Memory Efficient Max Flow for Multi-Label Submodular MRFs
Multi-label submodular Markov Random Fields (MRFs) have been shown to be solvable using max-flow based on an encoding of the labels proposed by Ishikawa, in which each variable <inline-formula><tex-math notation="LaTeX">$X_i$</tex-math><alternatives><inline-graphic xlink:href="ajanthan-ieq1-2819675.gif"/></alternatives></inline-formula> is represented by <inline-formula><tex-math notation="LaTeX">$\ell$</tex-math><alternatives><inline-graphic xlink:href="ajanthan-ieq2-2819675.gif"/></alternatives></inline-formula> nodes (where <inline-formula><tex-math notation="LaTeX">$\ell$</tex-math><alternatives><inline-graphic xlink:href="ajanthan-ieq3-2819675.gif"/></alternatives></inline-formula> is the number of labels) arranged in a column. However, this method in general requires <inline-formula><tex-math notation="LaTeX">$2\;\ell ^2$</tex-math><alternatives><inline-graphic xlink:href="ajanthan-ieq4-2819675.gif"/></alternatives></inline-formula> edges for each pair of neighbouring variables. This makes it inapplicable to realistic problems with many variables and labels, due to excessive memory requirement. In this paper, we introduce a variant of the max-flow algorithm that requires much less storage. Consequently, our algorithm makes it possible to optimally solve multi-label submodular problems involving large numbers of variables and labels on a standard computer.