Mean shift is a non-parametric feature-space mathematical analysis technique for locating the maxima of a density function, a so-called mode-seeking algorithm. Application domains include cluster analysis in computer vision and image processing.
History The mean shift procedure is usually credited to work by Fukunaga and Hostetler in 1975. It is, however, reminiscent of earlier work by Schnell in 1964.
Overview Mean shift is a procedure for locating the maxima—the modes—of a density function given discrete data sampled from that function. This is an iterative method, and we start with an initial estimate x {\displaystyle x} . Let a kernel function K ( x i − x ) {\displaystyle K(x_{i}-x)} be given. This function determines the weight of nearby points for re-estimation of the mean. Typically a Gaussian kernel on the distance to the current estimate is used, K ( x i − x ) = e − c | | x i − x | | 2 {\displaystyle K(x_{i}-x)=e^{-c||x_{i}-x||^{2}}} . The weighted mean of the density in the window determined by K {\displaystyle K} is
m ( x ) = ∑ x i ∈ N ( x ) K ( x i − x ) x i ∑ x i ∈ N ( x ) K ( x i − x ) {\displaystyle m(x)={\frac {\sum _{x_{i}\in N(x)}K(x_{i}-x)x_{i}}{\sum _{x_{i}\in N(x)}K(x_{i}-x)}}}
where N ( x ) {\displaystyle N(x)} is the neighborhood of x {\displaystyle x} , a set of points for which K ( x i − x ) ≠ 0 {\displaystyle K(x_{i}-x)\neq 0} . The difference m ( x ) − x {\displaystyle m(x)-x} is called mean shift in Fukunaga and Hostetler. The mean-shift algorithm now sets x ← m ( x ) {\displaystyle x\leftarrow m(x)} , and repeats the estimation until m ( x ) {\displaystyle m(x)} converges. Although the mean shift algorithm has been widely used in many applications, a rigid proof for the convergence of the algorithm using a general kernel in a high dimensional space is still not known. Aliyari Ghassabeh showed the convergence of the mean shift algorithm in one dimension with a differentiable, convex, and strictly decreasing profile function. However, the one-dimensional case has limited real world applications. Also, the convergence of the algorithm in higher dimensions with a finite number of the stationary (or isolated) points has been proved. However, sufficient conditions for a general kernel function to have finite stationary (or isolated) points have not been provided. Gaussian Mean-Shift is an Expectation–maximization algorithm.
Details Let data be a finite set S {\displaystyle S} embedded in the n {\displaystyle n} -dimensional Euclidean space, X {\displaystyle X} . Let K {\displaystyle K} be a flat kernel that is the characteristic function of the λ {\displaystyle \lambda } -ball in X {\displaystyle X} ,
In each iteration of the algorithm, s ← m ( s ) {\displaystyle s\leftarrow m(s)} is performed for all s ∈ S {\displaystyle s\in S} simultaneously. The first question, then, is how to estimate the density function given a sparse set of samples. One of the simplest approaches is to just smooth the data, e.g., by convolving it with a fixed kernel of width h {\displaystyle h} ,
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