Discrete Ricci Flow and Its Applications

Transcription

1 Discrete Ricci Flow and Its Applications State University of New York 1 Motivation Ricci flow is a powerful tool for designing Riemannian metrics, which lays down the theoretic foundation for many crucial engineering applications, such as surface matching, shape classification and analysis, which have broad applications in computer graphics, computer vision, geometric modeling, wireless sensor networking and medical imaging. The fundamental problems that can be tackled by Ricci flow include the followings: 1. Given a Riemannian metric on a surface with an arbitrary topology, determine the corresponding conformal structure. 2. Fix the conformal structure and the desired Gaussian curvature, compute the corresponding Riemannian metric. 3. Given the distortion between two conformal structures, compute the quasi-conformal mapping. 4. Conformal Welding, glue surfaces with various conformal modules, compute the conformal structure of the glued surface. In the past years, we have developed computational conformal geometry library, which is capable of computing conformal mappings among surfaces with arbitrary topologies; conformal modules; Riemannian metrics with prescribed curvatures; quasi-conformal mappings by solving Beltrami equations. 2 Theoretic Foundation Conformal Map Let φ : (S 1,g 1 ) (S 2,g 2 ) is a homeomorphism, φ is conformal if and only if φ g 2 = e 2u g 1. Conformal Mapping preserves angles and maps infinitesimal circles to infinitesimal circles as shown in figure. Suppose a surface Σ is with a Riemannian metric g, then it can be covered by isothermal coordinates (u,v), such that g = e 2λ(u,v) (du 2 + dv 2 ). The Gaussian curvature is given by K(u,v) = g λ = 1 λ(u,v), where = e 2λ(u,v) u 2 Suppose the Riemannian metric g is transformed by a conformal deformation to ḡ = e 2λ g, then the Gaussian curvature is changed to K = e 2λ ( g λ + K), this is called the Yamabe equation. Yamabe equation can be solved by Hamilton s Ricci flow. v 2. Hamilton s Surface Ricci Flow A closed surface with a Riemannian metric g, the Ricci flow on it is defined as dg ij = Kg ij. dt 1

2 Figure 1: The top row shows conformal mapping; the bottom row shows quasi-conformal mapping. If the total area of the surface is preserved during the flow, the Ricci flow will converge to a metric such that the Gaussian curvature is constant every where. Therefore, Hamilton s Ricci flow can be applied to prove the uniformization theorem. Theorem 2.1 (Poincaré Uniformization Theorem). Let (Σ, g) be a compact 2-dimensional Riemannian manifold. Then there is a metric g = e 2λ g conformal to g which has constant Gauss curvature. Spherical Euclidean Hyperbolic Figure 2: Uniformization for surfaces with boundaries. For surfaces with boundaries, the uniformization theorem still holds Circle Domain A domain in the Riemann sphere Ĉ is called a circle domain if every connected component of its boundary is either a circle or a point. Theorem 2.2. Any domain Ω in Ĉ, whose boundary Ω has at most countably many components, is conformally homeomorphic to a circle domain Ω in Ĉ. Moreover Ω is unique upto Möbius transformations, and every conformal automorphism of Ω is the restriction of a Möbius transformation. 2

3 Spherical Euclidean Hyperbolic Figure 3: Uniformization for surfaces with boundaries. 3 Discrete Algorithms The smooth surface Ricci flow theorem can be generalized to the discrete setting. The surfaces are represented as simplicial complex, each triangle is a Euclidean, spherical or hyperbolic triangle. The discrete Riemannian metric is the edge length function, the Gaussian curvature on a vertex is the angle deficit. Then we define discrete conformal factor function λ on each vertex. One way to define discrete conformal metric deformation is l ij e λ i l ij e λ j, where l ij is the length of edge [v i,v j ]. Then discrete curvature flow is given by dλ i dt = K i, which is the negative gradient flow of a convex energy i K idλ i. Figure 4 shows a Riemann mapping result using discrete curvature flow. Figure 4: Riemann mapping computed using discrete Ricci flow. 4 Applications Ricci flow has been applied for brain morphology study and virtual colonoscopy in medical imaging; spline fitting in geometric modeling; surface matching, tracking in computer vision; mesh parameterization in computer graphics; efficient and reliable routing in networking. It has more applications in a much broader range of fields in engineering and biomedicine. 3

4 Discrete Ricci Flow and Its Applications 1 1 Department of Computer Science University of New York at Stony Brook Tsinghua Sanya International Mathematics Forum

5 Thanks Thanks for the invitation.

6 Collaborators The work is collaborated with Shing-Tung Yau, Huai-Dong Cao, Feng Luo, Tony Chan, Ronald Lok Ming Lui, Paul Thompson, Yalin Wang, Hong Qin, Dimitris Samaras, Jie Gao, Arie Kaufman, and many other mathematicians, computer scientists and medical doctors.

