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Note that some of the horizontal (time) axes use a log scale to better visualize the results. ICM is omitted in the right plots, due to its poor performance. edu/MRF. A Comparative Study of Energy Minimization Methods for MRF’s 600% 25 180% 170% 500% 160% ICM BP 400% Swap 300% 150% 140% Expansion TRW-S 130% 200% 120% 100% 110% 0% 100% 1 10 100 1000 10000 4000% 1 10 100 1000 1000 300% 3500% 3000% ICM 250% 2500% BP Swap Expansion TRW-S 2000% 200% 1500% 1000% 150% 500% 0% 100% 1 10 100 1000 10000 1 10 100 1000 10000 Fig.

The same oscillation is seen in figure 2, though this time without as good performance. On the binary image segmentation problems, shown in figure 3, graph cuts are guaranteed to 26 R. Szeliski et al. 5 4 0 1 2 3 4 5 6 Fig. 3. Results on binary image segmentation benchmarks. Graph cuts are guaranteed to compute the global minimum, as is TRW-S. 04% error), but never actually attains it. Fig. 4. Results on “Panorama” benchmark. LBP output is shown at left, TRW-S in the middle, and expansion moves at right.

For a user of energy minimization methods, this lower bound can serve as a confidence measure, providing assurance that the solution obtained has near-optimal energy. Another area that needs investigation is the use of graph cut algorithms for wider classes of energy functions than the limited ones they were originally designed for. The benchmarks that were most challenging for the expansion move algorithm (such as “Venus”) use a V that is not a metric. Another important issue centers around the use of energy to compare energy minimization algorithms.

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