Mesh Resampling Wolfgang Knoll, Reinhard Russ, Cornelia Hasil 1 Institute of Computer Graphics and 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 2 Cornelia Hasil

Motivation Simplification useful to make storage transmission computation display more efficient Can reduce memory requirements and can speed network transmission 3 Cornelia Hasil

Problem Statement Transform a given polygonal mesh into another with fewer faces, edges, and vertices: 4 Cornelia Hasil

Mesh Simplification Approaches Two basic concepts Vertex Clustering Incremental Decimation Example Incremental decimation with quadric error metric 5 Cornelia Hasil

Mesh Simplification Approaches Vertex Clustering Cluster Generation Computing a representative Fast and effective Poor quality Uniform 3D grid Map 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 + P Pk)/k 6 Cornelia Hasil

Mesh Simplification Approaches Incremental decimation General Repeat pick mesh region apply decimation operator Until no further reduction possible 7 Cornelia Hasil

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 8 Cornelia Hasil

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, or 2. ||v1 − v2|| < t for some threshold t Benefits Can join unconnected components Can result in much nicer approximations 9 Cornelia Hasil

Approximating Error With Quadrics For each vertex vi store a symmetric 4x4 matrix Qi Error (v) at v = [vx vy vz 1] T is v T Q v v 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 10 Cornelia Hasil

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 = 0 which is equivalent to 11 Cornelia Hasil

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 12

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? Endpoint Optimization 14 Reinhard Russ

Wolfgang Knoll 15 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 angle Surrounding area 16 Reinhard Russ

Sample distance as Error Metrics Squared distance function Cluster distance function 17 Reinhard Russ

Cluster distance function 18 Reinhard Russ

Sample distance as Error Metrics Projection to closest point Restricted projection 19 Reinhard Russ

Error Metric based on Curvature Curvature 20 Reinhard Russ

Error Metric based on Curvature Curvature Tensor Field How many lines should be traced on the surface? Compute local density Spacing between two lines of curvature Cross section of the surface (normal to the lines) 21 Reinhard Russ

Error Metric based on Valence function Valence function 22 Reinhard Russ

Quadric Error Metric Based on point-to-plane distance (instead of point-to-point distance) Minimize sum of squared distance to all planes at vertex 23 Reinhard Russ

Quadric Error Metric Construct a quadric Q for every vertex Compute error of collapsing Compute quadric for new vertex 24 Reinhard Russ

Quadric Error Metric Adaptations Originally for ECP-based algorithms Adaptations for VDP-based and FCP-based algorithms Originally for Polygonal Models Adaptations for Point Clouds 25 Reinhard Russ

Feature Sensitive Metric Consider the field of unit normal vectors as a vector- valued image 26 Reinhard Russ

Metric Classification Wolfgang Knoll Institute of Computer Graphics and Algorithms Vienna University of Technology

Wolfgang Knoll 28 Groupings  Goal  Simplification/Minimization  Quality Improvement  Topology/Feature Preservation  Application Area  Approach

Wolfgang Knoll 29 Simplification/Minimization  Less vertices, triangles, faces etc. than before → Smaller Mesh  Either:  Given amount  Minimal under error boundary  Combinable with other goals

Wolfgang Knoll 30 Simplification/Minimization  Use of metrics depending on the methods.

Wolfgang Knoll 31 Simplification/Minimization  Use of metrics depending on the methods.  But: Almost every metric can be used for Minimization...

Wolfgang Knoll 32 Simplification/Minimization  Edge/Vertex/Region decimation based on simple Heuristics  Clustering in regards with Energy minimization  Etc...

Wolfgang Knoll 33 Quality Improvement  Improvement in regards with:  Vertex/Face distribution  Connectivity  Triangle shape  Keeping Quality in regards with:  Error-Metric  Topology/Feature Preservation

Wolfgang Knoll 34 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

Wolfgang Knoll 35 Quality Improvement Example: Redistribution

Wolfgang Knoll 36 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

Wolfgang Knoll 37 Topology/Feature Preservation Example: Feature preservation

Wolfgang Knoll 38 Application Area  Dependent on:  Operational area  Local  Global  Geometrical Element  Vertex  Edge  Face

Wolfgang Knoll 39 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

Wolfgang Knoll 40 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

Wolfgang Knoll 41 Application Area Example: global method

Wolfgang Knoll 42 Approach  Decimation Approaches  Energy minimization  Clustering  Other Approaches: Particle Simulation, Retiling etc.

Wolfgang Knoll 43 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

Wolfgang Knoll 44 Comparison Measures  Quantitative Goal:  Size: vertex/face number, # Bytes  Speed: computational performance  Qualitative Goal:  Error: max, avg  Triangle-angle

Wolfgang Knoll 45 Comparison Measures  Quality improvement often lowers performance  Size reduction often lowers quality

Wolfgang Knoll 46 Conclusion  Main goal is minimization! → reducing the numbers  Clustering and Decimation approaches  Curvature often used for quality improvement  Performance often goal dependent