Spectral Algorithms II - Computer...

Applications

Spectral Algorithms IIApplications

Slides based on “Spectral Mesh Processing” Siggraph 2010 course

ApplicationsApplications

• Shape retrievalShape retrieval

i i• Parameterization– 1D

– 2D

• Quad meshing

Shape RetrievalShape Retrieval

3D R it Q M h3D Repository Query Matches

Descriptor based shape retrievalDescriptor based shape retrieval

3D R it d i t Q d i t Cl t t h3D Repository descriptors Query descriptor Closest matches

Pose Invariant Shape Descriptorp p

“Similar” descriptors for shape in different poses

Cat Same cat Still the same cat

Spectral Shape DescriptorsSpectral Shape Descriptors

• Use pose invariant operatorsUse pose invariant operators– Matrix of geodesic distances– Laplace‐Beltrami operatorp p– Heat kernel

• Derive descriptors from eigen‐structure– Eigenvaluesg– Distance based descriptors on spectral embedding– Heat kernel signatureg

Geodesic Distances MatrixGeodesic Distances Matrix

• Operator: Matrix of Gaussian‐filtered• Operator: Matrix of Gaussian‐filtered

pair‐wise geodesic distances 2

2

2

2σji pp

ij eA−

−=

• Only take k << n samples

• Descriptor: eigenvalues of matrix [Jian and Zhang 06]

Geodesic Distances MatrixGeodesic Distances MatrixLFD: Light‐field descriptor [Chen et al. 03]

SHD: Spherical Harmonics descriptor [Kazhdan et al. 03]p p [ ]LFD‐S: LFD on spectral embedding

SHD‐S: SHD on spectral embedding

LFD

LFD‐S

SHD

SHD‐S

Retrieval on McGill Articulated Shape Database

EVD

LimitationsLimitations

• Geodesic distances sensitive to “shortcuts”Geodesic distances sensitive to shortcuts= small topological holes

Short circuit

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

Given a point p on the surface, define

– φi(p) value of the eigenfunction φi at the point p

– λi’s are the Laplace‐Beltrami eigenvalues

– Euclidean distance in GPS space =

commute time distance on the surface

GPS-based shape retrievalGPS based shape retrieval

• Use histogram of distances in• Use histogram of distances in

the GPS embeddings

– Invariance properties reflected

in GPS embeddings

– Less sensitive to topology– Less sensitive to topology

changes by using only low‐

frequency eigenfunctions Short circuitfrequency eigenfunctions

MDS on GPS

2D embedding that “almost” reproduces GPS distancesg p

Use for shape matching?Use for shape matching?

• Nope. Embedding sensitive to eigenvectorNope. Embedding sensitive to eigenvector “switching”

• Eigenvectors are not uniqueOnly defined up to sign– Only defined up to sign

– If repeating eigenvalues – any vector in subspace is eigenvectoris eigenvector

Heat Equation on a ManifoldHeat Equation on a Manifold

Heat kernel

: amount of heat transferred from to in time .

Heat KernelHeat Kernel

OrOr

Eigenvalues Eigenvalues ofof LL

EigenvectorsEigenvectorsofof LL

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of of LL of of LL

Heat KernelKt = Fundamental solution to

heat diffusion equation

Heat Kernel

heat diffusion equation

Prob. of reaching y from xafter t random steps

Heat Kernel Signature[Sun et al. 09]

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Heat Kernel SignatureHeat Kernel Signature

HKS(p) = : amount of heat left at p at time t.p pSignature of a point is a function of one variable.

Invariant to isometric deformations. Moreover complete:

Any continuous map between shapes that preserves HKS must y p p ppreserve all distances.

