An algorithm for triangulating multiple 3D polygons

Transcription

1 An algorithm for triangulating multiple 3D polygons Ming Zou 1, Tao Ju 1, Nathan Carr 2 1 Washington University In St. Louis, USA 2 Adobe, USA Eurographics SGP

2 Motivation Curves Surface 2

3 Motivation filling holes 3

4 Motivation filling holes surfacing parallel contours 4

5 Motivation filling holes surfacing parallel contours surfacing 3D sketches 5

6 Motivation filling holes surfacing parallel contours surfacing 3D sketches 6

7 filling holes surfacing parallel contours surfacing 3D sketches Step 1: Identify Patch Boundaries 7

8 Step 1: Identify Patch Boundaries Step 2: Surfacing each patch 8

9 Step 1: Identify Patch Boundaries Step 2: Surfacing each patch 9

10 Problem Definition Input: Input: k 3D polygons (k 1) Output: a triangulation Consists of triangles connecting only the input polygon vertices Topologically equivalent to a sphere with k holes Initial mesh for further refinement output: 10

11 Problem Definition Input: k 3D polygons (k 1) Output: a triangulation Consists of triangles connecting only the input polygon vertices Topologically equivalent to a sphere with k holes Initial mesh for further refinement Monkey Saddle 11

12 Problem Definition Input: k 3D polygons (k 1) Output: a triangulation Consists of triangles connecting only the input polygon vertices Topologically equivalent to a sphere with k holes Initial mesh for further refinement 12

13 Problem Definition Input: k 3D polygons (k 1) Output: an optimal triangulation A triangulation that minimize the sum of certain metric 13

14 Problem Definition Input: k 3D polygons (k 1) Output: an optimal triangulation A triangulation that minimize the sum of certain metric Per-triangle metric Area Perimeter Bi-triangle metric Dihedral angle 14

15 Related Work - Single Polygon A 2D polygon Linear-time triangulation of a simple polygon [Chazelle 91] No optimality guarantee Dynamic programming for optimal triangulation [Gilbert 79, Klincsek 80] A 3D polygon Dynamic programming for optimal triangulation [Barequet et al. 95, 96] Extension of [Gilbert 79] and [Klincsek 80] Heuristic-based algorithms [Liepa 03] [Roth et al. 97] [Bessmeltsev et al. 12] No optimality guarantee 15

16 Related Work - Single Polygon A 2D polygon Linear-time triangulation of a simple polygon [Chazelle 91] No optimality guarantee Dynamic programming for optimal triangulation [Gilbert 79, Klincsek 80] A 3D polygon Dynamic programming for optimal triangulation [Barequet et al. 95, 96] Extension of [Gilbert 79] and [Klincsek 80] Heuristic-based algorithms [Liepa 03] [Roth et al. 97] [Bessmeltsev et al. 12] No optimality guarantee 16

17 Related Work - Multiple Polygons Planar polygons in 3D Dynamic programming for optimally triangulating 2 parallel planar polygons [Fuchs et al. 77] Heuristic-based algorithms for triangulating 2 planar polygons [Barequet et al. 04] [Boissonnat et al. 07] [Liu et al. 08] No optimality guarantees 17

18 Related Work - Multiple Polygons Planar polygons in 3D Dynamic programming for optimally triangulating 2 parallel planar polygons [Fuchs et al. 77] Heuristic-based algorithms for triangulating 2 planar polygons [Barequet et al. 04] [Boissonnat et al. 07] [Liu et al. 08] No optimality guarantees Non-planar polygons in 3D Heuristic-based algorithms No optimality guarantees [Rose et al. 07] Optimal algorithm No known literature 18

19 Contributions 1. The first algorithm for optimally triangulating multiple nonplanar 3D polygons Extension of the dynamic programming algorithm for single polygon [Barequet et al. 95, 96] Guarantee the manifoldness of the surface 2. A fast near-optimal variant for practical use 19

20 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area 20

21 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area 21

22 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area : spanning edge spanning edge 22

23 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area : spanning edge : spanning triangle t spanning triangle 23

24 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area Di : spanning edge : spanning triangle : sub-domain D1 t D2 24

25 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area polygon segment Di : spanning edge : spanning triangle : sub-domain D1 D1 t D2 spanning edge 25

26 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area Di : spanning edge : spanning triangle : sub-domain D1 t D2 D Cost(D) = Cost(D1)+Cost(D2)+Area(t) 26

27 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area Di : spanning edge : spanning triangle : sub-domain t Cost(D) = Cost(D1)+Cost(D2)+Area(t) 27

