Showing papers by "Herbert Edelsbrunner published in 2014"

A Short Course in Computational Geometry and Topology

Abstract: This monograph presents a short course in computational geometry and topology. In the first part the book covers Voronoi diagrams and Delaunay triangulations, then it presents the theory of alpha complexes which play a crucial role in biology. The central part of the book is the homology theory and their computation, including the theory of persistence which is indispensable for applications, e.g. shape reconstruction. The target audience comprises researchers and practitioners in mathematics, biology, neuroscience and computer science, but the book may also be beneficial to graduate students of these fields.

On the computational complexity of Betti numbers: reductions from matrix rank

Abstract: We give evidence for the difficulty of computing Betti numbers of simplicial complexes over a finite field. We do this by reducing the rank computation for sparse matrices with m non-zero entries to computing Betti numbers of simplicial complexes consisting of at most a constant times m simplices. Together with the known reduction in the other direction, this implies that the two problems have the same computational complexity.

The Morse Theory of Cech and Delaunay Filtrations

Abstract: Given a finite set of points in Rn and a positive radius, we study the Cech, Delaunay--Cech, alpha, and wrap complexes as instances of a generalized discrete Morse theory. We prove that the latter three complexes are simple-homotopy equivalent. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data.

The Classification of Endoscopy Images with Persistent Homology

Abstract: Aiming at the automatic diagnosis of tumors from narrow band imaging (NBI) magnifying endoscopy (ME) images of the stomach, we combine methods from image processing, computational topology, and machine learning to classify patterns into normal, tubular, vessel. Training the algorithm on a small number of images of each type, we achieve a high rate of correct classifications. The analysis of the learning algorithm reveals that a handful of geometric and topological features are responsible for the overwhelming majority of decisions.

Stable Length Estimates of Tube-Like Shapes

Abstract: Motivated by applications in biology, we present an algorithm for estimating the length of tube-like shapes in 3-dimensional Euclidean space. In a first step, we combine the tube formula of Weyl with integral geometric methods to obtain an integral representation of the length, which we approximate using a variant of the Koksma---Hlawka Theorem. In a second step, we use tools from computational topology to decrease the dependence on small perturbations of the shape. We present computational experiments that shed light on the stability and the convergence rate of our algorithm.

Voronoi and Delaunay Diagrams

Abstract: The goal of this book is the introduction of the Voronoi diagram and the Delaunay triangulation of a finite set of points in the plane, and an elucidation of their dual relationship.

Functionals on Triangulations of Delaunay Sets

Abstract: We study densities of functionals over uniformly bounded triangulations of a Delaunay set of vertices, and prove that the minimum is attained for the Delaunay triangulation if this is the case for finite sets.

An Algorithm for Cartographic Generalization that Preserves Global Topology

Abstract: We propose an algorithm for the generalization of cartographic objects that can be used to represent maps on different scales.

The Voronoi Functional is Maximized by the Delaunay Triangulation in the Plane

Abstract: We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.