Asymptotic stability of equilibrium points of mean shift algorithm

The mean shift (MS) algorithm is a popular non-parametric technique that has been widely used in statistical pattern recognition and machine learning. The algorithm iteratively tries to find modes of an estimated probability density function. These modes play an important role in many applications, such as clustering, image segmentation, feature extraction, and object tracking. The modes are fixed points of a discrete, nonlinear dynamical system. Although the algorithm has been successfully used in many applications, a theoretical study of its convergence is still missing in the literature. In this paper, we first consider the iteration index as a continuous variable and, by introducing a Lyapunov function, show that the equilibrium points are asymptotically stable. We also show that the proposed function can be considered as a Lyapunov function for the discrete case with the isolated stationary points. The availability of a Lyapunov function for continuous and discrete cases shows that if the MS iterations start in a neighborhood of an equilibrium point, the generated sequence remains close to that equilibrium point and finally converges to it.