Mesh and Mesh Simplification

Transcription

1 Slide Credit: Mirela Ben-Chen Mesh and Mesh Simplification Qixing Huang Mar. 21 st 2018

2 Mesh DataStructures

3 Data Structures What should bestored? Geometry: 3D coordinates Attributes e.g. normal, color, texture coordinate Per vertex, per face, per edge Connectivity Adjacency relationships

4 Data Structures What should itsupport? Rendering Geometry queries What are the vertices offace #2? Is vertex A adjacent to vertex H? Which faces are adjacent toface #1? Modifications Remove/add a vertex/face Vertex split, edge collapse

5 Data Structures How good is adata structure? Time to construct (preprocessing) Time to answer aquery Time to perform an operation Space complexity Redundancy

6 Mesh DataStructures Face Set Shared Vertex HalfEdge Face Based Connectivity Edge Based Connectivity AdjacencyMatrix Corner Table

7 FaceSet TRIANGLES Vertex coord. Vertex coord. Vertex coord. Simple STL File No connectivity Redundancy [ ] [ ] [10 4 3]...

8 SharedVertex TRIANGLES Vertex Index Vertex Index Vertex Index Connectivity No neighborhood... VERTICES Vertex Coord. [ ] [ ] [10 4 3]...

9 SharedVertex v 3 v 6 f 1 f 2 f 4 v 1 v 4 TRIANGLES f v 1 f 3. v 2 v 5.. v 2 v 3 VERTICES [20 100] [19 200] [14 150]...

10 SharedVertex v 3 v 6 f 1 f 2 f 4 v 1 v 4 TRIANGLES f v 1 f 3. v 2 v 5.. v 2 v 3 VERTICES [20 100] [19 200] [14 150]... What are the vertices of facef 1? O(1) first triplet from face list

11 SharedVertex v 3 v 6 f 1 f 2 f 4 v 1 v 4 TRIANGLES f v 1 f 3. v 2 v 5.. v 2 v 3 VERTICES [20 100] [19 200] [14 150]... Are vertices v 1 and v 5 adjacent? Requires a full pass over allfaces

12 Half Edge DataStructure Vertex stores Position 1 outgoinghalfedge vertex outgoing halfedge

13 Half Edge DataStructure Halfedge stores 1 origin vertexindex 1 incident faceindex next, prev, twin halfedge indices halfedge twin origin vertex next

14 Half Edge DataStructure Face stores 1 adjacent halfedge index adjacenthalfedge face

15 Half Edge DataStructure Neighborhood Traversal a) b) c) d)

16 Face BasedConnectivity Vertex: position 1 adjacent face index Face: 3 vertex indices 3 neighboring face indices No (explicit) edgeinformation

17 Edge BasedConnectivity Vertex position 1 adjacent edgeindex Edge 2 vertex indices 2 neighboring face indices 4 edges Face 1 edgeindex No edge orientationinformation

18 AdjacencyMatrix v 3 v 6 f 1 f 2 f 4 v 1 v 4 A= f 3 v 2 v 5 v 1 v 2 v 3 v 4 v 5 v 6 v v v v v v Adjacency Matrix A If there is an edge between v i & v j then A ij = 1

19 AdjacencyMatrix Symmetric for undirected simple graphs (A n ) ij = # paths of length n from v i to v j Pros: Cons: Can represent non manifoldmeshes No connection between a vertex and its adjacent faces

20 CornerTable Corner is a vertexwith one of its indicent triangles v 3 c3 c 9 c 4 f 3 c7 c12 v 6 v 1 c f 1 1 f 2 c 6 c f 10 4 v 4 c 2 c 5 c 11 v 2 c 8 v 5

21 CornerTable Corner is a vertexwith one of its indicent triangles v v 6 3 Corner c c c 3 c 4 9 f 3 c 7 c12 c 1 f c 1 8 v f 2 f 1 c 6 c 10 4 v 4 c 2 c 5 c 11 v 2 v 5

