For a number of biomedical image segmentation tasks, including topological knowledge on the inter-class relationships, such as containment/nesting, can greatly improve the results. However, most `out-of-the-box' CNN models are still blind to this type of prior information. In this paper, we address the case of nested classes, and propose a novel approach to encode this information in the network, stepping away from the paradigmatic soft-max activation. Instead, we introduce a multi-level activation layer, as a generalization of logistic regression, and propose three compatible losses. We benchmark all of them on nuclei segmentation in bright-field microscopy cell images from the 2018 Data Science Bowl challenge, offering an exemplary segmentation task with cells and nested intra-cellular structures. Our scheme significantly outperforms standard multi-class classification with soft-max activation and a previously proposed method stemming from it, improving the Dice score significantly (p-values<0.002). On top of being conceptually simple, our approach is easy to implement and can be integrated in any CNN architecture. It can be generalized to a higher number of classes, with or without further relations of containment.
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