Local Smoothing Neighborhood Filters
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Denoising.......... ............. ............ ............. ............. ........... Analysis of Neighborhood Filter as a Denoising Algorithm. .. . .. .. . .. . .. . .. . .. . Neighborhood Filter Extension: The NL-Means Algorithm. .. . . . .. . . .. . . . .. . . .. Extension to Movies.... ..... .... ..... .... ..... .... ..... .... ..... ..... .... ..... .... .....
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Asymptotic......... ........... ........... ........... ........... ............ ...... PDE Models and Local Smoothing Filters. .. .. ... .. .. .. ... .. .. .. .. ... .. .. .. .. ... .. . Asymptotic Behavior of Neighborhood Filters (Dimension ). . . . . . . . . . . . . . . . . . The Two-Dimensional Case. .. .. . .. .. .. . .. .. . .. .. .. . .. .. . .. .. .. . .. .. .. . .. .. . .. .. .. . .. A Regression Correction of the Neighborhood Filter.. ... ... ... ... .... ... ... ... .. The Vector-Valued Case. .. . .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. . .. Interpretation.. .. ... .. ... .. ... .. ... .. ... .. ... .. ... ... .. ... .. ... .. ... .. ... .. ... .. ... .. ...
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Variational and Linear Diffusion. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . Linear Diffusion: Seed Growing. .... .... ... .... .... .... ... .... .... .... .... ... .... .... Linear Diffusion: Histogram Concentration . .. . .. . .. . .. . .. .. . .. . .. . .. . .. . .. .. . .. . .
Otmar Scherzer (ed.), Handbook of Mathematical Methods in Imaging, DOI ./---_, © Springer Science+Business Media LLC
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Local Smoothing Neighborhood Filters
Introduction
The neighborhood filter or sigma filter is attributed to J.S. Lee [] (in ) but goes back to L. Yaroslavsky and the Sovietic image processing theory []. This filter is introduced in a denoising framework for the removal of additive white noise: v(x) = u(x) + n(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 σ . 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 N noise values are averaged, the noise standard deviation is divided by N. Thus, one should, for example, find for each pixel nine other pixels in the image with the same color (up to the fluctuations due to noise) in order to reduce the noise by a factor . A first idea might be to chose the closest
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