Laplace-Beltrami: The Swiss Army Knife of Geometry Processing

Transcription

1 Laplace-Beltrami: The Swiss Army Knife of Geometry Processing (SGP 2014 Tutorial July ) Justin Solomon / Stanford University Keenan Crane / Columbia University Etienne Vouga / Harvard University

2 INTRODUCTION

3 Introduction Laplace-Beltrami operator ( Laplacian ) provides a basis for a diverse variety of geometry processing tasks.

4 Introduction Laplace-Beltrami operator ( Laplacian ) provides a basis for a diverse variety of geometry processing tasks. Remarkably common pipeline:

5 Introduction Laplace-Beltrami operator ( Laplacian ) provides a basis for a diverse variety of geometry processing tasks. Remarkably common pipeline: 1 simple pre-processing (build f )

6 Introduction Laplace-Beltrami operator ( Laplacian ) provides a basis for a diverse variety of geometry processing tasks. Remarkably common pipeline: 1 simple pre-processing (build f ) 2 solve a PDE involving the Laplacian (e.g., Du = f )

7 Introduction Laplace-Beltrami operator ( Laplacian ) provides a basis for a diverse variety of geometry processing tasks. Remarkably common pipeline: 1 simple pre-processing (build f ) 2 solve a PDE involving the Laplacian (e.g., Du = f ) 3 simple post-processing (do something with u)

8 Introduction Laplace-Beltrami operator ( Laplacian ) provides a basis for a diverse variety of geometry processing tasks. Remarkably common pipeline: 1 simple pre-processing (build f ) 2 solve a PDE involving the Laplacian (e.g., Du = f ) 3 simple post-processing (do something with u) Expressing tasks in terms of Laplacian/smooth PDEs makes life easier at code/implementation level.

9 Introduction Laplace-Beltrami operator ( Laplacian ) provides a basis for a diverse variety of geometry processing tasks. Remarkably common pipeline: 1 simple pre-processing (build f ) 2 solve a PDE involving the Laplacian (e.g., Du = f ) 3 simple post-processing (do something with u) Expressing tasks in terms of Laplacian/smooth PDEs makes life easier at code/implementation level. Lots of existing theory to help understand/interpret algorithms, provide analysis/guarantees.

10 Introduction Laplace-Beltrami operator ( Laplacian ) provides a basis for a diverse variety of geometry processing tasks. Remarkably common pipeline: 1 simple pre-processing (build f ) 2 solve a PDE involving the Laplacian (e.g., Du = f ) 3 simple post-processing (do something with u) Expressing tasks in terms of Laplacian/smooth PDEs makes life easier at code/implementation level. Lots of existing theory to help understand/interpret algorithms, provide analysis/guarantees. Also makes it easy to work with a broad range of geometric data structures (meshes, point clouds, etc.)

11 Goals of this tutorial: Introduction

12 Goals of this tutorial: Introduction Understand the Laplacian in the smooth setting. (Etienne)

13 Goals of this tutorial: Introduction Understand the Laplacian in the smooth setting. (Etienne) Build the Laplacian in the discrete setting. (Keenan)

14 Goals of this tutorial: Introduction Understand the Laplacian in the smooth setting. (Etienne) Build the Laplacian in the discrete setting. (Keenan) Use Laplacian to implement a variety of methods. (Justin)

15 SMOOTH THEORY

16 The Interpolation Problem f = 1 given: region W R 2 with boundary W function f on W fill in f as smoothly as possible W W f = 1

17 The Interpolation Problem f = 1 W given: region W R 2 with boundary W function f on W fill in f as smoothly as possible (what does this even mean?) W f = 1

18 The Interpolation Problem f = 1 W W f = 1 given: region W R 2 with boundary W function f on W fill in f as smoothly as possible (what does this even mean?) smooth: constant functions linear functions

19 The Interpolation Problem f = 1 W W f = 1 given: region W R 2 with boundary W function f on W fill in f as smoothly as possible (what does this even mean?) smooth: constant functions linear functions not smooth: f not continuous

20 The Interpolation Problem f = 1 W W f = 1 given: region W R 2 with boundary W function f on W fill in f as smoothly as possible (what does this even mean?) smooth: constant functions linear functions not smooth: f not continuous large variations over short distances (krf k large)

21 Dirichlet Energy non-smooth f (x) E(f )= R W krf k2 da properties: nonnegative zero for constant functions measures smoothness krf k 2

22 Dirichlet Energy non-smooth f (x) E(f )= R W krf k2 da properties: nonnegative zero for constant functions measures smoothness solution to interpolation problem is minimizer of E krf k 2

23 Dirichlet Energy non-smooth f (x) E(f )= R W krf k2 da properties: nonnegative zero for constant functions measures smoothness solution to interpolation problem is minimizer of E how do we find minimum? krf k 2

24 Dirichlet Energy E(f )= R W krf k2 da it can be shown that: E(f )=C R W f Df da non-smooth f (x) krf k 2

25 Dirichlet Energy non-smooth f (x) E(f )= R W krf k2 da it can be shown that: E(f )=C R W f Df da 2Df is the gradient of Dirichlet energy Df

26 Dirichlet Energy non-smooth f (x) E(f )= R W krf k2 da it can be shown that: E(f )=C R W f Df da 2Df is the gradient of Dirichlet energy f minimizes E if Df = 0 Df

27 Dirichlet Energy non-smooth f (x) E(f )= R W krf k2 da it can be shown that: E(f )=C R W f Df da 2Df is the gradient of Dirichlet energy f minimizes E if Df = 0 PDE form (Laplace s Equation): Df (x) =0 f (x) =f 0 (x) x 2 W x 2 W solution Df = 0

28 Dirichlet Energy non-smooth f (x) E(f )= R W krf k2 da it can be shown that: E(f )=C R W f Df da 2Df is the gradient of Dirichlet energy f minimizes E if Df = 0 PDE form (Laplace s Equation): Df (x) =0 f (x) =f 0 (x) x 2 W x 2 W solution Df = 0 physical interpretation: temperature at steady state

29 f = 1 On a Surface f = 1 boundary conditions nonsmooth f (x)

30 f = 1 On a Surface f = 1 boundary conditions nonsmooth f (x) krf k 2 can still define Dirichlet energy E(f )= R krf k2 M

31 f = 1 On a Surface f = 1 boundary conditions nonsmooth f (x) Df = 0 can still define Dirichlet energy E(f )= R krf k2 M re(f )= Df, now D is the Laplace-Beltrami operator of M

32 f = 1 On a Surface f = 1 boundary conditions nonsmooth f (x) Df = 0 can still define Dirichlet energy E(f )= R krf k2 M re(f )= Df, now D is the Laplace-Beltrami operator of M also works in higher dimensions, on discrete graphs/point clouds,...

33 Existence and Uniqueness Laplace s equation Df (x) =0 f (x) =f 0 (x) x 2 M x 2 M has a unique solution for all reasonable 1 surfaces M 1 e.g. compact, smooth, with piecewise smooth boundary

34 Existence and Uniqueness Laplace s equation Df (x) =0 f (x) =f 0 (x) x 2 M x 2 M has a unique solution for all reasonable 1 surfaces M physical interpretation: apply heating/cooling f 0 to the boundary of a metal plate. Interior temperature will reach some steady state 1 e.g. compact, smooth, with piecewise smooth boundary

35 Existence and Uniqueness Laplace s equation Df (x) =0 f (x) =f 0 (x) x 2 M x 2 M has a unique solution for all reasonable 1 surfaces M physical interpretation: apply heating/cooling f 0 to the boundary of a metal plate. Interior temperature will reach some steady state gradient descent is exactly the heat or diffusion equation df (x) =Df (x). dt 1 e.g. compact, smooth, with piecewise smooth boundary

36 time Heat Equation Illustrated

37 Boundary Conditions f 0 = 1 W N g 0 = 0 W can specify rf ˆn on boundary instead of f : Df (x) =0 x 2 W f (x) =f 0 (x) x 2 W D (Dirichlet bdry) rf ˆn = g 0 (x) x 2 W N (Neumann bdry) W D f 0 = 1

38 Boundary Conditions f 0 = 1 W N g 0 = 0 W can specify rf ˆn on boundary instead of f : Df (x) =0 x 2 W f (x) =f 0 (x) x 2 W D (Dirichlet bdry) rf ˆn = g 0 (x) x 2 W N (Neumann bdry) W D f 0 = 1 usually: g 0 = 0(natural bdry conds)

39 Boundary Conditions f 0 = 1 W N g 0 = 0 W can specify rf ˆn on boundary instead of f : Df (x) =0 x 2 W f (x) =f 0 (x) x 2 W D (Dirichlet bdry) rf ˆn = g 0 (x) x 2 W N (Neumann bdry) W D f 0 = 1 usually: g 0 = 0(natural bdry conds) physical interpretation: free boundary through which heat cannot flow

40 Interpolation with D in Practice in geometry processing: positions displacements vector fields parameterizations... you name it Joshi et al Eck et al Sorkine and Cohen-Or

41 Heat Equation with Source f = 1 what if you add heat sources inside W? g = 1

42 Heat Equation with Source f = 1 what if you add heat sources inside W? g = 1 df (x) =g(x)+df (x) dt

43 Heat Equation with Source f = 1 what if you add heat sources inside W? g = 1 df (x) =g(x)+df (x) dt PDE form: Poisson s equation Df (x) =g(x) f (x) =f 0 (x) x 2 W x 2 W

