Move-based algorithms for the optimization of an isotropic gradient MRF model

We consider an isotropic gradient model for the regularization terms in a multi-label MRF lattice. The isotropic gradient is modeled by considering 3- cliques in an 8-connected lattice. Of interest here are iterative move algorithms like alpha- expansion and alpha-beta swap, which try to minimize the energy function by solving a series of binary labeling problems. Such algorithms mainly differ in their update policy for the binary move at each iteration. Here, the aim is to study the submodularity of the binary move at each stage for a general update policy. We give the necessary and sufficient condition for the submodularity of a general update policy for the two major types of labels, namely ordered labels and unordered labels.