Title:  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 (persistence barcodes) of label (ground truth) and prediction (output of the neural network). 

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. Moreover, we demonstrate how induced matchings can be used to train segmentation networks and improve the topological correctness of the segmentations across all 6 baseline datasets while preserving volumetric segmentation performance. 