44 Heat Equation with Source f = 1 what if you add heat sources inside W? g = 1 df (x) =g(x)+df (x) dt PDE form: Poisson s equation Df (x) =g(x) f (x) =f 0 (x) x 2 W x 2 W common variational problem: Z min krf f M vk 2 da

45 Heat Equation with Source f = 1 what if you add heat sources inside W? g = 1 df (x) =g(x)+df (x) dt PDE form: Poisson s equation Df (x) =g(x) f (x) =f 0 (x) x 2 W x 2 W common variational problem: Z min krf f M vk 2 da becomes Poisson problem, g = r v

46 Essential Algebraic Properties I linearity: D (f (x) + ag(x)) = Df (x) + adg(x)

47 Essential Algebraic Properties I linearity: constants in kernel: Da = 0 D (f (x) + ag(x)) = Df (x) + adg(x)

48 Essential Algebraic Properties I linearity: D (f (x) + ag(x)) = Df (x) + adg(x) constants in kernel: Da = 0 for functions that vanish on M: Z Z self-adjoint: f Dg da = Z M negative: f Df da apple 0 M M Z hrf, rgi da = M gdf da

49 Essential Algebraic Properties I linearity: D (f (x) + ag(x)) = Df (x) + adg(x) constants in kernel: Da = 0 for functions that vanish on M: Z Z self-adjoint: f Dg da = Z M negative: f Df da apple 0 M M Z hrf, rgi da = M gdf da (intuition: D an -dimensional negative-semidefinite matrix)

50 Solving Poisson s Equation with Green s Functions the Green s function G on R 2 solves Df = g for g = d d(x) G(x)

51 Solving Poisson s Equation with Green s Functions the Green s function G on R 2 solves Df = g for g = d linearity: if g = Â a i d(x x i ), f = Â a i G(x x i ) d(x) G(x)

52 Solving Poisson s Equation with Green s Functions the Green s function G on R 2 solves Df = g for g = d linearity: if g = Â a i d(x x i ), f = Â a i G(x x i ) for any g, f = G g d(x) G(x)

53 Essential Algebraic Properties II a function f : M! R with Df = 0 is called harmonic. Properties: f is smooth and analytic some harmonic f (x, y)

54 Essential Algebraic Properties II a function f : M! R with Df = 0 is called harmonic. Properties: f is smooth and analytic f (x) is the average of f over any disk around x: f (x) = 1 Z pr 2 f (y) da B(x,r)

55 Essential Algebraic Properties II a function f : M! R with Df = 0 is called harmonic. Properties: f is smooth and analytic f (x) is the average of f over any disk around x: f (x) = 1 Z pr 2 f (y) da B(x,r) maximum principle: f has no local maxima or minima in M

56 Essential Algebraic Properties II a function f : M! R with Df = 0 is called harmonic. Properties: f is smooth and analytic f (x) is the average of f over any disk around x: f (x) = 1 Z pr 2 f (y) da B(x,r) maximum principle: f has no local maxima or minima in M (can have saddle points)

57 Essential Geometric Properties I for a curve g(u) =(x[u], y[u]) : R! R 2 total Dirichlet energy R krxk 2 + kryk 2 is arc length

58 Essential Geometric Properties I for a curve g(u) =(x[u], y[u]) : R! R 2 total Dirichlet energy R krxk 2 + kryk 2 is arc length Dg =(Dx, Dy) is gradient of arc length

59 Essential Geometric Properties I for a curve g(u) =(x[u], y[u]) : R! R 2 total Dirichlet energy R krxk 2 + kryk 2 is arc length Dg =(Dx, Dy) is gradient of arc length Dg is the curvature normal k ˆn

60 Essential Geometric Properties I for a curve g(u) =(x[u], y[u]) : R! R 2 total Dirichlet energy R krxk 2 + kryk 2 is arc length Dg =(Dx, Dy) is gradient of arc length Dg is the curvature normal k ˆn minimal curves are harmonic

61 Essential Geometric Properties I for a curve g(u) =(x[u], y[u]) : R! R 2 total Dirichlet energy R krxk 2 + kryk 2 is arc length Dg =(Dx, Dy) is gradient of arc length Dg is the curvature normal k ˆn minimal curves are harmonic (straight lines)

62 Essential Geometric Properties II for a surface r(u, v) =(x[u, v], y[u, v], z[u, v]) : R! R 3 total Dirichlet energy is surface area Dr =(Dx, Dy, Dz) is gradient of surface area

63 Essential Geometric Properties II for a surface r(u, v) =(x[u, v], y[u, v], z[u, v]) : R! R 3 total Dirichlet energy is surface area Dr =(Dx, Dy, Dz) is gradient of surface area Dr is the mean curvature normal 2H ˆn

64 Essential Geometric Properties II for a surface r(u, v) =(x[u, v], y[u, v], z[u, v]) : R! R 3 total Dirichlet energy is surface area Dr =(Dx, Dy, Dz) is gradient of surface area Dr is the mean curvature normal 2H ˆn minimal surfaces are harmonic! Images: Paul Nylander

65 D is intrinsic Essential Geometric Properties III

66 Essential Geometric Properties III D is intrinsic for W R 2, rigid motions of W don t change D

67 Essential Geometric Properties III D is intrinsic for W R 2, rigid motions of W don t change D for a surface W, isometric deformations of W don t change D

68 Signal Processing on a Line on line segment [0, 1]: recall Fourier basis: f i (x) =cos(ix)

69 Signal Processing on a Line on line segment [0, 1]: recall Fourier basis: f i (x) =cos(ix) can decompose f = Â a i f i

70 Signal Processing on a Line on line segment [0, 1]: recall Fourier basis: f i (x) =cos(ix) can decompose f = Â a i f i f i satisfies Df i = i 2 f i

71 Signal Processing on a Line on line segment [0, 1]: recall Fourier basis: f i (x) =cos(ix) can decompose f = Â a i f i f i satisfies Df i = i 2 f i Dirichlet energy of f : Â i 2 a i

72 Signal Processing on a Line on line segment [0, 1]: recall Fourier basis: f i (x) =cos(ix) can decompose f = Â a i f i f i satisfies Df i = i 2 f i Dirichlet energy of f : Â i 2 a i f (x) = N Â i=1 a i f i (x) {z } low-frequency base + Â i=n+1 a i f i (x) {z } high-frequency detail

73 Laplacian Spectrum f is a (Dirichlet) eigenfunction of D on M w/ eigenvalue l: Df(x) =lf(x), x 2 M 0 = f(x), x 2 M Z 1 = kfk da. M

74 Laplacian Spectrum f is a (Dirichlet) eigenfunction of D on M w/ eigenvalue l: Df(x) =lf(x), x 2 M 0 = f(x), x 2 M Z 1 = kfk da. M recall intuition: D as -dim negative-semidefinite matrix

75 Laplacian Spectrum f is a (Dirichlet) eigenfunction of D on M w/ eigenvalue l: Df(x) =lf(x), x 2 M 0 = f(x), x 2 M Z 1 = kfk da. M recall intuition: D as -dim negative-semidefinite matrix expect orthogonal eigenfunctions with negative eigenvalue

76 Laplacian Spectrum f is a (Dirichlet) eigenfunction of D on M w/ eigenvalue l: Df(x) =lf(x), x 2 M 0 = f(x), x 2 M Z 1 = kfk da. M recall intuition: D as -dim negative-semidefinite matrix expect orthogonal eigenfunctions with negative eigenvalue spectrum is discrete: countably many eigenfunctions, 0 l 1 l 2 l 3...

77 Laplacian Spectrum of Bunny f 2 f 3 f 6 f 18

78 Laplacian Spectrum: Signal Processing expand function f in eigenbasis: f (x) =Â a i f i (x) i Dirichlet energy of f : Z E(f )= M krf k 2 da = Z M f Df da = Â a 2 i ( l i) i

79 Laplacian Spectrum: Signal Processing expand function f in eigenbasis: f (x) =Â a i f i (x) i Dirichlet energy of f : Z E(f )= M krf k 2 da = large l i terms dominate Z M f Df da = Â a 2 i ( l i) i

80 Laplacian Spectrum: Signal Processing 10 modes 25 modes 50 modes large l i terms dominate f (x) = N Â i=1 a i f i (x) {z } low-frequency base + Â i=n+1 a i f i (x) {z } high-frequency detail

81 Laplacian Spectrum: Special Cases perhaps you ve heard of Fourier basis: M = R n spherical harmonics: M = sphere

82 Laplacian Spectrum: Special Cases perhaps you ve heard of Fourier basis: M = R n spherical harmonics: M = sphere Laplacian spectrum generalizes these to any surface

83 DISCRETIZATION

84 Discrete Geometry

85 Triangle Meshes approximate surface by triangles

86 Triangle Meshes approximate surface by triangles glued together along edges

87 Triangle Meshes approximate surface by triangles glued together along edges many possible data structures

88 Triangle Meshes approximate surface by triangles glued together along edges many possible data structures half edge, quad edge, corner table,...

