Topological Tracking of Connected Components in Image Sequences

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.

3D Well-composed Polyhedral Complexes

A binary three-dimensional (3D) image $I$ is well-composed if the boundary surface of its continuous analog is a 2D manifold. Since 3D images are not often well-composed, there are several voxel-based methods ("repairing" algorithms) for turning them into well-composed ones but these methods either do not guarantee the topological equivalence between the original image and its corresponding well-composed one or involve sub-sampling the whole image. In this paper, we present a method to locally "repair" the cubical complex $Q(I)$ (embedded in $\mathbb{R}^3$) associated to $I$ to obtain a polyhedral complex $P(I)$ homotopy equivalent to $Q(I)$ such that the boundary of every connected component of $P(I)$ is a 2D manifold. The reparation is performed via a new codification system for $P(I)$ under the form of a 3D grayscale image that allows an efficient access to cells and their faces.