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1、Chapter2MathematicalMRFModelsThischapterintroducesfoundationsofMRFtheoryanddescribesimportantmathematicalMRFmodelsformodelingimageproperties.TheMRFmodelswillbeusedinthesubsequentchapterstoderiveMAP-MRFimageanalysismodelsandforMRFparameterestimation.2.1MarkovRandomFie
2、ldsandGibbsDistributionsMarkovrandomfieldtheoryisabranchofprobabilitytheoryforanalyzingthespatialorcontextualdependenciesofphysicalphenomena.Itisusedinvisuallabelingtoestablishprobabilisticdistributionsofinteractinglabels.Thissectionintroducesnotationsandresultsrelate
3、dtoMRF’s.2.1.1NeighborhoodSystemandCliquesThesitesinSarerelatedtooneanotherviaaneighborhoodsystem(Sec-tion2.12).AneighborhoodsystemforSisdefinedasN={Ni
4、∀i∈S}(2.1)whereNiisthesetofsitesneighboringi.Theneighboringrelationshiphasthefollowingproperties:(1)Asiteisnotneighb
5、oringtoitself:i∈Ni.(2)Theneighboringrelationshipismutual:i∈Ni⇐⇒i∈Ni.S.Z.Li,MarkovRandomFieldModelinginImageAnalysis,21AdvancesinPatternRecognition,DOI:10.1007/978-1-84800-279-12,cSpringer-VerlagLondonLimited2009222.MathematicalMRFModels543450000421240x00x031x1300
6、004212454345(a)(b)(c)(d)(e)(f)(g)(h)Figure2.1:Neighborhoodandcliquesonalatticeofregularsites.ForaregularlatticeS,thesetofneighborsofiisdefinedasthesetofsites√withinaradiusofrfromiN={i∈S
7、[dist(pixel)]2≤r,i=i}(2.2)ii,pixeliwheredist(A,B)denotestheEuclideandistancebe
8、tweenAandB,andrtakesanintegervalue.Notethatsitesatorneartheboundarieshavefewerneighbors.Inthefirst-orderneighborhoodsystem,alsocalledthe4-neighborhoodsystem,every(interior)sitehasfourneighbors,asshowninFig.2.1(a)wherexdenotesthesiteconsideredandzerositsneighbors.Inthe
9、second-orderneighborhoodsystem,alsocalledthe8-neighborhoodsystem,thereareeightneighborsforevery(interior)site,asshowninFig.2.1(b).Thenumbersn=1,...,5showninFig.2.1(c)indicatetheoutermostneighboringsitesinthenth-orderneighborhoodsystem.Theshapeofaneighborsetmaybedescr
10、ibedasthehullenclosingallthesitesintheset.WhentheorderingoftheelementsinSisspecified,theneighborsetcanbedeterminedmoreexplicitly.For
