Graph cuts in computer vision: Difference between revisions

Revision as of 16:35, 5 July 2017

Revision as of 03:34, 15 November 2017

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== History ==

The theory of [[Cut (graph theory)|graph cuts]] 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), ''Exact maximum a posteriori estimation for binary images'', Journal of the Royal Statistical Society, Series B, '''51''', 271–279.</ref> of [[Durham University]]. 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), ''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 <math>k</math>[[Graph coloring|-colour problem]] remains unsolved 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), "[http://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), "[http://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. Greig, Porteous and Seheult approaches are often applied iteratively to a sequence of binary problems, usually yielding near optimal solutions.

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=== Implementation (approximation) ===

The Sim Cut algorithm <ref>P.J. Yim: "Method and System for Image Segmentation," United States Patent US8929636, January 6, 2016 </ref> approximates the graph cut by the solution of the set of simultaneous non-linear equations based on the analog electrical model of the flow network.<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 ==