Mesh Processing§

author:	David Coeurjolly

Mesh representation§

Overview§

Rough definition

Def.

A mesh is a piecewise linear geometrical structure embedded in $\mathbb{R}^d$

Usually, we assume

Meshes contain a combinatorial strucutre (no a soup of triangles)
The geometry is mostly defined by vertices coordinates

For instance, we can consider


Triangular meshes	Quadtrilateral meshes	Tetrahedral meshes

Overview (bis)§

Why meshes?

Provide topological information
Useful in many applications: modeling, rendering, numerical simulation, …

Formal characterization 1: Simplifical Complexes§

Cellular structure

Vertices (0-cell), edges (1-cell), triangles (2-cell), tetrahedon (3-cell) $\Rightarrow$ simplices

Def. Simplex

A k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 affinely independent vertices

Def. Simplicial Complex

A simplicial complex $\mathcal{K}$ is a set of simplices that satisfies the following conditions:

Any face of a simplex from $\mathcal{K}$ is also in $\mathcal{K}$
The intersection of any two simplices $\sigma_1$ , $\sigma_2 \in \mathcal{K}$ is a face of both $\sigma_1$ and $\sigma_2$

Formal characterization 1: Simplifical Complexes (bis)§

$\Rightarrow$ Strong topological structure (order, boundary, co-boundary, each simplex is topological ball)

$\Rightarrow$ Many topological operations on meshes can be described as k-simplices operations

$\Rightarrow$ Specific combinatoric makes implementation easy

Formal characterization 2: Cellular Complexes§

We remove the constraint that k-cells are defined by k+1-vertices

..but..

Cells are not topological balls anymore
$\Rightarrow$ restricted cases (CW-complexes)
$\Rightarrow$ combinatorial characterization (Combinatorial Maps)

Implementations§

Iteration on vertices
Neighboring vertices
no explicit representation of faces

Implementations§

direct access to faces
Mesh representation data structures on GPU
Optional backlink from vertices to faces
no easy access to faces neighbors

Implementations§

Neighboring faces for triangular meshes can be stored in a explicit structure

AdjF: $\quad f_i \rightarrow$ $\{f_a,f_b,f_c\}$

…but… not efficient for cellular meshes

Implementation: Winged-edge§

Implementation: Half-edge§

Double-linked list of edges

Orientation matters !
Faces can be made implicit from half-edges cycles
Geometry (vertices) is attached as membres of half-edges
For example, cycling around vertex can be done by sequences of {opposite/prev} operations on HS

cgal.org

Implementation: Half-edge (bis)§

Core topological operations maintaining the structure

cgal.org

Generic model: Combinatorial Maps§

Valid combinatorial model in any dimension

darts
involutions/permutations $\beta_i$ ( $i\in\{1\ldots n\}$ ) on dart labels to encode the structure
Formal constraints on involutions to ensure valid topological maps

cgal.org

Mesh processing examples§

Shape from mesh subdivision§

Principle

Start from a control mesh and apply local subdivision rules to generate finer object.

Subdivision rule

Several schemes
- 1 triangle -> 4 triangles
- 1 quad -> 4 quads
- …
Interpolation function for new vertices positions

Main idea: from the subdivison mechanism we can prove that the limit surface is ( $C^1$ , $C^2$ ,…)

Related to B-spline schemes

Examples§

(resp. Catmull-Clark, Loop, Doo-Sabin)

Mesh simplification§

Principle

Optimize the size of input mesh removing unnecessary triangles

More formally

Compute $\mathcal{M}'$ with $|\mathcal{M}'|<|\mathcal{M}|$ and

$d(\mathcal{M},\mathcal{M'}) < \epsilon$

What kind of shape metric d ?

Simple Example§

Local metric evaluation

For each vertex evaluate its distance to its tangent plane
If the distance is less than $\epsilon$ , we remove the point and triangulate its one-ring

Fast and easy implement but

No global metric guarantee
Topological must be ensured on local edits

Mesh parametrization§

Idea

Parametrized a complex mesh (2-manifold) on a 2D parameter space $[U,V]$

Issues

Planar embedding may not exist for some shapes
We would like the embedding to preserve some information (metric, area, angles,…)

Many applications

Texture mapping
Geometry processing
Remeshing
…

Mesh parametrization example§

Conformal Parametrization

Angle preserving transformation in complex plane

Discrete formulation

Defs.

Two meshes $\mathcal{M}$ $\mathcal{M}'$ with same topology are (conformally) equivalent iff $\exists \{u_i\}$ such that

$\tilde{l}_{ij} = e^{(u_i + u_j)/2} l_{ij}$

or equivalently (logspace)

$\tilde{\lambda}_{ij} = \lambda_{ij} + u_i + u_j$

$\Rightarrow$ linear formulation

Mesh parametrization example§

Topological issues

if $\mathcal{M}$ is homeomorphic to a disk, there exist a conformal map to plane [U,V]
if $\mathcal{M}$ is homeomorphic to a sphere, there exist a conformal map from $\mathcal{M}$ to a unit sphere
… but maps are harder to manipulate for high genus surfaces

Practical solution: Cut and Open

Mesh Compression§

Idea

Half-edge or Vertex-Edge data structure contains redundant information that can be compressed

We want to

compress the geometry (with or without loss)
compress the combinatorial structure
We may be interested in streamed decompression

Example: Triangular strip

<see blackboard>

Separate geometric + combinatorial encoders/decoders
Joint encoders from signal represnetation of the meshes (wavelets,…)
…