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

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Clough J. R., Byrne N., Öksüz İ., Zimmer V. A., Schnabel J. A., King A. P.

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, vol.44, no.12, pp.8766-8778, 2022 (SCI-Expanded) identifier identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 44 Issue: 12
  • Publication Date: 2022
  • Doi Number: 10.1109/tpami.2020.3013679
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, PASCAL, ABI/INFORM, Aerospace Database, Applied Science & Technology Source, Business Source Elite, Business Source Premier, Communication Abstracts, Compendex, Computer & Applied Sciences, EMBASE, INSPEC, MEDLINE, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.8766-8778
  • Keywords: Image segmentation, Topology, Shape, Training, Loss measurement, Neural networks, Network topology, Segmentation, persistent homology, topology, medical imaging, convolutional neural networks
  • Istanbul Technical University Affiliated: Yes


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.