Three Points Make a Triangle Or a Circle

Transcription

1 Three Points Make a Triangle Or a Circle Peter Schröder joint work with Liliya Kharevych, Boris Springborn, Alexander Bobenko 1

2 In This Section Circles as basic primitive it s all about the underlying geometry! Euclidean: triangles conformal: circles two examples conformal parameterization discrete curvature energies 2

3 Geometries The Erlangen program (1872) geometry through symmetries affine, perspective conformal CA CA =R 2 C A A Möbius xform 3

4 Conformal Mappings Piecewise Linear Surfaces map to domain discrete conformal preserve angles as well as possible circles as primitives 4

5 Paramaterizations An old problem can t have it all keep angles keep areas Our setup Mercator ( ); Frontispiece of Atlas Sive Cosmographicae ( ) USGS Map Projections Site find discrete conformal map from triangle mesh to Euclidean domain 5

6 Conformal Maps Mathematical basis Riemann mapping theorem unique (up to Möbius xforms) e* y e x x y 6

7 Conformal Maps Mathematical basis Riemann mapping theorem unique (up to Möbius xforms) Intrinsic angles in original mesh Texture coordinates 7

8 Sounds Great! You knew there was a catch Laplace problem Dirichlet or Neumann bndry. cond. 8

9 Sounds Great! You knew there was a catch Laplace problem Dirichlet or Neumann bndry. cond. too much control or too little 9

10 An Old Idea Riemann mapping theorem conformal maps map infinitesimal circles to infinitesimal circles Thurston (85) finite circles circle packing Ken Stephenson 10

11 An Old Idea Riemann mapping theorem conformal maps map infinitesimal circles to infinitesimal circles Thurston (85) finite circles circle packing Ken Stephenson 11

12 An Old Idea Riemann mapping theorem conformal maps map infinitesimal circles to infinitesimal circles Thurston (85) finite circles circle packing Stephenson s CirclePack Ken Stephenson 12

13 History Theory early: Koebe 36; Andreev 70 modern: Rudin & Sullivan 87 (hex packing); He & Schramm 98 (C convgce.) variational approaches de Verdière, Brägger, Rivin, Leibon, Bobenko & Springborn 13

14 Basic Setup Given a mesh local geometry around an edge 14

15 Circle Pattern Problem Rivin 94, Bobenko & Springborn 04 given a triangulation K an angle assignment sum conditions 15

16 Circle Pattern Problem Rivin 94, Bobenko & Springborn 04 given a triangulation K an angle assignment sum conditions 16

17 Circle Pattern Problem Uniquely realizable iff a coherent angle system exists linear feasibility problem 17

18 Geometry at an Edge Relationship between variables angle and radii 18

19 Energy Solution is unique minimum! convex energy in easy gradients and Hessians! 19

20 Algorithm Angle assignment quadratic program boundary curvatures free, prescribed Minimize energy Lay out circles angles and radii determine layout 20

21 Results Disk boundary Prescribed boundary Free boundary 37.9k F 39.6k F 12.6k F 28+8s 26+9s 7+2s 21

22 Boundary Control You get to control curvature 22

23 Quasi-Conformal Distr. 23

24 Quasi-Conformal Distr. 24

25 Robustness 25

26 Problems The price to pay want angles (nearly) preserved must suffer large area distortion 26

27 Piecewise Flat Back to first principles what does the mesh give us? everywhere flat with some exceptions Euclidean metric with cone singularities 27

28 Cone Singularities Circle pattern approach allows for cone singularities! set cone vertices rest of machinery works as before 28

29 Examples 29

30 Examples 30

31 Properties Circle patterns with cone sing. discrete conformal 31

32 Properties Circle patterns with cone sing. discrete conformal low area distortion arbitrary topology no cutting a priori! globally continuous 32

33 Summary Discrete conformal mappings formulate as circle pattern problem solution is min. of convex energy simple gradient and Hessian cone singularities no cutting a priori & arbitrary topology You can have both: area & angle! 33

