Local pixel grouping

In image Noise reduction, local pixel grouping is the algorithm to remove noise from images using principal component analysis (PCA).
Image denoising.
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. More over, efficiency is also taken into consideration.
Principal component analysis.
PCA was invented in 1901 by Karl Pearson, 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. 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 formula_1 and noise is denoted by formula_2, then the measured image will be formula_3. In order to denoising formula_4, first a train dataset formula_5 must be constructed using local pixel group. Using this formula_5 and apply PCA the noise in the image can be reduced.
Construct local pixel group.
For each pixel formula_7 in the image, select a formula_8 window centered at formula_7 denoted by 
and a training window centered at formula_7. The training window is formula_12. Take the pixels in each possible formula_8 block within the formula_14 training block yields formula_15 samples formula_16. If the distance between a sample and the center window formula_17 is smaller than some threshold, then accept the sample. So the train dataset formula_5 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 formula_5 and formula_20 is obtained. By computing the covariance matrix of formula_20 denoted by formula_22, the PCA transformation matrix formula_23 can be obtained. Apply formula_23 to formula_20 we have
The covariance matrix of formula_27 can also be calculated by
Shrink the coefficient of formula_29 by
and transform back to formula_32, 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.