2009 Fields of Experts(stefan roth et al)

IntJComputVis(2009)82:205–229DOI10.1007/s11263-008-0197-6FieldsofExpertsStefanRoth·eceived:22January2008/Accepted:17November2008/Publishedonline:24January2009©SpringerScience+BusinessMedia,LLC2009AbstractWedevelopaframeworkforlearninggeneric,expressiveimagepriorsthatcapturethestatistroachprovidesapracticalmethodforlearninghigh-orderMarkovrandomfield(MRF)modliquepotentialsaremodeledusingtheProduct-of-Expertsframeworkthatusesnon-linearfunc-tionsofmanylinearfirasttopreviousMRFapproachesallparameters,includingthelinearfiltersthemselves,nstratethecapabilitiesofthisField-of-Expertsmodelwithtwoex-ampleapplications,imagedenoisingandimageinpainting,whichareimplementedusingasimple,hemodelistrainedonagenericim-agedatabaseandisnottunedtowardaspecificapplica-tion,dsMarkovrandomfields·Low-levelvision·Imagemodeling·Learning·Imagerestoration1IntroductionTheneedforpriormodelsofimageorscenestructureoc-cursinmanymachinevisionandgraphicsproblemsinclud-ingstereo,opticalflow,denoising,super-resolution,image-basedrendering,volumetricsurfacereconstruction,eronehas“noise”oruncertainty,priormodelsofimages(ordepthmaps,flowfields,three-dimensionalvolumes,etc.)developamethodforlearningpriorsforlow-levelvisionproblemsthatcanbeusedinmanystandardvision,graphics,ideaistoformulatethesepriorsasahigh-orderMarkovrandomfield(MRF)defiultingFieldofExperts(FoE)mod-elsthepriorprobabilityofanimage,orotherlow-levelrep-resentation,intermsofarandomfieldwithoverlappingcliques,whosepotentialsarerepresentedasaProductofExperts(Hinton1999).Whilethismodelappliestoawiderangeoflow-levelrepresentations,rwork(RothandBlack2007b)wehavealreadystudiedtheapplicationtomodelingvector-valuedopticalflowfields;ytheapplicationofFieldsofExpertstomodelingnaturalimages,wetrainthemodelonastandarddatabaseofnaturalimages(Martinetal.2001)anddevelopadiffusionstratethepoweroftheFoEmodel,weuseitintwodifferentapplications:imagedenoisingandim-ageinpainting(Bertalmíoetal.2000)(i.e.,fillinginmissingpixelsinanimage).Despitethegenericnatureofthepriorandthesimplicityoftheapproximateinference,(