89 Triangle Meshes approximate surface by triangles glued together along edges many possible data structures half edge, quad edge, corner table,... for simplicity: vertex-face adjacency list

90 Triangle Meshes approximate surface by triangles glued together along edges many possible data structures half edge, quad edge, corner table,... for simplicity: vertex-face adjacency list (will be enough for our applications!)

91 Vertex-Face Adjacency List Example # xyz-coordinates of vertices v v v v # vertex-face adjacency info f f 1 4 2

92 Manifold

93 Nonmanifold

94 Manifold Triangle Mesh manifold () locally disk-like

95 Manifold Triangle Mesh manifold () locally disk-like Which triangle meshes are manifold?

96 Manifold Triangle Mesh manifold () locally disk-like Which triangle meshes are manifold? Two triangles per edge (no fins )

97 Manifold Triangle Mesh manifold () locally disk-like Which triangle meshes are manifold? Two triangles per edge (no fins ) Every vertex looks like a fan

98 Manifold Triangle Mesh manifold () locally disk-like Which triangle meshes are manifold? Two triangles per edge (no fins ) Every vertex looks like a fan Why? Simplicity.

99 Manifold Triangle Mesh manifold () locally disk-like Which triangle meshes are manifold? Two triangles per edge (no fins ) Every vertex looks like a fan Why? Simplicity. (Sometimes not necessary... )

100 Manifold Triangle Mesh manifold () locally disk-like Which triangle meshes are manifold? Two triangles per edge (no fins ) Every vertex looks like a fan Why? Simplicity. (Sometimes not necessary... )

101 (Assuming a manifold triangle mesh... ) The Cotangent Laplacian (Du) i 1 2A i  j2n (i) (cot a ij + cot b ij )(u i u j )

102 (Assuming a manifold triangle mesh... ) The Cotangent Laplacian (Du) i 1 2A i  j2n (i) (cot a ij + cot b ij )(u i u j ) The set N (i) contains the immediate neighbors of vertex i

103 (Assuming a manifold triangle mesh... ) The Cotangent Laplacian (Du) i 1 2A i  j2n (i) (cot a ij + cot b ij )(u i u j ) The set N (i) contains the immediate neighbors of vertex i The quantity A i is vertex area for now: 1/3rd of triangle areas

104 Many different ways to derive it Origin of the Cotan Formula?

105 Origin of the Cotan Formula? Many different ways to derive it piecewise linear finite elements (FEM)

106 Origin of the Cotan Formula? Many different ways to derive it piecewise linear finite elements (FEM) finite volumes

107 Origin of the Cotan Formula? Many different ways to derive it piecewise linear finite elements (FEM) finite volumes discrete exterior calculus (DEC)

108 Origin of the Cotan Formula? Many different ways to derive it piecewise linear finite elements (FEM) finite volumes discrete exterior calculus (DEC)...

109 Origin of the Cotan Formula? Many different ways to derive it piecewise linear finite elements (FEM) finite volumes discrete exterior calculus (DEC)... Re-derived in many different contexts: mean curvature flow [Desbrun et al., 1999]

110 Origin of the Cotan Formula? Many different ways to derive it piecewise linear finite elements (FEM) finite volumes discrete exterior calculus (DEC)... Re-derived in many different contexts: mean curvature flow [Desbrun et al., 1999] minimal surfaces [Pinkall and Polthier, 1993]

111 Origin of the Cotan Formula? Many different ways to derive it piecewise linear finite elements (FEM) finite volumes discrete exterior calculus (DEC)... Re-derived in many different contexts: mean curvature flow [Desbrun et al., 1999] minimal surfaces [Pinkall and Polthier, 1993] electrical networks [Duffin, 1959]

112 Origin of the Cotan Formula? Many different ways to derive it piecewise linear finite elements (FEM) finite volumes discrete exterior calculus (DEC)... Re-derived in many different contexts: mean curvature flow [Desbrun et al., 1999] minimal surfaces [Pinkall and Polthier, 1993] electrical networks [Duffin, 1959] Poisson equation [MacNeal, 1949]

113 Origin of the Cotan Formula? Many different ways to derive it piecewise linear finite elements (FEM) finite volumes discrete exterior calculus (DEC)... Re-derived in many different contexts: mean curvature flow [Desbrun et al., 1999] minimal surfaces [Pinkall and Polthier, 1993] electrical networks [Duffin, 1959] Poisson equation [MacNeal, 1949] (Courant? Frankel? Manhattan Project?)

114 Origin of the Cotan Formula? Many different ways to derive it piecewise linear finite elements (FEM) finite volumes discrete exterior calculus (DEC)... Re-derived in many different contexts: mean curvature flow [Desbrun et al., 1999] minimal surfaces [Pinkall and Polthier, 1993] electrical networks [Duffin, 1959] Poisson equation [MacNeal, 1949] (Courant? Frankel? Manhattan Project?) All these different viewpoints yield exact same cotan formula

115 Origin of the Cotan Formula? Many different ways to derive it piecewise linear finite elements (FEM) finite volumes discrete exterior calculus (DEC)... Re-derived in many different contexts: mean curvature flow [Desbrun et al., 1999] minimal surfaces [Pinkall and Polthier, 1993] electrical networks [Duffin, 1959] Poisson equation [MacNeal, 1949] (Courant? Frankel? Manhattan Project?) All these different viewpoints yield exact same cotan formula For three different derivations, see [Crane et al., 2013a]

116 MacNeal, 1949

117 Integrate over each dual cell C i Cotan-Laplacian via Finite Volumes

118 Cotan-Laplacian via Finite Volumes Integrate over each dual cell C i R C i Du = R C i f ( weak )

119 Cotan-Laplacian via Finite Volumes Integrate over each dual cell C i R C i Du = R C i f ( weak ) Right-hand side approximated as A i f i

120 Cotan-Laplacian via Finite Volumes Integrate over each dual cell C i R C i Du = R C i f ( weak ) Right-hand side approximated as A i f i Left-hand side becomes R C i r ru = R C i n ru (Stokes )

121 Cotan-Laplacian via Finite Volumes Integrate over each dual cell C i R C i Du = R C i f ( weak ) Right-hand side approximated as A i f i Left-hand side becomes R C i r ru = R C i n ru (Stokes ) Get piecewise integral over boundary  ej 2 C i n Re j j ru

122 Cotan-Laplacian via Finite Volumes Integrate over each dual cell C i R C i Du = R C i f ( weak ) Right-hand side approximated as A i f i Left-hand side becomes R C i r ru = R C i n ru (Stokes ) Get piecewise integral over boundary  ej 2 C i n Re j j ru After some trigonometry: 1 2  j2n (i)(cot a ij + cot b ij )(u i u j )

123 Cotan-Laplacian via Finite Volumes Integrate over each dual cell C i R C i Du = R C i f ( weak ) Right-hand side approximated as A i f i Left-hand side becomes R C i r ru = R C i n ru (Stokes ) Get piecewise integral over boundary  ej 2 C i n Re j j ru After some trigonometry: 1 2  j2n (i)(cot a ij + cot b ij )(u i u j ) (Can divide by A i to approximate pointwise value)

124 Triangle Quality Rule of Thumb good triangles bad triangles (For further discussion see Shewchuk, What Is a Good Linear Finite Element? )

125 Triangle Quality Delaunay Property Delaunay Not Delaunay

126 Local Mesh Improvement Some simple ways to improve quality of Laplacian

127 Local Mesh Improvement Some simple ways to improve quality of Laplacian E.g., if a + b > p, flip the edge; after enough flips, mesh will be Delaunay [Bobenko and Springborn, 2005]

128 Local Mesh Improvement Some simple ways to improve quality of Laplacian E.g., if a + b > p, flip the edge; after enough flips, mesh will be Delaunay [Bobenko and Springborn, 2005] Other ways to improve mesh (edge collapse, edge split,... )

129 Local Mesh Improvement Some simple ways to improve quality of Laplacian E.g., if a + b > p, flip the edge; after enough flips, mesh will be Delaunay [Bobenko and Springborn, 2005] Other ways to improve mesh (edge collapse, edge split,... ) Particular interest recently in interface tracking

130 Local Mesh Improvement Some simple ways to improve quality of Laplacian E.g., if a + b > p, flip the edge; after enough flips, mesh will be Delaunay [Bobenko and Springborn, 2005] Other ways to improve mesh (edge collapse, edge split,... ) Particular interest recently in interface tracking For more, see [Dunyach et al., 2013, Wojtan et al., 2011].

131 Meshes and Matrices 1 So far, Laplacian expressed as a sum: (Laplace matrix, ignoring weights!)

132 Meshes and Matrices 1 So far, Laplacian expressed as a sum: Â j2n (i)(cot a ij + cot b ij )(u j u i ) (Laplace matrix, ignoring weights!)

133 Meshes and Matrices 1 So far, Laplacian expressed as a sum: Â j2n (i)(cot a ij + cot b ij )(u j u i ) For computation, encode using matrices (Laplace matrix, ignoring weights!)

134 Meshes and Matrices 1 So far, Laplacian expressed as a sum: Â j2n (i)(cot a ij + cot b ij )(u j u i ) For computation, encode using matrices First, give each vertex an index 1,..., V (Laplace matrix, ignoring weights!)

