A topological loss function for deep-learning based image segmentation using persistent homology

James R Clough, Nicholas Byrne, Ilkay Oksuz, Veronika A Zimmer, Julia A Schnabel, Andrew P. King

Abstract: We introduce a method for training neural networks to perform image or volume segmentation in which prior knowledgeabout the topology of the segmented object can be explicitly provided and then incorporated into the training process. By using thedifferentiable properties of persistent homology, a concept used in topological data analysis, we can specify the desired topology ofsegmented objects in terms of their Betti numbers and then drive the proposed segmentations to contain the specified topologicalfeatures. Importantly this process does not require any ground-truth labels, just prior knowledge of the topology of the structure beingsegmented. We demonstrate our approach in four experiments: one on MNIST image denoising and digit recognition, one on leftventricular myocardium segmentation from magnetic resonance imaging data from the UK Biobank, one on the ACDC public challengedataset and one on placenta segmentation from 3-D ultrasound. We find that embedding explicit prior knowledge in neural networksegmentation tasks is most beneficial when the segmentation task is especially challenging and that it can be used in either asemi-supervised or post-processing context to extract a useful training gradient from images without pixelwise labels.