About the Incorporation of Topological Prescriptions in CNNs for Medical Image Semantic Segmentation

Incorporating prior knowledge into a segmentation task, whether it be under the form of geometrical constraints (area/volume penalisation, convexity enforcement, etc.) or of topological constraints (to preserve the contextual relations between objects, to monitor the number of connected components), proves to increase accuracy in medical image segmentation. In particular, it allows to compensate for the issue of weak boundary definition, of imbalanced classes, and to be more in line with anatomical consistency even though the data do not explicitly exhibit those features. This observation underpins the introduced contribution that aims, in a hybrid setting, to leverage the best of both worlds that variational methods and supervised deep learning approaches embody: (a) versatility and adaptability in the mathematical formulation of the problem to encode geometrical/topological constraints, (b) interpretability of the results for the former formalism, while (c) more efficient and effective processing models, (d) ability to become more proficient at learning intricate features and executing more computationally intensive tasks, for the latter one. To be more precise, a unified variational framework involving topological prescriptions in the training of convolutional neural networks through the design of a suitable penalty in the loss function is provided. These topological constraints are implicitly enforced by viewing the segmentation procedure as a registration task between the processed image and its associated ground truth under incompressibility conditions, thus making them homeomorphic. A very preliminary version (Lambert et al., in Calatroni, Donatelli, Morigi, Prato, Santacesaria (eds) Scale space and variational methods in computer vision, Springer, Berlin, 2023, pp. 363–375) of this work has been published in the proceedings of the Ninth International Conference on Scale Space and Variational Methods in Computer Vision, 2023. It contained neither all the theoretical results, nor the detailed related proofs, nor did it include the numerical analysis of the designed algorithm. Besides these more involved developments in the present version, a more complete, systematic and thorough analysis of the numerical experiments is also conducted, addressing several issues: (i) limited amount of labelled data in the training phase, (ii) low contrast or imbalanced classes exhibited by the data, and (iii) explainability of the results.