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As applied in the field of [[computer vision]], '''[[graph cut optimization]]''' can be employed to [[Polynomial time|efficiently]] solve a wide variety of low-level computer vision problems (''early vision''<ref>Adelson, Edward H., and James R. Bergen (1991), "[http://persci.mit.edu/pub_pdfs/elements91.pdf The plenoptic function and the elements of early vision]", Computational models of visual processing 1.2 (1991).</ref>), such as image [[image smoothing]], the stereo [[correspondence problem]], [[image segmentation]], [[object co-segmentation]], and many other computer vision problems that can be formulated in terms of [[energy minimization]]. Many of these energy minimization problems can be approximated by solving a [[maximum flow problem]] in a [[Graph (discrete mathematics)|graph]]<ref>Boykov, Y., Veksler, O., and Zabih, R. (2001), "[http://www.cs.cornell.edu/rdz/Papers/BVZ-pami01-final.pdfFast approximate energy minimization via graph cuts]," ''IEEE Transactions on Pattern Analysis and Machine Intelligence,'' 23(11): 1222-1239.</ref> (and thus, by the [[max-flow min-cut theorem]], define a minimal [[cut (graph theory)|cut]] of the graph). Under most formulations of such problems in computer vision, the minimum energy solution corresponds to the [[Bayesian estimation of templates in computational anatomy|maximum a posteriori estimate]] of a solution. Although many computer vision algorithms involve cutting a graph (e.g., normalized cuts), the term "graph cuts" is applied specifically to those models which employ a max-flow/min-cut optimization (other graph cutting algorithms may be considered as [[Graph partition|graph partitioning]] algorithms).
Many of these energy minimization problems can be approximated by solving a [[maximum flow problem]] in a [[Graph (discrete mathematics)|graph]]<ref>Boykov, Y., Veksler, O., and Zabih, R. (2001), "[https://www.cs.cornell.edu/rdz/Papers/BVZ-pami01-final.pdf Fast approximate energy minimization via graph cuts]," ''IEEE Transactions on Pattern Analysis and Machine Intelligence,'' 23(11): 1222-1239.</ref> (and thus, by the [[max-flow min-cut theorem]], define a minimal [[cut (graph theory)|cut]] of the graph).
"Binary" problems (such as denoising a binary image) can be solved exactly using this approach; problems where pixels can be labeled with more than two different labels (such as stereo correspondence, or denoising of a [[grayscale]] image) cannot be solved exactly, but solutions produced are usually near the global optimum. ▼
Under most formulations of such problems in computer vision, the minimum energy solution corresponds to the [[Maximum a posteriori estimation|maximum a posteriori estimate]] of a solution.
Although many computer vision algorithms involve cutting a graph (e.g., normalized cuts), the term "graph cuts" is applied specifically to those models which employ a max-flow/min-cut optimization (other graph cutting algorithms may be considered as [[graph partition]]ing algorithms).
▲"Binary" problems (such as [[denoising ]] a [[binary image ]]) can be solved exactly using this approach; problems where pixels can be labeled with more than two different labels (such as stereo correspondence, or denoising of a [[grayscale]] image) cannot be solved exactly, but solutions produced are usually near the global optimum.
== History ==
The foundational theory of [[Cut (graph theory)|graph cuts]] used as [[graph cut optimization|an optimization method]] was first applied in [[computer vision]] in the seminal paper by Greig, Porteous and Seheult<ref name="D.M. Greig, B.T 1989">D.M. Greig, B.T. Porteous and A.H. Seheult (1989), ''[https://classes.cs.uchicago.edu/archive/2006/fall/35040-1/gps.pdf Exact maximum a posteriori estimation for binary images]'', Journal of the Royal Statistical Society, Series B, '''51''', 271–279.</ref> of [[Durham University]]. Allan Seheult and Bruce Porteous were members of Durham's lauded statistics group of the time, leadled by [[Julian Besag]] and [[Peter Green (statistician)|Peter Green]], with the optimisation expert [[Margaret Greig]] notable as the first ever female member of staff of the Durham Mathematical Sciences Department.
In the [[Bayesian statistics|Bayesian]] statistical context of [[smoothing]] noisy (or corrupted) images, they showed how the [[MAP estimate|maximum a posteriori estimate]] of a [[binary image]] can be obtained ''exactly'' by maximizing the [[Flow network|flow]] through an associated image network, involving the introduction of a ''source'' and ''sink''. The problem was therefore shown to be efficiently solvable. Prior to this result, ''approximate'' techniques such as [[simulated annealing]] (as proposed by the [[Donald Geman|Geman brothers]]),<ref>D. Geman and S. Geman (1984), ''[httphttps://www.dam.brown.edu/people/documents/stochasticrelaxation.pdf Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images]'', IEEE Trans. Pattern Anal. Mach. Intell., '''6''', 721–741.</ref>), or [[iterated conditional modes]] (a type of [[greedy algorithm]] as suggested by [[Julian Besag]])<ref>J.E. Besag (1986), ''On the statistical analysis of dirty pictures (with discussion)'', [[Journal of the Royal Statistical Society]] Series B, '''48''', 259–302</ref> were used to solve such image smoothing problems.
