Convex multi-region probabilistic segmentation with shape prior in the isometric log-ratio transformation space

Image segmentation is often performed via the minimization of an energy function over a domain of possible segmentations. The effectiveness and applicability of such methods depends greatly on the properties of the energy function and its domain, and on what information can be encoded by it. Here we propose an energy function that achieves several important goals. Specifically, our energy function is convex and incorporates shape prior information while simultaneously generating a probabilistic segmentation for multiple regions. Our energy function represents multi-region probabilistic segmentations as elements of a vector space using the isometric log-ratio (ILR) transformation. To our knowledge, these four goals (convex, with shape priors, multi-region, and probabilistic) do not exist together in any other method, and this is the first time ILR is used in an image segmentation method. We provide examples demonstrating the usefulness of these features.