: Mesh Processing. Chapter 2
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1 : Mesh Processing Chapter 2
2 Data Structures Polygon/Triangle Soup Indexed Polygon/Triangle Set Winged-Edge Half-Edge Directed-Edge
3 List of faces Polygon Soup Each face represented independently If polygon soup, vertex count per face Polygon Soup: Polygons F k 1 : x 11, y 11, z 11,, x 1k1, y 1k1, z 1k1 k 2 : x 21, y 21, z 21,, x 2k1, y 2k1, z 2k1 k F : x F1, y F1, z F1,, x FkF, y FkF, z FkF Q: If the mesh is manifold, triangular with low genus, what is the storage cost, per vertex? (E-F+V=2-2g)
4 List of faces Triangle Soup Each face represented independently Triangle Soup: Triangles F x 11, y 11, z 11, x 12, y 12, z 12, x 13, y 13, z 13 x 21, y 21, z 21, x 22, y 22, z 22, x 23, y 23, z 23 x F1, y F1, z F1, x F2, y F2, z F2, x F3, y F3, z F3 Q: If the mesh is manifold, triangular with low genus, what is the storage cost, per vertex?
5 Polygon/Triangle Soup Advantage: General Disadvantages: Redundant storage No (chance for) connectivity information [Polygon] variable-size data structures
6 Structure of Meshes Common: Connected (small # of connected components) Two-manifolds Water-Tight (?) Orientable (?) Triangle meshes (?)
7 Indexed Polygon Set (Take 1) Lists of vertices, faces Each vertex stores geometry Each face points to its vertices Vertex List: Vertices V x 11, y 11, z 11 x V1, y V1, z V1 Polygon List: Polygons F k 1 : v 11,, v 1k1 k F : v F1,, v FkF Q: If the mesh is manifold, triangular with low genus, what is the storage cost, per vertex?
8 Indexed Triangles Set (Take 1) Lists of vertices, faces Each vertex stores geometry Each face points to its vertices Vertex List: Vertices V x 11, y 11, z 11 x V1, y V1, z V1 Triangle List: Triangles F v 11, v 12, v 13 v F1, v F2, v F3 Q: If the mesh is manifold with low genus, what is the storage cost, per vertex?
9 Indexed Triangles Set (Take 1) Lists of vertices, faces Each vertex stores geometry Each face points to its vertices Vertex List: Vertices V x 11, y 11, z 11 x V1, y V1, z V1 Triangle List: Triangles F v 11, v 12, v 13 v F1, v F2, v F3 Q: If the mesh is completely disconnected, what is the storage cost, per vertex?
10 Indexed Polygon/Triangle Set (Take 1) Advantage: Reduced storage Disadvantages: Doesn t leverage manifold structure No (explicit) connectivity information Can t store per-edge information [Polygon] variable-size data structures Non-Manifold Non-Orientable General Polygons Boundaries Indexed P. Set (1) Yes Yes Yes Yes
11 Indexed Triangles Set (Take 2) Lists of vertices, faces Each vertex stores geometry + neighbor face Each face points to its vertices + neighbor faces Vertex List: Vertices V x 11, y 11, z 11 ; f 1 x V1, y V1, z V1 ; f V Triangle List: Triangles F i 11, i 12, i 13 ; f 11, f 12, f 13 i F1, i F2, i F3 ; f F1, f F2, f F3 Q: If the mesh is manifold with low genus, what is the storage cost, per vertex?
