Surface Simplification Using Quadric Error Metrics

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Surface Simplification Using Quadric
Error Metrics

• Speaker Fengwei Zhang
• September 20. 2007

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Author
Michael Garland
A research scientist with NVIDIA and an adjunct
professor in the Department of Computer
Science at the University of Illinois,
Urbana-Champaign. Graduated with Ph.D. from the
Computer Science Department of Carnegie Mellon
University.
Paul S.Heckbert
Computer Science Professor at Carnegie Mellon
University from 1992-2001. Currently a 3D
Graphics Architect for Nvidia, living and
working in Pittsburgh
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Background

• Simple Introduction to Mesh
• Necessary to Simplification
• Mass data Livel Of Detail (LOD)

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Background

• Good simplification result
• Meet the given target criterion
• A face count
• A max tolerable error
• Good approximation of the original model
• Good algorithm
• Efficiency
• Quality (maintain high fidelity to the original
  model)
• Generality

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Related Work
1. Vertex Decimation

• References
• William J. Schroeder, Jonathan
  A. Zarge, and William E.Lorensen
• Decimation of triangle
  meshes.
• Computer Graphics(SIGGRAPH 92
  Proc.), 26(2)6570, July 1992.
• Solutions
• Select a vertex for removal
• Retriangulate the hole
• Property
• Limited to manifold surfaces

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Related Work
2. Vertex Clustering

• References
• Jarek Rossignac and Paul Borrel.
• Multi-resolution 3D approximations for rendering
  complex
• scenes
• Solutions
• A surrounding box, Divided into a
  grid, Clustering into a
• vertex
• Property
• 1.Generality and Fast
• 2.Can not provide a geometric error bound and
  low
• quality

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Related Work
3. Edge Contraction

• Solutions

• Property
• How to choose an edge to contract
• Be designed for use on manifold surfaces

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Compare the Solutions

• Vertex Decimation
• Provide reasonable efficiency and quality
• Limited to manifold surfaces
• Vertex Clustering
• Generality and Fast
• But bad control
• Edge Contraction
• Limited to manifold surfaces
• Not support aggregation

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New Solution
Pair Contraction
Edge contraction

Non-edge contraction

(V1,V2)-gtV
Pair Contraction
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Why Non-edge contraction

• When Overall shape is important than topology
• Less sensitive to the mesh connectivity repair
  this shortcoming of the initial mesh, when two
  faces meet at a vertex which is duplicated

A regular grid of spaced cubes
Non-edge contraction
Edge contraction

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Why Non-edge contraction

• Advantage of like the vertex clustering
• (V1,V2,Vk)-gtV
• Generates a large number of approximate models or
  a multiresolution representation
• (Mn,Mn-1,,Mg).

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Pair Selection
A valid pair for contraction
if 1. (v1, v2) is an edge or 2.v1-v2lt
t, where t is a threshold parameter

• Choosing t carefully
• Pairs are selected at initialization time and
• only consider these edges during the course
• of the algorithm
• Modify the topology

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Approximating Error With Quadrics

• How to get the new vertex
• V (V1,V2)-gtV
• How to give the cost of
• a contraction

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Approximating Error With Quadrics
The distance from the point to the set of surface
Vertex V Error matrix Q
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Approximating Error With Quadrics
Get the new vertex V
Why?
Minimize ?V
Find
?
?

• If the matrix is invertible, ()
• If not, find the optimal vertex along the segment
  (v1,v2)
• If failed, find the V amongst v1,v2,(v1v2)/2

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Approximating Error With Quadrics
1.Define a error matrix at each vertex Q
Step
2.Minimize ?V

• To get the new vertex V
• The cost of a contraction min value

• The matrix of the new vertex
• Required a 44 symmetric matrix (10 floating
• point numbers) at each matrix
• A single plane may be counted multipletimes, but
  
• at most 3 times

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New Algorithm
1. Compute the Q matrices for all the initial
vertices.
2. Select all valid pairs
4. Place all the pairs in a heap keyed on cost
with the minimum cost pair at the top.
5. Iteratively remove the pair (v1,v2) of least
cost from the heap,contract this pair,
and update the costs of all valid pairs
involving v1.
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Additional Details

• Preserving Boundaries
• Generate a perpendicular plane through
• the boundary edge
• Be weighted by a large penalty factor
• Preventing Mesh Inversion
• Compare the normal of each face before
• and after the contraction

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Experiment
Example
5,804
532
248
64
994

• Fast constructed in about a second
• High fidelity features such as horns and hooves
  only
• disappear in low levels of detail

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Experiment
Effect of optimal vertex placement
Error measurement
Example
Cow model (t0)
Fixedv1,v2,(v1v2)/2
Optimal Choosing an optimal position
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Experiment
the nature of the error quadrics
Given s vQvs is an ellipsoid around
the corresponding vertex
69451

• Conform to the shape of the model
• surface nicely
• large and flat on planar areas
• be elongated along lines

1000(15s)
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Experiment
Benefits of aggregation via pair contractions
Uniform Vertex Clustering (262) (1144 grid)
4,204 contain separate bone segments
Edge Contractions(250)
Pair Contractions(250) (t0.318) Toes are being
merged into larger solid components
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Experiment
Benefits of aggregation via pair contractions
The selection of t
Conclusion

• (tgt0) produce better
• approximations than are
• achieved by (t 0)
• Increasing t does not always
• improve the approximation

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Conclusion

• Pair contract
• edge contract Non-edge contract
• Tract the approximate error through Q
• Boundary preservation
• EfficiencyQualityGenerality

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Problem

• Measuring error as a distance to a set of planes
  
• only works well in a suitably local
  neighborhood
• Contraction created a non-manifold region

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Thank you!