text: We present a novel method to explicitly incorporate topological prior knowledge into deep learning based segmentation, which is, to our knowledge, the first work to do so. Our method uses the concept of persistent homology, a tool from topological data analysis, to capture high-level topological characteristics of segmentation results in a way which is differentiable with respect to the pixelwise probability of being assigned to a given class. The topological prior knowledge consists of the sequence of desired Betti numbers of the segmentation. As a proof-of-concept we demonstrate our approach by applying it to the problem of left-ventricle segmentation of cardiac MR images of 500 subjects from the UK Biobank dataset, where we show that it improves segmentation performance in terms of topological correctness without sacrificing pixelwise accuracy.
  @cite_0: Image segmentation based on convolutional neural networks is proving to be a powerful and efficient solution for medical applications. However, the lack of annotated data, presence of artifacts and variability in appearance can still result in inconsistencies during the inference. We choose to take advantage of the invariant nature of anatomical structures, by enforcing a semantic constraint to improve the robustness of the segmentation. The proposed solution is applied on a brain structures segmentation task, where the output of the network is constrained to satisfy a known adjacency graph of the brain regions. This criteria is introduced during the training through an original penalization loss named NonAdjLoss. With the help of a new metric, we show that the proposed approach significantly reduces abnormalities produced during the segmentation. Additionally, we demonstrate that our framework can be used in a semi-supervised way, opening a path to better generalization to unseen data.
  @cite_15: We propose a generic and efficient learning framework that is applicable to segment images in which individual objects are mainly discernible by boundary cues. Our approach starts by first hierarchically clustering the image and then explaining the image in terms of a cost-minimal subset of non-overlapping segments. The cost of a segmentation is defined as a weighted sum of features of the selected candidates. This formulation allows us to take into account an extensible set of arbitrary features. The maximally discriminative linear combination of features is learned from training data using a margin-rescaled structured SVM. At the core of our formulation is a novel and simple topology-based structured loss which is a combination of counts and geodesic distance of topological errors (splits, merges, false positives and false negatives) relative to the training set. We demonstrate the generality and accuracy of our approach on three challenging 2D cell segmentation problems, where we improve accuracy compared to the current state of the art.
summary: Other approaches have involved encouraging the correct adjacencies of various object classes, whether they were learned from the data as in @cite_15 or provided as a prior as in @cite_0 . Such methods allow the introduction of this simple topological feature into a loss function when performing image segmentation but cannot be easily generalised to any other kinds of higher-order feature such as the presence of holes, handles or voids.