34 More Fun with Circles Willmore energy of a surface vanishes iff surface is sphere 35

35 More Fun with Circles Willmore energy of a surface Möbius invariant of interest: minimizers theory of surfaces Conformal Geometry 36

36 More Fun with Circles Willmore energy of a surface of interest: minimizers theory of surfaces geometric modeling Conformal Geometry 37

37 More Fun with Circles Willmore energy of a surface of interest: minimizers theory of surfaces geometric modeling physical modeling Conformal Geometry 38

38 Discrete Willmore Flow Approach simplicial 2-manifolds preserve symmetries Möbius invariance non-linear formulation semi-implicit integration boundary conditions 39

39 Previous Work Discrete setting Discrete Differential Geometry 4 th order flows [SK01] [XPB05] [YB02] use existing lower order operators discretized continuous setting level set [TWBO03] [DR04] FEM [DDE03] [HGR01] [CDD*04] 40

40 Disc. Willmore Energy Definition [B05] object of conformal geometry use circles and angles at each vertex 41

41 Properties I Discrete Willmore energy vanishes iff co-spherical & convex 42

42 Properties I Discrete Willmore energy vanishes iff co-spherical & convex 43

43 Properties I Discrete Willmore energy vanishes iff co-spherical & convex 44

44 Properties II Discrete Willmore energy smooth limit discrete continuous independent of κ 1,2 45

45 Properties II Discrete Willmore energy smooth limit discrete continuous independent of κ 1,2 R 1 with equality for curvature line parameterizations 46

46 Properties III Evaluation symmetry 47

47 Properties III Evaluation symmetry derivatives gradient N x N linear system in (a,b,c,d) 48

48 Properties III Evaluation symmetry derivatives gradient N x N linear system in (a,b,c,d) 49

49 Properties III Evaluation symmetry derivatives gradient N x N linear system in (a,b,c,d) 50

50 Properties III Evaluation symmetry derivatives gradient N x N linear system in (a,b,c,d) 51

51 Properties III Evaluation symmetry derivatives gradient N x N linear system in (a,b,c,d) exact Hessian is 3N x 3N 52

52 Properties IV Gradient singularity what about csc(β)? direction of decrease Boundary conditions fixed: easy 53

53 Properties IV Gradient singularity what about csc(β)? direction of decrease Boundary conditions fixed: easy free: add vertex at boundary edge β 54

54 Properties IV Gradient singularity what about csc(β)? direction of decrease Boundary conditions fixed: easy free: add vertex at boundary edge β 55

55 Properties IV Gradient singularity what about csc(β)? direction of decrease Boundary conditions fixed: easy free: add vertex at boundary edge β 56

56 Results I Simple tests sphere 57

57 Results I Simple tests sphere boundaries 58

58 Results I Simple tests sphere boundaries 59

59 Results I Simple tests sphere boundaries 60

60 Results I Simple tests sphere boundaries 61

61 Results I Simple tests sphere boundaries 62

62 Results I Simple tests sphere boundaries 63

63 Results II Curvature alignment 64

64 Results III Hole filling 65

65 Results III Hole filling 66

66 Results III Hole filling 67

67 Results III Hole filling 68

68 Results III Hole filling 69

69 Results III Hole filling 70

70 Results III Hole filling Inpainting 71

71 Results III Hole filling Inpainting 72

72 Results III Hole filling Inpainting 73

73 Results III Hole filling Inpainting 74

74 Results IV Shrinkage free denoising 75

75 Summary Discrete Willmore flow preserve symmetries: Möbius semi-implicit time stepping relevance in many geo. proc. areas surface theory variational geometric modeling physical modeling 76

76 Outlook Future work more boundary conditions incorporate reference curvature control of triangle quality variational subdivision numerical robustness 77

77 Circle Summary Obey the geometry what geometry do the objects of interest belong to? conformal parameterization curvature energies circles and the angles they make with one another complete non-linear treatment 78