7 Motivation Ricci flow is a powerful tool for designing Riemannian metrics, which lays down the theoretic foundation for the following crucial engineering applications: Surface mapping Shape classification Shape analysis Applied in computer graphics, computer vision, geometric modeling, wireless sensor networking and medical imaging, and many other engineering, medical fields.

8 Fundamental Problems 1 Given a Riemannian metric on a surface with an arbitrary topology, determine the corresponding conformal structure. 2 Compute the complete conformal invariants (conformal modules), which are the coordinates of the surface in the Teichmuller shape space. 3 Fix the conformal structure, find the simplest Riemannian metric among all possible Riemannian metrics 4 Given desired Gaussian curvature, compute the corresponding Riemannian metric. 5 Given the distortion between two conformal structures, compute the quasi-conformal mapping. 6 Compute the extremal quasi-conformal maps. 7 Conformal welding, glue surfaces with various conformal modules, compute the conformal module of the glued surface.

9 Complete Tools Computational Library 1 Compute conformal mappings for surfaces with arbitrary topologies 2 Compute conformal modules for surfaces with arbitrary topologies 3 Compute Riemannian metrics with prescribed curvatures 4 Compute quasi-conformal mappings by solving Beltrami equation

10 Books The theory, algorithms and sample code can be found in the following books. You can find them in the book store.

11 Source Code Library Please me for updated code library on computational conformal geometry.

12 Conformal Mapping

13 biholomorphic Function Definition (biholomorphic Function) Suppose f : C C is invertible, both f and f 1 are holomorphic, then then f is a biholomorphic function. γ0 D 0 γ1 γ2 D 1

14 Conformal Map S 1 {(U α,φ α )} S 2 {(V β,τ β )} U α f V β φ α τ β z τ β f φ 1 α w The restriction of the mapping on each local chart is biholomorphic, then the mapping is conformal.

15 Conformal Mapping

16 Definition (Conformal Map) Let φ :(S 1,g 1 ) (S 2,g 2 ) is a homeomorphism, φ is conformal if and only if φ g 2 = e 2u g 1. Conformal Mapping preserves angles. θ θ

17 Conformal Mapping Conformal maps Properties Map a circle field on the surface to a circle field on the plane.

18 Quasi-Conformal Map Diffeomorphisms: maps ellipse field to circle field.

19 Computational Method - Surface Ricci Flow

20 Discrete Curvature Flow Isothermal Coordinates A surface Σ with a Riemannian metric g, a local coordinate system (u,v) is an isothermal coordinate system, if g=e 2λ(u,v) (du 2 + dv 2 ). Gaussian Curvature The Gaussian curvature is given by where Δ= 2 u v 2. K(u,v)= Δ g λ = 1 e 2λ(u,v)Δλ(u,v),

21 Conformal Metric Deformation Definition Suppose Σ is a surface with a Riemannian metric, ( ) g11 g g= 12 g 21 g 22 Suppose λ :Σ R is a function defined on the surface, then e 2λ g is also a Riemannian metric on Σ and called a conformal metric. λ is called the conformal factor. g e 2λ g Angles are invariant measured by conformal metrics. Conformal metric deformation.

22 Curvature and Metric Relations Yamabe Equation Suppose ḡ=e 2λ g is a conformal metric on the surface, then the Gaussian curvature on interior points are K = e 2λ ( Δ g λ + K), geodesic curvature on the boundary k g = e λ ( n λ + k g ).

