Fast approximate energy minimization via graph cuts

Y. Boykov, O. Veksler, R. Zabih
1999 Proceedings of the Seventh IEEE International Conference on Computer Vision  
AbstractÐMany tasks in computer vision involve assigning a label (such as disparity) to every pixel. A common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist, e.g., at object boundaries. These tasks are naturally stated in terms of energy minimization. In this paper, we consider a wide class of energies with various smoothness constraints. Global minimization of these energy functions is NP-hard even in the simplest
more » ... ntinuity-preserving case. Therefore, our focus is on efficient approximation algorithms. We present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves, namely expansion moves and swap moves. These moves can simultaneously change the labels of arbitrarily large sets of pixels. In contrast, many standard algorithms (including simulated annealing) use small moves where only one pixel changes its label at a time. Our expansion algorithm finds a labeling within a known factor of the global minimum, while our swap algorithm handles more general energy functions. Both of these algorithms allow important cases of discontinuity preserving energies. We experimentally demonstrate the effectiveness of our approach for image restoration, stereo and motion. On real data with ground truth, we achieve 98 percent accuracy. 2. In fact, we only assume V ; V ; in order to simplify the presentation. We can easily generalize all results in this paper to allow V ; T V ; . This generalization requires the use of directed graphs. 3. In special cases where the global minimum can be rapidly computed, it is possible to separate these issues. For example, [20] points out that the global minimum of a special case of Ising energy function is not necessarily the desired solution for image restoration. Blake [9], and Greig et al. [20] analyze the performance of simulated annealing in cases with a known global minimum. 4 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 23, NO. 11, NOVEMBER 2001 Fig. 2. Examples of standard and large moves from a given initial labeling (a). The number of labels is jvj 3. A standard move, (a) 3 (b), changes the label of a single pixel (in the circled area). Strong moves, --swap (a) 3 (c) and -expansion (a) 3 (d), allow large number of pixels to change their labels simultaneously.
doi:10.1109/iccv.1999.791245 dblp:conf/iccv/BoykovVZ99 fatcat:v6i6o2ql4fde7jaymrbrir4ila