Justin Solomon MIT, Spring 2017

Transcription

1 Fisher et al. Design of Tangent Vector Fields (SIGGRAPH 2007) Justin Solomon MIT, Spring 2017 Original version from Stanford CS 468, spring 2013 (Butscher & Solomon)

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3 Divergence Theorem Green s Theorem

4 Divergence Theorem Green s Theorem

5 Fundamental Theorem of Calculus

6 One equation, all of calculus

7 Extension of vector calculus to surfaces (and manifolds).

8 1. Exterior calculus Alternating k-forms, derivatives, and integration 2. Discrete exterior calculus All that, on a simplicial complex

9 Our goal: Semester course in 2.5 lectures

10 1. Exterior calculus Alternating k-forms, derivatives, and integration 2. Discrete exterior calculus All that, on a simplicial complex

11 Everything must be intrinsic! Vector fields are tangent!

12 Property: V, V * have same dimension.

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14 ω(v)=number of layers Needle in a 1-form onion

15 Row vs. column vectors

16 Sum over repeated indices!

17 Vector to covector (lowers index) Bloch, Schelomo

18 Covector to vector (raises index) Elgar, Cello Concerto

19 0-form

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21 No metric matrix g

22 What is a two-form?

23 Grid image from Wikipedia ω(v, w)= number of oriented cells

24 Explanation from: Work = force * distance

25 Work = force * distance

26 Bilinear (same as 1D): Flux through degenerate window:

27 Bilinear (same as 1D): Flux through degenerate window: Anti-symmetric (follows from properties above):

28 Bilinear: Flux through degenerate window: k-form: Same thing, k slots! Alternative equivalent definition: ω x Δx 1, Δx 2 = ω x (Δx 2, Δx 1 )(alternating)

29 One DOF 90 rotation Three DOFs DOFs agree with cross product!

30 For each point p on a surface: k vectors in the tangent space at p Differential k-form k-linear Alternating

31 Two relevant details: k = number of inputs n = dimension e.g. a 2-form over R 3 (k=2,n=3)

32 One-form: ω Δx = how much flux in direction Δx Two-form: ω Δx 1, Δx 2 = how much flux in parallelogram (Δx 1, Δx 2 )

33 On the board: Space of k-forms on R k is one-dimensional. On the board: k-forms on R n are zero when k > n.

34 The units change: Not product-like behavior: Dimensionality of cross product is variable: Cross product of vectors is weird!

35 Grid image from Wikipedia

36 Grid image from Wikipedia

37 Image courtesy K. Crane

38 Image courtesy K. Crane

39 For one-forms: How similar is parallelogram (a,b) to parallelogram (u,v)? Notice: All 2-forms are wedges of 1-forms.

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42 with no repeated indices.

43 Borrow from vectors

44 Example: Inner product of 2-forms over R 3 Again! How similar is parallelogram (v,w) to parallelogram (a,b)?

45 Image courtesy K. Crane

46 Image courtesy K. Crane

47 One differential form per tangent plane

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49 Suppose f: S R and take p S. For v T p S, choose a curve α: ε, ε S with α 0 = p and α 0 = v. Then the differential of f is df: T p S R with Does not depend on choice of α Linear map Following Curves and Surfaces, Montiel & Ros

50 Suppose f: S R and take p S. For v T p S, choose a curve α: ε, ε S with α 0 = p and α 0 = v. Then the differential of f is df: T p S R with Does not depend on choice of α Linear map Following Curves and Surfaces, Montiel & Ros

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52 Given a 1-form α, when is there a function f with α = df? Transforms d on 0-forms to d on 1-forms Iterate! Alternating!

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54 Images courtesy K. Crane

55 Measures amount of ω parallel to γ Integrate on k-dimensional objects

56 Image courtesy K. Crane

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58 Extra credit on homework

59 1. Exterior calculus Alternating k-forms, derivatives, and integration 2. Discrete exterior calculus All that, on a simplicial complex

60 Discrete version of exterior calculus.

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63 Store integrals of forms!

64 Discrete 0-form Store integrated quantities!

65 Discrete 1-form Store integrated quantities!

66 Discrete 2-form Store integrated quantities!

67 Stokes Theorem

68 consists of 1, 0, -1

69 consists of 1, 0, -1

70 consists of 1, 0, -1

71 Two different d matrices

72 Moves to dual mesh

73 Primal 2-form Dual 0-form Moves to dual mesh

74 Primal 1-form Dual 1-form Moves to dual mesh

75 Image courtesy K. Crane

76 Image courtesy K. Crane

77 Just triangle areas

78 Ratio of edge lengths Image courtesy K. Crane

79 Choice of dual: Circumcenter Image courtesy K. Crane

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81 Area of dual cell

82 Area/3 to each vertex

83 Slide courtesy M. Ben-Chen

84 Slide courtesy M. Ben-Chen

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87 Build up tons of matrices Multiply them together for complicated operators

88 Dot product: One primal, one dual. (Already integrated!)

89 grad div

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92 Discontinuous along edges! Image courtesy F. de Goes Interpolate one-form over triangle

93 Curl free

94 Curl free

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96 Conclusion: For λ 0, they re obtained by d and d of Laplacian eigenfunctions.

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99 Palit, Basu, Mandal. Applications of the Discrete Hodge Helmholtz Decomposition to Image and Video Processing. LNCS.

100 Stam. Stable Fluids. SIGGRAPH (and many others) Incompressible: No divergence

101 Tong et al. Discrete Multiscale Vector Field Decomposition. TOG 2003.

102 Stein and Nordlund. Realistic Solar Convection Simulations. Solar Physics 2000.

103 Bahl and Senthilkumaran. Helmholtz Hodge Decomposition of Scalar Optical Fields. J. Opt. Soc. Am. A 2012.

104 Macedo and Castro. Learning Divergence-Free and Curl-Free Vector Fields with Matrix-Valued Kernels.

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107 From Vector Field Processing on Triangle Meshes de Goes, Desbrun, and Tong (SIGAsia 2015)

108 Fisher et al. Design of Tangent Vector Fields (SIGGRAPH 2007) Justin Solomon MIT, Spring 2017 Original version from Stanford CS 468, spring 2013 (Butscher & Solomon)