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Efcient Inference in Fully Connected CRFs with Gaussian Edge Potentials Philipp Kr ahenb uhl Computer Science Department Stanford University philkrcs

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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

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Efcient Inference in Fully Connected CRFs with Gaussian Edge Potentials Philipp Kr ahenb uhl Computer Science Department Stanford University philkrcs






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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), ,and forthePottsmodel.w(1)canbefoundefcientlybyacombinationofexpectationmaximizationandhigh-dimensionalltering.Unfortunately,thekernelwidths and cannotbecomputedeffectivelywiththisapproach,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 References[1]A.Adams,J.Baek,andM.A.Davis.Fasthigh-dimensionallteringusingthepermutohedrallattice.ComputerGraphicsForum,29(2),2010.2,5[2]A.Adams,N.Gelfand,J.Dolson,andM.Levoy.Gaussiankd-treesforfasthigh-dimensionalltering.ACMTransactionsonGraphics,28(3),2009.2[3]M.Everingham,L.VanGool,C.K.I.Williams,J.Winn,andA.Zisserman.ThePASCALVisualObjectClasses(VOC)challenge.IJCV,88(2),2010.6,7[4]P.F.Felzenszwalb,R.B.Girshick,andD.A.McAllester.Cascadeobjectdetectionwithdeformablepartmodels.InProc.CVPR,2010.5[5]B.Fulkerson,A.Vedaldi,andS.Soatto.Classsegmentationandobjectlocalizationwithsuperpixelneighborhoods.InProc.ICCV,2009.1[6]C.Galleguillos,A.Rabinovich,andS.Belongie.Objectcategorizationusingco-occurrence,locationandappearance.InProc.CVPR,2008.1[7]S.Gould,J.Rodgers,D.Cohen,G.Elidan,andD.Koller.Multi-classsegmentationwithrelativelocationprior.IJCV,80(3),2008.1[8]X.He,R.S.Zemel,andM.A.Carreira-Perpinan.Multiscaleconditionalrandomeldsforimagelabeling.InProc.CVPR,2004.1[9]P.Kohli,L.Ladick´y,andP.H.S.Torr.Robusthigherorderpotentialsforenforcinglabelconsistency.IJCV,82(3),2009.1,2,6,7[10]D.KollerandN.Friedman.ProbabilisticGraphicalModels:PrinciplesandTechniques.MITPress,2009.3[11]V.KolmogorovandR.Zabih.Whatenergyfunctionscanbeminimizedviagraphcuts?PAMI,26(2),2004.2[12]S.KumarandM.Hebert.Ahierarchicaleldframeworkforuniedcontext-basedclassication.InProc.ICCV,2005.1[13]L.Ladick´y,C.Russell,P.Kohli,andP.H.S.Torr.Associativehierarchicalcrfsforobjectclassimagesegmentation.InProc.ICCV,2009.1,5[14]L.Ladick´y,C.Russell,P.Kohli,andP.H.S.Torr.Graphcutbasedinferencewithco-occurrencestatistics.InProc.ECCV,2010.1[15]J.D.Lafferty,A.McCallum,andF.C.N.Pereira.Conditionalrandomelds:Probabilisticmodelsforsegmentingandlabelingsequencedata.InProc.ICML,2001.3[16]S.ParisandF.Durand.Afastapproximationofthebilaterallterusingasignalprocessingapproach.IJCV,81(1),2009.2,4[17]N.PayetandS.Todorovic.(RF)2–randomforestrandomeld.InProc.NIPS.2010.1,2[18]A.Rabinovich,A.Vedaldi,C.Galleguillos,E.Wiewiora,andS.Belongie.Objectsincontext.InProc.ICCV,2007.1[19]J.Shotton,J.M.Winn,C.Rother,andA.Criminisi.Textonboostforimageunderstanding:Multi-classobjectrecognitionandsegmentationbyjointlymodelingtexture,layout,andcontext.IJCV,81(1),2009.1,3,5,6[20]S.W.Smith.Thescientistandengineer'sguidetodigitalsignalprocessing.CaliforniaTechnicalPub-lishing,1997.4[21]A.Torralba,K.P.Murphy,andW.T.Freeman.Sharingvisualfeaturesformulticlassandmultiviewobjectdetection.PAMI,29(5),2007.5[22]T.ToyodaandO.Hasegawa.Randomeldmodelforintegrationoflocalinformationandglobalinfor-mation.PAMI,30,2008.1[23]J.J.VerbeekandB.Triggs.Scenesegmentationwithcrfslearnedfrompartiallylabeledimages.InProc.NIPS,2007.19

Efcient Inference in Fully Connected CRFs with Gaussian Edge Potentials Philipp Kr ahenb uhl Computer Science Department Stanford University philkrcs - Description

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 Download Pdf

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