Computational Geometry: Digital Delaunay Triangulation and Applications§

author:	David Coeurjolly

Digital Delaunay Triangulation§

Description§

Input

Points from the digital contour of an object

local consistency of points

Points in $[0,N]^d$

specific integer based structures

Let’s start with good news

Exact InCircle predicates thanks to integer coordinates

… and the bad ones

On the regular lattice, we have many co-cyclic points
No $\epsilon$ perturbation can be done

Questions

Can we expect better bounds for Delaunay structure?

Digital Contour§

Delaunay triangulation from minimum spanning tree

Thm. [Devillers]

If the Euclidean Minimum Spanning Tree of the input point set, the whole triangulation can be constructed in expected time $O(n \log^* n)$

( $\log^* n=\inf\{k;\log(k)n\leq 1\}$ hence for $16<n\leq 65532$ , $\log^* n=4$ , $\log^* n= 5$ for $n<2^{65532}$ )

Why?

All edges of the EMST are Delaunay Edges
Even true for any spanning subgraph of the Delaunay structure with maximal degree d

Digital Contour (bis)§

Main observation

Thm.

The polyline defined from digital contour points is a Delaunay spanning graph with maximal degree 2

Thm.

Expected time for Delaunay construction for digital contour is in $O(n \log^* n)$

Deterministic algorithm exists in dimension 2

Digital Straight Segment Pattern§

Observation

For digital straight segment patterns, can we recover the Delaunay structure from arthimetic properties?

$\Rightarrow$ Yes! [Roussillon, Lachaud]

Digital Points§

Setting

Digital points in $M\times M$ domain

Main Result

Thm. [Chan]

$O(n \sqrt{\log M})$ expected randomized time for Delaunay Triangulation construction

Key Data Structure: Van Emde Boas Tree

Associative array with M-bit integer keys
Insert/Delete/Find/../ in $O(\log M)$ !!

Delaunay/Voronoi Applications: Reconstruction and Differential Estimators§

Surface Reconstruction§

Settings

Set $S$ with $n$ points sampling/approximating a smooth 2-manifold C can I reconstruct a discrete manifold M such that

$d(M,C)<\epsilon$ for some metric d (e.g. Haussdorff)
M is homeomorphic to C

Variants

How to control the sampling?

Example of theorem statement If sample set $S$ has good sampling properties: parametrized by $\epsilon_0$ (e.g. at least $d_H(S,C)< \epsilon_0$ ), then for samplings with $\epsilon<\epsilon_0$ Algorithm A produces a discrete structure homeomorphic to C

What kind of algorithmic tools or structure?

Example: Power Crust Reconstruction [Amenta]§

Sampling Definition

Def.

$S$ is an $\epsilon$ -sampling of $\partial C$ if $S\subset\partial C$ and $\forall x\in\partial C$ , $\exists p\in S$ such that $d(p,x)< \epsilon\cdot lfs(x)$ .

with lfs(x) being the local feature size at x: $lfs(x)= d(x,MedialAxis(C))$

$\epsilon$ -samples are on $\partial C$
$\epsilon$ is be used to control the number of samples and its distribution.

Question what does $d(p,x)< \epsilon \cdot lfs(x)$ mean?

Example: Power Crust Reconstruction [Amenta] (bis)§

Compute the Voronoi Diagram of $S$

Example: Power Crust Reconstruction [Amenta] (bis)§

Compute the Voronoi Diagram of $S$
Extract the poles and polar balls

pole of a sample s: pair of power diagram vertices farthest from s on either the inside or outside of the “object”.

Example: Power Crust Reconstruction [Amenta] (bis)§

Compute the Voronoi Diagram of $S$
Extract the poles and polar balls
Compute the Power Diagram of such poles

Example: Power Crust Reconstruction [Amenta] (bis)§

Compute the Voronoi Diagram of $S$
Extract the poles and construct
Compute the Power Diagram of such poles
Extract the power crust

Example: Power Crust Reconstruction [Amenta] (ter)§

Thm.

Homotopy equivalence result for some $\epsilon < \epsilon_0$
Distance between power crust and C tends to 0 when $\epsilon \rightarrow 0$

Differential Estimation From the Voronoi Diagram§

Side-product of Power Crust

Thm.