Although the general [[Graph coloring|<math>k</math>[[Graph coloring|-colour problem]] remainsis NP unsolvedhard for <math>k > 2,</math> the approach of Greig, Porteous and Seheult<ref name="D.M. Greig, B.T 1989" /> has turned out<ref>Y. Boykov, O. Veksler and R. Zabih (1998), "[httphttps://www.cs.cornell.edu/~rdz/Papers/BVZ-cvpr98.pdf Markov Random Fields with Efficient Approximations]", ''International Conference on Computer Vision and Pattern Recognition (CVPR)''.</ref><ref name="boykov2001fast">Y. Boykov, O. Veksler and R. Zabih (2001), "[httphttps://www.cs.cornell.edu/~rdz/Papers/BVZ-pami01-final.pdf Fast approximate energy minimisation via graph cuts]", ''IEEE Transactions on Pattern Analysis and Machine Intelligence'', '''29''', 1222–1239.</ref> to have wide applicability in general computer vision problems. For general problems, Greig, Porteous and Seheult's approachesapproach areis often applied iteratively to a sequencesequences of related binary problems, usually yielding near optimal solutions.
In 2011, C. Couprie ''et al''.<ref>Camille Couprie, Leo Grady, Laurent Najman and Hugues Talbot, "[http://leogrady.net/wp-content/uploads/2017/01/couprie2011power.pdf Power Watersheds: A Unifying Graph-Based Optimization Framework]”, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 33, No. 7, pp. 1384-1399, July 2011</ref> proposed a general image segmentation framework, called the "Power Watershed", that minimized a real-valued [[indicator function]] from [0,1] over a graph, constrained by user seeds (or unary terms) set to 0 or 1, in which the minimization of the indicator function over the graph is optimized with respect to an exponent <math>p</math>. When <math>p=1</math>, the Power Watershed is optimized by graph cuts, when <math>p=0</math> the Power Watershed is optimized by shortest paths, <math>p=2</math> is optimized by the [[Randomrandom walker algorithm]] and <math>p=\infty</math> is optimized by the [[Watershed (image processing)|watershed]] algorithm. In this way, the Power Watershed may be viewed as a generalization of graph cuts that provides a straightforward connection with other energy optimization segmentation/clustering algorithms.
==Binary segmentation of images==
{{see also|Graph cut optimization}}
* Minimization is done using a standard minimum cut algorithm.
* Due to the [[Maxmax-flow min-cut theorem]] we can solve energy minimization by maximizing the flow over the network. The Max Flow[[max-flow problem]] consists of a directed graph with edges labeled with capacities, and there are two distinct nodes: the source and the sink. Intuitively, it's is easy to see that the maximum flow is determined by the bottleneck.
=== Implementation (exact) ===
{{Wikibooks|Algorithm Implementation|Graphs/Maximum flow/Boykov & Kolmogorov}}
The Boykov-Kolmogorov algorithm<ref>Yuri Boykov, Vladimir Kolmogorov: [http://discovery.ucl.ac.uk/13383/1/13383.pdf An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision]. IEEE Trans. Pattern Anal. Mach. Intell. 26(9): 1124–1137 (2004)</ref> is an efficient way to compute the max-flow for computer vision -related graphgraphs.
=== Implementation (approximation) ===
The Sim Cut algorithm<ref>P.J. Yim: "[https://patentimages.storage.googleapis.com/2b/1e/e9/5834a9cc3312a0/US9214029.pdf Method and System for Image Segmentation]," United States Patent US8929636, January 6, 2016</ref> approximates the minimum graph cut. The algorithm implements a solution by simulation of an electrical network. This is the approach suggested by [[Cederbaum's maximum flow theorem]].<ref>{{Cite journal|last=Cederbaum|first=I.|date=1962-08-01|title=On optimal operation of communication nets|journal=Journal of the Franklin Institute|volume=274|issue=2|pages=130–141|doi=10.1016/0016-0032(62)90401-5|issn=0016-0032}}</ref><ref>I.T. Frisch, "On Electrical analogs for flow networks," Proceedings of IEEE, 57:2, pp. 209-210, 1969</ref> Acceleration of the algorithm is possible through [[parallel computing]].
== Software ==
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