  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 Betti
matching. We show how induced matchings guarantee the spatially correct
matching between barcodes in a segmentation setting. Furthermore, we propose an
efficient algorithm to compute the Betti matching of images. We show that the
Betti matching error 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 Betti matching 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. Our code is available in
https://github.com/nstucki/Betti-matching.