28 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area Di : spanning edge : spanning triangle : sub-domain t Cost(D) = Cost(D1)+Cost(D2)+Area(t) 28

29 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area Di : spanning edge : spanning triangle : sub-domain t Cost(D) = Cost(D1)+Cost(D2)+Area(t) 29

30 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area Di : spanning edge : spanning triangle : sub-domain t Cost(D) = Cost(D1)+Cost(D2)+Area(t) 30

31 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area Di : spanning edge : spanning triangle : sub-domain t Cost(D) = Cost(D1)+Cost(D2)+Area(t) 31

32 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area Di : spanning edge : spanning triangle : sub-domain t Cost(D) = Mint [Cost(D1)+Cost(D2)+Area(t)] 32

33 Triangulating A Single 3D Polygon [Barequet et. al 95, 96] Example: minimize the sum of triangle area Di : spanning edge : spanning triangle : sub-domain Cost(D) = Mint [Cost(D1)+Cost(D2)+Area(t)] 33

34 Triangulating Multiple 3D Polygons 34

35 Triangulating Multiple 3D Polygons Extension of Domain structure Di : spanning edge : spanning triangle : sub-domain 35

36 Triangulating Multiple 3D Polygons Extension of Domain structure Di : spanning edge : spanning triangle : sub-domain spanning edge spanning edge spanning edge 36

37 Triangulating Multiple 3D Polygons Extension of Domain structure polygon segment Di : spanning edge : polygon segment : spanning triangle : sub-domain polygon segment polygon segment 37

38 Triangulating Multiple 3D Polygons Extension of Domain structure Di : spanning edge : polygon segment : hole : spanning triangle : sub-domain hole hole 38

39 Triangulating Multiple 3D Polygons Extension of Domain structure Di : spanning edge : polygon segment : hole : spanning triangle : sub-domain an input polygon can appear at most once in a domain either as a polygon segment or a hole 39

40 Triangulating Multiple 3D Polygons Extension of Domain structure Di : spanning edge : polygon segment : hole : spanning triangle : sub-domain 1 # of polygon segments k 40

41 Triangulating Multiple 3D Polygons Example: minimize the sum of triangle area Di : spanning edge : polygon segment : hole : spanning triangle : sub-domain 41

42 Triangulating Multiple 3D Polygons Example: minimize the sum of triangle area Di : spanning edge : polygon segment : hole : spanning triangle : sub-domain 42

43 Triangulating Multiple 3D Polygons Example: minimize the sum of triangle area Di : spanning edge : polygon segment : hole : spanning triangle : sub-domain t 43

44 Triangulating Multiple 3D Polygons Example: minimize the sum of triangle area t t Case I split Case II split 44

45 Triangulating Multiple 3D Polygons Example: minimize the sum of triangle area D 1 t D 2 Case I split Case II split 45

46 Triangulating Multiple 3D Polygons Example: minimize the sum of triangle area D 1 t D 2 Cost(D,t)=Minx=1,2[ Cost(D x ) + Area(t) ] Case II split 46

47 Triangulating Multiple 3D Polygons Example: minimize the sum of triangle area D 1 D 1 1 D 1 2 D 2 1 D 2 2 D 2 Cost(D,t)=Minx=1,2[ Cost(D x ) + Area(t) ] D 3 1 D 3 2 D 4 1 D 4 2 Case II split 47

48 Triangulating Multiple 3D Polygons Example: minimize the sum of triangle area D 1 D 1 1 D 1 2 D 2 1 D 2 2 D 2 Cost(D,t)=Minx=1,2[ Cost(D x ) + Area(t) ] D 3 1 D 3 2 D 4 1 D 4 2 Cost(D,t)=Minx=1,2,3,4[ Cost(D x 1)+Cost(D x 2)+Area(t) ] 48

49 Triangulating Multiple 3D Polygons Example: minimize the sum of triangle area D 1 D 1 1 D 1 2 D 2 1 D 2 2 D 2 D 3 1 D 3 2 D 4 1 D 4 2 Cost(D) = Mint [ Cost(D,t) ] 49

