Spectral Algorithms II - Computer .Applications Spectral Algorithms II Slides based on “Spectral

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  • Applications

    SpectralAlgorithmsIIApplications

    SlidesbasedonSpectralMeshProcessingSiggraph 2010course

  • ApplicationsApplications

    Shape retrievalShaperetrieval

    i i Parameterization 1D 2D

    Quadmeshing

  • Shape RetrievalShapeRetrieval

    3D R it Q M h3DRepository Query Matches

  • Descriptor based shape retrievalDescriptorbasedshaperetrieval

    3D R it d i t Q d i t Cl t t h3DRepositorydescriptors Querydescriptor Closestmatches

  • PoseInvariantShapeDescriptorp p

    Similardescriptorsforshapeindifferentposes

    Cat Samecat Stillthesamecat

  • Spectral Shape DescriptorsSpectralShapeDescriptors

    Use pose invariant operatorsUseposeinvariantoperators Matrixofgeodesicdistances LaplaceBeltramioperatorp p Heatkernel

    Derivedescriptorsfromeigenstructure Eigenvaluesg Distancebaseddescriptorsonspectralembedding Heatkernelsignatureg

  • Geodesic Distances MatrixGeodesicDistancesMatrix

    Operator: Matrix of Gaussianfiltered Operator:MatrixofGaussianfilteredpairwisegeodesicdistances

    2

    22

    2ji pp

    ij eA

    =

    Onlytakek

  • Geodesic Distances MatrixGeodesicDistancesMatrixLFD:Lightfielddescriptor[Chenetal.03]

    SHD:SphericalHarmonicsdescriptor[Kazhdanetal.03]p p [ ]LFDS:LFDonspectralembedding

    SHDS:SHDonspectralembedding

    LFD

    LFDS

    SHD

    SHDS

    RetrievalonMcGillArticulatedShapeDatabase

    EVD

  • LimitationsLimitations

    Geodesic distances sensitive to shortcutsGeodesicdistancessensitiveto shortcuts=smalltopologicalholes

    Shortcircuit

  • Global Point Signatures [Rustamov 07]Global Point Signatures [Rustamov 07]

    Givenapointponthesurface,define

    i(p)valueoftheeigenfunction i atthepointp

    is aretheLaplaceBeltramieigenvalues

    EuclideandistanceinGPSspace=commutetimedistanceonthesurface

  • GPS-based shape retrievalGPS based shape retrieval

    Use histogram of distances in UsehistogramofdistancesintheGPSembeddings

    InvariancepropertiesreflectedinGPSembeddings

    Less sensitive to topology Lesssensitivetotopologychangesbyusingonlylow

    frequency eigenfunctions Shortcircuitfrequencyeigenfunctions

  • MDSonGPS

    2DembeddingthatalmostreproducesGPSdistancesg p

  • Use for shape matching?Useforshapematching?

    Nope. Embedding sensitive to eigenvectorNope.Embeddingsensitivetoeigenvectorswitching

    EigenvectorsarenotuniqueOnly defined up to sign Onlydefineduptosign

    Ifrepeatingeigenvalues anyvectorinsubspaceis eigenvectoriseigenvector

  • Heat Equation on a ManifoldHeatEquationonaManifold

    Heatkernel

    :amountofheattransferredfromtointime.

  • Heat KernelHeatKernel

    OrOr

    EigenvaluesEigenvaluesofof LL

    EigenvectorsEigenvectorsofof LL

    15

    ofofLL ofofLL

  • Heat KernelKt =Fundamentalsolutionto

    heat diffusion equation

    HeatKernel

    heatdiffusionequation

    Prob.ofreachingy fromxaftert randomsteps

    HeatKernelSignature[Sunetal.09]

    16

  • Heat Kernel SignatureHeatKernelSignature

    HKS(p)= :amountofheatleftatp attimet.p pSignatureofapointisafunctionofonevariable.

    Invarianttoisometricdeformations.Moreovercomplete:

    AnycontinuousmapbetweenshapesthatpreservesHKSmusty p p ppreservealldistances.