22 CornerTable Corner is a vertexwith one of its indicent triangles v v 6 3 Corner c 9 c 3 Triangle c.t c 4 f 3 c c 7 c12 c 1 f c 1 8 v f 2 f 1 c 6 c 10 4 v 4 c 2 c 5 c 11 v 2 v 5

23 CornerTable Corner is a vertexwith one of its indicent triangles v v 6 3 Corner c Triangle c.t Vertex c.v c c 3 c 4 9 f c 1 8 v c f 2 f 1 1 c 6 c 10 4 v 4 f 3 c 7 c12 c 2 c 5 c 11 v 2 v 5

24 CornerTable Corner is a vertexwith one of its indicent triangles v v 6 3 Corner c Triangle c.t Vertex c.v Next corner in c.t (ccw) c.n f c 1 f 8 v c f c 6 c 10 4 v 4 c c 3 c 4 9 f 3 c 7 c12 c 2 c 5 c 11 v 2 v 5

25 CornerTable Corner is a vertexwith one of its indicent triangles v v 6 3 Corner c Triangle c.t Vertex c.v Next corner in c.t (ccw) c.n Previous corner c.p (== c.n.n) c c 3c 4 9 f c 1 8 v c f 2 f 1 1 c 6 c 10 4 v 4 c 2 c 5 f 3 c 7 c12 c 11 v 2 v 5

26 CornerTable Corner is a vertexwith one of its indicent triangles v v 6 3 Corner c Triangle c.t Vertex c.v Next corner in c.t (ccw) c.n Previous corner c.p (== c.n.n) Corner opposite c c.o Edge E opposite c not incident on c.v Triangle T adjacent to c.t acrosse c.o.v vertex of T that is not incident one c c 3 c 4 9 f c 1 v c E 8 T f 1 1 c 6 c 10 4 v 4 c 2 c 5 f 3 c 7 c12 c 11 v 2 v 5

27 CornerTable Corner is a vertexwith one of its indicent triangles v v 6 3 Corner c Triangle c.t Vertex c.v Next corner in c.t (ccw) c.n Previous corner c.p (== c.n.n) Corner opposite c c.o c c 3 c 4 9 f c 1 8 v c f 2 f 1 1 c 6 c 10 4 v 4 c 2 c 5 f 3 c 7 c12 Edge E opposite c not incident on c.v Triangle T adjacent to c.t acrosse v 5 v c.o.v vertex of T that is not incident one 2 Rightcorner c.r corner oppositec.n(==c.n.o) c 11

28 CornerTable Corner is a vertexwith one of its indicent triangles v v 6 3 Corner c Triangle c.t Vertex c.v Next corner in c.t (ccw) c.n Previous corner c.p (== c.n.n) Corner opposite c c.o c c 3 c 4 9 f c 1 8 v c f 2 f 1 1 c 6 c 10 4 v 4 c 2 c 5 f 3 c 7 c12 Edge E opposite c not incident on c.v Triangle T adjacent to c.t acrosse v 5 v c.o.v vertex of T that is not incident one 2 Rightcorner c.r corner oppositec.n(==c.n.o) Left corner c.l (== c.p.o== c.n.n.o) c 11

29 CornerTable Corner is a vertexwith one of its indicent triangles Corner c Triangle c.t Vertex c.v Next corner in c.t (ccw) c.n Previous corner c.p (== c.n.n) Corner opposite c c.o Edge E opposite c not incident on c.v Triangle T adjacent to c.t acrosse c.o.v vertex of T that is not incident on E Rightcorner c.r corneroppositec.n(==c.n.o) Left corner c.l (== c.p.o== c.n.n.o)