23 Uniformization Theorem (Poincaré Uniformization Theorem) Let (Σ, g) be a compact 2-dimensional Riemannian manifold. Then there is a metric g=e 2λ g conformal to g which has constant Gauss curvature. Spherical Euclidean Hyperbolic

24 Surface Ricci Flow Definition (Hamilton s Surface Ricci Flow) A closed surface with a Riemannian metric g, the Ricci flow on it is defined as dg ij = Kg ij. dt If the total area of the surface is preserved during the flow, the Ricci flow will converge to a metric such that the Gaussian curvature is constant every where.

25 Key Idea Ricci flow on smooth Riemannian manifolds can be generalized to discrete Riemannian manifolds, (Riemannian metrics with cone singularities).

26 Generic Surface Model - Triangular Mesh Surfaces are represented as polyhedron triangular meshes. Isometric gluing of triangles in E 2. Isometric gluing of triangles in H 2,S 2.

27 Discrete Metrics Definition (Discrete Metric) A Discrete Metric on a triangular mesh is a function defined on the vertices, l : E ={all edges} R +, satisfies triangular inequality. A mesh has infinite metrics.

28 Discrete Curvature Definition (Discrete Curvature) Discrete curvature: K : V ={vertices} R 1. K(v)=2π i α i,v M;K(v)=π α i,v M i Theorem (Discrete Gauss-Bonnet theorem) K(v)+ K(v)=2πχ(M). v M v M v α 1 α 2 α 3 α 1 α2 v

29 Discrete Metrics Determines the Curvatures l j θ i v i v k v k θ k l i lj θ l j k li θ θ i θ j j v i l k v l j v k i v j θ i θ k θ j R 2 H2 S 2 l k v k l i v j Angle and edge length relations: cosine laws R 2,H 2,S 2 cosl i = cosθ i + cosθ j cosθ k sinθ j sinθ k (1) coshl i = coshθ i+ coshθ j coshθ k sinhθ j sinhθ k (2) 1 = cosθ i + cosθ j cosθ k sinθ j sinθ k (3)

30 Discrete Conformal Metric Deformation Conformal maps Properties transform infinitesimal circles to infinitesimal circles. preserve the intersection angles among circles. Idea - Approximate conformal metric deformation Replace infinitesimal circles by circles with finite radii.

31 Discrete Conformal Metric Deformation Circle Patterns There are many local settings for circle patterns. The radius is variable, the intersection angles do not change.

32 Circle Patterns i γi i Θki l ij θ i γ i Θij θi lki lij γk γ j θ j j l jk γ k l ki θ k k j θj γj ljk θk k Θjk i γ i i θ i l ij θ i l ki l ij l ki j θ j γ j l jk θ k γ k k j θ j l jk θ k k

33 Circle Patterns

34 Discrete Approach Circle Packing Thurston introduced circle packing metric for studying 3-manifolds in Sullivan and Rodin proved Thurston s circle packing conjecture in The first variational principle for CP metrics was founded by Colin de Verdiere (1991). Zheng-Xu He and O. Schramm proved the classical Riemannian mapping theorem in Chow and Luo built the connection between Ricci flow and circle packing in Springborn, Bobenko and Schroder s circle pattern in 2005.

35 Circle Packing Metric CP Metric We associate each vertex v i with a circle with radius γ i. On edge e ij, the two circles intersect at the angle of Φ ij. The edge lengths are φ12 φ r 1 v 31 1 e 12 e 31 e23 v 2 v 3 r 2 φ 23 r 3 l 2 ij = γ 2 i + γ 2 j + 2γ i γ j cosφ ij CP Metric (Σ,Γ,Φ), Σ triangulation, Γ={γ i v i },Φ={φ ij e ij }

36 Discrete Conformal Factor Conformal Factor Defined on each vertex u:v R, logγ i R 2 u i = logtanh γ i 2 H 2 logtan γ i 2 S 2 Properties Symmetry Discrete Laplace Equation K i u j = K j u i dk=δdu, Δ is a discrete Lapalce-Beltrami operator.

37 Discrete Yamabe Flow Feng Luo, Combinatorial Yamabe Flow on Surfaces, Commun.Contemp.Math., Vol.6 Num. 5, Pages , Boris, Springborn and Peter Schröder and Ulrich Pinkall, Conformal equivalence of triangle meshes, ACM Trans. Graph., vol.27 Num. 3, pages 1-11, 2008.

