meanShift2005 - IEEE TRANSACTIONS ON PATTERN ANALYSIS AND...

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Mean Shift Is a Bound Optimization Mark Fashing and Carlo Tomasi, Member , IEEE Abstract —We build on the current understanding of mean shift as an optimization procedure. We demonstrate that, in the case of piecewise constant kernels, mean shift is equivalent to Newton’s method. Further, we prove that, for all kernels, the mean shift procedure is a quadratic bound maximization. Index Terms —Mean shift, bound optimization, Newton’s method, adaptive gradient descent, mode seeking. æ 1I NTRODUCTION MEAN shift is a nonparametric, iterative procedure introduced by Fukunaga and Hostetler [1] for seeking the mode of a density function represented by a set S of samples. The procedure uses so- called kernels , which are decreasing functions of the distance from a given point t to a point s in S . For every point t in a given set T , the sample means of all points in S weighted by a kernel at t are computed to form a new version of T . This computation is repeated until convergence. The resulting set T contains estimates of the modes of the density underlying set S . The procedure will be reviewed in greater detail in Section 2. Cheng [2] revisited mean shift, developing a more general formulation and demonstrating its potential uses in clustering and global optimization. Recently, the mean shift procedure has met with great popularity in the computer vision community. Applica- tions range from image segmentation and discontinuity-preserving smoothing [3], [4] to higher level tasks like appearance-based clustering [5], [6] and blob tracking [7]. Despite the recent popularity of mean shift, few attempts have been made since Cheng [2] to understand the procedure theoretically. For example, Cheng [2] showed that mean shift is an instance of gradient ascent and also notes that, unlike naı¨ve gradient ascent, mean shift has an adaptive step size. However, the basis of step size selection in the mean shift procedure has remained unclear. We show that, in the case of piecewise constant kernels, the step is exactly the Newton step and, in all cases, it is a step to the maximum of a quadratic bound. Another poorly understood area is that of mean shift with an evolving sample set. Some variations on the mean shift procedure use the same set for samples and cluster centers. This causes the sample set to evolve over time. The optimization problem solved by this variation on mean shift has yet to be characterized. In this paper, we build on the current understanding of mean shift as an optimization procedure. Fukunaga and Hostetler [1] suggested that mean shift might be an instance of gradient ascent. Cheng [2] clarified the relationship between mean shift and optimization by introducing the concept of the shadow kernel and showed that mean shift is an instance of gradient ascent with an adaptive step size. We explore mean shift at a deeper level by examining not only the gradient, but also the Hessian of the shadow kernel density estimate. In doing so, we establish a connection between mean shift and the Newton step and we
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This note was uploaded on 11/27/2011 for the course MATH 3484 taught by Professor Staff during the Fall '10 term at University of Central Florida.

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meanShift2005 - IEEE TRANSACTIONS ON PATTERN ANALYSIS AND...

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