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

Abstract

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

Patch Complexity, Finite Pixel Correlations and Optimal Denoising

“…For example, BM3D [9] uses 8 × 8 windows at noise s.t.d below 40 and 12 × 12 windows at higher noise levels. Similarly, Bilateral filtering denoising algorithms [4] estimate a pixel as an adaptive average of its neighbors, where the neighbor weight is significantly reduced when an intensity discontinuity is observed. However, the discontinuity measure is relative to the noise level and only differences above the noise standard deviation actually reduce the neighbor weight.…”

Adaptive enhancement and noise reduction in very low light-level video

“…Since our current methodology is an extension of this technique we will present it in detail in Section 3. For a study of the connection between anisotropic diffusion, adaptive smoothing and bilateral filtering, see [2].…”

Image Filtering Driven by Level Curves

“…It should be pointed out that there exist methods such as adaptive filtering [4], [12] in which the weights in Equation 1 are obtained as w (2) (x j , y j ) = g(|∇I(x j , y j )|). These methods place more importance on the lower-gradient pixels of the neighborhood, but do not exploit level curve relationships in the way we do, and the choice of the weighting function does not have the geometric interpretation that exists in our technique.…”