A Primer on Laplacians. Max Wardetzky. Institute for Numerical and Applied Mathematics Georg-August Universität Göttingen, Germany

Transcription

1 A Primer on Laplacians Max Wardetzky Institute for Numerical and Applied Mathematics Georg-August Universität Göttingen, Germany

2 Warm-up: Euclidean case

3 Warm-up The Euclidean case Chladni s vibrating plates Plate vibrated by violin bow Sand settles on nodal curves

4 Warm-up The Euclidean case u n Basic properties: Z (u, v) = Z (u, u) = u = u ) 0 ru rv =( u, v) (Sym) ru ru 0 (Psd)

5 Warm-up The Euclidean case u n Basic properties: Z (u, v) = Z (u, u) = ru rv =( u, v) (Sym) ru ru 0 (Psd) u harmonic ) no strict loc. max. in (Max)

6 Riemannian case

7 Laplacians on Riemannian manifolds u = div ru Z ru X = Z u ( divx) Using exterior differentiation: Z U d = k-form (Stokes for k-forms) (k+1)-form

8 Laplacians on Riemannian manifolds u = div ru Z ru X = Z u ( divx) Using exterior differentiation: (, ) k := Z M g(, )vol g (inner product on k-forms) (d, ) k+1 =(,d ) k (adjoint to d-operator) := dd + d d (Laplacian on k-forms)

9 Laplacians on Riemannian manifolds u = div ru Z ru X = Z u ( divx) Using exterior differentiation: (, ) k := Z M g(, )vol g (inner product on k-forms) (d, ) k+1 =(,d ) k (adjoint to d-operator) u = d du (Laplacian on functions)

10 Laplacians on Riemannian manifolds u = d du (Laplacian on functions) Basic properties: Z (u, v) = Z (u, u) = M M g(du, dv) =( u, v) g(du, du) 0 u harmonic ) no strict loc. max. in (Sym) (Psd) (Max)

11 Why should we care? Three reasons

12 1. Topology Hodge-Helmholtz decomposition of k-forms: = dµ + d + h (unique and L 2 -orthogonal) h =0 Harmonic forms and cohomology: H k (M; R) = {h h =0}

13 2. Cheeger s constant Cheeger s isoperimetric constant: C := inf N vol n 1 (N) min(vol n (M 1 ), vol n (M 2 )) N M 1 M 2

14 2. Cheeger s constant Cheeger s isoperimetric constant: C := inf N vol n 1 (N) min(vol n (M 1 ), vol n (M 2 )) Cheeger-Buser: 2 C 4 apple 1 apple c(k C + 2 C) 1 st non-trivial eigenvalue of Laplacian

15 3. Rippa s theorem 15 Consider = function on finite point set (in plane)

16 3. Rippa s theorem 16 Consider = function on finite point set (in plane)

17 3. Rippa s theorem 17 Consider = function on finite point set (in plane)

18 3. Rippa s theorem 18 Ø Extend function piecewise linearly (hence continuous) over triangles. Ø Rippa s theorem: Among all possible triangulations, the Delaunay triangulation minimizes the Dirichlet energy E D [u] := 1 2 Z ru ru

19 Laplacian on simplicial meshes (mostly surfaces)

20 Discrete Laplacians many geometric applications 20 mesh denoising parameterization [Desbrun et al. 99] [Gu/Yau 03] mesh editing simulation [Sorkine et al. 04] [Bergou et al. 06]

21 21 Key properties of smooth Laplacians (Sym) (Loc) (Lin) (Psd) (Max) Ø (Sym) Symmetry: Ø (Loc) Locality: changing does not change Ø (Lin) Linear precision: Ø (Psd) Laplacians are positive (semi)definite Ø (Max) Maximum principle

22 22 Discrete Laplacians (Sym) (Loc) (Lin) (Psd) (Max) Discrete Laplace operators: Input: Output: = function on mesh vertices Properties of L are encoded by

23 23 Desired properties (Sym) (Loc) (Lin) (Psd) (Max) 1. (Sym) Symmetry: Motivation: ` Ø smooth symmetry Ø real eigenvalues & orthogonal eigenvectors

24 24 Desired properties (Sym) (Loc) (Lin) (Psd) (Max) 2. (Loc) Locality: if (ij) is not an edge Motivation: ` Ø smooth locality Ø diffusion: discrete: random walk probabilities along edges

25 25 Desired properties (Sym) (Loc) (Lin) (Psd) (Max) 3. (Lin) Linear precision: if mesh is in the plane and u is linear Motivation: ` Ø smooth linear precision Ø mesh denoising: no tangential vertex drift Ø mesh parameterization: planar vertices don t move

