A Topological Loss Function for Deep-Learning Based Image Segmentation Using Persistent Homology.

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

doi:10.1109/tpami.2020.3013679

Abstract

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

Highlights

S EGMENTATION is the process of assigning a meaningful label to each pixel in an image and is one of the fundamental tasks in image analysis
Our approach is compatible with any choice of network architecture, and the focus of our work is to introduce the topological loss function for segmentation, and not to assess the various convolutional neural networks (CNN) architectures that have been proposed in the literature
We have shown that by using a U-net-like network architecture and supplementing traditional pixel-wise loss functions with our topological loss function, the accuracy of the resulting segmentations can be improved in terms of the pixel-wise accuracy and that there are significant improvements in terms of their topological accuracy

Summary

Introduction

S EGMENTATION is the process of assigning a meaningful label to each pixel in an image and is one of the fundamental tasks in image analysis. For each case in the training set, the network is trained to minimise some loss function, typically a pixelwise measure of dissimilarity (such as the cross-entropy) between the predicted and the ground-truth segmentations. Errors in some regions of the image may be more significant than others, in terms of the segmented object’s interpretation, or for downstream calculation or modelling. In some cases this can be captured by alternative loss functions, such as the weighted cross-entropy or the generalised Dice loss [3] which can weight the contribution from rarer classes more strongly. Loss functions that only measure the degree of overlap between the predicted and the ground-truth segmentations are unable to capture the extent to which the large-scale structure of the predicted

Methods

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Discussion

Conclusion