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SIGGRAPH 2010

SIGGRAPH 2010. “ Spectral Mesh Processing” Bruno Lévy and Richard Hao Zhang. Spectral Mesh Processing Applications 2/2. Bruno Lévy INRIA - ALICE. Overview. 1D parameterization Surface quadrangulation Surface parameterization Surface characterization Green function Heat kernel.

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SIGGRAPH 2010

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  1. SIGGRAPH 2010 “Spectral Mesh Processing” Bruno Lévy and Richard Hao Zhang

  2. Spectral Mesh Processing Applications 2/2 Bruno Lévy INRIA - ALICE

  3. Overview • 1D parameterization • Surface quadrangulation • Surface parameterization • Surface characterization Green function Heat kernel 

  4. 1D surface parameterizationGraph Laplacian ai,j = wi,j > 0 if (i,j) is an edge ai,i = -S ai,j (1,1 … 1) is an eigenvector assoc. with 0 The second eigenvector is interresting [Fiedler 73, 75]

  5. 1D surface parameterizationFiedler vector Reorder with Fiedler vector FEM matrix, Non-zero entries

  6. 1D surface parameterizationFiedler vector Streaming meshes [Isenburg & Lindstrom]

  7. 1D surface parameterizationFiedler vector Streaming meshes [Isenburg & Lindstrom]

  8. F(u) = ½ ut A u Minimize 1D surface parameterizationFiedler vector F(u) = S wij (ui - uj)2

  9. F(u) = ½ ut A u Minimize 1D surface parameterizationFiedler vector F(u) = S wij (ui - uj)2 How to avoid trivial solution ? Constrained vertices ?

  10. F(u) = ½ ut A u Minimize 1D surface parameterizationFiedler vector F(u) = S wij (ui - uj)2 S ui = 0 subject to Global constraints are more elegant !

  11. F(u) = ½ ut A u Minimize 1D surface parameterizationFiedler vector F(u) = S wij (ui - uj)2 S ui = 0 subject to ½ S ui2 = 1 Global constraints are more elegant ! We need also to constrain the second mementum

  12. F(u) = ½ ut A u Minimize u L = A u - l11 - l2 u l1L = ut1 l2L = ½(ut u – 1) 1D surface parameterizationFiedler vector F(u) = S wij (ui - uj)2 S ui = 0 subject to ½ S ui2 = 1 L(u) = ½ ut A u - l1 ut1 - l2½ (utu - 1) u = eigenvector of A l1 = 0 l2 = eigenvalue

  13. 1D surface parameterizationFiedler vector Rem: Fiedler vector is also a minimizer of the Rayleigh quotient R(A,x) = xt A x xt x The other eigenvectors xi are the solutions of : minimize R(A,xi) subject to xit xj = 0 for j < i

  14. Overview • 1D parameterization • Surface quadrangulation • Surface parameterization • Surface characterization Green function Heat kernel 

  15. Surface quadrangulation Nodal sets are sets of curves intersecting at constant angles The N-th eigenfunction has at most N eigendomains

  16. Surface quadrangulation

  17. Surface quadrangulation [L 2006], [Vallet & L 2006]

  18. Surface quadrangulation Filtered morse complex One eigenfunction Morse complex [Dong and Garland 2006]

  19. Surface quadrangulation Reparameterization of the quads

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

  21. Overview • 1D parameterization • Surface quadrangulation • Surface parameterization • Surface characterization Green function Heat kernel

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

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

  24. Surface parameterization [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 solution Implementation: (1) assemble the matrix of the discrete conformal parameterization (2) compute its eigenvector associated with the first non-zero eigenvalue See http://alice.loria.fr/WIKI/ Graphite tutorials – Manifold Harmonics

  25. Overview • 1D parameterization • Surface quadrangulation • Surface parameterization • Surface characterization Green function Heat kernel 

  26. Surface characterization Green Function Solving Poisson equation: f = g f = G(x,y)f(y) dy Where G: Green function is defined by: G(x,y) = d(x-y) d : dirac