12 Indexed Triangles Set (Take 2) Q: Given a vertex, how do we enumerate all of the vertices in its one-ring?
13 Indexed Polygon/Triangle Set (Take 2) Advantages: Reduced storage Explicit connectivity information Disadvantages: Partially leverages manifold structure Messy connectivity information Can t store per-edge information [Polygon] variable-size data structures Non-Manifold Non-Orientable General Polygons Boundaries Indexed P. Set (1) Yes Yes Yes Yes Indexed P. Set (2) Yes/No Yes Yes Yes
14 Winged-Edge Mesh Lists of vertices, edges, faces Each vertex stores geometry + neighbor edge Each face stores neighbor edge Each edge points to vertices (x2), faces (x2), next edge (x2), and previous edge (x2) Vertex List: Vertices V x 11, y 11, z 11 ; e 1 x V1, y V1, z V1 ; e V Edge List: Edges E v 11, v 12 ; f 11, f 12 ; n 11, n 12, p 11, p 12 v E1, v E2 ; f E1, f E2 ; n E1, n E2, p E1, p E2 Q: If the mesh is manifold, triangular with low genus, what is the storage cost, per vertex? Face List: Faces F e 1 e F
15 Winged-Edge Mesh
16 Winged-Edge Mesh Example: e 4 e 1 e 3 e 5 e 7 e 2 e 6
17 Winged-Edge Mesh Example: Find vertices adjacent to v 2. e 4 e 1 e 3 e 5 e 7 e 2 e 6
18 Winged-Edge Mesh Example: Find vertices adjacent to v 2. e 4 e 1 e 3 e 5 e 7 e 2 e 6
19 Winged-Edge Mesh Example: Find vertices adjacent to v 2. e 4 e 1 e 3 e 5 e 7 e 2 e 6
20 Winged-Edge Mesh Example: Find vertices adjacent to v 2. e 4 e 1 e 3 e 5 e 7 e 2 e 6
21 Winged-Edge Mesh Example: Find vertices adjacent to v 2. e 4 e 1 e 3 e 5 e 7 e 2 e 6
22 Winged-Edge Mesh Example: Find vertices adjacent to v 2. e 4 e 1 e 3 e 5 e 7 e 2 e 6
23 Winged-Edge Mesh Example: Find vertices adjacent to v 2. e 4 e 1 e 3 e 5 e 7 e 2 e 6
24 Winged-Edge Mesh Example: Find vertices adjacent to v 2. e 4 e 1 e 3 e 5 e 7 e 2 e 6
25 Winged-Edge Mesh Example: Find vertices adjacent to v 2. e 4 e 1 e 3 e 5 e 7 e 2 e 6
26 Winged-Edge Mesh Example: Find vertices adjacent to v 2. e 4 e 1 e 3 e 5 e 7 e 2 e 6
27 Winged-Edge Mesh Advantages: Reduced storage Explicit connectivity information Can store per-edge information Fixed-size data-structures Disadvantage: Messy connectivity information Non-Manifold Non-Orientable General Polygons Boundaries Indexed P. Set (1) Yes Yes Yes Yes Indexed P. Set (2) Yes/No Yes Yes Yes Winged-Edge No Yes Yes Yes
28 Half-Edge Mesh (Take 1) Lists of vertices, half-edges, faces Each vertex stores geometry + half-edge Each face stores neighbor half-edge Each half-edge points to start vertex, face, opposite/next/previous half-edge Vertex List: Vertices V x 11, y 11, z 11 ; e 1 x V1, y V1, z V1 ; e V Half-Edge List: Edges E v 1 ; f 1 ; o 1, n 1, p 1 v E ; f E ; o E, n E, p E Face List: Faces F e 1 Q: If the mesh is manifold, triangular with low genus, what is the storage cost, per vertex? e F
29 Half-Edge Mesh (Take 1) Q: Given a vertex, how do we enumerate all of the vertices in its one-ring?
30 Half-Edge Mesh (Take 1) Advantages: Reduced storage Connectivity information is easy Fixed-size data structures Disadvantage: Cannot store per-edge information Non-Manifold Non-Orientable General Polygons Boundaries Indexed P. Set (1) Yes Yes Yes Yes Indexed P. Set (2) Yes/No Yes Yes Yes Winged-Edge No Yes Yes Yes Half-Edge (1) No No Yes Yes
31 Half-Edge Mesh (Take 2) Lists of vertices, pairs of half-edges, faces Each vertex stores geometry + half-edge Each face stores neighbor half-edge Each half-edge points to start vertex, face, and next half-edge Vertex List: Vertices V x 11, y 11, z 11 ; e 1 x V1, y V1, z V1 ; e V Half-Edge List: Edges E v 1 ; f 1 ; n 1 v E ; f E ; n E Face List: Faces F e 1 Q: If the mesh is manifold, triangular with low genus, what is the storage cost, per vertex? e F
32 Half-Edge Mesh (Take 2) Q: Given a vertex, how do we enumerate all of the vertices in its one-ring?
33 Half-Edge Mesh (Take 2) Advantage: Reduced storage Connectivity information is easy Can store per-edge information Fixed-size data structures Non-Manifold Non-Orientable General Polygons Boundaries Indexed P. Set (1) Yes Yes Yes Yes Indexed P. Set (2) Yes/No Yes Yes Yes Winged-Edge No Yes Yes Yes Half-Edge (1) No No Yes Yes Half-Edge (2) No No Yes Yes
34 Directed-Edge Triangle Mesh Lists of vertices, half-edges Each vertex stores geometry + half-edge Each half-edge points to start vertex and opposite half-edge Vertex List: Vertices V x 11, y 11, z 11 ; e 1 x V1, y V1, z V1 ; e V Half-Edge List: Edges E v 1 ; o 1 v E ; o E Q: If the mesh is manifold with low genus, what is the storage cost, per vertex?
35 Directed-Edge Triangle Mesh Q: Given a vertex, how do we enumerate all of the vertices in its one-ring?
36 Directed-Edge Triangle Mesh Advantages: Reduced storage Connectivity information is easy Disadvantage: Cannot store per-edge information Non-Manifold Non-Orientable General Polygons Boundaries Indexed P. Set (1) Yes Yes Yes Yes Indexed P. Set (2) Yes/No Yes Yes Yes Winged-Edge No Yes Yes Yes Half-Edge (1) No No Yes Yes Half-Edge (2) No No Yes Yes Directed-Edge No Yes No No
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