50 Triangulating Multiple 3D Polygons Topologically equivalent to a sphere with k holes 50

51 Manifold Guarantee 51

52 Manifold Guarantee 52

53 Manifold Guarantee manifold non-manifold 53

54 Manifold Guarantee 54

55 Manifold Guarantee t 55

56 Manifold Guarantee 56

57 Manifold Guarantee d c z e x w y b f a g 57

58 Manifold Guarantee c d e b f a g x x z w y 58

59 Manifold Guarantee c d e b f a g x x z w y 59

60 Manifold Guarantee c d e b f a g x t x z w y 60

61 Manifold Guarantee c d e b f a g x x z w y 61

62 Manifold Guarantee c d e b f a g x x z w y 62

63 Manifold Guarantee c d e b f a g x x z w y 63

64 Manifold Guarantee c d e b f a g x x z w y 64

65 Manifold Guarantee c d e b f a g x x z w y 65

66 Manifold Guarantee c d e a b Weak Edges: edges that connecting the ends of the polygon segments f g x x z w y 66

67 Manifold Guarantee c d e b f a g x t x z w y 67

68 Manifold Guarantee c d e b f a g x x z w y y 68

69 Manifold Guarantee c d e b f a g x x z w y y 69

70 Manifold Guarantee Compute optimal triangulation for each combination of weak edges, for each domain Only combine triangulations that do not share any weak edge c d e b f a g x x z w y y 70

71 71 a b c d e f g x y z x w y a b c d e f g x y z x w y a b c d e f g x y z x w y Opt triangulation that contains weak edge set {ax, ay, xy, gy} Opt triangulation that contains weak edge set {ax, ag, xg, gy}

72 Manifold Guarantee The triangulation computed by the algorithm is guarantee to be the triangulation with a minimum cost that has the right topology. Minimal set: a reduced set of weak edge combination. still guarantee optimality 72

73 Complexity Match the complexity of the algorithm for triangulating single 3D polygon [Barequet et al 95, 96] time space per-triangle metric bi-triangle metric O(n 2k+1 ) O(n 2k ) O(n 3k+1 ) O(n 3k ) n: # of total vertices; k: # of 3D polygons; 73

74 Performance - Single polygon Test on single polygon minimizing sun of dihedral angles practical complexities matches theoretical ones same complexity of [Barequet et al 95, 96] 74

75 Performance - Single polygon Test on single polygon minimizing sun of dihedral angles practical complexities matches theoretical ones same complexity of [Barequet et al 95, 96] 75

76 Performance - Single polygon Test on single polygon minimizing sun of dihedral angles practical complexities matches theoretical ones same complexity of [Barequet et al 95, 96] Theoretical: O(n 4 ); Practical: O(n 4.11 ) Theoretical: O(n 3 ); Practical: O(n 2.27 ) 76

77 Performance - Single polygon Test on single polygon minimizing sun of dihedral angles practical complexities matches theoretical ones same complexity of [Barequet et al 95, 96] Theoretical: O(n 4 ); Practical: O(n 4.11 ) Theoretical: O(n 3 ); Practical: O(n 2.27 ) 77

78 Performance - Multiple polygons Dataset: k (1 k 6) perturbed saddle curves on the corner of a regular octahedron 50~60 data for each k 78

79 Performance - Multiple polygons Dataset: k (1 k 6) perturbed saddle curves on the corner of a regular octahedron 50~60 data for each k 79

80 Performance - Multiple polygons 2 poly 3 poly 80

81 Performance - Multiple polygons 2 poly 3 poly 81

82 Improving efficiency 82

83 Improving efficiency All Triangles A Triangle Subset 83

84 Improving efficiency All Triangles Requirements of the subset Considerably smaller Big enough to contain a close-to-optimal triangulation A Triangle Subset 84

85 Improving efficiency All Triangles Requirements of the subset Considerably smaller Big enough to contain a close-to-optimal triangulation Delaunay Triangles 85

86 Improving efficiency All Triangles Requirements of the subset Considerably smaller Big enough to contain a close-to-optimal triangulation Delaunay Triangle Subset: Fewer: O(n 3 ) O(n 2 ) Good triangles No self-intersecting Delaunay Triangles 86

87 Performance - Single polygon 87

88 Performance - Single polygon 88

89 Performance - Multiple polygon poly poly poly poly poly 89

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92 How Optimal? 92

93 Optimality Close-to-optimal results on both smooth curves and random loops on mesh Minimizing total area near-optimal for single polygon less optimal for two polygons Minimizing average dihedral angle near-optimal for both single and two-polygon cases triangulation on all triangles triangulation on Delaunay triangles 93

94 Applications Sketch based modeling 94

95 Applications Hole filling 455 groups of holes (61 holes have interior islands) fill all the holes in 40 sec 95

96 Applications Hole filling 455 groups of holes (61 holes have interior islands) fill all the holes in 40 sec Islands 96

97 Conclusion The first optimal algorithm for triangulating multiple non-planar 3D polygons guarantee manifoldness of the surface fast near-optimal variation method Future work other triangle subsets besides Delaunay triangles more sophisticated weighting schemes automatic grouping of hole islands Code available online! 97

98 Thank you! Comments Suggestions Questions 98