Title: Localized Persistent Homologies for more Effective Deep Learning. Abstract: Persistent Homologies have been successfully used to increase the performance of deep networks trained to detect curvilinear structures and to improve the topological quality of the results. However, existing methods are very global and ignore the location of topological features. In this paper, we introduce an approach that relies on a new filtration function to account for location during network training. We demonstrate experimentally on 2D images of roads and 3D image stacks of neural processes that networks trained in this manner are better at recovering the topology of the curvilinear structures they extract.