When can l_p-norm objective functions be minimized via graph cuts?
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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≥ 1, thereby identifying a large class of l_p-norm objective functions that can be minimized via minimal graph cuts.
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