Local Smoothing Neighborhood Filters.

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 gray level is close enough to the one of the pixel in restoration. This very popular method unfortunately creates shocks and staircasing effects. It also excessivelly blurs texture and fine structures when noise dominates the signal. In this chapter, 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 PDEs proposed for image restoration. As a solution to the shock effect, 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 also present a generalization of neighborhood filters, the nonlocal means (NL-means) algorithm, addressing the preservation of structure in a digital image. The NL-means algorithm tries to take advantage of the high degree of redundancy of any natural image. By this, we simply mean that every small window in a natural image has many similar windows in the same image. Now in a very general sense inspired by the neighborhood filters, one can define as “neighborhood of a pixel” any set of pixels with a similar window around. All pixels in that neighborhood can be used for predicting its denoised value. We finally analyze the recently introduced variational formulations of neighborhood filters and their application to segmentation and seed diffusion.