stanfordedu Vladlen Koltun Computer Science Department Stanford University vladlencsstanfordedu Abstract Most stateoftheart techniques for multiclass image segmentation and labeling use conditional random 64257elds de64257ned over pixels or image reg ID: 4758
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structurethattilesthefeaturespacewithsimplicesarrangedalongd+1axes[1].ThepermutohedrallatticeexploitstheseparabilityofunitvarianceGaussiankernels.Thusweneedtoapplyawhiteningtransform~f=Uftothefeaturespaceinordertouseit.Thewhiteningtransformationisfoundus-ingtheCholeskydecompositionof(m)intoUUT.Inthetransformedspace,thehigh-dimensionalconvolutioncanbeseparatedintoasequenceofone-dimensionalconvolutionsalongtheaxesofthelattice.Theresultingapproximatemessagepassingprocedureishighlyefcientevenwithafullysequentialimplementationthatdoesnotmakeuseofparallelismorthestreamingcapabilitiesofgraphicshardware,whichcanprovidefurtheraccelerationifdesired.4LearningWelearntheparametersofthemodelbypiecewisetraining.First,theboostedunaryclassiersaretrainedusingtheJointBoostalgorithm[21],usingthefeaturesdescribedinSection5.Nextwelearntheappearancekernelparametersw(1), ,andforthePottsmodel.w(1)canbefoundefcientlybyacombinationofexpectationmaximizationandhigh-dimensionalltering.Unfortunately,thekernelwidths andcannotbecomputedeffectivelywiththisapproach,sincetheirgradientinvolvesasumofnon-Gaussiankernels,whicharenotamenabletothesameaccelerationtechniques.Wefoundittobemoreefcienttousegridsearchonaholdoutvalidationsetforallthreekernelparametersw(1), and.Thesmoothnesskernelparametersw(2)and donotsignicantlyaffectclassicationaccuracy,butyieldasmallvisualimprovement.Wefoundw(2)= =1toworkwellinpractice.Thecompatibilityparameters(a;b)=(b;a)arelearnedusingL-BFGStomaximizethelog-likelihood`(:I;T)ofthemodelforavalidationsetofimagesIwithcorrespondinggroundtruthlabelingsT.L-BFGSrequiresthecomputationofthegradientof`,whichisintractabletoestimateexactly,sinceitrequirescomputingthegradientofthepartitionfunctionZ.Instead,weusethemeaneldapproximationdescribedinSection3toestimatethegradientofZ.Thisleadstoasimpleapproximationofthegradientforeachtrainingimage:@ @(a;b)`(:I(n);T(n))�XiT(n)i(a)Xj6=ik(fi;fj)T(n)j(b)+XiQi(a)Xj6=ik(fi;fj)Qi(b);(6)where(I(n);T(n))isasingletrainingimagewithitsgroundtruthlabelingandT(n)(a)isabinaryimageinwhichtheithpixelT(n)i(a)hasvalue1ifthegroundtruthlabelattheithpixelofT(n)isaand0otherwise.AdetailedderivationofEquation6isgiveninthesupplementarymaterial.ThesumsPj6=ik(fi;fj)Tj(b)andPj6=ik(fi;fj)Qi(b)arebothcomputationallyexpensivetoeval-uatedirectly.AsinSection3.2,weusehigh-dimensionallteringtocomputebothsumsefciently.TheruntimeofthenallearningalgorithmislinearinthenumberofvariablesN.5ImplementationTheunarypotentialsusedinourimplementationarederivedfromTextonBoost[19,13].Weusethe17-dimensionallterbanksuggestedbyShottonetal.[19],andfollowLadick´yetal.[13]byaddingcolor,histogramoforientedgradients(HOG),andpixellocationfeatures.OurevaluationontheMSRC-21datasetusesthisextendedversionofTextonBoostfortheunarypotentials.FortheVOC2010datasetweincludetheresponseofboundingboxobjectdetectors[4]foreachobjectclassas20additionalfeatures.ThisincreasestheperformanceoftheunaryclassiersontheVOC2010from13%to22%.Wegainanadditional5%bytrainingalogisticregressionclassierontheresponsesoftheboostedclassier.Forefcienthigh-dimensionalltering,weuseapubliclyavailableimplementationofthepermuto-hedrallattice[1].Wefoundadownsamplingrateofonestandarddeviationtoworkbestforallourexperiments.Sampling-basedlteringalgorithmsunderestimatetheedgestrengthk(fi;fj)forverysimilarfeaturepoints.Propernormalizationcancanceloutmostofthiserror.Thepermutohedrallatticeallowsfortwotypesofnormalizations.Aglobalnormalizationbytheaveragekernelstrength5 Runtime Standardgroundtruth Accurategroundtruth Global Average Global Average Unaryclassiers � 84:0 76:6 83:21:5 80:62:3 GridCRF 1s 84:6 77:2 84:81:5 82:41:8 RobustPnCRF 30s 84:9 77:5 86:51:0 83:11:5 FullyconnectedCRF 0:2s 86:0 78:3 88:20:7 84:70:7 Figure3:QualitativeandquantitativeresultsontheMSRC-21dataset.rameters.Theunarypotentialswerelearnedonaseparatetrainingsetthatdidnotincludethe94accuratelyannotatedimages.WealsoadoptthemethodologyproposedbyKohlietal.[9]forevaluatingsegmentationaccuracyaroundboundaries.Specically,wecounttherelativenumberofmisclassiedpixelswithinanar-rowband(“trimap”)surroundingactualobjectboundaries,obtainedfromtheaccurategroundtruthimages.AsshowninFigure4,ouralgorithmoutperformspreviousworkacrossalltrimapwidths.PASCALVOC2010.DuetothelackofapubliclyavailablegroundtruthlabelingforthetestsetinthePASCALVOC2010,weusethetrainingandvalidationdataforallourexperiments.Werandomlypartitionedtheimagesinto3groups:40%training,15%validation,and45%testset.Seg-mentationaccuracywasmeasuredusingthestandardVOCmeasure[3].Theunarypotentialswerelearnedonthetrainingsetandyieldedanaverageclassicationaccuracyof27:6%.TheparametersforthePottspotentialsinthefullyconnectedCRFmodelwerelearnedonthevalidationset.The (a)Trimapsofdifferentwidths (b)SegmentationaccuracywithintrimapFigure4:Segmentationaccuracyaroundobjectboundaries.(a)Visualizationofthe“trimap”measure.(b)Percentofmisclassiedpixelswithintrimapsofdifferentwidths.7 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