135 Meshes and Matrices 1 So far, Laplacian expressed as a sum: Â j2n (i)(cot a ij + cot b ij )(u j u i ) For computation, encode using matrices First, give each vertex an index 1,..., V Weak Laplacian is matrix C 2 R V V (Laplace matrix, ignoring weights!)

136 Meshes and Matrices 1 So far, Laplacian expressed as a sum: Â j2n (i)(cot a ij + cot b ij )(u j u i ) For computation, encode using matrices First, give each vertex an index 1,..., V Weak Laplacian is matrix C 2 R V V Row i represents sum for ith vertex (Laplace matrix, ignoring weights!)

137 Meshes and Matrices 1 So far, Laplacian expressed as a sum: Â j2n (i)(cot a ij + cot b ij )(u j u i ) For computation, encode using matrices First, give each vertex an index 1,..., V Weak Laplacian is matrix C 2 R V V Row i represents sum for ith vertex C ij = 1 2 cot a ij + cot b ij for j 2N(i) (Laplace matrix, ignoring weights!)

138 Meshes and Matrices 1 So far, Laplacian expressed as a sum: Â j2n (i)(cot a ij + cot b ij )(u j u i ) For computation, encode using matrices First, give each vertex an index 1,..., V Weak Laplacian is matrix C 2 R V V Row i represents sum for ith vertex C ij = 1 2 cot a ij + cot b ij for j 2N(i) C ii = Â j2n (i) C ij (Laplace matrix, ignoring weights!)

139 Meshes and Matrices 1 So far, Laplacian expressed as a sum: Â j2n (i)(cot a ij + cot b ij )(u j u i ) For computation, encode using matrices First, give each vertex an index 1,..., V Weak Laplacian is matrix C 2 R V V Row i represents sum for ith vertex C ij = 1 2 cot a ij + cot b ij for j 2N(i) C ii = Â j2n (i) C ij All other entries are zero (Laplace matrix, ignoring weights!)

140 Meshes and Matrices 1 So far, Laplacian expressed as a sum: Â j2n (i)(cot a ij + cot b ij )(u j u i ) For computation, encode using matrices First, give each vertex an index 1,..., V Weak Laplacian is matrix C 2 R V V Row i represents sum for ith vertex C ij = 1 2 cot a ij + cot b ij for j 2N(i) C ii = Â j2n (i) C ij All other entries are zero (Laplace matrix, ignoring weights!) Use sparse matrices!

141 Meshes and Matrices 1 So far, Laplacian expressed as a sum: Â j2n (i)(cot a ij + cot b ij )(u j u i ) For computation, encode using matrices First, give each vertex an index 1,..., V Weak Laplacian is matrix C 2 R V V Row i represents sum for ith vertex C ij = 1 2 cot a ij + cot b ij for j 2N(i) C ii = Â j2n (i) C ij All other entries are zero (Laplace matrix, ignoring weights!) Use sparse matrices! (MATLAB: sparse, SuiteSparse: cholmod_sparse, Eigen: SparseMatrix)

142 Mass Matrix Matrix C encodes only part of Laplacian recall that (Du) i = 1 2A i  j2n (i) (cot a ij + cot b ij )(u j u i )

143 Mass Matrix Matrix C encodes only part of Laplacian recall that (Du) i = 1 2A i  j2n (i) (cot a ij + cot b ij )(u j u i ) Still need to incorporate vertex areas A i

144 Mass Matrix Matrix C encodes only part of Laplacian recall that (Du) i = 1 2A i  j2n (i) (cot a ij + cot b ij )(u j u i ) Still need to incorporate vertex areas A i For convenience, build diagonal mass matrix M 2 R V V : 2 A 1 6 M = 4... A V 3 7 5

145 Mass Matrix Matrix C encodes only part of Laplacian recall that (Du) i = 1 2A i  j2n (i) (cot a ij + cot b ij )(u j u i ) Still need to incorporate vertex areas A i For convenience, build diagonal mass matrix M 2 R V V : 2 A 1 6 M = 4... A V Entries are just M ii = A i (all other entries are zero)

146 Mass Matrix Matrix C encodes only part of Laplacian recall that (Du) i = 1 2A i  j2n (i) (cot a ij + cot b ij )(u j u i ) Still need to incorporate vertex areas A i For convenience, build diagonal mass matrix M 2 R V V : 2 A 1 6 M = 4... A V Entries are just M ii = A i (all other entries are zero) Laplace operator is then L := M 1 C

147 Mass Matrix Matrix C encodes only part of Laplacian recall that (Du) i = 1 2A i  j2n (i) (cot a ij + cot b ij )(u j u i ) Still need to incorporate vertex areas A i For convenience, build diagonal mass matrix M 2 R V V : 2 A 1 6 M = 4... A V Entries are just M ii = A i (all other entries are zero) Laplace operator is then L := M 1 C Applying L to a column vector u 2 R V implements the cotan formula shown above

148 Discrete Poisson / Laplace Equation Poisson equation Du = f becomes linear algebra problem: Lu = f

149 Discrete Poisson / Laplace Equation Poisson equation Du = f becomes linear algebra problem: Lu = f Vector f 2 R V is given data; u 2 R V is unknown.

150 Discrete Poisson / Laplace Equation Poisson equation Du = f becomes linear algebra problem: Lu = f Vector f 2 R V is given data; u 2 R V is unknown. Discrete approximation u approaches smooth solution u as mesh is refined (for smooth data, good meshes... ).

151 Discrete Poisson / Laplace Equation Poisson equation Du = f becomes linear algebra problem: Lu = f Vector f 2 R V is given data; u 2 R V is unknown. Discrete approximation u approaches smooth solution u as mesh is refined (for smooth data, good meshes... ). Laplace is just Poisson with zero on right hand side!

152 Discrete Heat Equation Heat equation du dt = Du must also be discretized in time

153 Discrete Heat Equation Heat equation du dt = Du must also be discretized in time Replace time derivative with finite difference: du dt ) u k+1 h u k, h > 0 {z } time step

154 Discrete Heat Equation Heat equation du dt = Du must also be discretized in time Replace time derivative with finite difference: du dt ) u k+1 h u k, h > 0 {z } time step How (or really, when ) do we approximate Du?

155 Discrete Heat Equation Heat equation du dt = Du must also be discretized in time Replace time derivative with finite difference: du dt ) u k+1 h u k, h > 0 {z } time step How (or really, when ) do we approximate Du? Explicit: (u k+1 u k )/h = Lu k (cheaper to compute)

156 Discrete Heat Equation Heat equation du dt = Du must also be discretized in time Replace time derivative with finite difference: du dt ) u k+1 h u k, h > 0 {z } time step How (or really, when ) do we approximate Du? Explicit: (u k+1 u k )/h = Lu k (cheaper to compute) Implicit: (u k+1 u k )/h = Lu k+1 (more stable)

157 Discrete Heat Equation Heat equation du dt = Du must also be discretized in time Replace time derivative with finite difference: du dt ) u k+1 h u k, h > 0 {z } time step How (or really, when ) do we approximate Du? Explicit: (u k+1 u k )/h = Lu k (cheaper to compute) Implicit: (u k+1 u k )/h = Lu k+1 (more stable) Implicit update becomes linear system (I hl)u k+1 = u k

158 Discrete Eigenvalue Problem Smallest eigenvalue problem Du = lu becomes Lu = lu for smallest nonzero eigenvalue l.

159 Discrete Eigenvalue Problem Smallest eigenvalue problem Du = lu becomes Lu = lu for smallest nonzero eigenvalue l. Can be solved using (inverse) power method: Pick random u 0 Until convergence: Solve Lu k+1 = u k Remove mean value from u k+1 u k+1 u k+1 / u k+1

160 Discrete Eigenvalue Problem Smallest eigenvalue problem Du = lu becomes Lu = lu for smallest nonzero eigenvalue l. Can be solved using (inverse) power method: Pick random u 0 Until convergence: Solve Lu k+1 = u k Remove mean value from u k+1 u k+1 u k+1 / u k+1 By prefactoring L, overall cost is nearly identical to solving a single Poisson equation!

161 Properties of cotan-laplace Always, always, always positive-semidefinite f T Cf 0 (even if cotan weights are negative!)

162 Properties of cotan-laplace Always, always, always positive-semidefinite f T Cf 0 (even if cotan weights are negative!) Why? f T Cf is identical to summing rf 2!

163 Properties of cotan-laplace Always, always, always positive-semidefinite f T Cf 0 (even if cotan weights are negative!) Why? f T Cf is identical to summing rf 2! No boundary ) constant vector in the kernel / cokernel

164 Properties of cotan-laplace Always, always, always positive-semidefinite f T Cf 0 (even if cotan weights are negative!) Why? f T Cf is identical to summing rf 2! No boundary ) constant vector in the kernel / cokernel Why does it matter? E.g., for Poisson equation:

165 Properties of cotan-laplace Always, always, always positive-semidefinite f T Cf 0 (even if cotan weights are negative!) Why? f T Cf is identical to summing rf 2! No boundary ) constant vector in the kernel / cokernel Why does it matter? E.g., for Poisson equation: solution is unique only up to constant shift

166 Properties of cotan-laplace Always, always, always positive-semidefinite f T Cf 0 (even if cotan weights are negative!) Why? f T Cf is identical to summing rf 2! No boundary ) constant vector in the kernel / cokernel Why does it matter? E.g., for Poisson equation: solution is unique only up to constant shift if RHS has nonzero mean, cannot be solved!

