Mesh Resampling

Wolfgang Knoll, Reinhard Russ, Cornelia Hasil

1 Institute of Computer Graphicsand Algorithms

Vienna University of Technology

Reducing number of faces while trying to keep overall shape, volume and boundaries

Oversampled 3D scan data

Fitting isosurfaces out of volume datasets

Motivation

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Motivation

Simplification useful tomake storagetransmissioncomputationdisplay more efficient

Can reduce memory requirementsand can speed networktransmission

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Problem Statement

Transform a given polygonal mesh into another with fewer faces, edges, and vertices:

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Mesh Simplification Approaches

Two basic concepts

Vertex Clustering

Incremental Decimation

Example

Incremental decimation with quadric error metric

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Mesh Simplification Approaches

Vertex ClusteringCluster GenerationComputing a representative Fast and effective Poor quality

Uniform 3D gridMap vertices to cluster cells

Remove degenerate triangular cells

Computing a representative:If P1, P2, ..., Pk are vertices in the same cell, then the representative is P = (P1 + P2 + ... + Pk)/k

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Mesh Simplification Approaches

Incremental decimationGeneral

Repeat pick mesh region apply decimation operator

Until no further reduction possible

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Example: Quadric Error Metrics

Surface Simplification Using Quadric Error Metrics

Iterative Pair Contraction with the Quadric Error Metric

Works on non-manifold geometry

Supports aggregation

Can be implemented efficiently

Produces high quality approximations

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Surface Simplification Using Quadric Error Metrics

Pair contraction: (v1 , v2 ) → v ̄

A pair of vertices (v1, v2) are valid for contraction if: 1. (v1, v2) is an edge, or2. ||v1 − v2|| < t for some threshold t

BenefitsCan join unconnected components Can result in much nicer approximations

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Approximating Error With Quadrics

For each vertex vi store a symmetric 4x4 matrix Qi

Error (v) at v = [vx vy vz 1]T is vTQvv The matrices Qi are called quadrics, because the level sets of (v) = ε form quadric surfaces (usually ellipsoids)

For a given contraction(v1 , v2 ) → v ̄ , let Q ̄ = Q1 + Q2

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Performing Contractions

To perform a contraction(v1 , v2 ) → v ̄ , we must find v ̄

Simple scheme: select v1, v2 or (v1 + v2)/2 with lowest value for (v ̄ )

We find minimum v ̄ by solving ∂ /∂x = ∂ /∂y = ∂ /∂z = 0which is equivalent to

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Algorithm Summary

Compute initial quadrics for each vertex

Select all valid pairs

Compute optimal contraction target for each pair and let its associated error be the cost of the contraction

Place all pairs in a keyed heap

Iteratively remove the pair with least cost from the heap, contract the pair, and update the cost of all valid pairs involving this contracted vertex

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Error Metrics in Mesh Simplification

Reinhard Russ

Institute of Computer Graphics and Algorithms

Vienna University of Technology

The purpose of using Error Metrics

Measurement for the introduced geometric error

What is the best contraction to perform?

What is the best position for the remaining vertices?EndpointOptimization

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Metrics Simple Heuristics

Edge length, Dihedral angle, area etc.

Sample Distance Squared distance function Cluster distance function

Curvature Valence function Quadratic Error Metric Feature Sensitive Metric

Simple Heuristics as Error Metrics

Edge length

Edge marking function

Dihedral angleSurrounding area

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Sample distance as Error Metrics

Squared distance function

Cluster distance function

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Cluster distance function

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Sample distance as Error Metrics

Projection to closest point

Restricted projection

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Error Metric based on Curvature

Curvature

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Error Metric based on Curvature

Curvature Tensor FieldHow many lines should be traced on the surface?

Compute local densitySpacing between two lines of curvatureCross section of the surface (normal to the lines)

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Error Metric based on Valence function

Valence function

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Quadric Error Metric

Quadric Error MetricBased on point-to-plane distance (instead of point-to-point distance)Minimize sum of squared distance to all planes at vertex

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Quadric Error Metric

Quadric Error MetricConstruct a quadric Q for every vertexCompute error of collapsingCompute quadric for new vertex

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Quadric Error Metric Adaptations

Originally for ECP-based algorithmsAdaptations for VDP-based and FCP-based algorithms

Originally for Polygonal ModelsAdaptations for Point Clouds

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Feature Sensitive Metric

Consider the field of unit normal vectors as a vector-valued image

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Metric Classification

Wolfgang Knoll

Institute of Computer Graphics and Algorithms

Vienna University of Technology

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Groupings Goal

Simplification/Minimization Quality Improvement Topology/Feature Preservation

Application Area Approach

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Simplification/Minimization Less vertices, triangles, faces etc. than before

→ Smaller Mesh

Either: Given amount Minimal under error boundary

Combinable with other goals

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Simplification/Minimization

Use of metrics depending on the methods.

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Simplification/Minimization

Use of metrics depending on the methods.

But:Almost every metric can be used for Minimization...

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Simplification/Minimization Edge/Vertex/Region decimation based on simple

Heuristics Clustering in regards with Energy minimization Etc...

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Quality Improvement Improvement in regards with:

Vertex/Face distribution Connectivity Triangle shape

Keeping Quality in regards with: Error-Metric Topology/Feature Preservation

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Quality Improvement Not always quantizable!

Metrics: Curvature based metrics QEM, squared distance function and other

metrics for minimizing error/energy(taking the best choice)

Valence function

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Quality Improvement

Example: Redistribution

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Topology/Feature Preservation Not necessary error minimization

Goal is to keep topology intact and/or maintain important features

More triangles at feature-areas

Rules for Topology preservation

Used metrics are often curvature-based

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Topology/Feature Preservation

Example: Feature preservation

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Application Area Dependent on:

Operational area Local Global

Geometrical Element Vertex Edge Face

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Application Area Local approaches are often based on decimation

approaches with simple heuristics

Global approaches: (Iterative) Energy minimization Clustering

→ both often use distance metrics Feature Sensitive Metric

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Application Area Distance metrics (obviously) use mostly the distance

between vertices/points

Clustering additionally can be extended with the Curvature around vertices and in an area/face

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Application Area

Example: global method

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Approach Decimation Approaches

Energy minimization

Clustering

Other Approaches:Particle Simulation, Retiling etc.

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Comparison Measures Used to analyze and compare a method

Often tightly tied with the method goal

Usually no direct comparison between proposed methods due to different measures & models

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Comparison Measures Quantitative Goal:

Size: vertex/face number, # Bytes Speed: computational performance

Qualitative Goal: Error: max, avg Triangle-angle

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Comparison Measures Quality improvement often lowers performance

Size reduction often lowers quality

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Conclusion Main goal is minimization!

→ reducing the numbers

Clustering and Decimation approaches

Curvature often used for quality improvement

Performance often goal dependent