Applications. Oversampled 3D scan data. ~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 extraction 3

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

5 Applications Adaptation to hardware capabilities

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 Computing a representative Average/median vertex position Error quadrics Mesh generation Topology changes 12

12 Computing a Representative Average vertex position 13

13 Computing a Representative Median vertex position 14

14 Computing a Representative Error quadrics 15

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

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

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

18 Comparison average median error quadric 19

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

20 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 ) Topology changes 21

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

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

23 Incremental Decimation 500K 50K 5K 0.5K 24

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

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

26 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

27 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

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

29 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

30 Vertex Removal Select a vertex to be eliminated 31

31 Vertex Removal Select all triangles sharing this vertex 32

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

33 Vertex Removal Fill the hole with new triangles 34

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

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

36 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

37 Half-Edge Collapse 38

38 Half-Edge Collapse 39

39 Half-Edge Collapse 40

40 Half-Edge Collapse 41

41 Half-Edge Collapse 42

42 Half-Edge Collapse 43

43 Half-Edge Collapse 44

44 Half-Edge Collapse 45

45 Half-Edge Collapse 46

46 Half-Edge Collapse flip! 47

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

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

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

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

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

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

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

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

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

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

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

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

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

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

61 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