A Generative Perspective on MRFs in Low-Level Vision Supplem(3)

2.ImageRestoration

Tofurtherillustratetheimagerestorationperformanceofourapproach,weprovidethefollowingadditionalresults: Table1repeatsTab.1ofthemainpaperandadditionallygivesthenumericalresultsofMAPestimationwithgraphcutsandα-expansion[1].Notethatinmostcases,α-expansionperformsslightlyworseintermsofPSNRthanconjugategradients,even(andinfactparticularly)fornon-convexpotentials.Also,usingaStudent-tpotential[3]doesnotshowfavorableresults. Table2showstheresultsofthesameexperimentasinTab.1,butreportstheperformanceintermsoftheperceptuallymorerelevantstructuralsimilarityindex(SSIM)[10].Notethatalloftheconclusionsreportedinthemainpaperalsoholdforthisperceptualqualitymetric. Table3repeatsTab.2ofthemainpaper,andadditionallyreportsstandarddeviationsaswellasSSIMperformance.TheSSIMsupportsthesameconclusionsaboutrelativeperformanceasthePSNR. Figs.1–6showdenoisingresultsfor6ofthe68images,forwhichtheaverageperformanceisreportedinTab.2ofthemainpaper.Notethatincontrasttothetestedpreviousapproaches,combiningourlearnedmodelswithMMSEleadstogoodperformanceonrelativelysmoothaswellasonstronglytexturedimages. Fig.7providesadifferentviewofthesummaryresultsinTab.2ofthepaper.Insteadoftheaverageperformance,weshowaper-imagecomparisonbetweenthedenoisingresultsofthediscriminativeapproachof[8](usingMAP)andtheresultsofourgeneratively-trained3×3FoE(usingMMSE).NotethatthePSNRandparticularlytheSSIMshowasubstantialperformanceadvantageforourapproach. Fig.8showsanuncroppedversionoftheinpaintingresultinFig.7ofthepaper.Additionally,oneotherinpaintingresultisprovidedasfurthervisualillustration.

3.SamplingthePriorandPosterior

Thefollowingadditionalresultsillustratepropertiesoftheauxiliary-variableGibbssampler.

Fig.9shows vesubsequentsamples(afterreachingtheequilibriumdistribution)fromallmodelslistedinTable1.Notehowsamplesfromcommonpairwisemodelsappeartoo“grainy”,whilethosefrompreviousFoEmodelsaretoosmoothandwithoutdiscontinuities. Fig.10showstwolargersamplesfromourlearnedmodels.Notethatourpairwisemodelleadstolocallyuniformsampleswithoccasionaldiscontinuitiesthatappearspatiallyisolated(“speckles”).Ourlearnedhigh-ordermodel,ontheotherhand,leadstosmoothlyvaryingsampleswithoccasionalspatiallycorrelateddiscontinuities,whichappearmorerealistic. Fig.11illustratestheconvergenceofthesamplingprocedureforthepriorandtheposterior(incaseofdenoising). Fig.12illustratestheadvantagesofrunningmultipleparallelsamplers.

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