Globally optimal regions and boundaries as minimum ratio weight cycle

We describe a new form of energy functional for the modelling and identification of regions in images. The energy is defined on the space of boundaries in the image domain and can incorporate very general combinations of modelling information both from the boundary (intensity gradients, etc.) and from the interior of the region (texture, homogeneity, etc.). We describe two polynomial-time digraph algorithms for finding the global minima of this energy. One of the algorithms is completely general, minimizing the functional for any choice of modelling information. It runs in a few seconds on a 256x256 image. The other algorithm applies to a subclass of functionals, but has the advantage of being extremely parallelizable. Neither algorithm requires initialization.