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# Non-local Sparse Models for Image Restoration - Département d ...

Non-local Sparse Models for Image Restoration - Département d ...

theyoverlap.However,thisapproximationmakesthecorrespondingoptimizationtractable.Indeed,althoughdictionarylearningistraditionallyconsideredasextremelycostly,onlineproceduressuchas[16]makeitpossibletoefficientlyprocessmillionsofpatches,allowingtheuseoflarge photographsand/orlargeimagedatabases. Oncethedictionary Dandcodes αihavebeenlearned, everypixeladmits mestimates(oneperpatchcontaining it),anditsvaluecanbecomputedbyaveragingthese: x = 1 m n� RiDαi, (4) i=1 where Riin R n×m isthebinarymatrixwhichplacespatch number iatitsproperpositionintheimage.Thisapproach learnsthedictionaryonthesetofoverlappingnoisypatches, therebyadaptingthedictionarytotheimageitself,whichis akeyelementinobtainingbetterresults. Howtochoosebetween p = 0or p = 1isnotaprioriclear. SolvingEq.(2)with p = 0isNPhard,leading toapproximatesolutionsobtainedwithagreedyalgorithm suchasforwardselection[30](alsoknownasorthogonal matchingpursuit[18]).When p = 1,theproblemisconvex andcanbesolvedefficientlywiththeLARSalgorithm[9]. FollowingEladandAharon[11],wehaveobservedexperimentallythat,givenafixeddictionary D,thereconstructed imageisingeneralofbetterqualitywhenusingthe ℓ0 pseudonormratherthanitsconvex ℓ1counterpart. However,wehavealsoobservedthatdictionarieslearnedwith the ℓ1normareusuallybetterfordenoising,evenwhenthe finalreconstructionisdonewiththe ℓ0pseudonorm. 2.3.BlockMatching3D(BM3D) Dabovetal.proposein[7]apatch-basedprocedurethat exploitsimageself-similaritiesandgivesstate-of-the-artresults. Asin[11],theyestimatethecodesofoverlapping patchesandaveragetheestimates.However,similartononlocalmeansfiltering[3],theyreconstructpatchesbyfinding similaronesintheimage(blockmatching),stackingthem togetherintoa3Dsignalblock,anddenoisingtheblock usinghardorsoftthresholding[8]witha3Dorthogonal dictionary(3Dfiltering).Inconjunctionwithafewheuristics, 4 thissimpleideahasproventobeveryefficientand givesbetterresultsthanregularnon-localmeans. Akey ideaofourpaperistoimplementasimilarjointdecompositionapproachinthecontextofsparsecodingwithlearned dictionaries,asexplainedinthenextsection. 3.ProposedFormulation Weshowinthissectionhowimageself-similaritiescan beusedtoimprovelearnedsparsemodelswithsimultane- 4 Namely,usingacombinationofweightedaveragesofoverlapping patches,Kaiserwindows,andWienerfilteringtofurtherimproveresults. Figure1.Sparsityvs.jointsparsity:Greysquaresrepresentsnonzerosvaluesinvectors(left)ormatrix(right). oussparsecoding,whichencouragessimilarpatchestoadmitsimilarsparsedecompositions. 3.1.SimultaneousSparseCoding A joint sparsity pattern—that is, a common set of nonzerocoefficients—canbeimposedtoasetofvectors α1,...,αlthroughagrouped-sparsityregularizeronthe matrix A = [α1,...,αl]in Rk×l (Figure1).Thisamounts torestrictingthenumberofnonzerorowsof A,orreplacing the ℓpvector(pseudo)norminEq.(3)bythe ℓp,q(pseudo) matrixnorm ||A||p,q △ = k� ||α i || p q, (5) i=1 where α i denotesthe i-throwof A.Inpractice,oneusually choosesforthepair (p,q)thevalues (1,2)or (0, ∞),the formerleadingtoaconvexnorm,whilethelatteractually countsthenumberofnonzerorowsandisonlyapseudo norm[28]. 3.2.PrincipleoftheFormulation Non-localmeansfilteringhasprovenveryeffectivein general,butitfailsinsomecases. Intheextreme,when apatchdoesnotlooklikeanyotheroneintheimage,it isimpossibletoexploitself-similaritiestodenoisethecorrespondingpixelvalue. Sparseimagemodelscanhandle suchsituationsbyexploitingtheredundancybetweenoverlappingpatches,buttheysufferfromanotherdrawback: Similarpatchessometimesadmitverydifferentestimates duetothepotentialinstabilityofsparsedecompositions(the ℓ0pseudonormis,afterall,piecewiseconstant,andits ℓ1 counterpartisonlypiecewisedifferentiable),whichcanresultinpracticeinnoticeablereconstructionartefacts.Inthis paper,weaddressthisproblembyforcingsimilarpatchesto admitsimilardecompositions.Concretely,letusdefinefor eachpatch y itheset Siofsimilarpatchesas Si △ = {j = 1,...,n s.t. ||y i − y j|| 2 2 ≤ ξ}, (6) where ξissomethreshold.Letusalsoconsiderforthemomentafixeddictionary Din Rm×k .Decomposingthepatch yiwithagrouped-sparsityregularizerontheset Siamounts tosolving min ||Ai||p,q s.t. Ai � ||yj − Dαij|| j∈Si 2 2 ≤ εi, (7)

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