38 Discrete Conformal Factor for Yamabe Flow Discrete conformal metric deformation: u 1 l 3 l 2 y3 θ 1 y 2 u 2 l 1 θ 3 θ 2 y 1 u 3 conformal factor y k2 = e u i l k 2 e u j R 2 sinh y k 2 = e u i sinh l k 2 e u j H 2 sin y k 2 = e u i sin l k 2 e u j S 2 Properties: K i u j = K j u i and dk=δdu.

39 Unified Framework of Discrete Curvature Flow Unified framework for both Discrete Ricci flow and Yamabe flow Curvature flow Energy E(u)= Hessian of E denoted as Δ, du dt = K K, ( K i K i )du i, i dk=δdu.

40 Hyperbolic Ricci Flow Computational results for genus 2 and genus 3 surfaces.

41 Hyperbolic Koebe s Method

42 Hyperbolic Volumetric Curvature Flow Suppose the given volume has complicated topology, such that the boundary surfaces are with high genus. Then we can compute the canonical hyperbolic Riemannian metric of the volume, and embed the universal covering space of the volume in three dimensional hyperbolic space H 3.

43 Hyperbolic Volumetric Curvature Flow 1 Triangulate the volume to truncated tetrahedra. 2 Compute the curvature on each edge of the tetrahedra mesh K(e ij )=2π θij kl, kl where θij kl is the dihedral angle on the edge (e ij in the tetrahedron [v i,v j,v k,v l ]. 3 Run curvature flow, dl ij = K dt ij where l ij is the edge length of e ij. Theorem If the input 3-manifold is a hyperbolic 3-manifold with complete geodesic boundary, then the curvature flow will converge to the canoical hyperbolic metric.

44 Hyperbolic Volumetric Curvature Flow a. input 3-manifold b. embedding of its UCS in H 3

45 Uniformization

46 Conformal Canonical Representations Theorem (Poincaré Uniformization Theorem) Let (Σ, g) be a compact 2-dimensional Riemannian manifold. Then there is a metric g=e 2λ g conformal to g which has constant Gauss curvature. Spherical David Euclidean Gu Hyperbolic

47 Uniformization of Open Surfaces Definition (Circle Domain) A domain in the Riemann sphere Ĉ is called a circle domain if every connected component of its boundary is either a circle or a point. Theorem Any domain Ω in Ĉ, whose boundary Ω has at most countably many components, is conformally homeomorphic to a circle domain Ω in Ĉ. Moreover Ω is unique upto Möbius transformations, and every conformal automorphism of Ω is the restriction of a Möbius transformation.

48 Uniformization of Open Surfaces Spherical Euclidean Hyperbolic

49 Conformal Canonical Representation Simply Connected Domains

50 Conformal Canonical Forms Topological Quadrilateral p 4 p 3 p 1 p 2 p 4 p 3 p 1 p 2

51 Conformal Canonical Forms Multiply Connected Domains

52 Conformal Canonical Forms Multiply Connected Domains

53 Conformal Canonical Forms Multiply Connected Domains

54 Conformal Canonical Forms Multiply Connected Domains

55 Conformal Canonical Representations Definition (Circle Domain in a Riemann Surface) A circle domain in a Riemann surface is a domain, whose complement s connected components are all closed geometric disks and points. Here a geometric disk means a topological disk, whose lifts in the universal cover or the Riemann surface (which is H 2, R 2 or S 2 are round. Theorem Let Ω be an open Riemann surface with finite genus and at most countably many ends. Then there is a closed Riemann surface R such that Ω is conformally homeomorphic to a circle domain Ω in R. More over, the pair (R,Ω ) is unique up to conformal homeomorphism.

56 Conformal Canonical Form Tori with holes

57 Conformal Canonical Form High Genus Surface with holes

58 Teichmüller Space

59 Teichmüller Theory: Conformal Mapping Definition (Conformal Mapping) Suppose (S 1,g 1 ) and (S 2,g 2 ) are two metric surfaces, φ : S 1 S 2 is conformal, if on S 1 g 1 = e 2λ φ g 2, where φ g 2 is the pull-back metric induced by φ. Definition (Conformal Equivalence) Suppose two surfaces S 1,S 2 with marked homotopy group generators, {a i,b i } and {α i,β i }. If there exists a conformal map φ : S 1 S 2, such that φ [a i ]=[α i ],φ [b i ]=[β i ], then we say two marked surfaces are conformal equivalent.

