Spectral Algorithms II

Transcription

1 Spectral Algorithms II Applications Slides based on Spectral Mesh Processing Siggraph 2010 course

2 Applications Shape retrieval Parameterization i 1D 2D Quad meshing

3 Shape Retrieval 3D Repository Query Matches

4 Descriptor based shape retrieval 3D Repository descriptors Query descriptor Closest matches th

5 Pose Invariant Shape Descriptor Similar descriptors for shape in different poses Cat Same cat Still the same cat

6 Spectral Shape Descriptors Use pose invariant operators Matrix of geodesic distances Laplace Beltrami operator Heat kernel Derive descriptors from eigen structure Eigenvalues Distance based descriptors on spectral embedding Heat kernel signature

7 Geodesic Distances Matrix Operator: Matrixof Gaussian filtered pair wise geodesic distances A ij = e p i p 2 σ j 2 2 Only take k << n samples Descriptor: eigenvalues of matrix [Jian and Zhang 06]

8 Geodesic Distances Matrix LFD: Light field descriptor [Chen et al. 03] SHD: Spherical Harmonics descriptor [Kazhdan et al. 03] LFD S: LFD on spectral embedding SHD S: SHD on spectral embedding LFD LFD S SHD SHD S EVD Retrieval on McGill Articulated Shape Database

9 Limitations Geodesic distances sensitive to shortcuts = small topological holes Short circuit

10 Global Point Signatures [Rustamov 07] Given a point p on the surface, define φ i (p) value of the eigenfunction φ i at the point p λ i s are the Laplace Beltrami eigenvalues Euclidean distance in GPS space = commute time distance on the surface

11 GPS-based shape retrieval Use histogram of distances in the GPS embeddings Invariance properties reflected in GPS embeddings Less sensitive to topology changes by using only low frequency eigenfunctions Short circuit

12 MDS on GPS 2D embedding that almost reproduces GPS distances

13 Use for shape matching? Nope. Embedding sensitive to eigenvector switching Eigenvectors are not unique Only defined up to sign If repeating eigenvalues any vector in subspace is eigenvector

14 Heat Equation on a Manifold Heat kernel : amount of heat transferred from to in time.

15 Heat Kernel Or Eigenvalues of L Eigenvectors of L 15

16 Heat Kernel K t = Fundamental solution to heat diffusion equation Prob. of reaching y from x after t random steps Heat Kernel Signature [Sun et al. 09] 16

17 Heat Kernel Signature HKS(p) p = : amount of heat left at p at time t. Signature of a point is a function of one variable. Invariant to isometric deformations. Moreover complete: Any continuous map between shapes that preserves HKS must preserve all distances. A Concise and Provably Informative, Sun et al., SGP 2009

18 Heat Kernel Column 18

19 Heat Kernel Signature 19

20 Diffusion wavelets [Coifman and Maggioni 06] Segmentation [degoes et al. 08] Heat kernel signature [Sun et al. 09] Heat kernel matching [Ovsjanikov et al. 10] Heat Kernel Applied 20

21 Applications Shape retrieval Parameterization i 1D 2D Quad meshing

22 1D surface parameterization Graph Laplacian a i,j = w i,j > 0 if (i,j) is an edge a Σ i,i = Σ a i,j (1,1 1) is an eigenvector assoc. with 0 The second eigenvector is interresting [Fiedler 73, 75]

23 1D surface parameterization Fiedler vector FEM matrix, Reorder with ih Non zero entries Fiedler vector

24 1D surface parameterization Fiedler vector Streaming meshes [Isenburg & Lindstrom]

25 1D surface parameterization Fiedler vector Streaming meshes [Isenburg & Lindstrom]

26 1D surface parameterization Fiedler vector F(u) = Σ w ij (u i -u j ) 2 Minimize F(u) = ½ u t A u

27 1D surface parameterization Fiedler vector F(u) = Σ w ij (u i -u j ) 2 Minimize F(u) = ½ u t A u How to avoid trivial solution? Constrained vertices?

28 1D surface parameterization Fiedler vector F(u) = Σ w ij (u i -u j ) 2 Σ u i = 0 Minimize F(u) = ½ u t A u subject to Global constraints are more elegant!

29 1D surface parameterization Fiedler vector F(u) = Σ w ij (u i -u j ) 2 Σ u i = 0 F(u) = ½ u t A u subject to Σ Minimize Global constraints are more elegant! We need also to constrain the second mementum ½ Σ u i2 = 1

30 1D surface parameterization Fiedler vector F(u) = Σ w ij (u i -u j ) 2 Σ u i = 0 F(u) = ½ u t A u subject to Σ Minimize L(u) = ½ u t A u - λ 1 u t 1 - λ 2 ½ (u t u - 1) ½ Σ u i2 = 1 u L = A u - λ λ 2 u λ1 L = u t 1 λ2 L = ½(u t u 1) u = eigenvector of A λ 1 = 0 λ 2 = eigenvalue

31 1D surface parameterization Fiedler vector Rem: Fiedler vector is also a minimizer of the Rayleigh quotient R(A,x) = x t A x x t x The other eigenvectors x i are the solutions of : minimize R(A,x i ) subject to x it x j = 0 for j < i

32 Surface parameterization Minimize Σ T v x v y - - u y u x 2 Discrete conformal mapping: [L, Petitjean, Ray, Maillot 2002] [Desbrun, Alliez 2002]

33 Surface parameterization Minimize Σ T v x v y - - u y u x 2 Discrete conformal mapping: [L, Petitjean, Ray, Maillot 2002] [Desbrun, Alliez 2002] Uses pinned points.

34 Sensitive to Pinned Vertices

35 Surface parameterization [Muellen, Tong, Alliez, Desbrun 2008] Use Fiedler vector, i.e. the minimizer of R(A,x) = x t A x / x t x that is orthogonal to the trivialconstant solution Implementation: (1) assemble the matrix of the discrete conformal parameterization (2) compute its eigenvector associated with the first non zero eigenvalue See Graphite tutorials Manifold Harmonics

36 Applications Shape retrieval Parameterization i 1D 2D Quad meshing

37 Chladni Patterns Nodal sets of eigenfunctions of Laplacian

38 Chladni Patterns

39 Quad Remeshing Nodal sets are sets of curves intersecting at constant angles The N-th eigenfunction has at most N eigendomains

40 Surface quadrangulation Oneeigenfunction eigenfunction Morsecomplex Filtered morsecomplex [Dong and Garland 2006]

41 Surface quadrangulation Reparameterization of the quads

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