167 Properties of cotan-laplace Always, always, always positive-semidefinite f T Cf 0 (even if cotan weights are negative!) Why? f T Cf is identical to summing rf 2! No boundary ) constant vector in the kernel / cokernel Why does it matter? E.g., for Poisson equation: solution is unique only up to constant shift if RHS has nonzero mean, cannot be solved! Exhibits maximum principle on Delaunay mesh

168 Properties of cotan-laplace Always, always, always positive-semidefinite f T Cf 0 (even if cotan weights are negative!) Why? f T Cf is identical to summing rf 2! No boundary ) constant vector in the kernel / cokernel Why does it matter? E.g., for Poisson equation: solution is unique only up to constant shift if RHS has nonzero mean, cannot be solved! Exhibits maximum principle on Delaunay mesh Delaunay: triangle circumcircles are empty

169 Properties of cotan-laplace Always, always, always positive-semidefinite f T Cf 0 (even if cotan weights are negative!) Why? f T Cf is identical to summing rf 2! No boundary ) constant vector in the kernel / cokernel Why does it matter? E.g., for Poisson equation: solution is unique only up to constant shift if RHS has nonzero mean, cannot be solved! Exhibits maximum principle on Delaunay mesh Delaunay: triangle circumcircles are empty Maximum principle: solution to Laplace equation has no interior extrema (local max or min)

170 Properties of cotan-laplace Always, always, always positive-semidefinite f T Cf 0 (even if cotan weights are negative!) Why? f T Cf is identical to summing rf 2! No boundary ) constant vector in the kernel / cokernel Why does it matter? E.g., for Poisson equation: solution is unique only up to constant shift if RHS has nonzero mean, cannot be solved! Exhibits maximum principle on Delaunay mesh Delaunay: triangle circumcircles are empty Maximum principle: solution to Laplace equation has no interior extrema (local max or min) NOTE: non-delaunay meshes can also exhibit max principle! (And often do.) Delaunay sufficient but not necessary. Currently no nice, simple necessary condition on mesh geometry.

171 Properties of cotan-laplace Always, always, always positive-semidefinite f T Cf 0 (even if cotan weights are negative!) Why? f T Cf is identical to summing rf 2! No boundary ) constant vector in the kernel / cokernel Why does it matter? E.g., for Poisson equation: solution is unique only up to constant shift if RHS has nonzero mean, cannot be solved! Exhibits maximum principle on Delaunay mesh Delaunay: triangle circumcircles are empty Maximum principle: solution to Laplace equation has no interior extrema (local max or min) NOTE: non-delaunay meshes can also exhibit max principle! (And often do.) Delaunay sufficient but not necessary. Currently no nice, simple necessary condition on mesh geometry. For more, see [Wardetzky et al., 2007]

172 Numerical Issues Symmetry Best case for sparse linear systems: symmetric positive-(semi)definite (A T = A, x T Ax 0 8x)

173 Numerical Issues Symmetry Best case for sparse linear systems: symmetric positive-(semi)definite (A T = A, x T Ax 0 8x) Many good solvers (Cholesky, conjugate gradient,... )

174 Numerical Issues Symmetry Best case for sparse linear systems: symmetric positive-(semi)definite (A T = A, x T Ax 0 8x) Many good solvers (Cholesky, conjugate gradient,... ) Discrete Poisson equation looks like M 1 Cu = f

175 Numerical Issues Symmetry Best case for sparse linear systems: symmetric positive-(semi)definite (A T = A, x T Ax 0 8x) Many good solvers (Cholesky, conjugate gradient,... ) Discrete Poisson equation looks like M 1 Cu = f C is symmetric, but M 1 C is not! Can easily be made symmetric: Cu = Mf

176 Numerical Issues Symmetry Best case for sparse linear systems: symmetric positive-(semi)definite (A T = A, x T Ax 0 8x) Many good solvers (Cholesky, conjugate gradient,... ) Discrete Poisson equation looks like M 1 Cu = f C is symmetric, but M 1 C is not! Can easily be made symmetric: Cu = Mf In other words: just multiply by vertex areas!

177 Numerical Issues Symmetry Best case for sparse linear systems: symmetric positive-(semi)definite (A T = A, x T Ax 0 8x) Many good solvers (Cholesky, conjugate gradient,... ) Discrete Poisson equation looks like M 1 Cu = f C is symmetric, but M 1 C is not! Can easily be made symmetric: Cu = Mf In other words: just multiply by vertex areas! Seemingly superficial change...

178 Numerical Issues Symmetry Best case for sparse linear systems: symmetric positive-(semi)definite (A T = A, x T Ax 0 8x) Many good solvers (Cholesky, conjugate gradient,... ) Discrete Poisson equation looks like M 1 Cu = f C is symmetric, but M 1 C is not! Can easily be made symmetric: Cu = Mf In other words: just multiply by vertex areas! Seemingly superficial change but makes computation simpler / more efficient

179 Numerical Issues Symmetry Best case for sparse linear systems: symmetric positive-(semi)definite (A T = A, x T Ax 0 8x) Many good solvers (Cholesky, conjugate gradient,... ) Discrete Poisson equation looks like M 1 Cu = f C is symmetric, but M 1 C is not! Can easily be made symmetric: Cu = Mf In other words: just multiply by vertex areas! Seemingly superficial change but makes computation simpler / more efficient

180 Numerical Issues Symmetry, continued Can also make heat equation symmetric

181 Numerical Issues Symmetry, continued Can also make heat equation symmetric Instead of (I hl)u k+1 = u k, use (M hc)u k+1 = Mu k

182 Numerical Issues Symmetry, continued Can also make heat equation symmetric Instead of (I hl)u k+1 = u k, use (M hc)u k+1 = Mu k What about smallest eigenvalue problem Lu = lu?

183 Numerical Issues Symmetry, continued Can also make heat equation symmetric Instead of (I hl)u k+1 = u k, use (M hc)u k+1 = Mu k What about smallest eigenvalue problem Lu = lu? Two options:

184 Numerical Issues Symmetry, continued Can also make heat equation symmetric Instead of (I hl)u k+1 = u k, use (M hc)u k+1 = Mu k What about smallest eigenvalue problem Lu = lu? Two options: 1 Solve generalized eigenvalue problem Cu = lmu

185 Numerical Issues Symmetry, continued Can also make heat equation symmetric Instead of (I hl)u k+1 = u k, use (M hc)u k+1 = Mu k What about smallest eigenvalue problem Lu = lu? Two options: 1 Solve generalized eigenvalue problem Cu = lmu 2 Solve M 1/2 CM 1/2 ũ = lũ, recover u = M 1/2 ũ

186 Numerical Issues Symmetry, continued Can also make heat equation symmetric Instead of (I hl)u k+1 = u k, use (M hc)u k+1 = Mu k What about smallest eigenvalue problem Lu = lu? Two options: 1 Solve generalized eigenvalue problem Cu = lmu 2 Solve M 1/2 CM 1/2 ũ = lũ, recover u = M 1/2 ũ Note: M 1/2 just means put 1/ p A i on the diagonal!

187 Numerical Issues Symmetry, continued Can also make heat equation symmetric Instead of (I hl)u k+1 = u k, use (M hc)u k+1 = Mu k What about smallest eigenvalue problem Lu = lu? Two options: 1 Solve generalized eigenvalue problem Cu = lmu 2 Solve M 1/2 CM 1/2 ũ = lũ, recover u = M 1/2 ũ Note: M 1/2 just means put 1/ p A i on the diagonal!

188 Numerical Issues Direct vs. Iterative Solvers Direct (e.g., LL T, LU, QR,...)

189 Numerical Issues Direct vs. Iterative Solvers Direct (e.g., LL T, LU, QR,...) pros: great for multiple right-hand sides; (can be) less sensitive to numerical instability; solve many types of problems, under/overdetermined systems.

190 Numerical Issues Direct vs. Iterative Solvers Direct (e.g., LL T, LU, QR,...) pros: great for multiple right-hand sides; (can be) less sensitive to numerical instability; solve many types of problems, under/overdetermined systems. cons: prohibitively expensive for large problems; factors are quite dense for 3D (volumetric) problems

191 Numerical Issues Direct vs. Iterative Solvers Direct (e.g., LL T, LU, QR,...) pros: great for multiple right-hand sides; (can be) less sensitive to numerical instability; solve many types of problems, under/overdetermined systems. cons: prohibitively expensive for large problems; factors are quite dense for 3D (volumetric) problems Iterative (e.g., conjugate gradient, multigrid,... )

192 Numerical Issues Direct vs. Iterative Solvers Direct (e.g., LL T, LU, QR,...) pros: great for multiple right-hand sides; (can be) less sensitive to numerical instability; solve many types of problems, under/overdetermined systems. cons: prohibitively expensive for large problems; factors are quite dense for 3D (volumetric) problems Iterative (e.g., conjugate gradient, multigrid,... ) pros: can handle very large problems; can be implemented via callback (instead of matrix); asymptotic running times approaching linear time (in theory... )

193 Numerical Issues Direct vs. Iterative Solvers Direct (e.g., LL T, LU, QR,...) pros: great for multiple right-hand sides; (can be) less sensitive to numerical instability; solve many types of problems, under/overdetermined systems. cons: prohibitively expensive for large problems; factors are quite dense for 3D (volumetric) problems Iterative (e.g., conjugate gradient, multigrid,... ) pros: can handle very large problems; can be implemented via callback (instead of matrix); asymptotic running times approaching linear time (in theory... ) cons: poor performance without good preconditioners; less benefit for multiple right-hand sides; best-in-class methods may handle only symmetric positive-(semi)definite systems

194 Numerical Issues Direct vs. Iterative Solvers Direct (e.g., LL T, LU, QR,...) pros: great for multiple right-hand sides; (can be) less sensitive to numerical instability; solve many types of problems, under/overdetermined systems. cons: prohibitively expensive for large problems; factors are quite dense for 3D (volumetric) problems Iterative (e.g., conjugate gradient, multigrid,... ) pros: can handle very large problems; can be implemented via callback (instead of matrix); asymptotic running times approaching linear time (in theory... ) cons: poor performance without good preconditioners; less benefit for multiple right-hand sides; best-in-class methods may handle only symmetric positive-(semi)definite systems No perfect solution! Each problem is different.

