Regional Hausdorff distance losses for medical image segmentation

Most medical image segmentation techniques rely on overlap-based metrics such as the Dice coefficient. In contrast, the Hausdorff Distance (HD) offers a more sensitive assessment of boundary discrepancies by explicitly capturing spatial misalignments. Despite its relevance, directly minimizing the HD during the training of convolutional neural networks for medical image segmentation remains challenging due to the non-differentiability of the conventional distance transform algorithms. Previous attempts of soft distance transforms are limited by numerical instability or require binary inputs, limiting their applicability. In this paper, we introduce novel regional Hausdorff Distance loss functions to optimize the HD without relying on any auxiliary losses. Specifically, we propose the maximum, modified, and average regional Hausdorff Distance losses. Central to our approach is a new method to compute a fully differentiable erosion-based distance function, which can be applied directly to probability maps. These functions accurately approximate the signed, unsigned, or positive distance maps while maintaining full differentiability. We validate our approach on multiple public medical image segmentation datasets, demonstrating that our HD losses achieve competitive performance, outperforming state-of-the-art methods.