Mesh Decimation Mark Pauly Applications Oversampled 3D scan data - - PowerPoint PPT Presentation

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Mesh Decimation Mark Pauly Applications Oversampled 3D scan data - - PowerPoint PPT Presentation

Mar 17, 2023 461 likes •1.24k views

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฀฀฀฀ ฀฀฀ ฀฀฀฀ ฀฀฀ ฀฀฀ ฀฀฀฀฀ ฀

  • Mesh Decimation

Mark Pauly

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Mark Pauly - ETH Zurich

Applications

  • Oversampled 3D scan data

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~150k triangles ~80k triangles

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Mark Pauly - ETH Zurich

Applications

  • Overtessellation: E.g. iso-surface extraction

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Mark Pauly - ETH Zurich

Applications

  • Multi-resolution hierarchies for

– efficient geometry processing – level-of-detail (LOD) rendering

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Applications

  • Adaptation to hardware capabilities
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Size-Quality Tradeoff

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error size

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Outline

  • Applications
  • Problem Statement
  • Mesh Decimation Methods

– Vertex Clustering – Iterative Decimation – Extensions – Remeshing – Variational Shape Approximation

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Problem Statement

  • Given:
  • Find: such that
  • 1. and is minimal, or
  • 2. and is minimal

M = (V, F) M = (V, F) |V| = n < |V| |V| M − M M − M < M M

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Mark Pauly - ETH Zurich 287

Problem Statement

  • Given:
  • Find: such that
  • 1. and is minimal, or
  • 2. and is minimal

M = (V, F) M = (V, F) |V| = n < |V| |V| M − M M − M <

hard! → look for sub-optimal solution

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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. M = (V, F) M = (V, F) |V| = n < |V| |V| M − M M − M <

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Outline

  • Applications
  • Problem Statement
  • Mesh Decimation Methods

– Vertex Clustering – Iterative Decimation – Extensions

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Vertex Clustering

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

  • Cluster Generation

– Uniform 3D grid – Map vertices to cluster cells

  • Computing a representative
  • Mesh generation
  • Topology changes
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Vertex Clustering

  • Cluster Generation

– Hierarchical approach – Top-down or bottom-up

  • Computing a representative
  • Mesh generation
  • Topology changes
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Vertex Clustering

  • Cluster Generation
  • Computing a representative

– Average/median vertex position – Error quadrics

  • Mesh generation
  • Topology changes
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Computing a Representative

Average vertex position → Low-pass filter

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Computing a Representative

Median vertex position → Sub-sampling

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Computing a Representative

Error quadrics

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Error Quadrics

  • Squared distance to plane

297

p = (x, y, z, 1)T , q = (a, b, c, d)T dist(q, p)2 = (qT p)2 = pT (qqT )p =: pT Qqp Qq =     a2 ab ac ad ab b2 bc bd ac bc b2 cd ad bd cd d2    

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Mark Pauly - ETH Zurich

  • Sum distances to vertex’ planes
  • Point that minimizes the error

Error Quadrics

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  • i

dist(qi, p)2 =

  • i

pT Qqip = pT

  • i

Qqi

  • p =: pT Qpp

    q11 q12 q13 q14 q21 q22 q23 q24 q31 q32 q33 q34 1     p∗ =     1    

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Mark Pauly - ETH Zurich

error quadric average median

Comparison

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Vertex Clustering

  • Cluster Generation
  • Computing a representative
  • Mesh generation

– Clusters p⇔{p0,...,pn}, q⇔{q0,...,qm} – Connect (p,q) if there was an edge (pi,qj)

  • Topology changes
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Vertex Clustering

  • Cluster Generation
  • Computing a representative
  • Mesh generation
  • Topology changes

– If different sheets pass through one cell – Not manifold

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Mark Pauly - ETH Zurich

Outline

  • Applications
  • Problem Statement
  • Mesh Decimation Methods

– Vertex Clustering – Iterative Decimation – Extensions

302

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Mark Pauly - ETH Zurich

Incremental Decimation

  • General Setup
  • Decimation operators
  • Error metrics
  • Fairness criteria
  • Topology changes

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General Setup

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

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Greedy Optimization

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

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Global Error Control

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

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Mark Pauly - ETH Zurich

Incremental Decimation

  • General Setup
  • Decimation operators
  • Error metrics
  • Fairness criteria
  • Topology changes

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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

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Vertex Removal

Select a vertex to be eliminated

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Vertex Removal

Select all triangles sharing this vertex

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Vertex Removal

Remove the selected triangles, creating the hole

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Vertex Removal

Fill the hole with triangles

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Decimation Operators

  • Remove vertex
  • Re-triangulate hole

– Combinatorial DOFs – Sub-sampling

Vertex Removal Vertex Insertion

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Decimation Operators

  • Merge two adjacent triangles
  • Define new vertex position

– Continuous DOF – Filtering

Vertex Split Edge Collapse

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Decimation Operators

  • Collapse edge into one end point

– Special vertex removal – Special edge collapse

  • No DOFs

– One operator per half-edge – Sub-sampling!

