MedicalImaging1996:ImageProcessing,M.H.Loew,ed.
ExploringthereliabilityofBayesianreconstructions
KennethM.HansonandGregoryS.Cunningham∗
LosAlamosNationalLaboratory,MSP940LosAlamos,NewMexico87545USAkmh@lanl.govcunning@lanl.gov
ABSTRACT
TheBayesianapproachallowsonetocombinemeasurementdatawithpriorknowledgeaboutmodelsof
realitytodrawinferencesaboutthevalidityofthosemodels.Theposteriorprobabilityquantifiesthedegreeofcertaintyonehasaboutthosemodels.Weproposeamethodtoexplorethereliability,oruncertainty,ofspecificfeaturesofaBayesiansolution.Ifonedrawsananalogybetweenthenegativelogarithmoftheposteriorandaphysicalpotential,thegradientofthispotentialcanbeinterpretedasaforcethatactsonthemodel.Asmodelparametersareperturbedfromtheirmaximumaposteriori(MAP)values,thestrengthoftherestoringforcethatdrivesthembacktotheMAPsolutionisdirectlyrelatedtotheuncertaintyinthoseparameterestimates.Thecorrelationsbetweentheuncertaintiesofparameterestimatescanbeelucidated.
Keywords:Bayesiananalysis,uncertaintyestimation,reliability,geometricmodels
1.INTRODUCTION
Bayesiananalysisprovidesthefoundationforarichenvironmentinwhichtoexploreinferencesaboutmodelsfrombothdataandpriorknowledgethroughtheposteriorprobability.Inanattempttoreduceananalysisproblemtoamanageablesize,theusualapproachistopresentasingleinstantiationoftheobjectmodelas“theanswer”,typicallythatwhichmaximizestheposterior(theMAPsolution).However,becauseofuncertaintiesinthemeasurementsand/orbecauseofalackofsufficientdatatodefineanunambiguousanswer(intheabsenceofregularizingpriors),1theremaybenouniqueanswertomanyrealanalysisproblems.Rather,innumerablesolutionsareallowable.Ofcourse,somesolutionsaremoreprobablethanothers.AprominentfeatureoftheBayesianapproachisthatitprovidestheprobabilityofeverypossiblesolution,which,inasense,ranksvarioussolutions.Theabilitytoascertaintheuncertaintyorreliabilityoftheanswerremainsapressingissue,particularlywhenthenumberofparametersinthemodelislarge.Thetraditionalapproachofspecifyinguncertainty,throughthecalculationofthecovarianceintheparameters,whichincludesthecorrelationbetweentheuncertaintiesinanytwoparameters,doesnotprovidemuchinsight.Skillingetal.2suggestedthatonedisplayasequenceofdistinctsolutionsdrawnfromtheposteriorprobabilitydistribution.Byviewingthisrandomwalkthroughtheposteriordistributionbymeansofavideoloop,onegetsafeelingfortheuncertaintyinaBayesiansolution.However,thecalculationalmethodusedinthatworkwasbasedonaGaussianapproximationoftheposteriorprobabilitydistributionintheneighborhoodoftheMAPsolution.LaterSkillingmadeprogressindealingwithnon-Gaussiandistributions.3WhiletheprobabilisticdisplayofSkillingetal.providesageneralimpressionoftheoveralldegreeofvariationpossibleinthesolution,wedesireameanstoprobetheuncertaintyinthesolutioninamoredirectedmanner.
∗
SupportedbytheUnitedStatesDepartmentofEnergyundercontractnumberW-7405-ENG-36.
WeproposeatechniquetoprobethereliabilityoftheMAPsolutioninafashionthatallowsonetoaskquestionsofparticularinterest.Theapproachwesuggestmakesuseofananalogybetweenthenegativelogarithmoftheposteriorandaphysicalpotential.TheuncertaintyofaparticularchangeoftheMAPsolutionisrevealedinatactilewayasaforcethattendstopullthesolutionbacktowardtheMAPsolution.Correlationsbetweentheperturbedsetofparametersandtheremainingparametersinthemodelarealsobroughttolight.ThisinnovativeBayesiantoolistangiblydemonstratedwithinthecontextofgeometrically-definedobjectmodelsusedfortomographicreconstructionfromverylimitedprojectiondata.
