Spring 2014 CSCI 599: Digital Geometry Processing 8.1 Decimation Hao Li http://cs599.hao-li.com � 1
Administrative • Today’s Office Hour from 2:00 to 3:00. • Exercise 4 this Thursday • Dr. Chongyang Ma will do next lecture � 2
Last Time Parameterization � � ⇥ 1 0 I ( u, v ) = • isometric 0 1 � ⇥ 1 0 • conformal I ( u, v ) = s ( u, v ) · 0 1 det( I ( u, v )) = 1 • equiareal � x T x T ⇥ u x u u x v I = x T x T u x v v x v � 3
Last Time Harmonic Maps � Z Z kr x k 2 = k x u k 2 + k x v k 2 d u d v • minimize Dirichlet energy: Ω Ω • Euler-Lagrange PDE ∆ x ( u, v ) = 0 Discrete Harmonic Maps Convex Combination Maps original � uniform � cotan � mean � mesh weights weights value � 4
Last Time fixed vs. open boundaries texture atlases cutting the mesh → disk topology constrained parameterization � 5
Mesh Optimization Smoothing � • Low geometric noise Fairing � • Simplest shape Decimation � • Low complexity Remeshing � • Triangle Shape � 6
Mesh Decimation Oversampled 3D scan data ~150k triangles ~80k triangles � 7
Mesh Decimation Over tesselation: e.g., Iso-surface extraction � 8
Mesh Decimation Multi-resolution hierarchies for � • efficient geometry processing • level-of-detail (LOD) rendering � 9
Mesh Decimation Adaptation to hardware capabilities � 10
Mesh Decimation Adaptation to hardware capabilities � 11
Size-Quality Tradeoff error size � 12
Problem Statement M � = ( V � , F � ) M = ( V , F ) Given , find such that � |V � | = n < |V| ⇤ M � M � ⇤ • and is minimal, or |V � | ⇤ M � M � ⇤ < � • and is minimal M � M � 13
Problem Statement M � = ( V � , F � ) M = ( V , F ) Given , find such that � |V � | = n < |V| ⇤ M � M � ⇤ • and is minimal, or |V � | ⇤ M � M � ⇤ < � • and is minimal NP hard � • Look for sub-optimal solution Respect additional fairness criteria � • Normal deviation, triangle shape, colors,… � 14
Outline Mesh Decimation methods • Vertex Clustering � • Iterative Decimation � 15
Vertex Clustering • Cluster Generation • Computing a representative • Mesh generation • Topology changes � 16
Vertex Clustering • Cluster Generation � • Uniform 3D grid • Map vertices to cluster cells � • Computing a representative • Mesh generation • Topology changes � 17
Vertex Clustering • Cluster Generation � • Hierarchical approach • Top-down or bottom-up � • Computing a representative • Mesh generation • Topology changes � 18
Vertex Clustering • Cluster Generation • Computing a representative � • Average/median vertex position • Error quadrics � • Mesh generation • Topology changes � 19
Computing a Representative average vertex position → low pass filter � 20
Computing a Representative median vertex position → sub-sampling � 21
Computing a Representative error quadrics → feature preservation � 22
Error Quadrics Squared distance to plane p = ( x, y, z, 1) T , q = ( a, b, c, d ) T dist( q , p ) 2 = ( q T p ) 2 = p T � p =: p T Q q p qq T ⇥ � 23
Error Quadrics Squared distance to plane p = ( x, y, z, 1) T , q = ( a, b, c, d ) T dist( q , p ) 2 = ( q T p ) 2 = p T � p =: p T Q q p qq T ⇥ a 2 ab ac ad b 2 ab bc bd Q q = c 2 ac bc cd d 2 ad bd cd � 24
Error Quadrics Sum of distances to vertex planes �⇤ ⇥ dist( q i , p ) 2 = p T Q q i p = p T p =: p T Q p p ⇤ ⇤ Q q i i i i � 25
Error Quadrics Sum of distances to vertex planes �⇤ ⇥ dist( q i , p ) 2 = p T Q q i p = p T p =: p T Q p p ⇤ ⇤ Q q i i i i Point that minimizes the error 0 q 11 q 12 q 13 q 14 0 p ∗ = q 21 q 22 q 23 q 24 0 q 31 q 32 q 33 q 34 0 0 0 1 1 � 26
Comparison average error quadric median � 27
Vertex Clustering • Cluster Generation • Computing a representative � • Mesh generation � • Clusters p ⇔ { p 0 , . . . , p n } , q ⇔ { q 0 , . . . , q n } • Connect if there was an edge � ( p i , q j ) ( p , q ) • Topology changes � 28
Vertex Clustering • Cluster Generation • Computing a representative � • Mesh generation � • Topology changes � • If different sheets pass through on cell • Can be non-manifold � 29