  27. Surface characterization Green Function Solving Poisson equation: f = g f = G(x,y)f(y) dy Where G: Green function is defined by: G(x,y) = d(x-y) d : dirac Proof:   G(x,y)g(y) dy = d(x-y)g(y)dy = g(x) =  f(x)

  28. Surface characterization Green Function Solving Poisson equation: f = g f = G(x,y)f(y) dy Where G: Green function is defined by: G(x,y) = d(x-y) d : dirac Proof:   G(x,y)g(y) dy = d(x-y)g(y)dy = g(x) =  f(x)  f(x) = g(x) =   G(x,y)g(y)dy=  (G(x,y)g(y)dy)

  29. Surface characterization Green Function Solving Poisson equation: f = g f = G(x,y)f(y) dy Where G: Green function is defined by: G(x,y) = d(x-y) d : dirac Proof:   G(x,y)g(y) dy = d(x-y)g(y)dy = g(x) =  f(x)  f(x) = g(x) =   G(x,y)g(y)dy=  (G(x,y)g(y)dy) f(x) = G(x,y)g(y)dy

  30. Surface characterization Green Function How to compute G ? G is defined by: G(x,y) = d(x-y) d : dirac d(x-y) =  i(x) i(y) (completeness of the eigen decomposition)

  31. Surface characterization Green Function How to compute G ? G is defined by: G(x,y) = d(x-y) d : dirac d(x-y) =  i(x) i(y) (completeness of the eigen decomposition)  i(x) i(y) Works ! Using G(x,y) = li (G(x,y) =  i(x) i(y) = d(x-y) ) Note: Convergence of G series needs to be proved (complicated)

  32. Surface characterization Green Function  i(x) i(y) G(x,y) = li Summary: Solution Poisson equation: f = g f = G(x,y)f(y) dy Pose-invariant embedding Connection with GPS embedding [Rustamov 2007] GPS(x) = [1(x)/l1 , 2(x)/l2 , … , i(x)/li , … ] G(x,y) = GPS(x) * GPS(y)

  33. Overview • 1D parameterization • Surface quadrangulation • Surface parameterization • Surface characterization Green function Heat kernel 

  34. Surface characterization Heat equation f(x) (heat) The heat equation:  f = - f  t x t = 0

  35. Surface characterization Heat equation f(x) (heat) The heat equation:  f = - f  t x t = 100

  36. Surface characterization Heat equation f(x) (heat) The heat equation:  f = - f  t x t = 1000

  37. Surface characterization Heat equation Heat kernel: K(t,x,y) =  e-lit i(x) i(y) The heat equation:  f Solution of the heat equation: f(t,x) =  K(t,x,y) f(0,y) dy = - f  t

  38. Surface characterization Heat equation - What is the meaning of K(t,x,y) ? How much heat do we get at y after t seconds ? K(t,x,y) Initial time: we inject a Dirac of heat at x y y x x

  39. Surface characterization Heat equation Heat Kernel Signature [Sun, Ovsjanikov and Gubas 09] Auto-diffusion [Gebal, Baerentzen, Anaes and Larsen 09] How much heat remains at x after t seconds ? ADF(t,x) = HKS(t,x) = K(t,x,x) =  e-lit i2(x) Applications: shape signature, segmentation using Morse decomposition, …

  40. Summary • Minimizing Rayleigh quotient instead of using « pinning » • enforces global constraints (moments) that avoid the trivial solution • fieldler vector for streaming meshes [Isenburg et.al] • Spectral conformal parameterization [Muellen et.al] • The notion of fundamental solution plays a … fundamental role. • Strong connections with spectral analysis (and this is what Fourier • invented Fourier analysis for !) • Green function / Poisson equation - GPS coordinates [Rustamov] • Heat kernel signature ,[Sun et.al] / auto-diffusion function [Gebal et.al] • More to explore: the Variational Principle (see Wikipedia)

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