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Register a new userThe Structure of Morphisms in Persistent Homology, I. Functorial Dualities
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Added by
Ulrich Bauer
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We derive the relationship between the persistent homology barcodes of two dual filtered CW complexes. Applied to greyscale digital images, we obtain an algorithm to convert barcodes between the two different (dual) topological models of pixel connectivity.
We consider generic curves in R^2, i.e. generic C^1 functions f from S^1 to R^2. We analyze these curves through the persistent homology groups of a filtration induced on S^1 by f. In particular, we consider the question whether these persistent homology groups uniquely characterize f, at least up to re-parameterizations of S^1. We give a partially positive answer to this question. More precisely, we prove that f=goh, where h:S^1-> S^1 is a C^1-diffeomorphism, if and only if the persistent homology groups of sof and sog coincide, for every s belonging to the group Sigma_2 generated by reflections in the coordinate axes. Moreover, for a smaller set of generic functions, we show that f and g are close to each other in the max-norm (up to re-parameterizations) if and only if, for every s belonging to Sigma_2, the persistent Betti numbers functions of sof and sog are close to each other, with respect to a suitable distance.
Persistent Topology studies topological features of shapes by analyzing the lower level sets of suitable functions, called filtering functions, and encoding the arising information in a parameterized version of the Betti numbers, i.e. the ranks of persistent homology groups. Initially introduced by considering real-valued filtering functions, Persistent Topology has been subsequently generalized to a multidimensional setting, i.e. to the case of $R^n$-valued filtering functions, leading to studying the ranks of multidimensional homology groups. In particular, a multidimensional matching distance has been defined, in order to compare these ranks. The definition of the multidimensional matching distance is based on foliating the domain of the ranks of multidimensional homology groups by a collection of half-planes, and hence it formally depends on a subset of $R^ntimesR^n$ inducing a parameterization of these half-planes. It happens that it is possible to choose this subset in an infinite number of different ways. In this paper we show that the multidimensional matching distance is actually invariant with respect to such a choice.
In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this metric, including its stability. In particular, we show that the computation of this distance is strongly related to suitable filtering functions associated with lines of slope 1, so underlining the key role of these lines in the study of bidimensional persistence. In order to prove these results, we introduce and study the concepts of extended Pareto grid for a normal filtering function as well as of transport of a matching. As a by-product, we obtain a theoretical framework for managing the phenomenon of monodromy in 2D persistent homology.
The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance d_T that represents a possible solution to this problem. Indeed, d_T is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the new pseudo-distance with respect to the change of the filtering function, expressed both with respect to the max-norm and to the natural pseudo-distance between topological spaces endowed with vector-valued filtering functions. Furthermore, we prove a result showing the relationship between d_T and the matching distance in the 1-dimensional case, when the homology coefficients are taken in a field and hence the comparison can be made.
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