~150k triangles ~80k triangles

Transcription

1 Mesh Simplification

2 Applications Oversampled 3D scan data ~150k triangles ~80k triangles 2

3 Applications Overtessellation: E.g. iso-surface surface extraction 3

4 Applications Multi-resolution hierarchies for efficient geometry processing level-of-detail (LOD) rendering 4

5 Applications Adaptation to hardware capabilities 5

6 Size-Quality Tradeoff error size 6

7 Problem Statement Given: Find: such that 1. and is minimal, or 2. and is minimal Respect additional fairness criteria normal deviation, triangle shape, scalar attributes, etc. 7

8 Mesh Decimation Methods Vertex clustering Incremental decimation Resampling Mesh approximation 8

9 Vertex Clustering Cluster Generation Computing a representative Mesh generation Topology changes 9

10 Vertex Clustering Cluster Generation Uniform 3D grid Map vertices to cluster cells Computing a representative Mesh generation Topology changes 10

11 Vertex Clustering Cluster Generation Hierarchical approach Top-down or bottom-up Computing a representative Mesh generation Topology changes 11

12 Vertex Clustering Cluster Generation Computing a representative Average/median vertex position Error quadrics Mesh generation Topology changes 12

13 Computing a Representative Average vertex position 13

14 Computing a Representative Median vertex position 14

15 Computing a Representative Error quadrics 15

16 Error Quadrics Patch is expected to be piecewise flat Minimize distance to neighboring g triangles planes 16

17 Error Quadrics Squared distance of point p to plane q: 17

18 Error Quadrics Sum distances to planes q i of vertex neighboring triangles: Point p* that minimizes the error satisfies: 18

19 Comparison average median error quadric di 19

20 Vertex Clustering Cluster Generation Computing a representative Mesh generation Clusters p {p 0,...,p n }, q {q 0,...,q m } Topology changes 20

21 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 j ) j Topology changes 21

22 Vertex Clustering Cluster Generation Computing a representative Mesh generation Topology changes If different sheets pass through one cell Canbenonmanifold non-manifold 22

23 Outline Applications Problem Statement Mesh Decimation Methods Vertex Clustering Incremental Decimation Extensions 23

24 Incremental Decimation 24

25 Incremental Decimation General Setup Decimation operators Error metrics Fairness criteria Topology changes 25

26 General Setup Repeat: pick mesh region apply decimation operator Until no further reduction possible 26

27 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 27

28 Global Error Control For each region evaluate quality after decimation enqeue(quality, region) Repeat: get best mesh region from queue if error < ε apply decimation operator update queue Until no further reduction possible 28

29 Incremental Decimation General Setup Decimation operators Error metrics Fairness criteria Topology changes 29

30 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 30

31 Vertex Removal Select a vertex to be eliminated 31

32 Vertex Removal Select all triangles sharing this vertex 32

33 Vertex Removal Remove the selected triangles, creating the hole 33

34 Vertex Removal Fill the hole with new triangles 34

35 Decimation Operators Vertex Removal Vertex Insertion Remove vertex Re-triangulate hole Combinatorial degrees of freedom 35

36 Decimation Operators Edge Collapse Vertex Split Merge two adjacent vertices Define new vertex position Continuous degrees of freedom Filter along the way 36

37 Decimation Operators Half-Edge Collapse Restricted Vertex Split Collapse edge into one end point Special case of vertex removal Special case of edge collapse No degrees of freedom Separates global optimization from local optimization 37

38 Half-Edge Collapse 38

39 Half-Edge Collapse 39

40 Half-Edge Collapse 40

41 Half-Edge Collapse 41

42 Half-Edge Collapse 42

43 Half-Edge Collapse 43

44 Half-Edge Collapse 44

45 Half-Edge Collapse 45

46 Half-Edge Collapse 46

47 Half-Edge Collapse flip! 47

48 Incremental Decimation General Setup Decimation operators Error metrics Fairness criteria Topology changes 48

49 Local Error Metrics Local distance to mesh Compute average plane No comparison to original geometry 49

50 Global Error Metrics Error quadrics Squared distance to planes at vertex No bound on true error p 1 p 2 Q 3 = Q 1 +Q 2 Q 1 Q 2 v 3 p it Q i p i = 0, i={1,2} solve v 3T Q 3 v 3 = min < ε? ok 50

51 Incremental Decimation General Setup Decimation operators Error metrics Fairness criteria Topology changes 51

52 Fairness Criteria Rate quality of decimation operation Approximation error Triangle shape Dihedral angles Valence balance... 52

53 Fairness Criteria Rate quality after decimation Approximation error Triangle shape Dihedral angles Valence balance... r 1 r 2 53

54 Fairness Criteria Rate quality after decimation Approximation error Triangle shape Dihedral angles Valence balance... 54

55 Fairness Criteria Rate quality after decimation Approximation error Triangle shape Dihedral angles Valence balance Color differences... 55

56 Fairness Criteria Rate quality after decimation Approximation error Triangle shape Dihedral angles Valence balance Color differences... 56

57 Fairness Criteria Rate quality after decimation Approximation error Triangle shape Dihedral angles Valence balance Color differences... 57

58 Fairness Criteria Rate quality after decimation Approximation error Triangle shape Dihedral angles Valence balance Color differences... 58

59 Incremental Decimation General Setup Decimation operators Error metrics Fairness criteria Topology changes 59

60 Topology Changes? Merge vertices across non-edges Changes mesh topology Need spatial neighborhood information Generates non-manifold meshes Vertex Contraction Vertex Separation 60

61 Topology Changes? Merge vertices across non-edges Changes mesh topology Need spatial neighborhood information Generates non-manifold meshes manifold non-manifold 61

62 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 quadric error metrics good trade-off between mesh quality and speed explicit control over mesh topology restricting normal deviation improves mesh quality 62