Numerische Mathematik

, Volume 105, Issue 1, pp 1–34 | Cite as

Neighborhood filters and PDE’s

Article

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.

Mathematics Subject Classification

68U10 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Antoni Buades
    • 1
    • 2
  • Bartomeu Coll
    • 1
  • Jean-Michel Morel
    • 2
  1. 1.University of Balearic IslandsPalma de MallorcaSpain
  2. 2.CMLACachanFrance

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