60 Teichmüller Space Definition (Teichmüller Space) Fix the topology of a marked surface S, all conformal equivalence classes sharing the same topology of S, form a manifold, which is called the Teichmüller space of S. Denoted as T S. Each point represents a class of surfaces. A path represents a deformation process from one shape to the other. The Riemannian metric of Teichmüller space is well defined.

61 Teichmüller Space Topological Quadrilateral p 4 p 3 p 1 p 2 Conformal module: h w. The Teichmüller space is 1 dimensional.

62 Teichmüller Space Multiply Connected Domains Conformal Module : centers and radii, with Möbius ambiguity. The Teichmüller space is 3n 3 dimensional, n is the number of holes.

63 Topological Pants Genus 0 surface with 3 boundaries is conformally mapped to the hyperbolic plane, such that all boundaries become geodesics.

64 Teichmüller Space Topological Pants γ 1 γ 2 γ 0 γ 0 τ 2 γ 0 2 τ0 γ1 2 γ1 τ0 γ 1 2 τ 1 τ 22 γ2 2 τ1 τ2 2 γ2 2 τ1 τ 0 τ 22 γ0 2 γ0 2 γ 1 2 τ 1 τ 2 γ 0 2

65 Teichmüller Space Topological Pants Decomposition - 2g 2 pairs of Pants P1 γ p1 p2 P2 θ

66 Compute Teichmüller coordinates Step 1. Compute the hyperbolic uniformization metric. Step 2. Compute the Fuchsian group generators.

67 Compute Teichmüller Coordinates Step 3. Pants decomposition using geodesics and compute the twisting angle.

68 Teichmüller Coordinates Compute Teichmüller coordinates Compute the pants decomposition using geodesics and compute the twisting angle.

69 Quasi-Conformal Maps

70 Quasi-Conformal Map Most homeomorphisms are quasi-conformal, which maps infinitesimal circles to ellipses.

71 Beltrami-Equation Beltrami Coefficient Let φ : S 1 S 2 be the map, z,w are isothermal coordinates of S 1, S 2, Beltrami equation is defined as µ < 1 φ z = µ(z) φ z

72 Diffeomorphism Space Theorem Given two genus zero metric surface with a single boundary, {Diffeomorphisms} = {Beltrami Coefficient}. {Mobius}

73 Solving Beltrami Equation The problem of computing Quasi-conformal map is converted to compute a conformal map. Solveing Beltrami Equation Given metric surfaces (S 1,g 1 ) and (S 2,g 2 ), let z,w be isothermal coordinates of S 1,S 2,w = φ(z). Then g 1 = e 2u 1 dzd z (4) g 2 = e 2u 2 dwd w, (5) φ :(S 1,g 1 ) (S 2,g 2 ), quasi-conformal with Beltrami coefficient µ. φ :(S 1,φ g 2 ) (S 2,g 2 ) is isometric φ g 2 = e u 2 dw 2 = e u 2 dz+ µddz 2. φ :(S 1, dz+ µddz 2 ) (S 2,g 2 ) is conformal.

74 Quasi-Conformal Map Examples D3 D2 D0 p q p q D1 D3 D2 D0 p q D1

75 Quasi-Conformal Map Examples

76 Applications

77 Medical Imaging Application Medical Imaging Quantitatively measure and analyze the surface shapes, to detect potential abnormality and illness. Shape reconstruction from medical images. Compute the geometric features and analyze shapes. Shape registration, matching, comparison. Shape retrieval.

78 Conformal Brain Mapping Brain Cortex Surface Conformal Brain Mapping for registration, matching, comparison.

79 Conformal Brain Mapping Using conformal module to analyze shape abnormalities. Brain Cortex Surface

80 Automatic sulcal landmark Tracking With the conformal structure, PDE on Riemann surfaces can be easily solved. Chan-Vese segmentation model is generalized to Riemann surfaces to detect sulcal landmarks on the cortical surfaces automatically

81 Abnormality detection on brain surfaces The Beltrami coefficient of the deformation map detects the abnormal deformation on the brain.

82 Abnormality detection on brain surfaces The brain is undergoing gyri thickening (commonly observed in Williams Syndrome) The Beltrami index can effectively measure the gyrification pattern of the brain surface for disease analysis.

