Besides per-pixel accuracy, topological correctness is also crucial for the segmentation of images with fine-scale structures, e.g., satellite images and biomedical images. In this paper, by leveraging the theory of digital topology, we identify locations in an image that are critical for topology. By focusing on these critical locations, we propose a new homotopy warping loss to train deep image segmentation networks for better topological accuracy. To efficiently identity these topologically c…