195 Solving Equations in Linear Time Is solving Poisson, Laplace, etc., truly linear time in 2D?

196 Solving Equations in Linear Time Is solving Poisson, Laplace, etc., truly linear time in 2D? Jury is still out, but keep inching forward: [Vaidya, 1991] use spanning tree as preconditioner [Alon et al., 1995] use low-stretch spanning trees [Spielman and Teng, 2004] first nearly linear time solver [Krishnan et al., 2013] practical solver for graphics Lots of recent activity in both preconditioners and direct solvers (e.g., [Koutis et al., 2011], [Gillman and Martinsson, 2013])

197 Solving Equations in Linear Time Is solving Poisson, Laplace, etc., truly linear time in 2D? Jury is still out, but keep inching forward: [Vaidya, 1991] use spanning tree as preconditioner [Alon et al., 1995] use low-stretch spanning trees [Spielman and Teng, 2004] first nearly linear time solver [Krishnan et al., 2013] practical solver for graphics Lots of recent activity in both preconditioners and direct solvers (e.g., [Koutis et al., 2011], [Gillman and Martinsson, 2013]) Best theoretical results may lack practical implementations!

198 Solving Equations in Linear Time Is solving Poisson, Laplace, etc., truly linear time in 2D? Jury is still out, but keep inching forward: [Vaidya, 1991] use spanning tree as preconditioner [Alon et al., 1995] use low-stretch spanning trees [Spielman and Teng, 2004] first nearly linear time solver [Krishnan et al., 2013] practical solver for graphics Lots of recent activity in both preconditioners and direct solvers (e.g., [Koutis et al., 2011], [Gillman and Martinsson, 2013]) Best theoretical results may lack practical implementations! Older codes benefit from extensive low-level optimization

199 Solving Equations in Linear Time Is solving Poisson, Laplace, etc., truly linear time in 2D? Jury is still out, but keep inching forward: [Vaidya, 1991] use spanning tree as preconditioner [Alon et al., 1995] use low-stretch spanning trees [Spielman and Teng, 2004] first nearly linear time solver [Krishnan et al., 2013] practical solver for graphics Lots of recent activity in both preconditioners and direct solvers (e.g., [Koutis et al., 2011], [Gillman and Martinsson, 2013]) Best theoretical results may lack practical implementations! Older codes benefit from extensive low-level optimization Long term: probably indistinguishable from O(n)

200 Boundary Conditions PDE (Laplace, Poisson, heat equation, etc.) determines behavior inside domain W

201 Boundary Conditions PDE (Laplace, Poisson, heat equation, etc.) determines behavior inside domain W Also need to say how solution behaves on boundary W

202 Boundary Conditions PDE (Laplace, Poisson, heat equation, etc.) determines behavior inside domain W Also need to say how solution behaves on boundary W Often trickiest part (both mathematically & numerically)

203 Boundary Conditions PDE (Laplace, Poisson, heat equation, etc.) determines behavior inside domain W Also need to say how solution behaves on boundary W Often trickiest part (both mathematically & numerically) Very easy to get wrong!

204 Dirichlet () prescribe values Dirichlet Boundary Conditions

205 Dirichlet Boundary Conditions Dirichlet () prescribe values E.g., f(0) =a, f(1) =b

206 Dirichlet Boundary Conditions Dirichlet () prescribe values E.g., f(0) =a, f(1) =b (Many possible functions in between! )

207 Neumann () prescribe derivatives Neumann Boundary Conditions

208 Neumann Boundary Conditions Neumann () prescribe derivatives E.g., f 0 (0) =u, f 0 (1) =v

209 Neumann Boundary Conditions Neumann () prescribe derivatives E.g., f 0 (0) =u, f 0 (1) =v (Again, many possible solutions.)

210 Both Neumann & Dirichlet Or: prescribe some values, some derivatives

211 Both Neumann & Dirichlet Or: prescribe some values, some derivatives E.g., f 0 (0) =u, f(1) =b

212 Both Neumann & Dirichlet Or: prescribe some values, some derivatives E.g., f 0 (0) =u, f(1) =b (What about f 0 (1) =v, f(1) =b?)

213 Laplace w/ Dirichlet Boundary Conditions (1D) 1D Laplace: 2 f/ x 2 = 0

214 Laplace w/ Dirichlet Boundary Conditions (1D) 1D Laplace: 2 f/ x 2 = 0 Solutions: f(x) = cx + d (linear functions)

215 Laplace w/ Dirichlet Boundary Conditions (1D) 1D Laplace: 2 f/ x 2 = 0 Solutions: f(x) = cx + d (linear functions) Can we always satisfy Dirichlet boundary conditions?

216 Laplace w/ Dirichlet Boundary Conditions (1D) 1D Laplace: 2 f/ x 2 = 0 Solutions: f(x) = cx + d (linear functions) Can we always satisfy Dirichlet boundary conditions? Yes: a line can interpolate any two points

217 Laplace w/ Neumann Boundary Conditions (1D) What about Neumann boundary conditions?

218 Laplace w/ Neumann Boundary Conditions (1D) What about Neumann boundary conditions? Solution must still be a line: f(x) =cx + d

219 Laplace w/ Neumann Boundary Conditions (1D) What about Neumann boundary conditions? Solution must still be a line: f(x) =cx + d Can we prescribe the derivative at both ends?

220 Laplace w/ Neumann Boundary Conditions (1D) What about Neumann boundary conditions? Solution must still be a line: f(x) =cx + d Can we prescribe the derivative at both ends? No! A line can have only one slope!

221 Laplace w/ Neumann Boundary Conditions (1D) What about Neumann boundary conditions? Solution must still be a line: f(x) =cx + d Can we prescribe the derivative at both ends? No! A line can have only one slope! In general: solutions to PDE may not exist for given BCs

222 Laplace w/ Dirichlet Boundary Conditions (2D) 2D Laplace: Df = 0

223 Laplace w/ Dirichlet Boundary Conditions (2D) 2D Laplace: Df = 0 Can we always satisfy Dirichlet boundary conditions?

224 Laplace w/ Dirichlet Boundary Conditions (2D) 2D Laplace: Df = 0 Can we always satisfy Dirichlet boundary conditions? Yes: Laplace is steady-state solution to heat flow d dt f = Df Dirichlet data is just heat along boundary

225 Laplace w/ Neumann Boundary Conditions (2D) What about Neumann boundary conditions?

226 Laplace w/ Neumann Boundary Conditions (2D) What about Neumann boundary conditions? Still want to solve Df = 0 Want to prescribe normal derivative n rf

227 Laplace w/ Neumann Boundary Conditions (2D) What about Neumann boundary conditions? Still want to solve Df = 0 Want to prescribe normal derivative n rf Wasn t always possible in 1D...

228 Laplace w/ Neumann Boundary Conditions (2D) What about Neumann boundary conditions? Still want to solve Df = 0 Want to prescribe normal derivative n rf Wasn t always possible in 1D... In 2D, we have divergence theorem: Z Z Z Z 0 =! Df = r rf = n rf W W W W

229 Laplace w/ Neumann Boundary Conditions (2D) What about Neumann boundary conditions? Still want to solve Df = 0 Want to prescribe normal derivative n rf Wasn t always possible in 1D... In 2D, we have divergence theorem: Z Z Z Z 0 =! Df = r rf = n rf W W W W Conclusion: can only solve Df = 0 if Neumann BCs have zero mean!

230 Discrete Boundary Conditions - Dirichlet Suppose we want to solve Du = f s.t. u W = g (Poisson equation w/ Dirichlet boundary conditions)

231 Discrete Boundary Conditions - Dirichlet Suppose we want to solve Du = f s.t. u W = g (Poisson equation w/ Dirichlet boundary conditions) Discretized Poisson equation as Cu = Mf

232 Discrete Boundary Conditions - Dirichlet Suppose we want to solve Du = f s.t. u W = g (Poisson equation w/ Dirichlet boundary conditions) Discretized Poisson equation as Cu = Mf Let I, B denote interior, boundary vertices, respectively. Get " #" # " #" # C II C IB u I M = II 0 f I C BI C BB u B 0 M BB f B

233 Discrete Boundary Conditions - Dirichlet Suppose we want to solve Du = f s.t. u W = g (Poisson equation w/ Dirichlet boundary conditions) Discretized Poisson equation as Cu = Mf Let I, B denote interior, boundary vertices, respectively. Get " #" # " #" # C II C IB u I M = II 0 f I C BI C BB u B 0 M BB f B Since u B is known (boundary values), solve just C II u I = M II f I C IB u B for u I (right-hand side is known).

