A Fast Continuous Max-Flow Approach to Non-convex Multi-labeling Problems

This paper studies continuous image labeling problems with an arbitrary data term and a total variation regularizer, where the labels are constrained to a finite set of real numbers. Inspired by Ishikawa’s multi-layered graph construction for the same labeling problem over a discrete image domain, we propose a novel continuous max-flow model and build up its duality to a convex relaxed formulation of image labeling under a new variational perspective. Via such continuous max-flow formulations, we show that exact and global optimizers can be obtained to the original non-convex labeling problem. We also extend the studies to problems with continuous-valued labels and introduce a new theory to this problem. Finally, we show the proposed continuous max-flow models directly lead to new fast flow-maximization algorithmic schemes which outperform previous approaches in terms of efficiency. Such continuous max-flow based algorithms can be validated by convex optimization theories and accelerated by modern parallel computational hardware.