Learning Probabilistic Topological Representations Using Discrete Morse Theory

Xiaoling Hu, Dimitris Samaras, Chao Chen

Keywords: Topological Representation, Discrete Morse Theory, Persistent Homology

TL;DR: We use discrete Morse theory and persistent homology to construct an one-parameter family of structures as the topological/structural representation space to perform inference tasks.

Abstract: Accurate delineation of fine-scale structures is a very important yet challenging problem. Existing methods use topological information as an additional training loss, but are ultimately making pixel-wise predictions. In this paper, we propose a novel deep learning based method to learn topological/structural. We use discrete Morse theory and persistent homology to construct a one-parameter family of structures as the topological/structural representation space. Furthermore, we learn a probabilistic model that can perform inference tasks in such a topological/structural representation space. Our method generates true structures rather than pixel-maps, leading to better topological integrity in automatic segmentation tasks. It also facilitates semi-automatic interactive annotation/proofreading via the sampling of structures and structure-aware uncertainty.

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