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, 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 of Durham University.

In the Bayesian statistical context of smoothing noisy (or corrupted) images, Greig, Porteous and Seheult 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), or iterated conditional modes (a type of greedy algorithm as suggested by Julian Besag) 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 has turned out to have wide applicability in general computer vision problems. See Boykov, Veksler and Zabih. 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 at al for a recent application.