Isotopic Approximation within a Tolerance Volume

Transcription

1 Isotopic Approximation within a Tolerance Volume Manish Mandad David Cohen-Steiner Pierre Alliez Inria Sophia Antipolis - 1

2 Goals and Motivation - 2

3 Goals and Motivation Input: Tolerance volume of a surface geometry - 3

4 Goals and Motivation Input: Tolerance volume of a surface geometry Output: Surface triangle mesh Properties: Within tolerance volume Motivation: control global approximation error - 4

5 Goals and Motivation Input: Tolerance volume of a surface geometry Output: Surface triangle mesh Properties: Within tolerance volume Intersection free Motivation: simulation, machining, printing etc. - 5

6 Goals and Motivation Input: Tolerance volume of a surface geometry Output: Surface triangle mesh Properties: Within tolerance volume Intersection free Low vertex count Motivation: get an approximated mesh, low polygon count - 6

7 Goals and Motivation A Condition for Isotopic Approximation [Chazal, Cohen-Steiner 04] Constructive algorithm for this theoretical result - 7

8 Goals and Motivation A Condition for Isotopic Approximation [Chazal, Cohen-Steiner 04] Ω : Topological thickening of S Ω Ω 1 Ω 2-8

9 Goals and Motivation A Condition for Isotopic Approximation [Chazal, Cohen-Steiner 04] Ω : Topological thickening of S If S is included and separates sides of Ω, S is connected and Genus of S does not exceed genus of S Then, S and S are isotopic. Ω S - 9

10 Related Work - 10

11 Related Work Simplification Envelopes Cohen et al Multiresolution decimation based on global error Ciampalini et al Adaptively Sampled Distance Fields Frisken et al Permission Grids: Practical, Error-Bounded Simplification Zelinka Garland 2002 Intersection Free Simplification Gumhold et al GPU-based Tolerance Volumes for Mesh Processing Simplification of surface mesh using Hausdorff envelope Botsch et al Borouchaki Frey

12 Related Work Simplification Envelopes [Cohen et al. 96] - 12

13 Related Work Simplification Envelopes [Cohen et al. 96] Very good for simple geometry Input dependent Tolerance hinders simplification - 13

14 Overview Sampling Refinement Tolerance Volume Ω Simplification Output - 14

15 Algorithm - 15

16 Algorithm (Initialization) S: Point sample of Ω with labels f = -1 f = +1 σ radii balls at S cover Ω Initialization - 16

17 Algorithm (Initialization) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box f = f = +1 f = Initialization - 17

18 Algorithm (Initialization) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box f = f T f = +1 f = Initialization - 18

19 Algorithm (Initialization) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box f = -1 s Error µ(s)= f(s) f T (s) f T f = +1 f = Initialization - 19

20 Algorithm (coarse-to-fine) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Sample point (s S) with maximum error Refinement - 20

21 Algorithm (coarse-to-fine) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T f = -1 f T = 0 Zero-set f = +1 f = +1 Refinement - 21

22 Algorithm (coarse-to-fine) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Refinement - 22

23 Algorithm (coarse-to-fine) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Refinement - 23

24 Algorithm (coarse-to-fine) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Refinement - 24

25 Algorithm (coarse-to-fine) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Boundary of simplicial tolerance ( Ɣ) Refinement - 25

26 Algorithm (coarse-to-fine) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Extension to 3D? Guarantees? Refinement - 26

27 Refinement (conditions) Zero set may cross Ω due to flat tetrahedra - 27

28 Refinement (conditions) Zero set may cross Ω due to flat tetrahedra Ensure interpolated function is Lipschitz - 28

29 Refinement (conditions) Zero set may cross Ω due to flat tetrahedra Ensure interpolated function is Lipschitz μ(s) < 1 α, given some α [0, 1] μ(s) < 1 Classify samples with a margin - 28

30 Refinement (conditions) Zero set may cross Ω due to flat tetrahedra Ensure interpolated function is Lipschitz μ(s) < 1 α, given some α [0, 1] h > 2σ/α - 29

31 Refinement (conditions) What else do we need to fix? - 30

32 Refinement (conditions) Misoriented elements - 31

33 Refinement (conditions) Misoriented elements Interpolated function should classify well local geometry f T defined on ΔABC should classify well samples of S nearest (orange) to a shrunk triangle (green) - 32

34 Refinement (conditions) S not classified : μ(s)>1 { } μ(s) 1 α given some α [0;1] h 2σ/α f does not classifies local geometry - 33