83 Virtual Colonoscopy Colon cancer is the 4th killer for American males. Virtual colonosocpy aims at finding polyps, the precursor of cancers. Conformal flattening will unfold the whole surface.

84 Virtual Colonoscopy Supine and prone registration. The colon surfaces are scanned twice with different postures, the deformation is not conformal.

85 Virtual Colonoscopy Supine and prone registration. The colon surfaces are scanned twice with different postures, the deformation is not conformal.

86 Computer Vision Application Vision Compute the geometric features and analyze shapes. Shape registration, matching, comparison. Tracking.

87 Surface Matching Isometric deformation is conformal. The mask is bent without stretching.

88 Surface Matching Facial expression change is not-conformal.

89 Surface Matching 3D surface matching is converted to image matching by using conformal mappings. f φ 1 φ 2 f

90 Face Surfaces with Different Expressions are Matched

91 Face Surfaces with Different Expressions are Matched

92 Face Expression Tracking

93 Face Expression Tracking

94 2D Shape Space-Conformal Welding { Diffeomorphism on S 1 } {Conformal Module} {2D Contours} = {Mobius Transformation}

95 Computer Graphics Application Graphics Surface Parameterization, texture mapping Texture synthesis, transfer Vector field design Shape space and retrieval.

96 Surface Parameterization Map the surfaces onto canonical parameter domains

97 Surface Parameterization Applied for texture mapping.

98 n-rosy Field Design Design vector fields on surfaces with prescribed singularity positions and indices.

99 n-rosy Field Design Convert the surface to knot structure using smooth vector fields.

100 Texture Transfer Transfer the texture between high genus surfaces.

101 Polycube Map Compute polycube maps for high genus surfaces.

102 Geometric Modeling Application: Manifold Spline Manifold Spline Convert scanned polygonal surfaces to smooth spline surfaces. Conventional spline scheme is based on affine geometry. This requires us to define affine geometry on arbitrary surfaces. This can be achieved by designing a metric, which is flat everywhere except at several singularities (extraordinary points). The position and indices of extraordinary points can be fully controlled.

103 Manifold Spline Extraordinary Points Fully control the number, the index and the position of extraordinary points. For surfaces with boundaries, splines without extraordinary point can be constructed. For closed surfaces, splines with only one singularity can be constructed.

104 Manifold Spline F cafa cbfb fab faua fa fb fbub Z ua ub M

105 Manifold Spline Converting a polygonal mesh to TSplines with multiple resolutions.

106 Manifold Spline Converting scanned data to spline surfaces.

107 Manifold Spline Converting scanned data to spline surfaces, the control points, knot structure are shown.

108 Manifold Spline Converting scanned data to spline surfaces, the control points, knot structure are shown.

109 Manifold Spline Polygonal mesh to spline, control net and the knot structure.

110 Manifold Spline

111 Manifold Spline volumetric spline.

112 Wireless Sensor Network Application Wireless Sensor Network Detecting global topology. Routing protocol. Load balancing. Isometric embedding.

113 Greedy Routing Given sensors on the ground, because of the concavity of the boundaries, greedy routing doesn t work.

114 Greedy Routing Map the network to a circle domain, all boundaries are circles, greedy routing works.

115 Load Balancing Schoktty Group - Circular Reflection

116 Computational Topology Application Canonical Homotopy Class Representative Under hyperbolic metric, each homotopy class has a unique geodesic, which is the representative of the homotopy class. γ γ Γ Γ

117 Computational Topology Application Homotopy Detection Problem Verify whether two loops are homotopic equivalent. γ γ Γ Γ

118 Computational Topology Application Shortest Word Problem Find the shortest representation of a homotopy class in the fundamental group.

119 Graph Theory Planar Planar Graph Embedding.

120 Graph Theory Graph isomorphism Detect if two planar graphs are isomorphic.

121 Summary Ricci flow is a powerful tool for designing Riemannian metrics using curvatures, and extremely valuable for many fields in computer science. Computer Graphics - Parameterization Computer Vision - Surface registration, tracking, analysis Geometric Modeling - Spline construction Networking - Routing, load balancing Medical Imaging - Brain mapping, Virtual Colonoscopy

122 Future Works Generalize the computational methods for Ricci flow to high dimensional manifolds.

123 Thanks For more information, please to Thank you!