Vertex Clustering • Cluster Generation • Computing a representative � • Mesh generation � • Topology changes � • If different sheets pass through on cell • Can be non-manifold � 30
Outline Mesh Decimation methods • Vertex Clustering • Iterative Decimation � 31
Example � 32
Incremental Decimation • General Setup � • Decimation operators � • Error metrics � • Fairness criteria • Topology changes � 33
General Setup Repeat: pick mesh region apply decimation operator Until no further reduction possible � 34
Greedy Optimization For each region evaluate quality after decimation enqueue(quality, region) � Repeat: pick best mesh region apply decimation operator update queue Until no further reduction possible � 35
Global Error Control For each region evaluate quality after decimation enqueue(quality, region) � Repeat: pick best mesh region if error < ε apply decimation operator update queue Until no further reduction possible � 36
Incremental Decimation • General Setup • Decimation operators � • Error metrics � • Fairness criteria • Topology changes � 37
Decimation Operators • What is a “region”? � • What are the DOFs for re-triangulation? • Classification • topology-changing vs. topology-preserving • subsampling vs. filtering • inverse operation → progressive meshes � 38
Vertex Removal Select a vertex to be eliminated � 39
Vertex Removal Select all triangles sharing this vertex � 40
Vertex Removal Remove the selected triangles, creating a hole � 41
Vertex Removal Fill the hole with triangles � 42
Decimation Operators • Remove vertex • Re-triangulate hole • Combinatorial DOFs • Sub-sampling Vertex Removal Vertex Insertion � 43
Decimation Operators • Merge two adjacent triangles • Define new vertex position • Continuous DOF • Filtering Edge Collapse Vertex Split � 44
Decimation Operators • Collapse edge into one end point • Special vertex removal • Special edge collapse • No DOFs • One operator per half-edge • Sub-sampling H. Hoppe: Progressive Meshes Halfedge Collapse Restricted Vertex Split � 45
Edge Collapse � 46
Edge Collapse � 47
Edge Collapse � 48
Edge Collapse � 49
Edge Collapse � 50
Edge Collapse � 51
Edge Collapse � 52
Edge Collapse � 53
Edge Collapse � 54
Edge Collapse (Flip!) � 55
Application: Progressive Meshes � 56
Priority Queue Updating � 57
Incremental Decimation • General Setup • Decimation operators • Error metrics � • Fairness criteria • Topology changes � 58
Local Error Metrics Local distance to mesh [Schröder et al. ‘92] � • Compute average plane • No comparison to original geometry � 59
Global Error Metrics Simplification envelopes [Cohen al. ‘96] � • Compute (non-intersecting) offset surfaces • Simplification guarantees to stay within bounds � 60
Global Error Metrics (Two-sided) Hausdorff distance: Maximum distance between two shapes � � d ( A, B ) := max a ∈ A min b ∈ B ⇥ a � b ⇥ � d ( A, B ) 6 = d ( B, A ) • In general • Computationally involved A d(A,B) B d(B,A) � 61
Global Error Metrics Scan data: One-sided Hausdorff distance sufficient � • From original vertices to current surface � 62
Global Error Metrics Error quadrics [Garland, Heckbert 97] � • Squared distance to planes at vertex • No bound on true error Q 3 = Q 1 + Q 2 Q 1 Q 2 p 3 p 1 p 2 solve p 3T Q 3 p 3 = min p i T Q i p i = 0 , < ε ? → ok i ={1,2} � 63
Global Error Metrics Initialization: � • Assign each vertex the quadric built from all its incident triangles’ planes Decimation: � ( p 1 , p 2 ) → p 3 • After collapsing edge , simply add the Q 3 = Q 1 + Q 2 corresponding quadrics: Memory consumption � • Quasi-global error metric with 10 floats per vertex � 64
Complexity • number of vertices N = • Priority Queue for half edges 6 N log(6 N ) • • Error control • Local global O (1) ⇒ O ( N ) O ( N 2 ) • Local global O ( N ) ⇒ � 65
Incremental Decimation • General Setup • Decimation operators • Error metrics • Fairness criteria � • Topology changes � 66
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