Medi-cine, graduating first in his class in 1980.Nuclear medicine training then followed attheA

Universite de Montreal and McGill Uni-versity, with certifications from the Physi-cians’ College of the Province of Quebec,from the Royal College of Physicians and

Surgeons of Canada, and from the American Board of NuclearMedicine in 1984. He then spent a year as a fellow of the McLaugh-lin Foundation in Service Hospitalier Frederic-Joliot, Orsay, France,doing cerebral blood flow studies in stroke and dementia patientsunder the supervision of Dr. Claude Raynaud. Dr. Soucy has prac-ticedA

Nuclear Medicine at the Hopital Notre-Dame, Montreal, laterincorporated into the Universite de Montreal Health Sciences Center(CHUM), from 1985 to 2000. He is still on staff at that hospital butsince September of 2000 he has practiced at The Ottawa Hospital.He was Chief of the Department of Nuclear Medicine at CHUM from1997 to 2000, and Chief of Residency training in nuclear medicine attheA

University of Montreal from 1994 to 2000. He is a professor ofmedicine, division of nuclear medicine, at the Faculty of Medicine oftheA

Ottawa University, and a professor of radiology (clinical) at theFaculty of Medicine of the University of Montreal.


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Christian Janicki received his BSc degreein physics from the University of Montrealin 1981, his MSc degree in physics in1982, and his PhD in 1988. He joined theCentre Canadien de fusion magnetique

A(CCFM) in 1988, where he worked on x-raytomographicA

imaging of the tokamakplasma, heat transport, and bremsstrah-lung emission from relativistic electrons.He joined the biomedical physics group attheA

Centre Hospitalier de l’Universite deMontreal (CHUM) in 1995. His current research interests include 3DSPECT imaging, beta and gamma dosimetry modeling, and intra-vascular@ brachytherapy.

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