Graph cuts in computer vision

As applied in the field of computer vision, graph cuts can be employed to efficiently solve a wide variety of low-level computer vision problems ("early vision"), such as image smoothing, the stereo correspondence problem, and many other computer vision problems that can be formulated in terms of energy minimization. Such energy minimization problems can be reduced to instances of the maximum flow problem in a graph (and thus, by the max-flow min-cut theorem, define a minimal cut of the graph). Under most formulations of such problems in computer vision, the minimum energy solution corresponds to the maximum a posteriori estimate of a solution.

"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 theory of graph cuts was first applied in computer vision in the paper by Greig, Porteous and Seheult [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.] of Durham University. In the Bayesian statistical context of smoothing noisy (or corrupted) images, they showed how the maximum a posteriori estimate of a binary image can be obtained "exactly" by maximising the 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 Geman brothers [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.] ), or iterated conditional modes (a type of greedy algorithm as suggested by Julian Besag) [J.E. Besag (1986), "On the statistical analysis of dirty pictures (with discussion)", Journal of the Royal Statistical Society Series B, 48, 259–302] ) were used to solve such image smoothing problems.

Although the general $k$-colour problem remains unsolved for $k\; >\; 2,$ the approach of Greig, Porteous and Seheult [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.] has turned out [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)".] [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.] 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; see the article by Funka-Lea et al. [Gareth Funka-Lea, Yuri Boykov, Charles Florin, M. P. Jolly, Romain Moreau-Gobard, R. Ramaraj and D. Rinck (2006), "Automatic heart isolation for CT coronary visualization using graph cuts", IEEE International Symposium on Biomedical Imaging, 614–617.] for a recent application.

References