2.TRADITIONALAPPROACHTOUNCERTAINTY
Bayesiananalysisisbasedontheposteriorprobabilityp(a|d)ofaparametricmodel,wherethemodelparametersarerepresentedbythevectoraandthedatabyd.Theposteriorp(a|d)incorporatesthedatathroughthelikelihoodp(d|a),i.e.theprobabilityoftheobserveddatagiventheparameters,andpriorinformationthroughapriorprobabilityontheparametersp(a).Bayes’slawgivestheposteriorasp(a|d)∝p(d|a)p(a).Itisconvenienttodealwiththenegativelogarithmoftheposterior:
−log[p(a|d)]=ϕ(a)=Λ(a)+Π(a)+C,
(1)
whereΛandΠarethenegativelogarithmsofthelikelihoodandtheprior,respectively,bothofwhichdependontheparameters,andCisanormalizationconstantthatdoesnotdependontheparameters.ThemosttypicaluseofBayesiananalysisistofindtheparametervaluesthatmaximizetheposterior,calledtheMAPsolution.Ofcourse,theMAPestimateisfoundbyminimizingϕwithrespecttothe
∂ϕˆ.TheconditionfortheMAPsolutionis∂aparameters,yieldingtheestimatedparametervaluesa=0fori
allparametersai,providingtherearenoconstraintsontheparametersthemselves.
Thestandardtextbooksondataanalysisusuallydealwithalikelihoodanalysisofparameterizedmodels.UndertheassumptionthatthemeasurementsaresubjecttoGaussian,additivenoise,theminus
−2
ˆi)2,thesumthesquaredχ2=1iσi(di−dlog-likelihoodishalfofthefamiliarchisquared,Λ(a)=122residuals(thedifferencebetweenanobservedmeasurementsandtheirvaluespredictedbythetheestimated
ˆ)dividedbytheestimatedvarianceofthenoise,σ2.The“best”solutionfortheparametersparametersa
isfoundbyminimizingχ2,correspondingtotheBayesianMAPsolutionintheabsenceofacontributionfromtheprior.
Usingtheresultsofthetraditionalapproachtotheestimationofuncertaintyinalikelihoodanalysis,4
ˆ,thecovariancesoftheparameteruncertaintiesarederivedfromthecurvaturematrixofϕ,calculatedata
∂2ϕ
.Kij=
∂ai∂aj
(2)
Sincethismatrixisevaluatedattheminimumofϕ,itmustbepositivesemi-definite,i.e.(∆a)TK∆a≥0forany∆a.IntheGaussianapproximationtotheposterior,theerrormatrixE,whichgivesthecovariancesbetweentheparameters,
[E]ij=(ai−aˆi)(aj−aˆj),(3)wherethebracketsindicateanensembleaverage,istheinverseofthecurvaturematrix,
E=K−1.
(4)
Thisresultprovidesthesecondmomentoftheparameteruncertaintiesandtheircorrelationsunderthe
assumptionofGaussianprobabilitydistributions.Itsuffersfromnotbeingveryilluminatingintermsof
417
itsconsequencesfortheparametricmodel,particularlyintermsofthecorrelationsintheuncertaintiesofvariousparameters.Furthermore,forthe105–106pixelamplitudesthataretypicallyneededtodescribea2Dimage,thefullerrormatrixcontains1010–1012elements,whichcanneitherbepracticallycalculatednorstored.Weproposeanapproachtoprovideamoretangibleindicationofthedegreeofuncertaintyintheinferredmodelaswellastheabilitytodirectlyprobetheuncertaintyofspecificfeaturesofthemodel.
3.BAYESIANMECHANICS
Ifonedrawsananalogybetweenϕandaphysicalpotential,thenthegradientofϕisanalogoustoaforce,
∂ϕjustasinphysics.Theforcefi=−∂aisroughlyinthedirectionofthelocalminimumofϕ,undersuitablei
assumptionsconcerningthesmoothnessofthedependenceofϕontheparameters.TheconditionfortheMAPsolution,∇aϕ=0,canbeinterpretedasstatingthatattheMAPoperatingpointtheforcesonallthevariablesintheproblembalance:thenetforceoneachvariableiszero.Further,whenthevariableaiisperturbedslightlyfromtheMAPsolution,theforcefipullsaibacktowardstheMAPsolution.Thephrase“forceofthedata”takesonrealmeaninginthiscontext.
AquadraticapproximationtoϕintheneighborhoodoftheMAPsolutionimpliesalinearforcelaw,i.e.therestoringforceisproportionaltothedisplacementfromequilibrium,asinasimplespring.InthisquadraticapproximationthecurvatureofϕisproportionaltothecovarianceoftheMAPestimate.Ahighcurvatureisanalogoustoastiffspringandthereforerepresentsa“rigid”,reliablesolution.
