Invariance properties of the multidimensional matching distance in Persistent Topology and Homology

Abstract. 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 Rnvalued 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 Rn

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Andrea Cerri, Patrizio Frosini
• Keywords: CORR 2010, Education, Homology Groups, Multidimensional Homology Groups, Multidimensional Matching Distance