234 Discrete Boundary Conditions - Dirichlet Suppose we want to solve Du = f s.t. u W = g (Poisson equation w/ Dirichlet boundary conditions) Discretized Poisson equation as Cu = Mf Let I, B denote interior, boundary vertices, respectively. Get " #" # " #" # C II C IB u I M = II 0 f I C BI C BB u B 0 M BB f B Since u B is known (boundary values), solve just C II u I = M II f I C IB u B for u I (right-hand side is known). Can skip matrix multiply and compute entries of RHS directly: A i f i  j2n (i)(cot a ij + cot b ij )u j Here N (i) denotes neighbors of i on the boundary

235 Discrete Boundary Conditions - Neumann Integrate both sides of Du = f over cell C i ( finite volume ) Z Z Z Z! f = Du = r ru = n ru C i C i C i C i Gives usual cotangent formula for interior vertices; for boundary vertex i, yields A ii! = 1 2 (g a + g b )+ 1 2 Â j2n int (cot a ij + cot b ij )(u j u i ) Here g a, g b are prescribed normal derivatives; just subtract from RHS and solve Cu = Mf as usual

236 Discrete Boundary Conditions - Neumann Other possible boundary conditions (e.g., Robin)

237 Discrete Boundary Conditions - Neumann Other possible boundary conditions (e.g., Robin) Dirichlet, Neumann most common implementation of other BCs will be similar

238 Discrete Boundary Conditions - Neumann Other possible boundary conditions (e.g., Robin) Dirichlet, Neumann most common implementation of other BCs will be similar When in doubt, return to smooth equations and integrate!

239 Discrete Boundary Conditions - Neumann Other possible boundary conditions (e.g., Robin) Dirichlet, Neumann most common implementation of other BCs will be similar When in doubt, return to smooth equations and integrate!... and make sure your equation has a solution!

240 Discrete Boundary Conditions - Neumann Other possible boundary conditions (e.g., Robin) Dirichlet, Neumann most common implementation of other BCs will be similar When in doubt, return to smooth equations and integrate!... and make sure your equation has a solution! Solver will NOT always tell you if there s a problem!

241 Discrete Boundary Conditions - Neumann Other possible boundary conditions (e.g., Robin) Dirichlet, Neumann most common implementation of other BCs will be similar When in doubt, return to smooth equations and integrate!... and make sure your equation has a solution! Solver will NOT always tell you if there s a problem! Easy test? Compute the residual r := Ax b. If the relative residual r / b is far from zero (e.g., greater than in double precision), you did not actually solve your problem!

242 Discrete Boundary Conditions - Neumann Other possible boundary conditions (e.g., Robin) Dirichlet, Neumann most common implementation of other BCs will be similar When in doubt, return to smooth equations and integrate!... and make sure your equation has a solution! Solver will NOT always tell you if there s a problem! Easy test? Compute the residual r := Ax b. If the relative residual r / b is far from zero (e.g., greater than in double precision), you did not actually solve your problem!

243 Alternative Discretizations Have spent a lot of time on triangle meshes...

244 Alternative Discretizations Have spent a lot of time on triangle meshes plenty of other ways to describe a surface!

245 Alternative Discretizations Have spent a lot of time on triangle meshes plenty of other ways to describe a surface! E.g., points are increasingly popular (due to 3D scanning)

246 Alternative Discretizations Have spent a lot of time on triangle meshes plenty of other ways to describe a surface! E.g., points are increasingly popular (due to 3D scanning) Also: more accurate discretization on triangle meshes

247 Quad, Polygon Meshes Quads popular alternative to triangles. Why?

248 Quad, Polygon Meshes Quads popular alternative to triangles. Why? capture principal curvatures of a surface

249 Quad, Polygon Meshes Quads popular alternative to triangles. Why? capture principal curvatures of a surface nice bases can be built via tensor products

250 Quad, Polygon Meshes Quads popular alternative to triangles. Why? capture principal curvatures of a surface nice bases can be built via tensor products see [Bommes et al., 2013] for further discussion

251 Quad, Polygon Meshes Quads popular alternative to triangles. Why? capture principal curvatures of a surface nice bases can be built via tensor products see [Bommes et al., 2013] for further discussion More generally: meshes with quads and triangles and...

252 Quad, Polygon Meshes Quads popular alternative to triangles. Why? capture principal curvatures of a surface nice bases can be built via tensor products see [Bommes et al., 2013] for further discussion More generally: meshes with quads and triangles and... Nice discretization: [Alexa and Wardetzky, 2011]

253 Quad, Polygon Meshes Quads popular alternative to triangles. Why? capture principal curvatures of a surface nice bases can be built via tensor products see [Bommes et al., 2013] for further discussion More generally: meshes with quads and triangles and... Nice discretization: [Alexa and Wardetzky, 2011] Can then solve all the same problems (Laplace, Poisson, heat,... )

254 Point Clouds Real data often point cloud with no connectivity (plus noise, holes... )

255 Point Clouds Real data often point cloud with no connectivity (plus noise, holes... ) Can still build Laplace operator!

256 Point Clouds Real data often point cloud with no connectivity (plus noise, holes... ) Can still build Laplace operator! Rough idea: use heat flow to discretize D

257 Point Clouds Real data often point cloud with no connectivity (plus noise, holes... ) Can still build Laplace operator! Rough idea: use heat flow to discretize D d dtu = Du =) Du (u(t) u(0))/t

258 Point Clouds Real data often point cloud with no connectivity (plus noise, holes... ) Can still build Laplace operator! Rough idea: use heat flow to discretize D d dt u = Du =) Du (u(t) u(0))/t How do we get u(t)? Convolve u with (Euclidean) heat kernel 1 4pT er2 /4T

259 Point Clouds Real data often point cloud with no connectivity (plus noise, holes... ) Can still build Laplace operator! Rough idea: use heat flow to discretize D d dt u = Du =) Du (u(t) u(0))/t How do we get u(t)? Convolve u with (Euclidean) heat kernel 1 4pT er2 /4T Converges with more samples, T goes to zero (under certain conditions!)

260 Point Clouds Real data often point cloud with no connectivity (plus noise, holes... ) Can still build Laplace operator! Rough idea: use heat flow to discretize D d dtu = Du =) Du (u(t) u(0))/t How do we get u(t)? Convolve u with 1 (Euclidean) heat kernel 4pT er2 /4T Converges with more samples, T goes to zero (under certain conditions!) Details: [Belkin et al., 2009, Liu et al., 2012]

261 Point Clouds Real data often point cloud with no connectivity (plus noise, holes... ) Can still build Laplace operator! Rough idea: use heat flow to discretize D d dtu = Du =) Du (u(t) u(0))/t How do we get u(t)? Convolve u with 1 (Euclidean) heat kernel 4pT er2 /4T Converges with more samples, T goes to zero (under certain conditions!) Details: [Belkin et al., 2009, Liu et al., 2012] From there, solve all the same problems! (Again.)

262 Dual Mesh barycentric circumcentric (superimposed) Earlier saw Laplacian discretized via dual mesh

263 Dual Mesh barycentric circumcentric (superimposed) Earlier saw Laplacian discretized via dual mesh Different duals lead to operators with different accuracy

264 Dual Mesh barycentric circumcentric (superimposed) Earlier saw Laplacian discretized via dual mesh Different duals lead to operators with different accuracy Space of orthogonal duals explored by [Mullen et al., 2011]

265 Dual Mesh barycentric circumcentric (superimposed) Earlier saw Laplacian discretized via dual mesh Different duals lead to operators with different accuracy Space of orthogonal duals explored by [Mullen et al., 2011] Leads to many applications in geometry processing [de Goes et al., 2012, de Goes et al., 2013, de Goes et al., 2014]

266 Volumes / Tetrahedral Meshes Same problems (Poisson, Laplace, etc.) can also be solved on volumes

267 Volumes / Tetrahedral Meshes Same problems (Poisson, Laplace, etc.) can also be solved on volumes Popular choice: tetrahedral meshes (graded, conform to boundary,... )

268 Volumes / Tetrahedral Meshes Same problems (Poisson, Laplace, etc.) can also be solved on volumes Popular choice: tetrahedral meshes (graded, conform to boundary,... ) Many ways to get Laplace matrix

269 Volumes / Tetrahedral Meshes Same problems (Poisson, Laplace, etc.) can also be solved on volumes Popular choice: tetrahedral meshes (graded, conform to boundary,... ) Many ways to get Laplace matrix One nice way: discrete exterior calculus (DEC) [Hirani, 2003, Desbrun et al., 2005]

270 Volumes / Tetrahedral Meshes Same problems (Poisson, Laplace, etc.) can also be solved on volumes Popular choice: tetrahedral meshes (graded, conform to boundary,... ) Many ways to get Laplace matrix One nice way: discrete exterior calculus (DEC) [Hirani, 2003, Desbrun et al., 2005] Just incidence matrices (e.g., which tets contain which triangles?) & primal / dual volumes (area, length, etc.).

