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Local Smoothing Neighborhood Filters

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Handbook of Mathematical Methods in Imaging

Introduction

The neighborhood filter or sigma filter is attributed to J.S. Lee [48] (in 1983) but goes back to L. Yaroslavsky and the Sovietic image processing theory [76]. This filter is introduced in a denoising framework for the removal of additive white noise:

$$v(\mathbf{x}) = u(\mathbf{x}) + n(\mathbf{x}),$$

where x indicates a pixel site, v(x) is the noisy value, u(x) is the “true” value at pixel x, and n(x) is the noise perturbation. When the noise values n(x) and n(y) at different pixels are assumed to be independent random variables and independent of the image value u(x), one talks about “white noise.” Generally, n(x) is supposed to follow a Gaussian distribution of zero mean and standard deviation \(\sigma \).

Lee and Yaroslavsky proposed to smooth the noisy image by averaging only those neighboring pixels that have a similar intensity. Averaging is the principle of most denoising methods. The variance law in probability theory ensures that if Nnoise values are averaged,...

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Author information

Authors and Affiliations

  1. École Normale Supérieure de Cachan, Cachan, France

    Jean-Michel Morel

  2. Ecole Polytechnique, Palaiseau, France

    Antoni Buades

  3. Universitat de les Illes Balears, Palma-Illes Balears, Spain

    Tomeu Coll

Authors
  1. Jean-Michel Morel
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  2. Antoni Buades
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  3. Tomeu Coll
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Editors and Affiliations

  1. Computational Science Center, University of Vienna, Nordbergstrasse 15, Vienna, Austria

    Otmar Scherzer

  2. RICAM, Austrian Academy of Sciences, Linz, Austria

    Otmar Scherzer

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Morel, JM., Buades, A., Coll, T. (2011). Local Smoothing Neighborhood Filters. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-0-387-92920-0_26

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