When Convexity Helps Collapsing Complexes

This paper illustrates how convexity hypotheses help collapsing simplicial complexes. We first consider a collection of compact convex sets and show that the nerve of the collection is collapsible whenever the union of sets in the collection is convex. We apply this result to prove that the Delaunay complex of a finite point set is collapsible. We then consider a convex domain defined as the convex hull of a finite point set. We show that if the point set samples sufficiently densely the domain, then both the Cech complex and the Rips complex of the point set are collapsible for a well-chosen scale parameter. A key ingredient in our proofs consists in building a filtration by sweeping space with a growing sphere whose center has been fixed and studying events occurring through the filtration. Since the filtration mimics the sublevel sets of a Morse function with a single critical point, we anticipate this work to lay the foundations for a non-smooth, discrete Morse Theory

Geometry-driven collapses for converting a Cech complex into a triangulation of a nicely triangulable shape

Given a set of points that sample a shape, the Rips complex of the data points is often used in machine-learning to provide an approximation of the shape easily-computed. It has been proved recently that the Rips complex captures the homotopy type of the shape assuming the vertices of the complex meet some mild sampling conditions. Unfortunately, the Rips complex is generally high-dimensional. To remedy this problem, it is tempting to simplify it through a sequence of collapses. Ideally, we would like to end up with a triangulation of the shape. Experiments suggest that, as we simplify the complex by iteratively collapsing faces, it should indeed be possible to avoid entering a dead end such as the famous Bing's house with two rooms. This paper provides a theoretical justification for this empirical observation. We demonstrate that the Rips complex of a point-cloud (for a well-chosen scale parameter) can always be turned into a simplicial complex homeomorphic to the shape by a sequence of collapses, assuming the shape is nicely triangulable and well-sampled (two concepts we will explain in the paper). To establish our result, we rely on a recent work which gives conditions under which the Rips complex can be converted into a Cech complex by a sequence of collapses. We proceed in two phases. Starting from the Cech complex, we first produce a sequence of collapses that arrives to the Cech complex, restricted by the shape. We then apply a sequence of collapses that transforms the result into the nerve of some robust covering of the shape.Comment: 24 pages, 9 figure

Epsilon-covering is NP-complete

International audienceConsider the dilation and erosion of a shape S by a ball of radius ε. We call ε-covering of S any collection of balls whose union lies between the dilation and erosion of S. We prove that finding an ε-covering of minimum cardinality is NP-complete, using a reduction from vertex cover

Recognizing shrinkable complexes is NP-complete

International audienceWe say that a simplicial complex is shrinkable if there exists a sequence of admissible edge contractions that reduces the complex to a single vertex. We prove that it is NP-complete to decide whether a (three-dimensional) simplicial complex is shrinkable. Along the way, we describe examples of contractible complexes which are not shrinkable

Reconstruction en grandes dimensions

Dans cette thèse, nous cherchons à reconstruire une approximation d'une variété connue seulement à partir d'un nuage de points de grande dimension l'échantillonnant. Nous nous efforçons de trouver des méthodes de reconstructions efficaces et produisant des approximations ayant la même topologie que la variété échantillonnée. Une attention particulière est consacrée aux flag-complexes et particulièrement aux complexes de Rips. Nous montrons que le complexe de Rips capture la topologie d'une variété échantillonnée en supposant de bonnes conditions d'échantillonnage. En tirant avantage de la compacité des flags-complexes qui peuvent être représentés de manière compacte avec un graphe, nous présentons une structure de données appelée squelette/bloqueurs pour complexes simpliciaux. Nous étudions ensuite deux opérations de simplifications, la contraction d'arête et le collapse simplicial, qui s'avèrent utiles pour réduire un complexe simplicial sans en changer sa topologie.In this thesis, we look for methods for reconstructing an approximation of a manifold known only through a high-dimensional point cloud. Especially, we are interested in efficient methods that produce approximations that share the same topology as the sampled manifold. A particular attention is devoted to flag-complexes and more specially to Rips complexes due to their compactedness. We show that the Rips complex shares the topology of a sampled manifold under good sampling conditions. By taking advantage of the compactedness of flag-complexes, we present a data structure for simplicial complexes called skeleton/blockers. We then study two simplification operations, the edge contraction and the simplicial collapse, that turn out to be useful for reducing a simplicial complex without changing its topology.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF

