A Mayer–Vietoris Formula for Persistent Homology with an Application to Shape Recognition in the Presence of Occlusions

Abstract

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 ∩ 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 ∩ 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 ∩ 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.

Persistent homology and partial similarity of shapes

“…A Mayer-Vietoris formula is a formula relating the ranks of the homology groups of spaces X, A, B,C when X = A ∪ B and C = A ∩ B. A Mayer -Vietoris formula for ordinary persistence has been obtained in (Di Fabio and Landi, 2010). Similar formulas also for relative and extended persistence are a novel contribution of this paper.…”

“…This fact is illustrated by an example in Table 1, and implicitly stated in the following result from (Di Fabio and Landi, 2010).…”

Alexander duality for functions

“…• Can our results be extended to the more general setting of a decomposition X = A ∪ B of a topological space X, as considered by Di Fabio and Landi [4,5].…”

“…The General Shore Theorem can also be proven with algebraic arguments: Di Fabio and Landi [4,5] consider decompositions X = A ∪ B, for a general topological space X, and relate the ordinary, relative, and extended (horizontal plus vertical) persistent Betti numbers of a function restricted to these sets with each other. These relationships depend on the ranks of maps between certain absolute and relative homology groups; see Theorems 3.1, 3.3, and 3.4 in [5] for details.…”

Object-oriented persistent homology

“…7, 22, 23, 26, 50 Some of these persistent homology algorithms have been implemented in many software packages, namely Perseus, 50, 52 JavaPlex 71 and Dionysus. In the past few years, persistent homology has been applied to image analysis, 5, 9, 58, 67 image retrieval, 30 chaotic dynamics verification, 42, 49 sensor networks, 66 complex networks, 40, 45 data analysis, 8, 47, 53, 60, 73 computer vision, 67 shape recognition 24 and computational biology. 21, 31, 43, 86 …”