Nonlocal Graph-PDEs and Riemannian Gradient Flows for Image Labeling

On the example of Optical Coherence Tomography (OCT), which is the mostly used non-invasive acquisition method of large volumetric scans of human retinal tissues, it will be shown how incorporation of constraints on the geometry of statistical manifold results in a novel purely data driven geometric approach for order-constrained segmentation of volumetric data in any metric space. By introducing a new formulation of ordered distributions, the first major contribution of this work comprises a fully automated segmentation algorithm that comes up with a high segmentation accuracy and a high level of built-in-parallelism that as opposed to many established retinal layer segmentation methods, only takes local information as input without incorporation of additional global shape priors. As a second main contribution we introduce a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs that is derived as nonlocal reparametrization of the assignment flow approach. Due to this parameterization, solving the G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with respect to a nonconvex potential. In addition, by introducing a entropy-regularized difference-of-convex-functions (DC) decomposition it will be shown how the basic geometric Euler scheme for integrating the assignment flow is equivalent to solving the G-PDE by an established DC programming scheme.