On the convergence of the mean shift algorithm in the one-dimensional space

• A proof for the convergence of the mean shift algorithm in the one-dimensional space is given.
• It has been discussed that the previously given proofs for the mean shift algorithm are not correct.
• It was shown that the mode estimate sequence in the one-dimensional space is a monotone sequence.
• Theoretical results are confirmed through simulations.

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.