The mean shift algorithm is a widely used method for finding local maxima in feature spaces. Mean shift algorithms have been shown in the literature to be equivalent to a gradient ascent optimization of a kernel density estimate. This paper describes a novel, globally optimal optimization method and compares the suboptimal mean shift solutions with the globally optimal solutions derived by the new algorithm. Experimental results on both simulated and real data show that the new algorithm yields solutions that are often significantly better than the suboptimal solutions identified by the mean shift algorithm, and that it scales better to large sample sizes and is more robust to noise levels.