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1、MarkovRandomFieldsandApplicationsClicktoeditMastersubtitlestyleSoumyaGhosh09年3月25日Outline•MRFbasics–BayesianImageAnalysis–MarkovRandomFieldstheory–Gibbs–MarkovEquivalence–Inference–Learning•Application–ImageSegmentation09年3月25日BayesianImageAnalysisNoiseTransmissionOriginalImageDegraded(observ
2、ed)Image09年3月25日BayesianImageAnalysis•LetXbetheobservedimage={x1,x2…xmn}•LetYbethetrueimage={y1,y2…ymn}•Goal:findY=y*={y1*,y2*…}suchthatP(Y=y*
3、X)ismaximum.•LabelingproblemwithasearchspaceofLmn–Listhesetoflabels.–m*nobservations.3/25/09FirstGuess•Assumethelabelsyiareindependent.P(Y
4、X)=∏mni=1P(
5、yi
6、xi)MaximizingP(Y
7、X)boilsdowntosimplymaximizingtheindividualP(yi)s.3/25/09UnfortunatelyObservedSVMMRFImage3/25/09MarkovRandomFields•Introducedinthe1960s,aprincipledapproachforincorporatingcontextinformation.•Incorporatingdomainknowledge.•WorkswithintheBayesianframework.•Widelyworkedoninthe7
8、0sdisappearedoverthe80sandfinallymadeabigcomebackinthelate90s.3/25/09MarkovRandomField•RandomField:LetbeafamilyF={F,F,...,F}12MofrandomvariablesdefinedonthesetS,inwhicheachrandomvariabletakesavalueinalabelsetL.ThefamilyFiscalledarandomfield.Ffii•MarkovRandomField:FissaidtobeaMarkovrandomfield
9、 onSwithrespecttoaneighborhoodsystemNifandonlyifthefollowingtwoconditionsaresatisfied:Possitivity:P(f)>0,"fÎFMarkovianity:P(f
10、S-{i})=P(f
11、f)iiNi3/25/09ImageAnalysis•Wecouldthusrepresentboththeobservedimage(X)andthetrueimage(Y)asMarkovrandomfields.X–observedimageY–trueimage•AndinvoketheBayesian
12、frameworktofindP(Y
13、X)3/25/09Details•P(Y
14、X)proportionaltoP(X
15、Y)P(Y)–P(X
16、Y)isthedatamodel.–P(Y)modelsthelabelinteraction.•NextweneedtocomputethepriorP(Y=y)andthelikelihoodP(X
17、Y).3/25/09Cliques•AcliqueisdefinedasasubsetofsitesinF,whereeverypairofsitesareneighborsofeachother.Thecollectionsofsingl
18、e-site,double-site,andtriple-sitecliquesaredenotedbyC1,C2,C3and…Cn3/25/09Markov–GibbsEquivalence•GibbsDistribution:P(f)=(1/Z)e(-1/T)U(f)whereU(f)=∑cinCVc(f)=∑iinC1V1(fi)+∑iinC2V2(fi,fj)+…Z–NormalizingFactoroverthespaceofallconfigurations.•Ham