26 26 Desired Four desired properties properties (Sym) (Loc) (Lin) (Psd) (Pos) (Max) 4. (Pos) Positivity: (Psd) + (Max) Motivation: ` Ø positive (semi)-definiteness [Gortler/Gotsman/Thurston 05] Ø parameterization: no flipped triangles (locally) Ø barycentric coordinates (maximum principle)

27 27 Four examples (Sym) (Loc) (Lin) (Pos) 1. Combinatorial Laplacians [Tutte 63, ] [Karni/Gotsman 00]

28 Graph Laplacian cont. (Sym) (Loc) (Lin) (Pos) Cheeger constant: vol( 1, 1) := X C := min vol( 1, 1) min(vol( 1 ), vol( 1)) i2v 1,j /2V 1! ij and vol( 1 ):= X i2v 1,j2V 1! ij Γ 1 Γ 1 28

29 Graph Laplacian cont. (Sym) (Loc) (Lin) (Pos) Cheeger constant: vol( 1, 1) := X C := min vol( 1, 1) min(vol( 1 ), vol( 1)) i2v 1,j /2V 1! ij and vol( 1 ):= X i2v 1,j2V 1! ij Discrete Cheeger inequality: 2 C 2 apple 1 apple 2 C 1 st non-trivial eigenvalue of (normalized) graph Laplacian 29

30 30 Four examples (Sym) (Loc) (Lin) (Pos) 2. Mean-value coordinates [Floater 03, ] [Floater 03]

31 31 Four examples (Sym) (Loc) (Lin) (Pos) 3. cotan weights [Pinkall/Poltier 93, McNeal 49, ] [Hildebrandt/Polthier 05]

32 32 cotan Laplacian cont. (Sym) (Loc) (Lin) (Pos) C k = simplicial cochains (dual to simplicial k-chains) C 0 : real values at vertices C 1 : dual to oriented edges C 2 : dual to oriented triangles simplicial coboundary operator: inner product on k-cochains: simplicial codifferential: (, : C k! C k+1 ) k (, ) k =(, ) k+1 L := +

33 33 cotan Laplacian cont. (Sym) (Loc) (Lin) (Pos) L := + Hodge-Helmholtz decomposition of k-cochains: = µ + + h Lh =0 Harmonic forms and cohomology: H k (M; R) = {h Lh =0} Ø Whitney-map (lin. interpolation) & L 2 inner product lead to cotan Laplacian

34 34 Four examples (Sym) (Loc) (Lin) (Pos) 4. intrinsic Delaunay [Bobenko/Sprinborn 05, ] [Fisher et al. 06] [intrinsic edge flips]

35 Putting four things together 35 (Sym) (Loc) (Lin) (Pos) mean value ø ü ü ü intrinsic Delaunay ü ø ü ü combinatorial ü ü ø ü cotan ü ü ü ø on general irregular meshes!

36 Putting four things together 36 No-free-lunch-theorem (preliminary version) General meshes do not allow for discrete Laplacians with (Sym)+(Loc)+(Lin)+(Pos).

37 Sketch of proof

38 Proof: planar stress frameworks (Sym)+(Loc) & stress frameworks in the plane contracting

39 Proof: planar stress frameworks (Sym)+(Loc) & stress frameworks in the plane expanding

40 Proof: planar stress frameworks (Sym)+(Loc)+(Lin) inner vertices are in force balance [e.g., use cotan weights] fixed boundary vertices contracting edges expanding edges

41 Proof: planar stress frameworks (Sym)+(Loc)+(Lin)+(Pos) Ø contracting forces: net torque Ø need negative weights Ø not all meshes allow for perfect Laplacians [Schönhardt polytope] QED

42 Which meshes allow for perfect Laplacians?

43 Which meshes allow for perfect Laplacians? 43 Theorem (Maxwell-Cremona 1864) A stress framework in the plane is in force-balance iff there exists an orthogonal dual graph. primal edge dual edge orthogonal crossing Example: Delaunay triangulation & Voronoi dual

44 44 Which meshes allow for perfect Laplacians? Theorem (Maxwell-Cremona 1864) (Sym)+(Loc)+(Lin) orthogonal duals Proof: 1) Given (Sym)+(Loc)+(Lin), observe that

45 45 Which meshes allow for perfect Laplacians? Theorem (Maxwell-Cremona 1864) (Sym)+(Loc)+(Lin) orthogonal duals Proof: 1) Given (Sym)+(Loc)+(Lin), observe that

46 46 Which meshes allow for perfect Laplacians? Theorem (Maxwell-Cremona 1864) (Sym)+(Loc)+(Lin) orthogonal duals Proof: 1) Given (Sym)+(Loc)+(Lin), observe that

