Gaussian Mean-Shift Is an EM Algorithm

Abstract

The mean-shift algorithm, based on ideas proposed by Fukunaga and Hostetler [16], is a hill-climbing algorithm on the density defined by a finite mixture or a kernel density estimate. Mean-shift can be used as a nonparametric clustering method and has attracted recent attention in computer vision applications such as image segmentation or tracking. We show that, when the kernel is Gaussian, mean-shift is an expectation-maximization (EM) algorithm and, when the kernel is non-Gaussian, mean-shift is a generalized EM algorithm. This implies that mean-shift converges from almost any starting point and that, in general, its convergence is of linear order. For Gaussian mean-shift, we show: 1) the rate of linear convergence approaches 0 (superlinear convergence) for very narrow or very wide kernels, but is often close to 1 (thus, extremely slow) for intermediate widths and exactly 1 (sublinear convergence) for widths at which modes merge, 2) the iterates approach the mode along the local principal component of the data points from the inside of the convex hull of the data points, and 3) the convergence domains are nonconvex and can be disconnected and show fractal behavior. We suggest ways of accelerating mean-shift based on the EM interpretation.

Mean shift: An information theoretic perspective

“…This results in a beautiful pattern in the image space where whole objects which are similar are merged together in an intuitive manner. This phenomenon is again observed as we move from 6 segments to 4 where GMS puts all the gray objects in one cluster thus putting together three full objects of similar intensity in one group.…”

“…Thus the dataset consists of 8030 points in the feature space. In order to use an isotropic kernel we prescale the intensity value such that they fall in the same range as the spatial features as done in [4]. All the values reported are in pixel units.…”

“…Note that for this image the best number of segments is 5 to 6 segments as seen in the image itself. Many researcher have tried to do this using various methods [4], [16]. Fig.…”

“…Carreira-Perpiñán [4] also showed that GMS is an Expectation-Maximization (EM) algorithm and thus has a linear convergence rate.…”

Half Quadratic Analysis for Mean Shift: with Extension to A Sequential Data Mode-Seeking Method

“…Theoretical understanding and extension of mean shift procedure has received much attention recently [8,18,3] …”