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 combinatorial structure, 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.

Cagliari F, Ferri M, Gualandri L, Landi C (2012). Persistence modules, shape description, and completeness. HEIDELBERG : Springer [10.1007/978-3-642-30238-1_16].

Persistence modules, shape description, and completeness

CAGLIARI, FRANCESCA;FERRI, MASSIMO;GUALANDRI, LUCIANO;LANDI, CLAUDIA
2012

Abstract

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 combinatorial structure, 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.
2012
Computational Topology in Image Context
148
156
Cagliari F, Ferri M, Gualandri L, Landi C (2012). Persistence modules, shape description, and completeness. HEIDELBERG : Springer [10.1007/978-3-642-30238-1_16].
Cagliari F; Ferri M; Gualandri L; Landi C
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/122670
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