30 CornerTable Corner is a vertexwith one of its indicent triangles v 3 c 3 c 9 v 6 c7 c12 corner c.v c.t c.n c.p c.o c.r c.l c 4 f 3 c 1 v 1 f 1 c 2 c 3 c 6 c 2 v 2 f 1 c 3 c 1 c 6 f 1 v c f 2 f 1 1 c 6 c 10 4 v 4 c 2 c 5 c 11 c 8 v 2 v 5 c 3 v 3 f 1 c 1 c 2 c 6 c 4 v 3 f 2 c 5 c 6 c 7 c 1 c 5 v 2 f 2 c 6 c 4 c 7 c 1 c 6 v 4 f 2 c 4 c 5 c 1 c 7

31 CornerTable Store: Corner table For each vertex a list of all its corners Corner numberj*3 2, j*3 1 and j*3 match face number j

32 CornerTable What are the vertices offace #3? Check c.v of corners 9, 8,7 v 3 c3 c 9 c 4 f 3 c7 c12 v 6 f 1 v c f c 6 c 10 v 2 c 8 v 4 f 4 c 2 c 5 c 11 v 5

33 CornerTable Are vertices 2 and 6adjacent? Scan all corners of vertex 2, check if c.p.v or c.n.v are 6 v 3 c3 c 9 c 4 f 3 c7 c12 v 6 f 1 v c f c 6 c 10 v 2 c 8 v 4 f 4 c 2 c 5 c 11 v 5

34 CornerTable Which faces are adjacent to vertex3? Check c.t of all corners ofvertex 3 v 3 c 3 c 9 c7 c12 v 6 c 4 f 3 f 1 v c f c 6 c 10 v 2 c 8 v 4 f 4 c 2 c 5 c 11 v 5

35 CornerTable One ring neighbors of vertexv 4? Get the corners c 6 c 8 c 10 of thisvertex Go to c i.n.v and c i.p.v for i = 6, 8, 10. Remove duplicates v 3 c 3 c 9 c7 c12 v 6 c 4 f 3 c 1 f 1 v f 2 f 1 c 6 c 10 4 v 4 c 8 c 2 c 5 c 11 v 2 v 5

36 CornerTable Pros: All queries in O(1)time Most operations areo(1) Convenient for rendering Cons: Only triangular, manifold meshes Redundancy

37 Multiple Simplification

38 Applications Oversampled 3D scan data ~150k triangles ~80k triangles

39 Applications Overtessellation: E.g. iso-surface extraction

40 Applications Multi-resolution hierarchies for efficient geometry processing level-of-detail (LOD) rendering

41 Applications Adaptation to hardware capabilities

42 Size-Quality Tradeoff error size

43 Problem Statement Given: M = (V,F) Find: M = (V,F ) such that V = n < V and d(m,m ) is minimal, or d(m,m ) < eps and V is minimal Respect additional fairness criteria Normal deviation, triangle shape, scalar attributes, etc.

44 Mesh Decimation Methods Vertex clustering Incremental decimation Remeshing

45 Vertex Clustering Cluster Generation Computing a representative Mesh generation Topology changes

46 Vertex Clustering Cluster Generation Uniform 3D grid Map vertices to cluster cells Computing a representative Mesh generation Topology changes

47 Vertex Clustering Cluster Generation Hierarchical approach Top-down or bottom-up Computing a representative Mesh generation Topology changes

48 Vertex Clustering Cluster Generation Computing a representative Average/median vertex position Error quadrics Mesh generation Topology changes

49 Computing a Representative Average vertex position

50 Computing a Representative Median vertex position

51 Computing a Representative Error quadrics

52 Error Quadrics Patch is expected to be piecewise flat Minimize distance to neighboring triangles planes

53 Error Quadrics Squared distance of point p to plane q

54 Error Quadrics Sum distances to planes q i of vertex neighboring triangles Point p* that minimizes the error satisfies:

55 Comparison

56 Vertex Clustering Cluster Generation Computing a representative Mesh generation Clusters p<-> {p 0,, p n }, q<-> {q 0,, q m } Topology changes

57 Vertex Clustering Cluster Generation Computing a representative Mesh generation Clusters p<-> {p 0,, p n }, q<-> {q 0,, q m } Connect (p,q) if there was an edge (p i, q i ) Topology changes

