A branch and bound algorithm for finding the modes in kernel density estimates

Kernel density estimators are established tools in nonparametric statistics. Due to their flexibility and ease of use, these methods are popular in computer vision and pattern recognition for tasks such as object tracking in video or image segmentation. The most frequently used algorithm for finding the modes in such densities (the mean shift) is a gradient ascent rule, which converges to local maxima. We propose a novel, globally optimal branch and bound algorithm for finding the modes in kernel densities. We show in experiments on datasets up to dimension five that the branch and bound method is faster than local optimization and observe linear runtime scaling of our method with sample size. Quantitative experiments on simulated data show that our method gives statistically significantly more accurate solutions than the mean shift. The mode localization accuracy is about five times more precise than that of the mean shift for all tested parameters. Applications to color image segmentation on an established benchmark test set also show measurably improved results when using global optimization.