An algorithm for triangulating multiple 3D polygons
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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
90
91
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
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