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'''Non-local means''' is an algorithm in image processing for [[image denoising]]. Unlike "local smoothing" filters, non-local means does not update a pixel's value with an average those of the pixels around it - instead, it updates it using a weighted average of the pixels judged to be ''most similar'' to it, judged by its distance in colourspace. The weight of each pixel depends on the distance between its intensity grey level vector and that of the target pixel. Non-local means algorithm was published by Antoni Buades,et al. in 2005.<ref>Buades,A.; Coll,B.;Morel,J.-M." A non-local algorithm for image denoising",Computer Vision and Pattern Recognition, 2005.</ref>

If compared with other well-known denoising techniques, such as the [[Gaussian filter|Gaussian smoothing]] model, the [[anisotropic diffusion]] model, the [[total variation denoising]], the [[Neighbourhood system|neighborhood filters]] and an elegant variant, the [[Wiener filter|Wiener local empirical filter]], the translation invariant [[wavelet]] thresholding, the non-local means method noise looks more like [[white noise]].<ref>{{cite web|url=http://123seminarsonly.com/Seminar-Reports/029/42184313-10-1-1-100-81.pdf |title=On image denoising methods |publisher=123seminarsonly.com |accessdate=2013-10-27}}</ref>

==Definition==
Suppose <math>\Omega</math> is the area of an image <math>I</math>, <math>x</math> is the location inside <math>\Omega</math>, <math>u(x)</math> and <math> v(x)</math> are the clean and observed noisy image value at location <math>x</math> respectively. Then the non-local means algorithm can be defined as 

<math>NL \left [ u \right ] (x) = {1 \over C(x)}\int_\Omega e^{-{{(G_a* \left \vert v(x+.)-v(y+.)\right \vert ^2)(0)}\over h^2}}v(y)dy.</math>

where <math>G_a</math> is a Gaussian function with standard deviation <math>a</math>,<br /> 
<math>{C(x)}={\int_\Omega e^{-{{(G_a* \left \vert v(x+.)-v(y+.)\right \vert ^2)(0)}\over h^2}}}dz</math> and <math>h</math> is the filtering parameter.

==Discrete Algorithm==

If image <math>I</math> is discrete, the non-local means algorithm can be represented as

<math>NL \left [ u \right ](i)= \sum_{j \in I}w(i,j)v(j),</math>

where the weight <math> w(i,j)</math> depends on the distance between observed gray level vectors at points <math>i</math> and <math>j</math>. Such distance can be represented as

<math> d={\lVert v(\mathcal{N}_i)-v(\mathcal{N}_j)\rVert}_{2,a}^2 .</math> 

So the weight can be defined as 

<math>w(i,j)= {1 \over Z(i)}e^{-{{\lVert v(\mathcal{N}_i)-v(\mathcal{N}_j)\rVert}_{2,a}^2\over h^2}},</math>

where <math> Z(i)= \sum_j e^{-{{\lVert v(\mathcal{N}_i)-v(\mathcal{N}_j)\rVert}_{2,a}^2\over h^2}}.</math>


==See also==
* [[Total variation denoising]]
* [[Anisotropic diffusion]]
* [[Signal Processing]]
* [[Digital Image Processing]]
* [[Noise reduction]]

==References==
{{Reflist}}

==External links==
*[http://www.stanford.edu/~slansel/tutorial/summary.htm Recent trends in denoising tutorial]{{dead link|date=September 2013}}
*[http://www.ipol.im/pub/algo/bcm_non_local_means_denoising/ Non-local image denoising, with code and online demonstration]

{{Noise|state=uncollapsed}}

[[Category:Noise reduction]]
[[Category:Image processing]]
[[Category:Image noise reduction techniques]]


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