271 Volumes / Tetrahedral Meshes Same problems (Poisson, Laplace, etc.) can also be solved on volumes Popular choice: tetrahedral meshes (graded, conform to boundary,... ) Many ways to get Laplace matrix One nice way: discrete exterior calculus (DEC) [Hirani, 2003, Desbrun et al., 2005] Just incidence matrices (e.g., which tets contain which triangles?) & primal / dual volumes (area, length, etc.). Added bonus: play with definition of dual to improve accuracy [Mullen et al., 2011].

272 ...and More! Covered some standard discretizations

273 ...and More! Covered some standard discretizations Many possibilities (level sets, hex meshes... )

274 ...and More! Covered some standard discretizations Many possibilities (level sets, hex meshes... ) Often enough to have gradient G and inner product W.

275 ...and More! Covered some standard discretizations Many possibilities (level sets, hex meshes... ) Often enough to have gradient G and inner product W. (weak!) Laplacian is then C = G T WG (think Dirichlet energy)

276 ...and More! Covered some standard discretizations Many possibilities (level sets, hex meshes... ) Often enough to have gradient G and inner product W. (weak!) Laplacian is then C = G T WG (think Dirichlet energy) Key message: build Laplace; do lots of cool stuff.

277 APPLICATIONS

278 Remarkably Common Pipeline {simple pre-processing}!d( 1)! {simple post-processing}

279 Common Refrain Our method boils down to backslash in Matlab!

280 Reminder: Model Equations Df = 0 Laplace equation Linear solve Df = g Poisson equation Linear solve f t = Df Df i = l i f i Heat equation ODE time-step Vibration modes Eigenproblem

281 Look here! Reminder: Model Equations Df = 0 Laplace equation Linear solve Df = g Poisson equation Linear solve f t = Df Df i = l i f i Heat equation ODE time-step Vibration modes Eigenproblem

282 Reminder: Model Equations Df = 0 Laplace equation Linear solve Df = g Poisson equation Linear solve f t = Df Df i = l i f i Heat equation ODE time-step Vibration modes Eigenproblem

283 Df = 0 Reminder: Variational Interpretation min f (x) RS krf (x)k2 da l <calculus> Df (x) =0

284 Df = 0 Reminder: Variational Interpretation min f (x) RS krf (x)k2 da l <calculus> Df (x) =0 The (inverse) Laplacian wants to make functions smooth. Elliptic regularity

285 Df = 0 Application: Mesh Parameterization Want smooth f : M! R 2.

286 Df = 0 Variational Approach min f :M!R 2 R krf k 2 Does this work?

287 Df = 0 Variational Approach min f :M!R 2 R krf k 2 Does this work? f (x) const.

288 Df = 0 Harmonic Parameterization min f :M!R 2 f M fixed R krf k 2 [Eck et al., 1995]

289 Df = 0 Harmonic Parameterization min f :M!R 2 f M fixed R krf k 2 [Eck et al., 1995] Df = 0 in M\ M, with f M fixed

290 Reminder: Model Equations Df = 0 Laplace equation Linear solve Df = g Poisson equation Linear solve f t = Df Df i = l i f i Heat equation ODE time-step Vibration modes Eigenproblem

291 Df = g Recall: Green s Function Dg p = d p for p 2 M

292 Df = g Application: Biharmonic Distances d b (p, q) kg p g q k 2 [Lipman et al., 2010], formula in [Solomon et al., 2014]

293 Df = g Hodge Decomposition ~v(x) =R 90 rg + rf +~ h(x) Divergence-free part: R 90 rg Curl-free part: rf Harmonic part: ~ h(x) (= ~ 0 if surface has no holes)

294 Df = g Computing the Curl-Free Part min f (x) RS krf (x) ~v(x)k2 da l <calculus> Df (x) =r ~v(x) Get divergence-free part as~v(x) rf (x) (when~ h ~ 0)

295 Df = g Application: Vector Field Design Df = K! ~v(x) =rf (x) [Crane et al., 2010, de Goes and Crane, 2010]

296 Df = g Application: Earth Mover s Distance Z min k~ J(x)k ~ J(x) M such that~ J = R 90 rg + rf +~ h(x) Df = r 1 r 0 [Solomon et al., 2014]

297 Reminder: Model Equations Df = 0 Laplace equation Linear solve Df = g Poisson equation Linear solve f t = Df Df i = l i f i Heat equation ODE time-step Vibration modes Eigenproblem

298 f t = Df Generalizing Gaussian Blurs Gradient descent on R krf (x)k 2 dx: f (x,t) t = D x f (x, t) with f (,0) f 0 ( ). Image by M. Bottazzi

299 f t = Df Application: Implicit Fairing Idea: Take f 0 (x) to be the coordinate function.

300 f t = Df Application: Implicit Fairing Idea: Take f 0 (x) to be the coordinate function. Detail: D changes over time. [Desbrun et al., 1999]

301 Df = g Alternative: Screened Poisson Smoothing Simplest incarnation of [Chuang and Kazhdan, 2011]: min f (x) a 2 kf f 0 k 2 + krf k 2 l (a 2 I D)f = a 2 f 0

302 f t = Df! Df = g Interesting Connection (Semi-)Implicit Euler: (I hl)u k+1 = u k Screened Poisson: (a 2 I D)f = a 2 f 0

303 f t = Df! Df = g Interesting Connection (Semi-)Implicit Euler: (I hl)u k+1 = u k Screened Poisson: (a 2 I D)f = a 2 f 0 One time step of implicit Euler is screened Poisson.

304 f t = Df! Df = g Interesting Connection (Semi-)Implicit Euler: (I hl)u k+1 = u k Screened Poisson: (a 2 I D)f = a 2 f 0 One time step of implicit Euler is screened Poisson. Accidentally replaced one PDE with another!

305 f t = Df and Df = g Application: The Heat Method Eikonal equation for geodesics: krfk 2 = 1 =) Need direction of rf.

306 f t = Df and Df = g Application: The Heat Method Eikonal equation for geodesics: krfk 2 = 1 =) Need direction of rf. Idea: Find u such that ru is parallel to geodesic.

307 f t = Df and Df = g Application: The Heat Method 1 Integrate u 0 = ru (heat equation) to time t 1. 2 Define vector field X ru kruk 2. 3 Solve least-squares problem rf X () Df = r X. Blazingly fast! [Crane et al., 2013b]

308 Reminder: Model Equations Df = 0 Laplace equation Linear solve Df = g Poisson equation Linear solve f t = Df Df i = l i f i Heat equation ODE time-step Vibration modes Eigenproblem

309 Df i = l i f i Laplace-Beltrami Eigenfunctions Image by B. Vallet and B. Lévy Use eigenvalues and eigenfunctions to characterize shape.

310 Df i = l i f i Intrinsic Laplacian-Based Descriptors All computable from eigenfunctions! HKS(x; t) = i e l it f i (x) 2 [Sun et al., 2009] GPS(x) = f1 (x) p l1, p f 2(x) l2,... [Rustamov, 2007] WKS(x; e) =C e  i f i (x) 2 exp 1 2s 2 (e log( l i )) [Aubry et al., 2011] Many others or learn a function of eigenvalues! [Litman and Bronstein, 2014]

311 f t = Df Example: Heat Kernel Signature Heat diffusion encodes geometry for all times t 0! HKS(x; t) k t (x, x) Amount of heat diffused from x to itself over at time t. [Sun et al., 2009] Signature of point x is a function of t 0 Intrinsic descriptor

312 Df i = l i f i HKS via Laplacian Eigenfunctions Df i = l i f i, f 0 (x) =Â a i f i (x) i f (x, t) t = Df with f (x,0) f 0 (x)

313 Df i = l i f i HKS via Laplacian Eigenfunctions Df i = l i f i, f 0 (x) =Â a i f i (x) i f (x, t) t = Df with f (x,0) f 0 (x) =) f (x, t) =Â a i e lit f i (x) i

314 Df i = l i f i HKS via Laplacian Eigenfunctions Df i = l i f i, f 0 (x) =Â a i f i (x) i f (x, t) t = Df with f (x,0) f 0 (x) =) f (x, t) =Â a i e lit f i (x) i =) HKS(x; t) k t (x, x) = Â e l i t f i (x) 2 i

315 Df i = l i f i Application: Shape Retrieval Solve problems like shape similarity search. Shape DNA [Reuter et al., 2006]: Identify a shape by its vector of Laplacian eigenvalues

316 Df i = l i f i Different Application: Quadrangulation Connect critical points (well-spaced) of f i in Morse-Smale complex. [Dong et al., 2006]

317 Other Ideas I Mesh editing: Displacement of vertices and parameters of a deformation should be smooth functions along a surface [Sorkine et al., 2004, Sorkine and Alexa, 2007] (and many others)

318 Other Ideas II Surface reconstruction: Poisson equation helps distinguish inside and outside [Kazhdan et al., 2006] Regularization for mapping: To compute f : M 1! M 2, ask that f D 1 D 2 f [Ovsjanikov et al., 2012]