58 Vertex Clustering Cluster Generation Computing a representative Mesh generation Topology changes If different sheets pass through one cell Can be non-manifold

59 Incremental Decimation

60 Incremental Decimation

61 Incremental Decimation General Setup Decimation operators Error metrics Fairness criteria

62 General Setup Repeat: Pick mesh region Apply decimation operator Until no further reduction possible

63 Greedy Optimization For each region evaluate quality after decimation enqeue(quality, region) Repeat: get best mesh region from queue apply decimation operator update queue Until no further reduction possible

64 Global Error Control For each region evaluate quality after decimation enqeue(quality, region) Repeat: get best mesh region from queue If error < eps Apply decimation operator Update queue Until no further reduction possible

65 Incremental Decimation General Setup Decimation operators Error metrics Fairness criteria

66 Decimation Operators What is a region? What are the DOF for re-triangulation? Classification Topology-changing vs. topology-preserving Subsampling vs. filtering Inverse operation -> progressive meshes [Hoppe et al.]

67 Vertex Removal Select a vertex to be eliminated

68 Vertex Removal Select all triangles sharing this vertex

69 Vertex Removal Remove the selected triangles, creating the hole

70 Vertex Removal Fill the hole with new triangles

71 Decimation Operators Remove vertex Re-triangulate hole Combinatorial degrees of freedom

72 Decimation Operators Merge two adjacent vertices Define new vertex position Continuous degrees of freedom

73 Decimation Operators Collapse edge into one end point Special case of vertex removal Special case of edge collapse No degrees of freedom

74 Fairness Criteria Rate quality of decimation operation Approximation error Triangle shape Dihedral angles Valence balance

75 Fairness Criteria Rate quality of decimation operation Approximation error Triangle shape Dihedral angles Valence balance

76 Incremental Decimation General Setup Decimation operators Error metrics Fairness criteria

77 Local Error Metrics Local distance to mesh Compute average plane No comparison to original geometry

78 Global Error Metrics Error quadrics Squared distance to planes at vertex No bound on true error

79 Incremental Decimation General Setup Decimation operators Error metrics Fairness criteria

80 Fairness Criteria Rate quality of decimation operation Approximation error Triangle shape Dihedral angles Valence balance

81 Fairness Criteria Rate quality of decimation operation Approximation error Triangle shape Dihedral angles Valence balance

82 Fairness Criteria Rate quality of decimation operation Approximation error Triangle shape Dihedral angles Valence balance

83 Fairness Criteria Rate quality of decimation operation Approximation error Triangle shape Dihedral angles Valence balance

84 Fairness Criteria Rate quality of decimation operation Approximation error Triangle shape Dihedral angles Valence balance

85 Comparison Vertex clustering fast, but difficult to control simplified mesh Topology changes, non-manifold meshes Global error bound, but often not close to optimum Incremental decimation with quadratic error metrics good trade-off between mesh quality and speed explicit control over mesh topology restricting normal deviation improves mesh quality

86 Remeshing

87 Remeshing Given a 3D mesh, find a better discrete representation of the underlying surface

88 What is a good mesh? Equal edge lengths Equilateral triangles Valence close to 6

89 What is a good mesh? Equal edge lengths Equilateral triangles Valence close to 6 Uniform vs. adaptive sampling

90 What is a good mesh? Equal edge lengths Equilateral triangles Valence close to 6 Uniform vs. adaptive sampling Feature preservation

91 What is a good mesh? Equal edge lengths Equilateral triangles Valence close to 6 Uniform vs. adaptive sampling Feature preservation Alignment to curvature lines Isotropic vs. anisotropic

92 What is a good mesh? Equal edge lengths Equilateral triangles Valence close to 6 Uniform vs. adaptive sampling Feature preservation Alignment to curvature lines Isotropic vs. anisotropic Triangles vs. quadranges