A Mayer-Vietoris formula for persistent homology with an application to shape recognition in the presence of occlusions

Di Fabio, Barbara ; Landi, Claudia (2009) A Mayer-Vietoris formula for persistent homology with an application to shape recognition in the presence of occlusions. [Preprint]

This is the most updated version of the document.

Full text available as:
[thumbnail of amsActaDiFabioLandi.pdf]
Download (1MB) | Preview


In algebraic topology it is well-known that, using the Mayer-Vietoris sequence, the homology of a space $X$ can be studied splitting $X$ into subspaces $A$ and $B$ and computing the homology of $A$, $B$, $A\cap B$. A natural question is to which an extent persistent homology benefits of a similar property. In this paper we show that persistent homology has a Mayer-Vietoris sequence that in general is not exact but only of order two. However, we obtain a Mayer-Vietoris formula involving the ranks of the persistent homology groups of $X$, $A$, $B$ and $A\cap B$ plus three extra terms. This implies that topological features of $A$ and $B$ either survive as topological features of $X$ or are hidden in $A\cap B$. As an application of this result, we show that persistence diagrams are able to recognize an occluded shape by showing a common subset of points.

Document type
Di Fabio, Barbara
Landi, Claudia
Additional Information
MSC (2010): 55N05, 68U05
\v{C}ech homology, Mayer-Vietoris sequence, persistence diagram, partial matching, shape occlusion
Deposit date
04 Aug 2010 10:36
Last modified
16 May 2011 12:14

Other metadata

Available versions of this document

This work may be freely consulted and used, may be reproduced on a permanent basis in a digital format (i.e. saving) and can be printed on paper with own personal equipment (without availing of third -parties services), for strictly and exclusively personal, research or teaching purposes, with express exclusion of any direct or indirect commercial use, unless otherwise expressly agreed between the user and the author or the right holder. It is also allowed, for the same purposes mentioned above, the retransmission via telecommunication network, the distribution or sending in any form of the work, including the personal redirection (e-mail), provided it is always clearly indicated the complete link to the page of the Alma DL Site in which the work is displayed. All other rights are reserved.



Staff only: View the document