Extending α-expansion to a larger set of regularization functions

Many problems of image processing lead to the minimization of an energy, which is a function of one or several given images, with respect to a binary or multi-label image. When this energy is made of unary data terms and of pairwise regularization terms, and when the pairwise regularization term is a metric, the multi-label energy can be minimized quite rapidly, using the so-called α-expansion algorithm. α-expansion consists in decomposing the multi-label optimization into a series of binary sub-problems called move. Depending on the chosen decomposition, a different condition on the regularization term applies. The metric condition for α-expansion move is rather restrictive. In many cases, the statistical model of the problem leads to an energy which is not a metric. Based on the enlightening article [1], we derive another condition for β-jump move. Finally, we propose an alternated scheme which can be used even if the energy fulfills neither the α-expansion nor β-jump condition. The proposed scheme applies to a much larger class of regularization functions, compared to α-expansion. This opens many possibilities of improvements on diverse image processing problems. We illustrate the advantages of the proposed optimization scheme on the image noise reduction problem.