    AConciseandProvablyInformative,Sunetal.,SGP2009

  • HeatKernelColumn

    18

  • HeatKernelSignature

    19

  • Heat Kernel Applied

    Diffusionwavelets

    HeatKernelApplied

    [Coifman andMaggioni 06]

    Segmentation[deGoes et al 08][deGoes etal. 08]

    Heatkernelsignature[Sunetal.09][ ]

    Heatkernelmatching[Ovsjanikov etal.10]

    20

  • ApplicationsApplications

    Shape retrievalShaperetrieval

    i i Parameterization 1D 2D

    Quadmeshing

  • 1D surface parameterizationGraph Laplacian

    ai,j =wi,j >0if(i,j)isanedge

    a = aai,i = ai,j(1,1 1) is an eigenvector assoc. with 0(1,11)isaneigenvectorassoc.with0

    Thesecondeigenvectorisinterresting[Fiedler73,75]

  • 1D surface parameterizationFiedler vector

    FEM i R d i hFEMmatrix,Nonzero entries

    Reorder withFiedler vector

  • 1D surface parameterizationFiedler vector

    Streamingmeshes[Isenburg & Lindstrom][Isenburg&Lindstrom]

  • 1D surface parameterizationFiedler vector

    Streamingmeshes[Isenburg & Lindstrom][Isenburg&Lindstrom]

  • 1D surface parameterizationFiedler vector

    F( ) ( )2

    F(u) = ut A uMinimize

    F(u) = wij (ui - uj)2

    ( )

  • 1D surface parameterizationFiedler vector

    F( ) ( )2

    F(u) = ut A uMinimize

    F(u) = wij (ui - uj)2

    ( )

    Howtoavoidtrivialsolution?Constrainedvertices?

  • 1D surface parameterizationFiedler vector

    F( ) ( )2

    ui = 0F(u) = ut A uMinimize subjecttoF(u) = wij (ui - uj)2

    ( ) j

    Global constraintsaremoreelegant!

  • 1D surface parameterizationFiedler vector

    F( ) ( )2

    ui = 0

    F(u) = ut A uMinimize subjectto

    F(u) = wij (ui - uj)2

    ui2 = 1( ) j

    Global constraintsaremoreelegant!Weneedalsotoconstrainthesecondmementum

  • 1D surface parameterizationFiedler vector

    F( ) ( )2

    ui = 0

    F(u) = ut A uMinimize subjectto

    F(u) = wij (ui - uj)2

    ui2 = 1( ) j

    L(u) = ut A u - 1 ut 1 - 2 (utu - 1)( ) 1 2 ( )

    u L = A u - 1 1 - 2 u u = eigenvectorofA1L = ut 12L = (ut u 1)

    1 = 02 = eigenvalue

  • 1D surface parameterizationFiedler vector

    Rem:FiedlervectorisalsoaminimizeroftheRayleighquotient

    R(A,x)=

    xt x

    xt Ax

    Theothereigenvectorsxi arethesolutionsof:

    minimizeR(A,xi)subjecttoxit xj =0forj

  • Surface parameterization

    Minimize

    22 uu yy--

    vv xx Discreteconformalmapping:Minimize

    uu xx

    vv yy

    yy--

    TT [L,Petitjean,Ray,Maillot2002][Desbrun,Alliez2002]

  • Surface parameterization

    Discreteconformalmapping:Minimize

    22 uu yy--

    vv xx

    [L,Petitjean,Ray,Maillot2002][Desbrun,Alliez2002]

    Minimize uu xx

    vv yy

    yy--

    TT

    Usespinned points.

  • Sensitive to Pinned VerticesSensitivetoPinnedVertices

  • Surface parameterization

    [Muellen,Tong,Alliez,Desbrun2008]

    UseFiedlervector,i.e.theminimizerofR(A,x)=xt Ax/xt xthat is orthogonal to the trivial constant solutionthatisorthogonaltothetrivialconstantsolution

    Implementation:(1)assemblethematrixofthediscreteconformalparameterization( ) d h h f l(2)computeitseigenvectorassociatedwiththefirstnonzeroeigenvalue

    Seehttp://alice.loria.fr/WIKI/ Graphitetutorials ManifoldHarmonics

  • ApplicationsApplications

    Shape retrievalShaperetrieval

    i i Parameterization 1D 2D

    Quadmeshing

  • Chladni PatternsChladni Patterns

    Nodal sets of eigenfunctions of LaplacianNodalsetsofeigenfunctions ofLaplacian

  • Chladni PatternsChladni Patterns

  • Quad RemeshinggNodal sets are sets of curves intersecting at constant anglesNodal sets are sets of curves intersecting at constant angles

    The NThe N th eigenfunction has at most N eigendomainsth eigenfunction has at most N eigendomainsThe NThe N--th eigenfunction has at most N eigendomainsth eigenfunction has at most N eigendomains

  • Surface quadrangulationg

    One eigenfunction Morse complex Filtered morse complexOneeigenfunction Morsecomplex Filteredmorsecomplex

    [DongandGarland2006]

  • Surface quadrangulationg

    Reparameterizationofthequads

  • Surface quadrangulationg

    Improvementin[Huang,Zhang,Ma,Liu,KobbeltandBao2008],takesaguidancevectorfieldintoaccount.