A common framework for nonlinear diffusion, adaptive smoothing, bilateral filtering and mean shift

In this paper, a common framework is outlined for nonlinear diffusion, adaptive smoothing, bilateral filtering and mean shift procedure. Previously, the relationship between bilateral filtering and the nonlinear diffusion equation was explored by using a consistent adaptive smoothing formulation. However, both nonlinear diffusion and adaptive smoothing were treated as local processes applying a 3 £ 3 window at each iteration. Here, these two approaches are extended to an arbitrary window, showing their equivalence and stressing the importance of using large windows for edge-preserving smoothing. Subsequently, it follows that bilateral filtering is a particular choice of weights in the extended diffusion process that is obtained from geometrical considerations. We then show that kernel density estimation applied in the joint spatial–range domain yields a powerful processing paradigm—the mean shift procedure, related to bilateral filtering but having additional flexibility. This establishes an attractive relationship between the theory of statistics and that of diffusion and energy minimization. We experimentally compare the discussed methods and give insights on their performance. q 2003 Elsevier B.V. All rights reserved.