35 Algorithm (fine-to-coarse) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Collapse edges of Ɣ Simplification - 34

36 Simplification (conditions) - 36

37 Simplification (conditions) Validity of Triangulation Combinatorial Topology [Dey et al. 98] Link condition Valid embedding Visibility kernel of 1-ring [Lee-Preparata 79] Edge PQ to be collapsed Valid embedding kernel - 37

38 Simplification (conditions) Validity of Triangulation Combinatorial Topology [Dey et al. 98] Valid embedding Visibility kernel of 1-ring [Lee-Preparata 79] Edge PQ to be collapsed Valid embedding kernel - 38

39 Simplification (conditions) Validity of Triangulation Combinatorial Topology [Dey et al. 98] Valid embedding Visibility kernel of 1-ring [Lee-Preparata 79] Preserve classification of S Non convex problem Edge PQ to be collapsed Valid embedding kernel Region that preserve classification of S - 39

40 Simplification (conditions) Validity of Triangulation Combinatorial Topology [Dey et al. 98] Valid embedding Visibility kernel of 1-ring [Lee-Preparata 79] Preserve classification of S Non convex problem Optimal location : Minimizes sum of squared distances between target vertex and 2-ring planes - 40

41 Simplification (conditions) Validity of Triangulation Combinatorial Topology [Dey et al. 98] Valid embedding Visibility kernel of 1-ring [Lee-Preparata 79] Preserve classification of S Non convex problem Optimal location : Minimizes sum of squared distances between target vertex and 2-ring planes Faithful normals (same as in refinement) - 42

42 Algorithm (fine-to-coarse) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Collapse edges of Ɣ Simplification - 40

43 Algorithm (fine-to-coarse) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Collapse edges of Ɣ Simplification - 41

44 Algorithm (fine-to-coarse) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Collapse edges of Ɣ Simplification - 42

45 Algorithm (fine-to-coarse) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Collapse edges of Ɣ Insert Z in T Simplification - 43

46 Algorithm (fine-to-coarse) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Collapse edges of Ɣ Insert Z in T Collapse edges of Z Simplification - 44

47 Algorithm (fine-to-coarse) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Collapse edges of Ɣ Insert Z in T Collapse edges of Z Simplification - 45

48 Algorithm (fine-to-coarse) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Collapse edges of Ɣ Insert Z in T Collapse edges of Z Simplification - 46

49 Algorithm (fine-to-coarse) S: Point sample of Ω with labels Delaunay Triangulation T of loose bounding-box While (S not classified) Insert Steiner point to T Collapse edges of Ɣ Insert Z in T Collapse edges of Z Final output Output - 47

50 Recap - 48

51 Recap - 48

52 Recap - 48

53 Recap - 48

54 Proof - 55

55 Proof Termination Topology Only assumes separability condition on Ω - 56

56 Results - 62

57 Blade Input Ω Refinement Simplification on Ɣ Simplification on Z Simplification on all edges - 63

58 Varying Tolerance (fertility) - 64

59 Varying Tolerance (vase lion) - 65

60 Progress of Algorithm tolerance - 66

61 Robustness to input dataset and noise - 67

62 Robustness to input dataset and noise - 68

63 Robustness to input dataset and noise - 69

64 Robustness to input dataset and noise [Chazal et al. 11] - 70

65 Comparisons - 71

66 Comparison (Simplification Envelopes) Input Simplification Envelopes H = 60% Our Algorithm H = 10% - 72

67 Comparisons Hausdorff distance (output to input) - 73

68 Comparisons Hausdorff distance (output to input) - 74

69 Comparisons Hausdorff distance (input to output) - 75

70 Comparisons Hausdorff distance (input to output) - 76

71 Comparisons Hausdorff distance (input to output) - 77

72 Comparisons Hausdorff distance (input to output) - 78

73 Comparisons Hausdorff distance (output to input) dots: self intersection - 79

74 Comparisons Hausdorff distance (input to output) - 80

75 Anisotropy - 81

76 Anisotropy - 82

77 Extensions - 83

78 Non-closed Surfaces 2δ rδ During Simplification : Hausdorff distance between Z and boundary is enforced to be δ. - 84

79 Non-closed Surfaces Input preserving holes repairing holes - 85

80 Non-manifold - 86

81 Limitations Tolerance volume dependence Compute and memory intensive Slow for small tolerance - 87

82 Conclusions Algorithm for isotopic approximation Always intersection free and within tolerance Low mesh complexity - 88

83 Further Work Out-of-core Progressive algorithm - 89

84 Questions Thank you. - 90