Local pixel grouping

In image Noise reduction, local pixel grouping is the algorithm to remove noise from images using principal component analysis (PCA).

## Contents

* 1 Image denoising
* 2 Principal component analysis
* 2.1 Construct local pixel group
* 2.2 Denoising using local pixel group
* 3 References
* 4 External links

## Image denoising

Main article: Noise reduction

Sensors such as CCD, CMOS or ultrasonic probe may encapsulate noise signal. Noise reduction is commonly used to improve quality of the image. However, techniques such as smoothing filters and many other algorithms may lose local structure of image while denoising the image.[1] More over, efficiency is also taken into consideration.

## Principal component analysis

Main article: Principal component analysis

PCA was invented in 1901 by Karl Pearson,[2] to transform original dataset into linearly uncorrelated PCA domain. PCA works in the way that principal components with larger possible variance are preserved while discarding low variance components.

Image denoising by principal component analysis with local pixel grouping(LPG-PCA) was developed by Lei et. in 2010.[3] It is based on the assumption that the energy of a signal will concentrate on a small subset of the PCA transformed dataset, while the energy of noise will evenly spread over the whole dataset. Assume original image is denoted by I and noise is denoted by v, then the measured image will be Iv=I+v. In order to denoising Iv, first a train dataset Xv must be constructed using local pixel group. Using this Xv and apply PCA the noise in the image can be reduced.

### Construct local pixel group

For each pixel px in the image, select a K×K window centered at px denoted by

x=[x1...xm]T,m=K2

and a training window centered at px. The training window is L×L,L>K. Take the pixels in each possible K×K block within the L×L training block yields (L−K+1)2 samples xi->v. If the distance between a sample and the center window x0->v is smaller than some threshold, then accept the sample. So the train dataset Xv is acquired by put all the accepted sample together as column vectors into a matrix.

### Denoising using local pixel group

First step of this part is centralize Xv and Xv‾ is obtained. By computing the covariance matrix of Xv‾ denoted by Ωx‾, the PCA transformation matrix Px‾ can be obtained. Apply Px‾ to Xv‾ we have

Yv‾=Px‾Xv‾

The covariance matrix of Yv‾ can also be calculated by

Ωyv‾=1nYv‾ Yv‾T

Shrink the coefficient of Ωy‾v by

Y‾->^k=wk Y‾->vk
wk=Ωy‾(k,k)Ωy‾(k,k)+Ωvy(k,k)

and transform back to X‾^, the noise in that pixel is reduced. Apply this to all the pixels in the image and the denoised image can be obtained. Experiments by Lei show that LGP-PCA can effectively preserve the image fine structures while smoothing noise. The solution is competitive compared with other algorithms such as Block-matching algorithm.

## References

1. ↑ Buades, A.; Coll, B.; Morel, J. M. (2005). "A Review of Image Denoising Algorithms, with a New One". Multiscale Modeling & Simulation 4 (2): 490. doi:10.1137/040616024.
2. ↑ Pearson, K. (1901). "On Lines and Planes of Closest Fit to Systems of Points in Space". Philosophical Magazine 2 (11): 559–572. doi:10.1080/14786440109462720. http://stat.smmu.edu.cn/history/pearson1901.pdf.
3. ↑ Zhang, L.; Dong, W.; Zhang, D.; Shi, G. (2010). "Two-stage image denoising by principal component analysis with local pixel grouping". Pattern Recognition 43 (4): 1531. doi:10.1016/j.patcog.2009.09.023.

## External links

* Lei's homepage and matlab code of LPG-PCA

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