Estimating persistent Betti numbers for discrete shape analysis

Cavazza, Niccolò (2011) Estimating persistent Betti numbers for discrete shape analysis, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in Matematica, 23 Ciclo. DOI 10.6092/unibo/amsdottorato/3468.
Documenti full-text disponibili:
Documento PDF (English) - Richiede un lettore di PDF come Xpdf o Adobe Acrobat Reader
Download (923kB) | Anteprima


Persistent Topology is an innovative way of matching topology and geometry, and it proves to be an effective mathematical tool in shape analysis. In order to express its full potential for applications, it has to interface with the typical environment of Computer Science: It must be possible to deal with a finite sampling of the object of interest, and with combinatorial representations of it. Following that idea, the main result claims that it is possible to construct a relation between the persistent Betti numbers (PBNs; also called rank invariant) of a compact, Riemannian submanifold X of R^m and the ones of an approximation U of X itself, where U is generated by a ball covering centered in the points of the sampling. Moreover we can state a further result in which, this time, we relate X with a finite simplicial complex S generated, thanks to a particular construction, by the sampling points. To be more precise, strict inequalities hold only in "blind strips'', i.e narrow areas around the discontinuity sets of the PBNs of U (or S). Out of the blind strips, the values of the PBNs of the original object, of the ball covering of it, and of the simplicial complex coincide, respectively.

Tipologia del documento
Tesi di dottorato
Cavazza, Niccolò
Dottorato di ricerca
Scuola di dottorato
Scienze matematiche, fisiche ed astronomiche
Settore disciplinare
Settore concorsuale
Parole chiave
persistent homology simplicial complex ball covering shape analysis blind strips
Data di discussione
6 Giugno 2011

Altri metadati

Statistica sui download

Gestione del documento: Visualizza la tesi