Stability of multidimensional persistent homology with respect to domain perturbations

Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer vision and computer graphics, we extend results concerning the stability of persistent homology with respect to function perturbations to results concerning the stability with respect to domain perturbations. Domain perturbations can be measured in a number of different ways. An important method to compare domains is the Hausdorff distance. We show that by encoding sets using the distance function, the multidimensional matching distance between rank invariants of persistent homology groups is always upperly bounded by the Hausdorff distance between sets. Moreover, we prove that our construction maintains information about the original set. Other well known methods to compare sets are considered, such as the symmetric difference distance between classical sets and the sup-distance between fuzzy sets. Also in these cases we present results stating that the multidimensional matching distance between rank invariants of persistent homology groups is upperly bounded by these distances. An experiment showing the potential of our approach concludes the paper. Introduction Persistent topology is a theory for studying objects related to computer vision and computer graphics, by analyzing the qualitative and quantitative behavior of real-valued functions defined over topological spaces and measuring the shape properties of the topological space under study (e.g., roundness, elongation, bumpiness, color). More precisely, persistent topology studies the sequence of nested lower level sets of the considered measuring functions and encodes at which scale a topological feature (e.g., a connected component, a tunnel, a void) is created, and when it is annihilated along this filtration. At the very beginning of the development of persistent topology, this encoding captured only the connected component changes in the lower level sets of a real valued function, and took the name of size function [13, 16]. Some years later, it was extended to consider all homotopy groups of the lower level sets of a vector-valued function, under the name of size homotopy groups [14]. Nowadays we have a wide choice of variants for this encoding, ranging from persistent homology groups capturing the homology of a one-parameter increasing family of spaces [12], to multidimensional persistent homology groups extending the previous concept to a multi-parameter setting [4], to vineyards coping with changes in the function over time [7], to interval persistence [9], just to cite a few. In this paper we focus on multidimensional persistent homology groups. For application 2010 Mathematics Subject Classification. Primary: 55N35; Secondary: 68T10, 68U05, 55N05.