A spatially continuous max-flow and min-cut framework for binary labeling problems

We propose and investigate novel max-flow models in the spatially continuous setting, with or without i priori defined supervised constraints, under a comparative study of graph based max-flow/min-cut. We show that the continuous max-flow models correspond to their respective continuous min-cut models as primal and dual problems. In this respect, basic conceptions and terminologies from discrete max-flow/min-cut are revisited under a new variational perspective. We prove that the associated nonconvex partitioning problems, unsupervised or supervised, can be solved globally and exactly via the proposed convex continuous max-flow and min-cut models. Moreover, we derive novel fast max-flow based algorithms whose convergence can be guaranteed by standard optimization theories. Experiments on image segmentation, both unsupervised and supervised, show that our continuous max-flow based algorithms outperform previous approaches in terms of efficiency and accuracy.