TOPOLOGICALLY FAITHFUL IMAGE SEGMENTATION VIA INDUCED MATCHING OF PERSISTENCE BARCODES

Abstract

Image segmentation is a largely researched field where neural networks find vast applications in many facets of technology. Some of the most popular approaches to train segmentation networks employ loss functions optimizing pixel-overlap, an objective that is insufficient for many segmentation tasks. In recent years, their limitations fueled a growing interest in topology-aware methods, which aim to recover the correct topology of the segmented structures. However, so far, none of the existing approaches achieve a spatially correct matching between the topological features of ground truth and prediction. In this work, we propose the first topologically and feature-wise accurate metric and loss function for supervised image segmentation, which we term TopoMatch. We show how induced matchings guarantee the spatially correct matching between barcodes in a segmentation setting. Furthermore, we propose an efficient algorithm to compute TopoMatch for images. We show that TopoMatch is an interpretable metric to evaluate the topological correctness of segmentations, which is more sensitive than the well-established Betti number error. Moreover, the differentiability of the TopoMatch loss enables its use as a loss function. It improves the topological performance of segmentation networks across six diverse datasets while preserving the volumetric performance.

1. INTRODUCTION

Topology studies properties of shapes that are related to their connectivity and that remain unchanged under deformations, translations, and twisting. Some topological concepts, such as cubical complexes, homology and Betti numbers, form interpretable descriptions of shapes in space that can be efficiently computed. Naturally, the topology of physical structures is highly relevant in machine learning tasks, where the preservation of its connectivity is crucial, a prominent example being image segmentation. Recently, a number of methods have been proposed to improve topology preservation in image segmentation for a wide range of applications. However, none of the existing concepts take the spatial location of the topological features (e.g. connected components or cycles) within their respective image into account. Evidently, spatial correspondence of these features is a critical property of segmentations, see Fig. 1 . We match cycles between label and prediction for a CREMI image and denote matched pairs in the same color. We visualize only six (randomly selected out of the total 23 matches for both methods) matched pairs for presentation clarity. Note that TopoMatch always matches spatially correctly while the Wasserstein matching gets most matches wrong. Our contribution In this work, we introduce a rigorous framework for faithfully quantifying the preservation of topological properties in the context of image segmentation, see Fig. 2 . Our method



Figure 1: Motivation -comparison of our TopoMatch and Wasserstein matching (Hu et al. (2019)).We match cycles between label and prediction for a CREMI image and denote matched pairs in the same color. We visualize only six (randomly selected out of the total 23 matches for both methods) matched pairs for presentation clarity. Note that TopoMatch always matches spatially correctly while the Wasserstein matching gets most matches wrong. Our contribution In this work, we introduce a rigorous framework for faithfully quantifying the preservation of topological properties in the context of image segmentation, see Fig.2. Our method