Restricted Vertex Split Half-Edge Collapse

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Edge Collapse

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Edge Collapse

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Edge Collapse

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Edge Collapse

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Edge Collapse

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Edge Collapse

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Edge Collapse

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Edge Collapse

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Edge Collapse

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Edge Collapse

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Priority Queue Updating

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Incremental Decimation

  • General Setup
  • Decimation operators
  • Error metrics
  • Fairness criteria
  • Topology changes

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  • Local distance to mesh [Schroeder et al. 92]

– Compute average plane – No comparison to original geometry

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Local Error Metrics

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  • Simplification envelopes [Cohen et al. 96]

– Compute (non-intersecting) offset surfaces – Simplification guarantees to stay within bounds

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Global Error Metrics

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  • (Two-sided) Hausdorff distance: Maximum

distance between two shapes

– In general d(A,B) ≠ d(B,A) – Computationally involved

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Global Error Metrics

A B d(A,B) d(B,A)

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Global Error Metrics

  • Scan data: One-sided Hausdorff distance

sufficient

– From original vertices to current surface

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Global Error Metrics

  • Error quadrics [Garland, Heckbert 97]

– Squared distance to planes at vertex – No bound on true error

p1 p2 solve v3TQ3v3 = min Q3 = Q1+Q2 Q2 Q1

piTQipi = 0, i={1,2}

< ε ? → ok v3

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Complexity

  • N = number of vertices
  • Priority queue for half-edges

– 6 N * log ( 6 N )

  • Error control

– Local O(1) ⇒ global O(N) – Local O(N) ⇒ global O(N2)

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Incremental Decimation

  • General Setup
  • Decimation operators
  • Error metrics
  • Fairness criteria
  • Topology changes

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  • Rate quality of decimation operation

– Approximation error – Triangle shape – Dihedral angles – Valence balance – Color differences – ...

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Fairness Criteria

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  • Rate quality after decimation

– Approximation error – Triangle shape – Dihedral angles – Valence balance – Color differences – ...

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Fairness Criteria

r1 e1 r2

r1 e1 < r2 e2

e2

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  • Rate quality after decimation

– Approximation error – Triangle shape – Dihedral angles – Valence balance – Color differences – ...

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Fairness Criteria

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  • Rate quality after decimation

– Approximation error – Triangle shape – Dihedral angles – Valence balance – Color differences – ...

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Fairness Criteria

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  • Rate quality after decimation

– Approximation error – Triangle shape – Dihedral angles – Valence balance – Color differences – ...

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Fairness Criteria

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  • Rate quality after decimation

– Approximation error – Triangle shape – Dihedral angles – Valence balance – Color differences – ...

340

Fairness Criteria

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  • Rate quality after decimation

– Approximation error – Triangle shape – Dihedral angles – Valence balance – Color differences – ...

341

Fairness Criteria

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Fairness Criteria

  • Rate quality after decimation

– Approximation error – Triangle shape – Dihedral angles – Valance balance – Color differences – ...

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Incremental Decimation

  • General Setup
  • Decimation operators
  • Error metrics
  • Fairness criteria
  • Topology changes

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  • Merge vertices across non-edges

– Changes mesh topology – Need spatial neighborhood information – Generates non-manifold meshes

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Topology Changes ?

Vertex Contraction Vertex Separation

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  • Merge vertices across non-edges

– Changes mesh topology – Need spatial neighborhood information – Generates non-manifold meshes

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Topology Changes ?

manifold non-manifold

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Comparison

  • Vertex clustering

– fast, but difficult to control simplified mesh – topology changes, non-manifold meshes – global error bound, but often not close to optimum

  • Iterative decimation with quadric error metrics

– good trade-off between mesh quality and speed – explicit control over mesh topology – restricting normal deviation improves mesh quality

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Outline

  • Applications
  • Problem Statement
  • Mesh Decimation Methods

– Vertex Clustering – Iterative Decimation – Extensions

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Out-of-core Decimation

  • Handle very large data sets that do not fit into

main memory

  • Key: Avoid random access to mesh data structure

during simplification

  • Examples

– Garland, Shaffer: A Multiphase Approach to Efficient Surface Simplification, IEEE Visualization 2002 – Wu, Kobbelt: A Stream Algorithm for the Decimation

  • f Massive Meshes, Graphics Interface 2003

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Multiphase Simplification

  • 1. Phase: Out-of-core clustering
  • compute accumulated error quadrics and vertex

representative for each cell of uniform voxel grid

  • 2. Phase: In-core iterative simplification
  • compute fundamental quadrics
  • iteratively contract edge of smallest cost

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Multiphase Simplification

  • 1. Phase: Out-of-core clustering
  • compute accumulated error quadrics and vertex

representative for each cell of uniform voxel grid

  • 2. Phase: In-core iterative simplification
  • compute fundamental quadrics
  • use accumulated quadrics from clustering phase
  • iteratively contract edge of smallest cost

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→ achieves a coupling between the two phases

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Multiphase Simplification

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Original (8 million triangles) Uniform clustering (1157 triangles) Multiphase (1000 triangles) Garland, Shaffer: A Multiphase Approach to Efficient Surface Simplification, IEEE Visualization 2002

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Multiphase Simplification

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Garland, Shaffer: A Multiphase Approach to Efficient Surface Simplification, IEEE Visualization 2002

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Out-of-core Decimation

  • Streaming approach based on edge collapse
  • perations using QEM
  • Pre-sorted input stream allows fixed-sized active

working set independent of input and output model complexity

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Wu, Kobbelt: A Stream Algorithm for the Decimation of Massive Meshes, Graphics Interface 2003

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Out-of-core Decimation

  • Randomized multiple choice optimization avoids

global heap data structure

  • Special treatment for boundaries required

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Wu, Kobbelt: A Stream Algorithm for the Decimation of Massive Meshes, Graphics Interface 2003