On the Convergence of the Mean Shift Algorithm in the One-Dimensional Space

TL;DR

This paper provides a rigorous proof of the convergence of the mean shift algorithm in one-dimensional space, establishing its monotonicity and convergence properties.

Contribution

It offers the first rigorous proof of convergence for the mean shift algorithm in one-dimensional space.

Findings

The mean shift sequence is monotone.

The mean shift sequence converges in one-dimensional space.

The proof fills a gap in theoretical understanding.

Abstract

The mean shift algorithm is a non-parametric and iterative technique that has been used for finding modes of an estimated probability density function. It has been successfully employed in many applications in specific areas of machine vision, pattern recognition, and image processing. Although the mean shift algorithm has been used in many applications, a rigorous proof of its convergence is still missing in the literature. In this paper we address the convergence of the mean shift algorithm in the one-dimensional space and prove that the sequence generated by the mean shift algorithm is a monotone and convergent sequence.