47 47 Which meshes allow for perfect Laplacians? Theorem (Maxwell-Cremona 1864) (Sym)+(Loc)+(Lin) orthogonal duals Proof: 1) Given (Sym)+(Loc)+(Lin), observe that

48 48 Which meshes allow for perfect Laplacians? Theorem (Maxwell-Cremona 1864) (Sym)+(Loc)+(Lin) orthogonal duals Proof: 1) Given (Sym)+(Loc)+(Lin), observe that

49 49 Which meshes allow for perfect Laplacians? Theorem (Maxwell-Cremona 1864) (Sym)+(Loc)+(Lin) orthogonal duals Proof: 1) Given (Sym)+(Loc)+(Lin), observe that

50 50 Which meshes allow for perfect Laplacians? Theorem (Maxwell-Cremona 1864) (Sym)+(Loc)+(Lin) orthogonal duals Proof: 1) Define dual edges by Get closed dual cycles.

51 Which meshes allow for perfect Laplacians? 51 Theorem (Maxwell-Cremona 1864) (Sym)+(Loc)+(Lin) orthogonal duals Proof: 2) Vice-versa, given orthogonal dual, define Closed dual cycles give (Lin).

52 52 Which meshes allow for perfect Laplacians? Theorem (Maxwell-Cremona 1864) (Sym)+(Loc)+(Lin) orthogonal duals & Theorem (Aurenhammer 1987) Orthogonal duals w/ pos. weights [not regular] regular triangulations No-free-lunch-theorem (W., Mathur, Kälberer, Grinspun) (Sym)+(Loc)+(Lin)+(Pos) regular triangulations

53 Regular triangulations 53 Regular triangulations: Ø Delaunay Ø more generally: weighted Delaunay [Edelsbrunner/Shah 92, Bobenko/Springborn 05, Glickenstein 05] Intrinsic weighted-delaunay-laplacians: Ø break (Loc) wrt. input mesh

54 Taxonomy on literature 54 Laplacian Zoo Ø dropping (Loc): weighted Delaunay Laplacians Ø dropping (Sym): barycentric coordiantes Ø dropping (Lin): combinatorial Laplacians Ø dropping (Pos): cotan weights and generalizations no free lunch!

55 Application: Geodesic distance computation

56 Problem formulation 56

57 Problem formulation 57

58 Problem formulation 58

59 Problem formulation 59

60 Challenges Challenges [Dijkstra 1959, Mitchell et al 1987, Chen & Han 1990, Sethian & Kimmel 1998, Surazhsky et al ]

61 Challenges Challenges No reuse of information! [Dijkstra 1959, Mitchell et al 1987, Chen & Han 1990, Sethian & Kimmel 1998, Surazhsky et al ]

62 Challenges Challenges

63 Challenges Challenges Highly specialized!

64 The Heat Method [Crane, Weischedel, W. 2013] Solve two standard linear equations heat equation Poisson equation Fast, general, simple parallelize prefactor any spatial discretization easy to implement

65 Eikonal equation distance to source distance changes at one meter per meter

66 Varadhan s formula Varadhan s Formula distance to source heat kernel [Varadhan 1967]

67 Just apply Varadhan s formula? Just Apply Varadhan s Formula?

68 Normalizing the gradient Normalizing the Gradient Eikonal:

69 Recovering distance Recovering Distance

70 The Heat Method Linear Easy Title Linear

71 Temporal Discretization Temporal Discretization heat equation backward Euler linear elliptic equation

72 Choosing t Optimal t

73 Choosing t mean error Optimal t m

74 Spatial Discretization Triangle Meshes [MacNeal 1949] Point Clouds [Liu et al 2012] Regular Grids [Newton 1693] Polygon Meshes [Alexa & W. 2011] Tet Meshes [Desbrun et al 2008]

75 Convergence Rate of Convergence

76 Prefactorization Prefactorization

77 Prefactorization Prefactorization

78 Performance Performance

79 Visual Comparison of Accuracy Visual Comparison of Accuracy

80 Visual Comparison of Accuracy Visual Comparison of Accuracy fast marching heat method exact polyhedral

81 Visual Comparison of Accuracy Visual Comparison of Accuracy fast marching heat method exact polyhedral

82 Medial Axis Medial Axis fast marching heat method

83 Example: Distance to boundary Example: Distance to Boundary

84 Example: Robustness Example: Robustness

85 Example: Point Cloud Example: Point Cloud

86 Example: Polygonal Mesh Example: Polygonal Mesh

87 Example: Regular Grid Example: Regular Grid

88 Example: Noise Noise

89 Challenge Challenges Show convergence of heat method under refinement.

90 Thank you!