Surface Simplification Using Quadric Error Metrics

Speaker: Fengwei ZhangSeptember 20. 2007

Author:

Michael Garland:

Paul S.Heckbert:

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.

Computer Science Professor at Carnegie Mellon University from 1992-2001. Currently a 3D Graphics Architect for Nvidia, living and working in Pittsburgh

Background

Simple Introduction to Mesh

Necessary to Simplification: Mass data; Livel Of Detail (LOD)

Good simplification resultMeet the given target criterion:

A face count A max tolerable error

Good approximation of the original model

Good algorithmEfficiencyQuality (maintain high fidelity to the original model) Generality

Background

Related Work

1. Vertex DecimationReferences: William J. Schroeder, Jonathan A. Zarge, and William E.Lorensen Decimation of triangle meshes. Computer Graphics(SIGGRAPH ’92 Proc.), 26(2):65–70, July 1992.Solutions:

Select a vertex for removalRetriangulate the hole

Property: Limited to manifold surfaces

2. Vertex Clustering

Related Work

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 vertexProperty:

1.Generality and Fast 2.Can not provide a geometric error bound and low quality

3. Edge Contraction

Related Work

Solutions

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

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

New Solution:

Pair Contraction

Edge contraction

Non-edge contraction

+

=

(V1,V2)->V

Pair 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

Edge contraction Non-edge contraction

Why Non-edge contraction

Advantage of like the vertex clustering (V1,V2,…Vk)->V

Generates a large number of approximate models or a multiresolution representation

(Mn,Mn-1,…,Mg).

Why Non-edge contraction

Pair Selection

if 1. (v1, v2) is an edge or 2.||v1−v2||< t, where t is a threshold parameter

A valid pair for contraction:

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

Approximating Error With Quadrics

How to get the new vertex V: (V1,V2)->VHow to give the cost of a contraction

The distance from the point to the set of surface:

Vertex V Error matrix: Q=

Approximating Error With Quadrics

Get the new vertex: VMinimize △V= Why?

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,(v1+v2)/2

*

Approximating Error With Quadrics

Approximating Error With Quadrics

1.Define a error matrix at each vertex Q Step:2.Minimize △V=To get the new vertex: VThe cost of a contraction: min value

The matrix of the new vertexRequired a 4×4 symmetric matrix (10 floating point numbers) at each matrixA single plane may be counted multipletimes, but at most 3 times

New Algorithm1. 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.

Additional DetailsPreserving Boundaries

Generate a perpendicular plane through the boundary edge Be weighted by a large penalty factor

Preventing Mesh InversionCompare the normal of each face before and after the contraction

Experiment

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

5,804 994

532 248

64

Example

Experiment

Fixed:v1,v2,(v1+v2)/2

Effect of optimal vertex placement

Optimal: Choosing an optimal position

Cowmodel (t=0)

Error measurement:

Example:

Experiment

69451

1000(15s)

the nature of the error quadrics

Given s: vQv=s is an ellipsoid around the corresponding vertex

Conform to the shape of the model surface nicely:

large and flat on planar areas; be elongated along lines;

ExperimentBenefits of aggregation via pair contractions

4,204 contain separate bone segmentsUniform Vertex Clustering (262)

(11×4×4 grid)

Edge Contractions(250) Pair Contractions(250) (t=0.318)Toes are being merged into larger solid components

ExperimentBenefits of aggregation via pair contractions

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

Conclusion:

The selection of t:

Conclusion

Pair contract = edge contract + Non-edge contractTract the approximate error through Q Boundary preservationEfficiency;Quality;Generality

Problem

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

Thank you!