@article {Szeliski2008,
	title = {A comparative study of energy minimization methods for Markov random fields with smoothness-based priors},
	journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
	volume = {30},
	number = {6},
	year = {2008},
	pages = {1068{\textendash}1080},
	publisher = {Springer-Verlag},
	abstract = {Among the most exciting advances in early vision has been the development of efficient energy minimization algorithms for pixel-labeling tasks such as depth or texture computation. It has been known for decades that such problems can be elegantly expressed as Markov random fields, yet the resulting energy minimization problems have been widely viewed as intractable. Recently, algorithms such as graph cuts and loopy belief propagation (LBP) have proven to be very powerful: for example, such methods form the basis for almost all the top-performing stereo methods. However, the tradeoffs among different energy minimization algorithms are still not well understood. In this paper we describe a set of energy minimization benchmarks and use them to compare the solution quality and running time of several common energy minimization algorithms. We investigate three promising recent methods graph cuts, LBP, and tree-reweighted message passing in addition to the well-known older iterated conditional modes (ICM) algorithm. Our benchmark problems are drawn from published energy functions used for stereo, image stitching, interactive segmentation, and denoising. We also provide a general-purpose software interface that allows vision researchers to easily switch between optimization methods. Benchmarks, code, images, and results are available at http://vision.middlebury.edu/ MRF/. {\textcopyright} 2008 IEEE.},
	keywords = {Belief propagation, Global optimization, Graph Cuts, Markov random fields, Performance evaluation},
	issn = {01628828},
	doi = {10.1109/TPAMI.2007.70844},
	url = {http://vision.middlebury.edu/MRF.},
	author = {Szeliski, Richard and Zabih, Ramin and Scharstein, Daniel and Veksler, Olga and Kolmogorov, Vladimir and Agarwala, Aseem and Tappen, Marshall and Carsten Rother}
}
@article {Szeliski2008a,
	title = {A comparative study of energy minimization methods for Markov random fields with smoothness-based priors},
	journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
	volume = {30},
	number = {6},
	year = {2008},
	month = {jun},
	pages = {1068{\textendash}1080},
	abstract = {Among the most exciting advances in early vision has been the development of efficient energy minimization algorithms for pixel-labeling tasks such as depth or texture computation. It has been known for decades that such problems can be elegantly expressed as Markov random fields, yet the resulting energy minimization problems have been widely viewed as intractable. Recently, algorithms such as graph cuts and loopy belief propagation (LBP) have proven to be very powerful: for example, such methods form the basis for almost all the top-performing stereo methods. However, the tradeoffs among different energy minimization algorithms are still not well understood. In this paper we describe a set of energy minimization benchmarks and use them to compare the solution quality and running time of several common energy minimization algorithms. We investigate three promising recent methods graph cuts, LBP, and tree-reweighted message passing in addition to the well-known older iterated conditional modes (ICM) algorithm. Our benchmark problems are drawn from published energy functions used for stereo, image stitching, interactive segmentation, and denoising. We also provide a general-purpose software interface that allows vision researchers to easily switch between optimization methods. Benchmarks, code, images, and results are available at http://vision.middlebury.edu/ MRF/. {\textcopyright} 2008 IEEE.},
	keywords = {Belief propagation, Global optimization, Graph Cuts, Markov random fields, Performance evaluation},
	issn = {01628828},
	doi = {10.1109/TPAMI.2007.70844},
	author = {Szeliski, Richard and Zabih, Ramin and Scharstein, Daniel and Veksler, Olga and Kolmogorov, Vladimir and Agarwala, Aseem and Tappen, Marshall and Carsten Rother}
}
@booklet {Kolmogorov2007,
	title = {Minimizing nonsubmodular functions with graph cuts - A review},
	howpublished = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
	volume = {29},
	number = {7},
	year = {2007},
	pages = {1274{\textendash}1279},
	abstract = {Optimization techniques based on graph cuts have become a standard tool for many vision applications. These techniques allow to minimize efficiently certain energy functions corresponding to pairwise Markov Random Fields (MRFs). Currently, there is an accepted view within the computer vision community that graph cuts can only be used for optimizing a limited class of MRF energies (e.g., submodular functions). In this survey, we review some results that show that graph cuts can be applied to a much larger class of energy functions (in particular, nonsubmodular functions). While these results are well-known in the optimization community, to our knowledge they were not used in the context of computer vision and MRF optimization. We demonstrate the relevance of these results to vision on the problem of binary texture restoration. {\textcopyright} 2007 IEEE.},
	keywords = {energy minimization, Markov random fields, Min cut/max flow, Quadratic pseudo-Boolean optimization, Texture restoration},
	issn = {01628828},
	doi = {10.1109/TPAMI.2007.1031},
	author = {Kolmogorov, Vladimir and Carsten Rother}
}
