Nonobtuse Remeshing and Mesh Decimation

John Y.S. Li and Richard (Hao) ZhangGrUVi lab, Simon Fraser University, Canada

Introduction

Goal Nonobtuse = at most 90o

Motivation Delaunay Triangulation

Finite Element Method

Geodesic Computation

Planar Mesh Embedding

Nonobtuse Remeshing and Mesh Decimation SGP 2006 Cagliari, Italy

I changed spacing

Related work

Open problem. [GY03] – a bit misleading Nonobtuse triangulation of polygons. [BMR94]

30o - 120o angle bound. [Che93]

Minimum 30o angle bound. [CS05]

3D domain tiling with acute tetrahedra. [ESU04]

Nonobtuse Remeshing and Mesh Decimation SGP 2006 Cagliari, Italy

Overview of our Approach

1 2 3

Generate initial nonobtuse mesh

Deform-to-fit optimization

Nonobtuse decimation

Nonobtuse Remeshing and Mesh Decimation SGP 2006 Cagliari, Italy

Generality and flexibility of our framework to allow … ??? I think this is quite important.

Modified Marching Cubes

Input: Signed distance field Based on midpoint-based Ma

rching Cubes (MC) Some new vertices are inserted

inside the cubes

Guaranteed nonobtusity

No angles less than 30o

Handling open mesh is a bit tricky, but possible

Nonobtuse Remeshing and Mesh Decimation SGP 2006 Cagliari, Italy

1

Input is mesh and convert to SDF?

Optimization

Objective

Quadric term Associate closest triangle T to v

Average squared distances Smoothing:

Better angle quality

Nonobtuse Remeshing and Mesh Decimation SGP 2006 Cagliari, Italy

2

Mark T, v, etc. below

Clearly separate error reduction and smoothing

Optimization Constraints

feasible region of an edge

feasible region of a vertex

Nonobtuse Remeshing and Mesh Decimation SGP 2006 Cagliari, Italy

2

Optimization Quadratic Programming Iterative optimization

Priority Queue

Updating quadrics Local search of closest triangle T to v ― a heuristic

Stopping condition

Smoothing

Nonobtuse Remeshing and Mesh Decimation SGP 2006 Cagliari, Italy

2

Indicate alternating between …

You need to elaborate a bit on this slide.

Nonobtuse Decimation Similar framework for nonobtuse edge collapse

Optimal location for the unified vertex

Cost of edge collapse

Updating quadrics

Nonobtuse Remeshing and Mesh Decimation SGP 2006 Cagliari, Italy

3

If you are short on time, this slide can be simplified. The framework is quite standard.

Results Different sampling grids in Modified MC

Nonobtuse Remeshing and Mesh Decimation SGP 2006 Cagliari, Italy

Results Effects of smoothness term and smoothing

Nonobtuse Remeshing and Mesh Decimation SGP 2006 Cagliari, Italy

Results Angle distribution

Nonobtuse Remeshing and Mesh Decimation SGP 2006 Cagliari, Italy

No slide on speed?

Future work

Better angle boundRecently achieved 30o – 90o bound

Nonobtuse meshing from point cloud Handling meshes with boundaries Feature preservation Guaranteed acute remeshing

Nonobtuse Remeshing and Mesh Decimation SGP 2006 Cagliari, Italy

Improving speed?

Thank you!

Any questions?