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

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.