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A sufficient condition for the convergence of the mean shift algorithm with Gaussian kernel

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  • Aliyari Ghassabeh, Youness

Abstract

The mean shift (MS) algorithm is a non-parametric, iterative technique that has been used to find modes of an estimated probability density function (pdf). Although the MS algorithm has been widely used in many applications, such as clustering, image segmentation, and object tracking, a rigorous proof for its convergence is still missing. This paper tries to fill some of the gaps between theory and practice by presenting specific theoretical results about the convergence of the MS algorithm. To achieve this goal, first we show that all the stationary points of an estimated pdf using a certain class of kernel functions are inside the convex hull of the data set. Then the convergence of the sequence generated by the MS algorithm for an estimated pdf with isolated stationary points will be proved. Finally, we present a sufficient condition for the estimated pdf using the Gaussian kernel to have isolated stationary points.

Suggested Citation

  • Aliyari Ghassabeh, Youness, 2015. "A sufficient condition for the convergence of the mean shift algorithm with Gaussian kernel," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 1-10.
    • Handle: RePEc:eee:jmvana:v:135:y:2015:i:c:p:1-10
    DOI: 10.1016/j.jmva.2014.11.009
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    Cited by:

    1. José E. Chacón, 2019. "Mixture model modal clustering," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 13(2), pages 379-404, June.
    2. Federico Ferraccioli & Giovanna Menardi, 2023. "Modal clustering of matrix-variate data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 17(2), pages 323-345, June.
    3. Chen, Ting-Li & Fujisawa, Hironori & Huang, Su-Yun & Hwang, Chii-Ruey, 2016. "On the weak convergence and Central Limit Theorem of blurring and nonblurring processes with application to robust location estimation," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 165-184.

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