Complexity of the Delaunay triangulation of points on polyhedral surfaces

It is well known that the complexity of the Delaunay triangulation of $n$ points in $R ^d$, i.e. the number of its simplices, can be $\Omega (n^\lceil \frac{d{2}\rceil })$. In particular, in $R ^3$, the number of tetrahedra can be quadratic. Differently, if the points are uniformly distributed in a cube or a ball, the expected complexity of the Delaunay triangulation is only linear. The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface. In this paper, we bound the complexity of the Delaunay triangulation of points distributed on the boundary of a given polyhedron. Under a mild uniform sampling condition, we provide deterministic asymptotic bounds on the complexity of the 3D Delaunay triangula- tion of the points when the sampling density increases. More precisely, we show that the complexity is $O(n^1.8)$ for general polyhedral surfaces and O(n\sqrtn) for convex polyhedral surfaces. Our proof uses a geometric result of independent interest that states that the medial axis of a surface is well approximated by a subset of the Voronoi vertices of the sample points. The proof extends easily to higher dimensions, leading to the first non trivial bounds for the problem when $d>3$

Construction d'iso-surfaces sous contraintes de delaunay, codage par squelettes et filtrage

Les iso-surfaces calculées à l'aide de l'algorithme de marching-cubes fournissent une triangulation des objets présents dans une image volumétrique. Dans cet article, nous construisons un nouveau type d'iso-surfaces ayant la propriété d'être incluses dans la triangulation de Delaunay de leurs sommets. Pour chaque 8-cube, un ensemble de boucles est calculé, dépendant uniquement de la connexité choisie pour les voxels intérieurs et extérieurs. Puis, chaque boucle est triangulée selon sa géométrie. Nous montrons que cette triangulation sous contrainte de Delaunay est toujours possible, et ceci même pour des voxels anisotropes. L'inclusion de l'iso-surface dans la triangulation de Delaunay a des conséquences importantes. Elle permet de déduire un modèle volumique des objets formés de tétraèdres. Elle permet également d'accéder au squelette de l'objet. Des applications au calcul du squelette sont présentées

A Linear Bound on the Complexity of the Delaunay triangulation of points on polyhedral surfaces

Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time therefore depends on the complexity of the Delaunay triangulation of such point sets. Although the Delaunay triangulation of points in ^3 can be quadratic in the worst-case, we show that, under some mild sampling condition, the complexity of the 3D Delaunay triangulation of points distributed on a fixed number of facets of ^3 (e.g. the facets of a polyhedron) is linear. Our bound is deterministic and the constants are explicitly given

Classification topologique locale d'images 3D

L'objectif de ce travail est de proposer une méthode d'analyse locale des formes des objets contenus dans une image 3D. Nous nous intéressons plus particulièrement aux formes de type cylindre ou plaque. Notre approche est basée sur l'analyse des points du squelette 3D et se déroule en deux étapes. Premièrement, 4 types de points du squelette sont identifiés : régulier, arc, bord et multiple. Un point du squelette est classé en fonction des propriétés topologique d'une région d'intérêt locale autour de ce point. La taille de cette région est réglée en fonction de l'épaisseur locale de la structure en ce point. Ensuite, la réversibilité du squelette est utilisée pour en déduire une classification du volume entier. Après avoir obtenu des résultats sur des images simulées 3D, nous présentons une application de la méthode dans l'identification des structures osseuses à partir d'images tomographiques hautes- résolution 3D

Stability and Computation of Medial Axes: a State-of-the-Art Report

International audienceThe medial axis of a geometric shape captures its connectivity. In spite of its inherent instability, it has found applications in a number of areas that deal with shapes. In this survey paper, we focus on results that shed light on this instability and use the new insights to generate simplified and stable modifications of the medial axis