Persistent Homology with Improved Locality Information for more Effective Delineation

TL;DR

This paper introduces a new filtration function for persistent homology that incorporates local information, improving the topological accuracy of deep network reconstructions of structures like roads and neurons.

Contribution

It proposes a novel filtration method combining thresholding and height functions, enhancing the locality information in persistent homology for better network training.

Findings

Improved topological reconstruction of road networks.

Enhanced neuronal process connectivity in results.

Demonstrated superiority over existing PH-based loss functions.

Abstract

Persistent Homology (PH) has been successfully used to train networks to detect curvilinear structures and to improve the topological quality of their results. However, existing methods are very global and ignore the location of topological features. In this paper, we remedy this by introducing a new filtration function that fuses two earlier approaches: thresholding-based filtration, previously used to train deep networks to segment medical images, and filtration with height functions, typically used to compare 2D and 3D shapes. We experimentally demonstrate that deep networks trained using our PH-based loss function yield reconstructions of road networks and neuronal processes that reflect ground-truth connectivity better than networks trained with existing loss functions based on PH. Code is available at https://github.com/doruk-oner/PH-TopoLoss.