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Graph cuts method such as a-expansion [4] and fusion moves [22] have been successful at solving many optimization problems in computer vision. Higher-order Markov Random Fields (MRF's), which are important for numerous applications, have proven to be very difficult, especially for multilabel MRF's (i.e. more than 2 labels). In this paper we propose a new primal-dual energy minimization method for arbitrary higher-order multilabel MRF's. Primal-dual methods provide guaranteed approximation bounds, and can exploit information in the dual variables to improve their efficiency. Our algorithm generalizes the PD3 [19] technique for first-order MRFs, and relies on a variant of max-flow that can exactly optimize certain higher-order binary MRF's [14]. We provide approximation bounds similar to PD3 [19], and the method is fast in practice. It can optimize non-submodular MRF's, and additionally can in- corporate problem-specific knowledge in the form of fusion proposals. We compare experimentally against the existing approaches that can efficiently handle these difficult energy functions [6, 10, 11]. For higher-order denoising and stereo MRF's, we produce lower energy while running significantly faster.

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