Minisymposium: Topologie appliquée et numérique
Multiparameter persistent homology for shape comparison: from continuous to discrete
Kaczynski, Tomasz
Université de Sherbrooke

The theory of multiparameter persistent homology was initially developed in the discrete setting of filtered simplicial complexes. Stability of persistence was proved for topological spaces filtered by continuous vector-valued functions, that is, for continuous data. This talk aims to provide a bridge between the continuous setting, where stability properties hold, and the discrete setting, where actual computations are carried out. The existence of this bridge is not obvious due to the phenomenon of structural gap between the two settings, called topological aliasing, which appears in the multiparameter case when using the standard piecewise linear interpolation of the discrete model. We solve the problem by introducing an adapted axis-wise linear interpolation and develop a stability preserving method for comparing rank invariants of vector functions obtained from discrete data. These advances support the choice of multiparameter persistent homology as a tool for shape comparison in computer vision. This is a joint work with M. Ethier, P. Frosini, and C. Landi.

Mercredi, 19 juin, 15h30
Salle Du Jardin