When Can \(l_p\)-norm Objective Functions Be Minimized via Graph Cuts?
Techniques based on minimal graph cuts have become a standard tool for solving combinatorial optimization problems arising in image processing and computer vision applications. These techniques can be used to minimize objective functions written as the sum of a set of unary and pairwise terms, provided that the objective function is submodular. This can be interpreted as minimizing the \(l_1\)-norm of the vector containing all pairwise and unary terms. By raising each term to a power p, the same technique can also be used to minimize the \(l_p\)-norm of the vector. Unfortunately, the submodularity of an \(l_1\)-norm objective function does not guarantee the submodularity of the corresponding \(l_p\)-norm objective function. The contribution of this paper is to provide useful conditions under which an \(l_p\)-norm objective function is submodular for all \(p\ge 1\), thereby identifying a large class of \(l_p\)-norm objective functions that can be minimized via minimal graph cuts.
KeywordsMinimal graph cuts \(l_p\) norm Submodularity