AninterestingaspectofthisinterpretationisthepossibilityofdecomposingtheforcesactingontheMAPsolutionintotheirvariouscomponents.Forexample,theforcederivedfromalldata(throughthelikelihood),orevenaselectedsetofdata,maybecomparedtotheforcederivedfromtheprior.Inthiswayitispossibletoexaminetheinfluenceofthepriorsonthesolutionaswellasdeterminewhichdatahavethelargesteffectonaparticularfeatureofthesolution.
Wenotethatthenotionofapplyingforcestomodelparametersintheprecedingdiscussionmustulti-matelybestatedintermsofpressures,thatis,forcesappliedoverregions,actingonphysicallymeaningfulquantities.Thefirstreasonisthatthephysicalworld,whichweusuallymodel,existsasacontinuum:thephysicalquantitiesofinterestaretypicallydensities,whichareafunctionofcontinuousspatialortemporalcoordinates.Thusmeaningfulquestionsaboutrealityshouldreallybestatedintermsaveragesoverregions,notaspointvalues.Secondly,physicallyfeasiblemeasurementscanonlyprobephysicalquan-titiesoverfinite-sizedregions.Pointsamplingisfundamentallyimpossible.Asanexample,aradiographicmeasurementinwhichtheattenuationofanx-raybeamismeasuredisalwayssubjecttotheeffectsofablurringprocessthatarisesfromafinitespotsizeforthesourceofxraysandthefiniteresolutionofthex-raydetector.Thusthemeasuredattenuationisnecessarilyanaverageoveracylinderinspace.Intruth,radiographicmeasurementscannotprovidelineintegralsofanattenuationcoefficientthroughanobject,asisoftenassumedasanapproximationtotherealprocess.Putsuccinctly,allphysicalmeasurementshavelimitedspatialortemporalresolutionthatrenderasmeaninglessquestionsaboutwhathappensinaninfinitesimallysmallregion.Asaresult,uncertaintiesinanestimatedphysicalquantitycanonlybeaddressedintermsoftheaverageofthatquantityoverafiniteregion.AstheconceptsofBayesiananalysismature,wewilllearntodealonlywithphysicalquantitiesthatarefunctionsofcontinuousindependentvariablesandwewillavoidreferencingdirectlytheunderlyingdiscreteparametersofthemodels.Oneneedstobeawarethatanyfiniterepresentation,whichweareforcedtouseincomputermodels,hasalimitedresolution.Thuswhenoneexploresthemodelatascalefinerthantheinherentresolutionofthemodel,themodelcanonlyrespondbyinterpolationoftheunderlyingdiscretemodel.5Onecanonlymeaningfullyexplorethemodelatresolutionscoarserthanthis.
418
4.PERTURBATIONFROMEQUILIBRIUM
Weproposetoexploittheabovephysicalanalogytofacilitatetheexplorationoftheuncertaintyina
ˆ,ϕiswellMAPsolution.ForthepresentwewillassumethatintheneighborhoodoftheMAPpointa
approximatedbyaquadraticexpansion:
ϕ=1(∆a)TK∆a+ϕ0,
2(5)
ˆisthedisplacementfromtheMAPpointandϕ0=ϕ(ˆwhere∆a=a−aa).Supposethatwestartfrom
ˆanddisplacetheparametervaluesbyasmallamount∆a.Thenthegradientofϕ,−∇aϕ,representsa
aforcethatpullstheparametersbacktowardtheMAPpoint.Theunitsoftheforcearethoseofthereciprocaloftheparameter.Thecurvature,andhencethereciprocalofthevariance,inthedirectionof
ˆ+∆a,o|∆a|,forvanishinglysmalldisplacements.∆aisgivenbytheratioof|∇aϕ|,evaluatedata
Insteadofdirectlydisplacingparameters,wemayperturbthembyapplyinganexternalforcetothe
parameters.Supposethatonepullsontheparameterswithaforcef.Notethatthisforcecanactonjustoneparameteroronmany.Fromthephysicalanalogy,itiseasytowritedownthenewpotential;
ϕ=1(∆a)TK∆a−∆aTf+ϕ0.
2Thenewminimumofϕoccurswhen
∇aϕ=0=K∆a−f.
(6)(7)
SolvingforthedisplacementinaandusingEq.(4),
∆a=K−1f=Ef.
