Persistence modules are algebraic constructs that can be used to describe the shape of an object starting from a geometric representation of it. As shape descriptors, persistence modules are not complete, that is they may not distinguish non-equivalent shapes. In this paper we show that one reason for this is that homomorphisms between persistence modules forget the geometric nature of the problem. Therefore we introduce geometric homomorphisms between persistence modules, and show that in some cases they perform better. A combinatorialstructure, the H0-tree, is shown to be an invariant for geometric isomorphism classes in the case of persistence modules obtained through the 0th persistent homology functor.

Persistence Modules, Shape Description, and Completeness / F., Cagliari; M., Ferri; L., Gualandri; Landi, Claudia. - STAMPA. - 7309(2012), pp. 148-156. ((Intervento presentato al convegno 4th International Workshop on Computational Topology in Image Context (CTIC 2012) tenutosi a Bertinoro (Italy) nel May 28-30, 2012.

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