Tracking a generator by persistence

Abstract

The persistent homology provides a mathematical tool to describe "features" in a principled manner. The persistence algorithm proposed by Edelsbrunner et al. [5] can compute not only the persistent homology for a filtered simplicial complex, but also representative generating cycles for persistent homology groups. However, if there are dynamic changes either in the filtration or in the underlying simplicial complex, the representative generating cycle can change wildly. In this paper, we consider the problem of tracking generating cycles with temporal coherence. Specifically, our goal is to track a chosen essential generating cycle so that the changes in it are "local". This requires reordering simplices in the filtration. To handle reordering operations, we build upon the matrix framework proposed by Cohen-Steiner et al. [3] to swap two consecutive simplices, so that we can process a reordering directly. We present an application showing how our algorithm can track an essential cycle in a complex constructed out of a point cloud data. © 2010 Springer-Verlag Berlin Heidelberg.