(8)
IfthecurvaturematrixK,andhencethecovariancematrixE,isnotdiagonal,theresultingdisplacementisnotinthedirectionoftheappliedforce.Thisphenomenondemonstratesthecorrelationsbetweentheuncertaintiesinalltheparameters.Thecomponentof∆ainthedirectionoftheappliedforcedividedbythemagnitudeoftheforce,i.e.∆aTf/|f|2,istheeffectivevarianceintheparametersinthatdirection.
Althoughweassumedabovethatϕisquadratic,thisapproachcanbeusefulevenwhenitisnon-quadratic.Whileitmaynotbefeasibletoexpresstheresultsanalytically,weobtainafeelingfortheuncertaintyin∆aandthecorrelationsbetween∆aandtheotherparameters.Anyconstraintsontheparameterscanbeexplicitlyseen.Fornonquadraticϕtheplotofthevalueofϕversustheappliedforceprovidesthemeanstovisualizetheuncertaintyin∆a.
5.USEWITHDEFORMABLEGEOMETRICMODELS
Theaboveapproachtakesonapoignantinterpretationwhenthereconstructedobjectisdefinedintermsofitsgeometricshape.Theprioronthegeometryisdefinedintermsofthedefaultshapetogetherwithaprescriptionofhowtoassesstheprobabilityofotherpossibleshapes.ThelatterissimplydonebyusingaGibbsformfortheprobabilitygivenasexp(−βW),whereWisthedeformationenergy,i.e.theenergyrequiredtodeformthegeometryfromthedefaultshapeintoanewshape.6–10Theparameterβregulatesthestrengthoftheprioronthegeometry.
Figure1showsapolygondefinedintermsofits20verticesornodes.Thusthereare40parametersinthismodelcorrespondingtothetwocoordinatesneededtospecifyeachvertexofthepolygon.Weassumethattwosetsofparallelprojections,oneverticalandonehorizontal,areavailableandthattheyaresubjecttoaverysmallamountofmeasurementnoise.Forsimplicityweignoretheprioronthedeformationdescribedabove.Startingfromtheknownoriginalpolygon,aforceisappliedtotheleftmost
419
Figure1.Anexampleofhowapolygon(solidline)canbedistortedbyeitherpushingnodeAinward(dashedline)oroutward(dottedline),assumingthatthemeasurementsconsistoftwoorthogonalpro-jections.Notetheeffectontheoverallshapeoftheobject,whichindicatesthecorrelationsbetweenthepolygonvertices.
node(nodeA),pullingitoutward.TheplotoftheappliedforceandtheresultinghorizontaldisplacementofthenodeisshowninFig.2.ForpositiveforcesnodeAmovesoutwardsteadilyuptoabreakpoint(atadisplacementof0.18),whichwecallpointB.Thedotted-linefigureinFig.1showstheconfigurationofthepolygonatthatpoint.WenotethattheactofdisplacingnodeAoutwardcontradictstheverticalprojections,whichindicatethatthereisprobablynomaterialtotheleftoftheoriginalpositionofthenode.BeyondpointBtheslopeofthecurvedecreasessubstantially,principallybecausenewconfigurationsofthepolygonarepossible,whichcanreducetheexcessiveprojectionvaluestotheleftoftheoriginalpositionofnodeA.
Applyingtheforceinward(negativeforcevalues)resultsinquiteadifferentbehavior.Withasmallinwardpush,thedisplacementreachesabreakpoint,pointCinFig.2.TheconfigurationofthepolygonatthispointisshownFig.1asthedashedfigure.NodeAhasjustreachedthelineconnectingitsneighbors,oneofwhichhasmovedoutwardtotakeitsplaceinsupplyingtheproperverticalprojection.PushingharderonlymakesnodeAslidedownthatline,whichrequiresonlyalittleforcetoachievealargedisplacement.ThepositionofnodeAisnotwelldeterminedinthisregion.Wenoticethattheshapeoftheobjectdoesnotchangeduringthisprocess.Theresultsforthissituationarecorrect,butmaynotbewhatonehasinmindwhenspecifyingtheforce.Itseemsdesirabletoavoidapplyingtheforcedirectlytotheparameters,inthiscase,tothepositionofthenodesofthepolygon.Theforceshouldinsteadbeappliedtotheobjectanditseffecttranslatedtotheparameters.AlsoweobservethattheonlyreasonpointCisnotclosertotheoriginisthatthecoarsenessofourpolygonobjectmodellimitstheflexibilityoftheobjecttorespond.Withmanymoredegreesoffreedom,wewouldexpectneighboringsectionsoftheobjectboundarytomoveouttotaketheplaceofnodeAinresponsetoaslightinwardforce.
