Object Learning and Convex Cardinal Shape Composition

This work mainly focuses on a novel segmentation and partitioning scheme, based on learning the principal elements of the optimal partitioner in the image. The problem of interest is characterizing the objects present in an image as a composition of matching elements from a dictionary of prototype shapes. The composition model allows set union and difference among the selected elements, while regularizing the problem by restricting their count to a fixed level. This is a combinatorial problem addressing which is not in general computationally tractable. Convex cardinal shape composition (CSC) is a recent relaxation scheme presented as a proxy to the original problem. From a theoretical standpoint, this paper improves the results presented in the original work, by deriving the general sufficient conditions under which CSC identifies a target composition. We also provide qualitative results on who well the CSC outcome approximates the combinatorial solution. From a computational standpoint, two convex solvers, one supporting distributed processing for large-scale problems, and one cast as a linear program are presented. Applications such as multi-resolution segmentation and recovery of the principal shape components are presented as the experiments supporting the proposed ideas.