Capacity Scaling for Graph Cuts in Vision

Olivier Juan, Yuri Boykov

Abstract: Capacity scaling is a hierarchical approach to graph representation that can improve theoretical complexity and practical efficiency of max-flow/min-cut algorithms. Introduced by Edmonds, Karp, and Dinic in 1972, capacity scaling is well known in the combinatorial optimization community. Surprisingly, this major performance improving technique is overlooked in computer vision where graph cut methods typically solve energy minimization problems on huge N-D grids and algorithms' efficiency is a widely studied issue. Unlike some earlier hierarchical methods addressing efficiency of graph cuts in imaging, e.g. (H. Lombaert, 2005), capacity scaling preserves global optimality of the solution. This is the main motivation for our work studying capacity scaling in the context of vision. We show that capacity scaling significantly reduces non-polynomial theoretical time complexity of the max-flow algorithm in (Y. Boykov and V. Kolmorogorov, 2004) to weakly polynomial O(m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> log(U)) where U is the largest edge weight. While (Y. Boykov and V. Kolmorogorov, 2004) is the fastest method for many applications in vision, capacity scaling gives several folds speed-ups for problems with large number of local minima. The effect is particularly strong in 3D applications with denser neighborhoods.