420
Figure2.PlotoftheforceappliedtonodeAofthepolygoninFig.1versustheresultingdisplace-mentofthatnode.Thenonlinearnatureoftheforce-displacementlawforthisproblemisdramaticallydemonstrated.TheconfigurationsshowninFig.1areatthetwobreakpointsinthecurve:thedashedlinecorrespondstoaforceof-0.006(inward)atpointCandthedottedlinetoaforceof0.080(outward)atB.ThecorrelationbetweentheuncertaintyinthepositionofnodeAandthepositionsoftheothernodesinthepolygonisdemonstratedinFig.1.Weobservethatthenodesontherightsideofthepolygonmovetomaintainthemeasuredhorizontalprojection.Ofcourse,theconstraintsoftheverticalprojectionalsofigureintotheproblem,makingtheoverallmovementofthesidesofthepolygonrathercomplex.Thisapproachnicelyhandlesthecomplexinteractionbetweenalltheconstraintsarisingfrommeasurementsandpriorknowledge.
Foranobjectmodeledintermsofitsgeometry,poorreliabilityoftheMAPestimatemeansthattheobjectissoftorsquishy,pliable.Goodreliabilityoftheestimatemeansthattheobjectisfirm.Therefore,“truth”ishardorrigid.
Asanaside,thisexamplecomesfromaworkstationapplicationthatwearedevelopingtofacilitatetheuseoftheBayesianapproachtosolvechallengingtomographicreconstructionproblems.11–13Thisapplication,calledtheBayesInferenceEngine,providesanintuitiveenvironmentforsettinguptheobjectmodelandthemeasurementmodelbymeansofdata-flowdiagramthatmaybeprogrammedgraphicallybytheuser.Theobjectsbeingreconstructedcanbemodeledusinggeometricmodelssuchasthatpresentedhere.Theoptimizationprocedureemployedisbasedonasteepestdescentmethod.Weusewhatappearstobealittleknowntechniquecalledadjointdifferentiationtoefficientlycalculatetherequiredderivativesoftheoptimizationfunctionwithrespecttotheparameters.14
421
6.DISCUSSION
Inthefutureitmaybepossibletousethetoolsofvirtualreality,coupledtoturbocomputation,toexplorethereliabilityofaBayesiansolutionofcomplexproblemsthroughdirectmanipulationofthecomputermodel.Forcefeedbackwillpermitonetoactually“feel”thestiffnessofamodel.Higherdimensionalcorrelationsmightbe“felt”throughone’svarioussenses.
ToreiteratethecommentsmadeinSect.3,wesuggestthatqueriesregardingphysicalquantitiesshouldbemadeintermsofaveragesoverregionsratherthanintermsoftheirvaluesatpoints.Furthermore,theuncertaintiesofindividualparametersthat,asacollection,aremeanttodescribeaphysicalquantityasafunctionofcontinuouscoordinates,mayhavelittlemeaning.Inregardtoanimagerepresentedasagridofpixels,thequestion“whatisthermserrorinapixelvalue?,”isimpossibletoanswerwithoutaclearunderstandingofwhatapixelvaluerepresents,e.g.theaveragevalueovertheareaofthepixel.Moremeaningfulquestionscanbemadeforareaslargerthanthatofasinglepixel.Furthermore,thecorrelationsbetweentheaveragevaluewithinaregionandtherestoftheimagemustbeconsidered.Consequently,ourlanguagemustchange.Insteadofapplyingforcestoprobethereliabilityofindividualparametersthatareusedtodescribeanobject,weshouldspeakofapplyingpressuresoverregionsoftheobject.Anditmustbeunderstoodthatwhenweaskaboutregionswhosesizeisontheorderof,orsmaller,thantheresolutionofthediscretemodeloftheobject,wewillonlylearnabouttheinterpolationpropertiesofthemodel.
