Topological Tracking of Connected Components in Image Sequences

TL;DR

This paper introduces a novel algorithm for tracking connected components over time in 2D binary image sequences using spatiotemporal barcodes, respecting the unidirectional nature of time and enabling backward component tracking.

Contribution

We develop a new algorithm that computes spatiotemporal paths directly for both foreground and background components, extending previous work and setting the stage for higher-dimensional feature tracking.

Findings

Algorithm effectively tracks components backward in time.

Works for both foreground and background components.

Provides a foundation for future higher-dimensional topological tracking.

Abstract

Persistent homology provides information about the lifetime of homology classes along a filtration of cell complexes. Persistence barcode is a graphical representation of such information. A filtration might be determined by time in a set of spatiotemporal data, but classical methods for computing persistent homology do not respect the fact that we can not move backwards in time. In this paper, taking as input a time-varying sequence of two-dimensional (2D) binary digital images, we develop an algorithm for encoding, in the so-called {\it spatiotemporal barcode}, lifetime of connected components (of either the foreground or background) that are moving in the image sequence over time (this information may not coincide with the one provided by the persistence barcode). This way, given a connected component at a specific time in the sequence, we can track the component backwards in time until the moment it was born, by what we call a {\it spatiotemporal path}. The main contribution of this paper with respect to our previous works lies in a new algorithm that computes spatiotemporal paths directly, valid for both foreground and background and developed in a general context, setting the ground for a future extension for tracking higher dimensional topological features in $nD$ binary digital image sequences.