The Swap and Expansion moves revisited and fused

Many solutions to computer vision and image processing problems involve the minimization of multi-label energy functions with up to K variables in each term. In the minimization process, Swap and Expansion are two commonly used moves. This paper re-derives the optimal swap and expansion moves for K = 2 in a short manner by using the original pseudo-Boolean quadratic function minimization method as a “black box”. It is then revealed that the found minima w.r.t. expansion moves are in fact also minima w.r.t. swap moves. This process is repeated for K = 3. The minima-related result is extended to all functions under the condition that they are reduced into submodular ones, which makes it applicable to all expansion algorithms. These may explain the prevalent impression that expansion algorithms are more effective than swap algorithms. To make the search space larger, Exwap — a generalization of the expansion and the swap moves — is introduced. Efficient algorithms for minimizing w.r.t. it for K = 2 (including a ‘truncation’ procedure), K = 3, and the Pn Potts model are provided. The move is capable of reaching lower energies than those reached by the expansion algorithm, as demonstrated for several benchmark problems.