Introduc1on to Computa1onal Manifolds and Applica1ons
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1 Trimester Program on Computa1onal Manifolds and Applica1ons Introduc1on to Computa1onal Manifolds and Applica1ons Manifold Harmonics Luis Gustavo Nonato Depto Matemá3ca Aplicada e Esta9s3ca ICMC- USP- Brazil
2 Summary Today (Tuesday): Differen3al Operators on Surfaces - Differen3al operators in the parametric domain - Cotangent formula - Belkin s approach - SPH- based scheme Thursday: Manifold Harmonics and Applica3ons - Some theore3cal background - Mesh Filtering - Embedding in high- dimension - Fiedler tree - Heat Trace
3 Spectral Mesh Processing Although rela3vely recent in the context of Geometry Processing, spectral methods have already experienced a large development in the field of spectral graph theory.
4 Spectral Mesh Processing Although rela3vely recent in the context of Geometry Processing, spectral methods have already experienced a large development in the field of spectral graph theory. Those techniques rely on spectrum of a Laplacian- like matrix.
5 Laplacian Matrices
6 Laplacian Matrices
7 Laplacian Matrices Cotangent Formula!!
8 Short Review of Eigenvalues and Eigenvectors
9 Short Review of Eigenvalues and Eigenvectors eigenvalue eigenvector
10 Short Review of Eigenvalues and Eigenvectors ARPACK Large sparse matrices Lanczos algorithm (derived from the power method)
11 Short Review of Eigenvalues and Eigenvectors
12 Short Review of Eigenvalues and Eigenvectors
13 Spectral Mesh Processing There are three main steps involved in most spectral mesh processing methods: 1. Construc3on of the matrix L 2. Eigendecomposi3on of L. 3. Handling the eigendecomposi3on towards obtaining the desired results.
14 Spectral Mesh Processing
15 Spectral Mesh Processing
16 Spectral Mesh Processing eigenvalues eigenvectors
17 Spectral Mesh Processing eigenvalues eigenvectors Fourier Basis
18 Spectral Mesh Processing filter
19 Spectral Mesh Processing filter For surfaces, the spectrum of the Laplace operator behaves quite similarly to a Fourier basis, allowing for filtering func3ons defined on the surface.
20 Spectral Mesh Processing In par3cular, if the coordinates of the ver3ces of surface mesh are seem as func3ons defined on the surface, band- pass filtering can be performed. [Vallet and Levy, SGP 08]
21 Spectral Mesh Processing In par3cular, if the coordinates of the ver3ces of surface mesh are seem as func3ons defined on the surface, band- pass filtering can be performed
22 [Taubin, Siggraph 95] Spectral Mesh Processing
23 [Taubin, Siggraph 95] Spectral Mesh Processing
![[Taubin, Siggraph 95]](/docs-images/87/96350401/images/24-3.jpg "Spectral Mesh")
24 [Taubin, Siggraph 95] Spectral Mesh Processing
25 [Taubin, Siggraph 95] Spectral Mesh Processing
26 [Taubin, Siggraph 95] Spectral Mesh Processing
27 Spectral Mesh Processing [Taubin, Siggraph 95] Avoid to compute the spectrum
28 [Taubin, Siggraph 95]
29 Spectral Mesh Processing What about eigenvectors?
30 Spectral Mesh Processing Nodal Domain: The nodal set of an eigenfunc3on is the set of points where the eigenfunc3on is zero. The regions bounded by the nodal set are called nodal domains.
31 Spectral Mesh Processing Nodal Domain: The nodal set of an eigenfunc3on is the set of points where the eigenfunc3on is zero. The regions bounded by the nodal set are called nodal domains. An eigenfunc3on is built by interpola3ng the values of an eigenvector (defined on the ver3ces of a mesh) in each point of the surface.
32 Spectral Mesh Processing Courant's Nodal Theorem: Let the eigenvectors of the Laplace operator be labeled in ascending order according to the corresponding eigenvalues. Then, the k- th eigenfunc3on has at most k nodal domains, that is, the k- th eigenfunc3on can separate the surface into at most k connected components.
33 Spectral Mesh Processing Courant's Nodal Theorem: Let the eigenvectors of the Laplace operator be labeled in ascending order according to the corresponding eigenvalues. Then, the k- th eigenfunc3on has at most k nodal domains, that is, the k- th eigenfunc3on can separate the surface into at most k connected components.
34 Zero is an eigenvalue of the Laplace operator with a constant corresponding eigenvector.
35 Spectral Mesh Processing Eigenvectors capture symmetries of the model; Invariant by isometric transforma3on; Not sensi3ve to small topological and geometrical changes
36 Spectral Mesh Processing Eigenvectors capture symmetries of the model; Invariant by isometric transforma3on; Not sensi3ve to small topological and geometrical changes Powerful tool for many mesh processing tasks.
37 Spectral Mesh Processing Mesh Segmenta3on [O. Sidi et al., SigAsia 11]
38 Spectral Mesh Processing Global Point Signature [Rustamov., SGP 07] Euclidean distance in the GPS space is related to Green s func3on on the surface.
39 Spectral Mesh Processing Global Point Signature [Rustamov., SGP 07]
40 Spectral Mesh Processing Global Point Signature [Rustamov., SGP 07]
41 Global Point Signature Spectral Mesh Processing [Rustamov., SGP 07]
42 Global Point Signature Spectral Mesh Processing [Rustamov., SGP 07]
43 Diffusion Maps Spectral Mesh Processing [Goes, SGP 08] Euclidean distance in the DM space is related to diffusion distance on the surface.
44 Diffusion Maps Spectral Mesh Processing [Goes, SGP 08]
45 Diffusion Maps Spectral Mesh Processing [Goes, SGP 08]
46 Spectral Mesh Processing The eigenvector corresponding to the smallest non- zero eigenvalue is called Fiedler vector and it is characterized by:
47 Spectral Mesh Processing The eigenvector corresponding to the smallest non- zero eigenvalue is called Fiedler vector and it is characterized by: Will be minimum when adjacent ver3ces have similar values.
48 Spectral Mesh Processing The Fiedler vector also generates nodal domains with similar areas and minimal boundary curve
49 Fiedler Tree Spectral Mesh Processing [Berger, SMI 09]
50 Fiedler Tree Spectral Mesh Processing [Berger, SMI 09]
51 Fiedler Tree Spectral Mesh Processing [Berger, SMI 09]
52 Fiedler Tree Spectral Mesh Processing [Berger, SMI 09]
53 Some Interes3ng Results Spectral Mesh Processing
54 Some Interes3ng Results Spectral Mesh Processing Heat Trace
55 Some Interes3ng Results Spectral Mesh Processing Heat Trace
56 Spectral Mesh Processing
57 That is all Folks!!
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