TheapproachtoreliabilitytestingdescribedaboveisverygeneralandcanbeusedinvirtuallyanyotherkindofBayesiananalysis.Examplesofothercontextsareasfollows:
Spectralestimation:Intypicalspectralanalysisascalarvariablequantityisestimatedfordifferentdiscretefrequencyvalues.Normallyasinglespectrumisestimated.Skillingetal.3probedthevariabilitypossibleintheanswerthroughtheirprobabilisticdisplaytechnique.Thatdisplaygivesoneatruefeelingfortherangeofanswerspossibleforagivensetofinputdata.Withourtechnique,onecanaskdirectquestionsaboutthepoweroveraspecificrangeoffrequencies.Themodeofinteractionwiththespectrummightbethoughtofaspushingdownorpullinguponaregionofthespectrum.Inavirtualrealitysetting,wecanimaginethattheanalystwouldbeabletousehisfingerstopressupwardordownwardonvariousportionsofthespectrum.Theresistancetothisattemptedaction,fedbacktotheuser’sfingersasaforce,wouldindicatethedegreeofuncertaintyinthesolution.
Imagereconstruction:Thebasicproblemistoestimatetheamplitudesinimagepixelsfromdata,eachofwhichisacombinationofmanypixels,asintomographicreconstructionfromprojections(stripintegrals)throughtheimage,ordeconvolutionofblurredimages.Interactionwiththeimagecanbeprovidedbyallowingonetopushorpullontheamplitudesinanareaofinterest.Theconceptsbehindthistechniquecanbeusedtomakebinarydecisions,forexample,todecidewhetheranobjectispresentornot,ortodecidebetweentwodifferentsignals.15
REFERENCES
1.K.M.HansonandG.W.Wecksung.Bayesianapproachtolimited-anglereconstructionincomputedtomogra-phy.J.Opt.Soc.Amer.,73:1501–1509,1983.
2.J.Skilling,D.R.T.Robinson,andS.F.Gull.Probabilisticdisplays.InJr.W.T.GrandyandL.H.Schick,editors,MaximumEntropyandBayesianMethods,pages365–368.KluwerAcademic,1991.
3.J.Skilling.Clouds.PresentedattheWorkshoponMaximumEntropyandBayesianMethods,July19-24,1992,Paris.
4.P.R.Bevington.DataReductionandErrorAnalysisforthePhysicalSciences.McGraw-Hill,NewYork,1969.5.K.M.HansonandG.W.Wecksung.Localbasis-functionapproachtocomputedtomography.Appl.Opt.,24:4028–4039,1985.
422
6.R.Szeliski.Probabilisticmodelingofsurfaces.Proc.SPIE,1570:154–165,1991.
7.R.SzeliskiandD.Terzopoulos.Physically-basedandprobabilisticmodelsforcomputervision.Proc.SPIE,1570:140–152,1991.
8.K.M.Hanson.Reconstructionbasedonflexiblepriormodels.Proc.SPIE,vol.1652:183–191,1992.
9.K.M.Hanson.FlexiblepriormodelsinBayesianimageanalysis.InA.Mohammad-DjafariandG.Demoment,editors,MaximumEntropyandBayesianMethods,pages399–406.KluwerAcademic,1993.
10.K.M.Hanson.Bayesianreconstructionbasedonflexiblepriormodels.J.Opt.Soc.Amer.,A10:997–1004,1993.11.G.S.Cunningham,K.M.Hanson,G.R.Jennings,Jr.,andD.R.Wolf.Anobject-orientedimplementationofa
graphical-programmingsystem.MedicalImaging:ImageProcessing,ed.M.H.Loew,Proc.SPIE,2167:914–923,1994.
12.G.S.Cunningham,K.M.Hanson,G.R.Jennings,Jr.,andD.R.Wolf.AninteractivetoolforBayesian
inference.TobepublishedinReviewofProgressinQuantitativeNondestructiveEvaluation,21,1994.
13.K.M.Hanson,G.S.Cunningham,G.R.Jennings,Jr.,andD.R.Wolf.Tomographicreconstructionbasedon
flexiblegeometricmodels.Proc.IEEEInt.Conf.ImageProcessing,II:145–147,1994.
14.G.S.Cunningham,K.M.Hanson,G.R.Jennings,Jr.,andD.R.Wolf.Anobject-orientedoptimizationsystem.
Proc.IEEEInt.Conf.ImageProcessing,III:826–830,1994.
15.K.M.Hanson.Makingbinarydecisionsbasedontheposteriorprobabilitydistributionassociatedwithtomo-graphicreconstructions.InC.R.Smith,G.J.Erickson,andP.O.Neudorfer,editors,MaximumEntropyandBayesianMethods,pages313–326.KluwerAcademic,1991.
423
因篇幅问题不能全部显示，请点此查看更多更全内容