Non-local means

Non-local means is an algorithm in image processing for image denoising. Unlike "local mean" filters, which take the mean value of a group of pixels surrounding a target pixel to smooth the image, non-local means filtering takes a mean of all pixels in the image, weighted by how similar these pixels are to the target pixel. This results in much greater post-filtering clarity, and less loss of detail in the image compared with local mean algorithms.^[1]

If compared with other well-known denoising techniques, non-local means adds "method noise" (i.e. error in the denoising process) which looks more like white noise, which is desirable because it is typically less disturbing in the denoised product.^[2] Recently non-local means has been extended to other image processing applications such as deinterlacing^[3] and view interpolation.^[4]

Definition

Suppose $\Omega$ is the area of an image, and $p$ and $q$ are two points within the image. Then, the algorithm is:^[5]

u(p)={1 \over C(p)}\int _{\Omega }v(q)f(p,q)\,\mathrm {d} q.

where $u(p)$ is the filtered value of the image at point $p$ , $v(q)$ is the unfiltered value of the image at point $q$ , $f(p,q)$ is the weighting function, and the integral is evaluated over $\forall q\in \Omega$ .

$C(p)$ is a normalizing factor, given by:

C(p)=\int _{\Omega }f(p,q)\,\mathrm {d} q.

Common weighting functions

The purpose of the weighting function, $f(p,q)$ , is to determine how closely related the image at the point $p$ is to the image at the point $q$ . It can take many forms.

Gaussian

The Gaussian weighting function sets up a normal distribution with a mean, $\mu =B(p)$ and a variable standard deviation:^[6]

f(p,q)=e^{{-{{\left\vert B(q)-B(p)\right\vert ^{2}} \over h^{2}}}}

where $h$ is the filtering parameter (i.e., standard deviation) and $B(p)$ is the local mean value of the image point values surrounding $p$ .

Discrete algorithm

For an image, $\Omega$ , with discrete pixels, a discrete algorithm is required.

u(p)={1 \over C(p)}\sum _{{q\in \Omega }}v(q)f(p,q)

where $C(p)$ is given by:

C(p)=\sum _{{q\in \Omega }}f(p,q)

Then, for a Gaussian weighting function,

f(p,q)=e^{{-{{\left\vert B(q)-B(p)\right\vert ^{2}} \over h^{2}}}}

where $B(p)$ is given by:

B(p)={1 \over |R(p)|}\sum _{{i\in R(p)}}v(i)

where $R(p)\subseteq \Omega$ and is a square region of pixels surrounding $p$ and $|R(p)|$ is the number of pixels in the region $R$ .

References

↑ Buades, Antoni (20–25 June 2005). "A non-local algorithm for image denoising". Computer Vision and Pattern Recognition, 2005. 2: 60–65. doi:10.1109/CVPR.2005.38.
↑ Buades, Antoni. "On image denoising methods" (PDF). 123 Seminars Only.
↑ Dehghannasiri, R.; Shirani, S. (2012). "A novel de-interlacing method based on locally-adaptive Nonlocal-means".
↑ Dehghannasiri, R.; Shirani, S. (2013). "A view interpolation method without explicit disparity estimation".
↑ Buades, Antoni. "Non-Local Means Denoising". Image Processing On Line.
↑ Buades, Antoni. "On image denoising methods (page 10)" (PDF). 123 Seminars Only.

External links

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[1] Buades, Antoni (20–25 June 2005). "A non-local algorithm for image denoising". Computer Vision and Pattern Recognition, 2005. 2: 60–65. doi:10.1109/CVPR.2005.38.

[2] Buades, Antoni. "On image denoising methods" (PDF). 123 Seminars Only.

[3] Dehghannasiri, R.; Shirani, S. (2012). "A novel de-interlacing method based on locally-adaptive Nonlocal-means".

[4] Dehghannasiri, R.; Shirani, S. (2013). "A view interpolation method without explicit disparity estimation".

[5] Buades, Antoni. "Non-Local Means Denoising". Image Processing On Line.

[6] Buades, Antoni. "On image denoising methods (page 10)" (PDF). 123 Seminars Only.