After [10, 15, 12, 2, 4] minimum cut/maximum ow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in low-level vision. The combinatorial optimization literature provides many min-cut/max-ow algorithms with dierent polynomial time complexity. Their practical eciency, however, has to date been studied mainly outside the scope of computer vision.

Combinatorial min-cut algorithms on graphs emerged as an increasingly useful tool for problems in vision. Typically, the use of graphcuts is motivated by one of the following two reasons. Firstly, graph-cuts allow geometric interpretation; under certain conditions a cut on a graph can be seen as a hypersurface in N-D space embedding the corresponding graph. Thus, many applications in vision and graphics use min-cut algorithms as a tool for computing optimal hypersurfaces. Secondly, graphcuts also work as a powerful energy minimization tool for a fairly wide class of binary and non-binary energies that frequently occur in early vision. In some cases graph cuts produce globally optimal solutions. More generally, there are iterative graph-cut based techniques that produce provably good approximations which (were empirically shown to) correspond to high-quality solutions in practice. Thus, another large group of applications use graph-cuts as an optimization technique for low-level vision problems based on global energy formulations. This chapter is intended as a tutorial illustrating these two aspects of graph-cuts in the context of problems in computer vision and graphics. We explain general theoretical properties that motivate the use of graph cuts, as well as, show their limitations.