Mesh Simplification

Plan

• Introduction

• Mesh Simplifications

• Current Techniques

• The Projet

• Results

Introduction

What’s a mesh?Vertices, edges, polygonesManifold (machine are able to make it)Non-manifold

Meshes usageScientific imaging, movies, games…

Common problemsStocking, rendering, data transferts…

Plan

• Introduction

• Mesh Simplification

• Current Techniques

• The Project

• Results

Mesh Simplification

IdeaRemove « negligeable » elements

CaracteristicsFidelity to original morphologyTopologyRe-exploitation

Plan

• Introduction

• Mesh Simplification

• Current Techniques

• The Project

• Results

Current Techniques

DecimationContractionClusteringProgressive MeshStatistics

Wavelets, fractals,…

Decimation

Contraction

Clustering

Plan

• Introduction

• Mesh Simplification

• Current Techniques

• The Project

• Results

The Project (1)

Studies on current techniquesSchroeder, Lorensen, ZargeGarland, Heckbert

ImplementationC/C++OpenGLGLUT, GLUI

The Project (2)

BoundsTriangulated meshes

Implemented triangulations methodsTriangle FanTriangle Strip

Optimal solutionDelaunay’s Triangulation

Triangulation : Triangle Fan

Triangulation : Triangle Strip

The Project (3)

Method 1: Edge ContractionEdge Length

• Distance criteria

Percentage• How many vertices to remove?

Cardinality ordering• Stronger• Weaker• Goal : avoid « dumb » mesh browsing

Cardinalities, examples

The Project (4)

Method 2: QEMQuadric Error Metric

• One error value per vertex• Create valid vertices pairs• Compute the contraction cost

Percentage• How many vertices to remove?

The Project (5)

Method 2: QEM

p = [a b c d] T , ax + by +cz +d = 0 Δ(v) = Σ p Є plans (v) (pT v) ² Δ(v) = Σ p Є plans (v) (vT p) (pT v)

= Σ p Є plans (v) vT (ppT) v

= vT [ Σ p Є plans (v) Kp] v

The Project (6)

Méthode 2: QEM (suite)

a² ab ac ad Kp = ppT = ab b² bc bd

ac bc c² cd

ad bd cd d²

Q = Σ p Є plans (v) Kp

Plan

• Introduction

• Mesh Simplification

• Current Techniques

• The Project

• Results

Results

Results

Results

Results

Results

Results

Results

Results

Results

Results

Results

Results

Results

Results

Results

Results

Reduction Time(s)

10% 0.390

50% 0.810

80% 1.051

« cow.obj » model(5804 faces) Method 1

Reduction Time(s)

10% 1.081

50% 3.044

80% 3.825

« cow.obj » model(5804 faces) QEM

Plan

• Introduction

• Mesh Simplification

• Current Techniques

• The Project

• Results

The

EOF ;)