Neighborhood filters and PDE’s

Abstract

Denoising images can be achieved by a spatial averaging of nearby pixels. However, although this method removes noise it creates blur. Hence, neighborhood filters are usually preferred. These filters perform an average of neighboring pixels, but only under the condition that their grey level is close enough to the one of the pixel in restoration. This very popular method unfortunately creates shocks and staircasing effects. In this paper, we perform an asymptotic analysis of neighborhood filters as the size of the neighborhood shrinks to zero. We prove that these filters are asymptotically equivalent to the Perona-Malik equation, one of the first nonlinear PDE's proposed for image restoration. As a solution, we propose an extremely simple variant of the neighborhood filter using a linear regression instead of an average. By analyzing its subjacent PDE, we prove that this variant does not create shocks: it is actually related to the mean curvature motion. We extend the study to more general local polynomial estimates of the image in a grey level neighborhood and introduce two new fourth order evolution equations.

From Local Kernel to Nonlocal Multiple-Model Image Denoising

“…the so-called "staircasing" or "shock" effects. In order to repair and avoid these undesirable effects, the nonlocal polynomial models of the Þrst and higher orders have been proposed by Buades, Coll, and Morel [11], [12]. Similar higher-order nonlocal algorithm has been reported in Chatterjee and Milanfar [16], where the polynomial approximations up to second order are used.…”

Image Filtering Driven by Level Curves

“…Unlike our method, it does not use areas between level sets as weights to explicitly perform a weighted averaging. Secondly as proved in [13], its limiting behavior when W r → 0 and |N (x, y)| → 0 resembles the geometric heat equation with a linear polynomial, and resembles higher order PDEs when the degree of the polynomial is increased. Our method is the true continuous form of the KDE-based filter from Equation 1.…”

“…Our method is the true continuous form of the KDE-based filter from Equation 1. This KDE-based filter limits to the Perona-Malik equation, as proved in [13].…”

“…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. There also exists an extension to the standard neighborhood filter in Equation 1 reported in [13], which performs a weighted least squares polynomial fit to the intensity values (of the pixels) in the neighborhood of a location (x, y). The value of this polynomial at (x, y) is then considered to be the smoothed intensity value.…”

Generalised Nonlocal Image Smoothing

“…This simple idea allows a real incorporation of nonlocal pixel interactions in the smoothing process, providing impressive denoising results. The NL-means filter belongs to the class of neighbourhood filters [51,86,70,75,16] that average similar pixels based on their photometric and spatial proximitieswhere the spatial distance does not play a role in NL-means. In particular, it can be seen as a bilateral filter [75] with a patch-based photometric similarity measure.…”