The Swap, Expansion and Exwap Moves ñ A Simple Derivation and Implementation

In the context of multi-label energy function minimization, the swap and the expansion are two types of moves that were introduced by Boykov et al. (3). They pro- posed efcient algorithms for nding local minima with respect to each of these two moves when each energy term depends on two variables at most. The minimization was carried out by a sequence of optimal moves that were calculated by seeking minimum cuts of specially constructed weighted graphs. In this paper these optimal swap and ex- pansion moves are obtained in a short and simple manner by incorporating the original algorithm by Greig et al. (5) as a ìblack box.î Our alternative derivation has three ad- vantages over the original one: 1. Given Greig et al.'s original solution as a black box, it is shorter, purely algebraic (that is, no graphs are involved) and, we believe, simpler to understand and implement. 2. It is derived under more general conditions. 3. It contains a proof that the found local minima with respect to expansion moves are actually also local minima with respect to swap moves ñ a point that seems to have been overlooked in previous work. All the results are extended for energy terms that depend on three variables by using Kolmogorov et al.'s binary minimization algorithm (9) as a black box. In addition, the exwap move type, a generalization of the expansion and the swap move types, is introduced and an efcient algorithm for minimizing with respect to it is derived.