Direction $(p_1,p_2)$ from poles $p_1$ and $p_2$ at a sample s is a convergent (w.r.t. $\epsilon$ ) estimation of the normal direction at s

Differential Estimation From the Voronoi Diagram§

…but very sensitive to noise or sampling conditions

keep in mind that in theorems, $S$ samples C exactly

Alternative solutions: use Voronoi cell covariance matrix [Alliez]

Robust Voronoi-based curvature and feature estimation [Mérigot..]§

Idea

Covariance matrix is still a key tool but it is evaluated on r-offest of the input set

Thm.

Eigenvalues/Eigenvectors of the covariance matrix at a point are related to principal curvature/principal curvature direction

Convergence results exist with Haussdorff hypothesis on the point set

Robust Voronoi-based curvature and feature estimation [Mérigot..]§

Robust tool for feature extraction

Context: Monte-Carlo Integration§

Idea

Estimate

$\int_{\Omega}f(\overline{\mathbf{x}}) \, d\overline{\mathbf{x}}$

from

$\frac{1}{N} \sum_{i=1}^N f(\overline{\mathbf{x}}_i)$

Many fields

Numerical analysis
Image rendering
…

Sampling Quality Evaluation§

Variance in the Monte-Carlo Integration process (uniform sampling)

$\mathrm{Var}(S_N) = \frac{V^2}{N^2} \sum_{i=1}^N \mathrm{Var}(f) = V^2\frac{\mathrm{Var}(f)}{N} = V^2\frac{\sigma_N^2}{N}$
Integration error :

$\delta S_N\approx\sqrt{\mathrm{Var}(S_N)}=V\frac{\sigma_N}{\sqrt{N}}$

Spectral properties

No high energy peaks
Control of the point sampling spectrum shape

Stochastic Approaches§

Uniform sampling

white noise
fast
rely on pseudo-number random generator

Jittered/Stratified sampling

Grid structure and uniform sampling in each cell
enhance local consistency
we loose some stochastic properties

Stochastic Approaches (bis)§

Poisson Disk

Throw darts with “forbidden” zones
Good spectral properties but hard to generate efficiently

Deterministic Approaches§

Low discrepancy sequences Quasi-Monte-Carlo approaches

Tiled based approaches

Voronoi Diagram based Approaches: Llyod’s relaxation§

Description Iterative algorithm

Generate N points using uniform sampling
Compute its Voronoi diagram V

while (not(stability))
{
   For each cell
     Compute its centroid
     Move site to the centroid
}

Converges to a stable structure (honeycomb) but if we stop the process, we obtain a reasonable point sampling

On 3D surfaces for remeshing§

[Cohen-Steiner et al]

Capacity Constrained Voronoi Diagram§

Equi-distribution of samples $\equiv$ Cells with same capacity

Isotropic influence zone of samples $\Rightarrow$ Energy model on cell shapes

$\Rightarrow$ iterative process to minimize global energy
$\Rightarrow$ related to power diagram

Discrete version

Many variants/alternatives§

Experimental comparison

Computational Geometry: Digital Delaunay Triangulation and Applications§

Digital Delaunay Triangulation§

Description§

Digital Contour§

Digital Contour (bis)§

Digital Straight Segment Pattern§

Digital Points§

Delaunay/Voronoi Applications: Reconstruction and Differential Estimators§

Surface Reconstruction§

Example: Power Crust Reconstruction [Amenta]§

Example: Power Crust Reconstruction [Amenta] (bis)§

Example: Power Crust Reconstruction [Amenta] (bis)§

Example: Power Crust Reconstruction [Amenta] (bis)§

Example: Power Crust Reconstruction [Amenta] (bis)§

Example: Power Crust Reconstruction [Amenta] (ter)§

Differential Estimation From the Voronoi Diagram§

Differential Estimation From the Voronoi Diagram§

Example§

Robust Voronoi-based curvature and feature estimation [Mérigot..]§

Robust Voronoi-based curvature and feature estimation [Mérigot..]§

Point Sampling§

Context: Monte-Carlo Integration§

Sampling Quality Evaluation§

Stochastic Approaches§

Stochastic Approaches (bis)§

Deterministic Approaches§

Voronoi Diagram based Approaches: Llyod’s relaxation§

On 3D surfaces for remeshing§

Capacity Constrained Voronoi Diagram§

CCVT Example§

Many variants/alternatives§

Many variants/alternatives§