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

Lexicographic optimal homologous chains and applications to point cloud triangulations

This paper considers a particular case of the Optimal Homologous Chain Problem (OHCP), where optimality is meant as a minimal lexicographic order on chains induced by a total order on simplices. The matrix reduction algorithm used for persistent homology is used to derive polynomial algorithms solving this problem instance, whereas OHCP is NP-hard in the general case. The complexity is further improved to a quasilinear algorithm by leveraging a dual graph minimum cut formulation when the simplicial complex is a strongly connected pseudomanifold. We then show how this particular instance of the problem is relevant, by providing an application in the context of point cloud triangulation

Regular triangulations as lexicographic optimal chains

We introduce a total order on n-simplices in the n-Euclidean space for which the support of the lexicographic-minimal chain with the convex hull boundary as boundary constraint is precisely the n-dimensional Delaunay triangulation, or in a more general setting, the regular triangulation of a set of weighted points. This new characterization of regular and Delaunay triangulations is motivated by its possible generalization to submanifold triangulations as well as the recent development of polynomial-time triangulation algorithms taking advantage of this order

The convex hull in a new model of computation

We present a new model of geometric computation which supports the design of robust algorithms for exact real number input as well as for input with uncertainty, i.e. partial input. In this framework, we show that the convex hull of N computable real points in R^d is indeed computable. We provide a robust algorithm which, given any set of N partial inputs, i.e. N dyadic or rational rectangles, approximating these points, computes the partial convex hull in time O(N log N) in 2d and 3d. As the rectangles are refined to the N points, the sequence of partial convex hulls converges effectively both in the Hausdorff metric and the Lebesgue measure to the convex hull of the N points

The reach, metric distortion, geodesic convexity and the variation of tangent spaces

International audienceIn this paper we discuss three results. The first two concern general sets of positive reach: We first characterize the reach by means of a bound on the metric distortion between the distance in the ambient Euclidean space and the set of positive reach. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the distance between the points and the reach

Normal Cone Approximation and Offset Shape Isotopy

This work adresses the problem of the approximation of the normals of the offsets of general compact sets in euclidean spaces. It is proven that for general sampling conditions, it is possible to approximate the gradient vector field of the distance to general compact sets. These conditions involve the $\mu$-reach of the compact set, a recently introduced notion of feature size. As a consequence, we provide a sampling condition that is sufficient to ensure the correctness up to isotopy of a reconstruction given by an offset of the sampling. We also provide a notion of normal cone to general compact sets which is stable under perturbation

The reach, metric distortion, geodesic convexity and the variation of tangent spaces

In this paper we discuss three results. The first two concern general sets of positive reach: We first characterize the reach by means of a bound on the metric distortion between the distance in the ambient Euclidean space and the set of positive reach. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the distance between the points and the reach

Regular triangulations as lexicographic optimal chains

We introduce a total order on n-simplices in the n-Euclidean space for which the support of the lexicographic-minimal chain with the convex hull boundary as boundary constraint is precisely the n-dimensional Delaunay triangulation, or in a more general setting, the regular triangulation of a set of weighted points. This new characterization of regular and Delaunay triangulations is motivated by its possible generalization to submanifold triangulations as well as the recent development of polynomial-time triangulation algorithms taking advantage of this order