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Optimization with graph cuts became very popular in recent years. Progress in problems such as stereo correspondence, image segmentation, etc., can be attributed, in part, to the development of efficient graph cut based optimization. Recent evaluation of optimization techniques shows that the popular expansion and swap graph cut algorithms perform extremely well for energies where the underlying MRF has the Potts prior, which corresponds to the assumption that the true labeling is piecewise constant. For more general priors, however, such as corresponding to piece-wise smoothness assumption, both swap and expansion algorithms do not perform as well. We develop several optimization algorithms for truncated convex priors, which imply piecewise smoothness assumption. Both expansion and swap algorithms are based on moves that give each pixel a choice of only two labels. Our insight is that to obtain …