  In the segmentation of fine-scale structures from natural and biomedical
images, per-pixel accuracy is not the only metric of concern. Topological
correctness, such as vessel connectivity and membrane closure, is crucial for
downstream analysis tasks. In this paper, we propose a new approach to train
deep image segmentation networks for better topological accuracy. In
particular, leveraging the power of discrete Morse theory (DMT), we identify
global structures, including 1D skeletons and 2D patches, which are important
for topological accuracy. Trained with a novel loss based on these global
structures, the network performance is significantly improved especially near
topologically challenging locations (such as weak spots of connections and
membranes). On diverse datasets, our method achieves superior performance on
both the DICE score and topological metrics.
