Combining Persistent Homology and Invariance Groups for Shape Comparison

Abstract

Persistent homology has proven itself quite efficient in the topological and qualitative comparison of filtered topological spaces, when invariance with respect to every homeomorphism is required. However, we can make the following two observations about the use of persistent homology for application purposes. On the one hand, more restricted kinds of invariance are sometimes preferable (e.g., in shape comparison). On the other hand, in several practical situations filtering functions are not just auxiliary technical tools that can be exploited to study a given topological space, but instead the main aim of our analysis. Indeed, most of the data is usually produced by measurements, whose results are quite often functions defined on a topological space. As a simple example we can consider a 3D laser scanning of a surface, where the result of each measurement can be seen as a real-valued function defined on the manifold that describes the positions of the rangefinder measuring the distances. In fact, in many applications the dataset of interest is seen as a collection $${\varPhi }$$? of real-valued functions defined on a given topological space X, instead of a family of topological spaces. As a natural consequence, in these cases observers can be seen as collections of suitable operators on $${\varPhi }$$?. Starting from these remarks, this paper proposes a way to combine persistent homology with the use of G-invariant non-expansive operators defined on $${\varPhi }$$?, where G is a group of self-homeomorphisms of X. Our goal is to give a method to study $${\varPhi }$$? in a way that is invariant with respect to G. Some theoretical results concerning our approach are proven, and two experiments are presented. An experiment illustrates the application of the proposed technique to compare 1D-signals, when the invariance is expressed by the group of affinities, the group of orientation-preserving affinities, the group of isometries, the group of translations and the identity group. Another experiment shows how our technique can be used for image comparison.