# Spectral Algorithms II - Computer .Applications Spectral Algorithms II Slides based on â€œSpectral

• View
215

• Download
0

Embed Size (px)

### Text of Spectral Algorithms II - Computer .Applications Spectral Algorithms II Slides based on...

• 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.

Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Software
Documents