A Concise and Provably Informative …,Sun et al., SGP 2009

Heat Kernel Column

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Heat Kernel Signature

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Heat Kernel Applied

• Diffusion wavelets

Heat Kernel Applied

[Coifman and Maggioni ‘06]

• Segmentation [deGoes et al ‘08][deGoes et al. 08]

• Heat kernel signature [Sun et al. ‘09][ ]

• Heat kernel matching [Ovsjanikov et al. ’10]

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– 2D

• Quad meshing

1D surface parameterizationGraph Laplacian

ai,j = wi,j > 0 if (i,j) is an edge

a = Σ aai,i = ‐Σ ai,j

(1,1 … 1) is an eigenvector assoc. with 0(1,1 … 1) is an eigenvector assoc. with 0

The second eigenvector is interresting[Fiedler 73, 75]

1D surface parameterizationFiedler vector

FEM i R d i hFEM matrix,Non‐zero entries

Reorder withFiedler vector


Streaming meshes [Isenburg & Lindstrom][Isenburg & Lindstrom]


F( ) Σ ( )2

F(u) = ½ ut A uMinimize

F(u) = Σ wij (ui - uj)2

( )


F( ) Σ ( )2

F(u) = ½ ut A uMinimize


( )

How to avoid trivial solution ?Constrained vertices ?


F( ) Σ ( )2

Σ ui = 0F(u) = ½ ut A uMinimize subject to


( ) j

Global constraints are more elegant !


F( ) Σ ( )2

Σ ui = 0

ΣF(u) = ½ ut A uMinimize subject to


½ Σ ui2 = 1

( ) j

Global constraints are more elegant !We need also to constrain the second mementum


F( ) Σ ( )2

Σ ui = 0

ΣF(u) = ½ ut A uMinimize subject to


½ Σ ui2 = 1

( ) j

L(u) = ½ ut A u - λ1 ut 1 - λ2 ½ (utu - 1)( ) 1 2 ( )

∇u L = A u - λ1 1 - λ2 u u = eigenvector of A∇λ1L = ut 1∇λ2L = ½(ut u – 1)

λ1 = 0λ2 = eigenvalue


Rem: Fiedler vector is also a minimizer of the Rayleigh quotient

R(A,x) =

xt x

xt A x

The other eigenvectors xi are the solutions of :

minimize R(A,xi) subject to xit xj = 0 for j < i ( , i) j i j j

Surface parameterization

Minimize

22∂∂ uu∂∂ yy--

∂∂ vv∂∂ xxΣΣ Discrete conformal mapping:

Minimize∂∂ uu∂∂ xx

∂∂ vv∂∂ yy

yy--ΣΣ

TT [L, Petitjean, Ray, Maillot 2002][Desbrun, Alliez 2002]


Discrete conformal mapping:Minimize

22∂∂ uu∂∂ yy--

∂∂ vv∂∂ xxΣΣ

[L, Petitjean, Ray, Maillot 2002][Desbrun, Alliez 2002]

Minimize∂∂ uu∂∂ xx

∂∂ vv∂∂ yy

yy--ΣΣ

TT

Usespinned points.

Sensitive to Pinned VerticesSensitive to Pinned Vertices


[Muellen, Tong, Alliez, Desbrun 2008]

Use Fiedler vector,i.e. the minimizer of R(A,x) = xt A x / xt x that is orthogonal to the trivial constant solutionthat is orthogonal to the trivial constant solution

Implementation: (1) assemble the matrix of the discrete conformal parameterization( ) d h h f l(2) compute its eigenvector associated with the first non‐zero eigenvalue

See http://alice.loria.fr/WIKI/ Graphite tutorials – Manifold Harmonics




– 2D

• Quad meshing

Chladni PatternsChladni Patterns

Nodal sets of eigenfunctions of LaplacianNodal sets of eigenfunctions of Laplacian

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 complexOne eigenfunction Morse complex Filtered morse complex

[Dong and Garland 2006]


Reparameterization of the quads


Improvement in [Huang, Zhang, Ma, Liu, Kobbelt and Bao 2008], takes a guidance vector field into account.

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