Dynamics of a mean-shift-like algorithm and its applications on clustering

The Mean-Shift (MS) algorithm and its variants have wide applications in pattern recognition and computer vision tasks such as clustering, segmentation, and tracking. In this paper, we study the dynamics of the algorithm with Gaussian kernels, based on a Generalized MS (GMS) model that includes the standard MS as a special case. First, we prove that the GMS has solutions in the convex hull of the given data points. By the principle of contraction mapping, a sufficient condition, dependent on a parameter introduced into Gaussian kernels, is provided to guarantee the uniqueness of the solution. It is shown that the solution is also globally stable and exponentially convergent under the condition. When the condition does not hold, the GMS algorithm can possibly have multiple equilibriums, which can be used for clustering as each equilibrium has its own attractive basin. Based on this, the condition can be used to estimate an appropriate parameter which ensures the GMS algorithm to have its equilibriums suitable for clustering. Examples are given to illustrate the correctness of the condition. It is also shown that the use of the multiple-equilibrium property for clustering, on the data sets such as IRIS, leads to a lower error rate than the standard MS approach, and the K-Means and Fuzzy C-Means algorithms. (c) 2012 Elsevier B.V. All rights reserved.