# confusion regarding correlation in noise?

I am reading online article about image processing at following link

https://www.electronicsforu.com/electronics-projects/software-projects-ideas/image-processing-using-matlab-basic-operations-part-2-4

I have also attached a snapshot of a paragraph of this link and i have highlighted confusing statements

As paragraph says

"In either case, the noise at different pixels can either be correlated or uncorrelated".

That give rise to two queries

1) What is difference between correlated noise and non correlated noise?

2)If an image has noise, all pixels will have same noise,correlated or non correlated, or is it possible that some pixels have correlated noise and others have non correlated noise?

Tl;DR Uncorrelated noise - a noise without any correlation between pixels. Correlated noise - a noise that has a correlation between different pixels or time correlation in the same pixel.

It is possible that a certain image pixel will be deviated by both correlated and uncorrelated noise. The mathematical modeling of this is a summation of the two noise processes.

Noise - In image processing, a deviation of the value of a pixel is called noise. An image is basically a table of values. A grayscale image is a single table with brightness values per pixel while a color (RGB) image has three tables, one for each color. Suppose that a single-pixel (grayscale for simplicity) is supposed to hold the value 119 but for some reason, the value changed into 121. There are many reasons why this could happen but commonly, we are discussing causes that affect many\all pixels. This will distort the image with respect to the severity of the cause.

Correlated noise - A noise that deviates the values in a correlated way. The correlation is pixel-wise. For example, think of the image acquiring process. the light goes through a lens in the process which has a shape designed to concentrate the light. Obviously, manufacturing methods are not perfect and some error occurs. The results would be a small smear of the image. If you will take a picture of a black point on a white background with such a lens it will spread into other pixels on the image plane, that is, instead of all pixels being 255 (white) and a single-pixel being 0 (blank), neighboring pixels will also have small values (10?20?). This is called the point spread function (PSF) of the lens and it is an important parameter we wish to be small. There will be a correlation in the noise, between neighboring pixels. In PSF it is modeled as a 2D gaussian:

The center will be most affected (red) and as we get farther away the effect will fade (blue).

Note - Correlation could be with pixels that are far away from each other (sparse correlated noise) or could refer to time correlation, a correlation of a single-pixel noise over time (in a video).

Uncorrelated noise - A noise that has no correlation between pixels. For example, deviations in the manufacturing process of different picture elements (pixels) can cause each one of them to record the light it is experiencing a little differently. For example, think of tow neighboring pixels that experience the same light intensity (we assume they supposed to hold the same value). One will record a certain value and the other will record a slightly different value. Current pixels are electrical components of a CCD which may record the light slightly different from one another due to differences between them. The way in which they differ could be an intuitive example of a random process that holds no correlation between two pixels. Such noise is uncorrelated.

Note - it is not a good example because the manufacturing process of CCD (photolithography) will probably have a correlation between 'messing up' neighboring pixels due to its nature. But still, it is a good example to understand uncorrelated noise. Another example could be an image wich was corrupted while sent over a communication channel (satellite?). Let us assume that the communication corrupted each pixel in an uncorrelated manner, hence, uncorrelated noise.

The different processes which cause the deviations of the values are acting together in a superposition manner. Is simpler words, you can calculate the effect of each noise separately and then sum up all the effects. This means that a pixel could have both correlated and uncorrelated noise.

• what do you mean by "Tl;DR" in start of your answer?
– engr
Apr 26 '20 at 18:14
• you wrote "In PSF it is modeled as a 2D gaussian:" but you drew a 3d plot,why difference?
– engr
Apr 26 '20 at 18:15
• In your last para,you wrote " Is simpler words, you can calculate the effect of each noise separately and then sum up all the effects" what do you mean by "each noise"? Do you mean correlated and uncorrelated? or do you mean "Gaussian noise" and "salt and pepper noise"?
– engr
Apr 26 '20 at 18:22
• TL;DR is short internet slang for Too long, didn't read. It represents that the first section answers your question briefly if you don't wish to keep reading the small details. Apr 27 '20 at 9:42
• What I mean is each modeled noise, correlated or otherwise. For example, Let's say my image was transmitted and I am modeling the noise in the transmission as white Gaussian noise. Also, it is obviously affected by the PSF noise. Lastly, let us say that it was taken through a glass with drops of water (however you model this noise) and in bed lighting (probably a gradient of lighting intensity over the picture). You can consider the effect of each one of the four noises separately, and then sum it up. Some of them will be correlated (bad lighting and PSF) and some will not (transmission). Apr 27 '20 at 9:50

Uncorrelatedness of WSS noise process is required to model the Noise as White Noise. And, Whiteness of noise is a desired property of noise process so that we can assume that the image spectrum is affected by the noise equally across the complete image spectrum. There is no frequency selective impact on the image.

Explanation :

You can understand it like this: Correlation of two random variable is $$0$$, if the covariance is $$0$$. Zero Covariance of 2 random variables means $$\mathbf EXY = \mathbf EX. \mathbf EY$$.

Now consider a WSS (Wide sense stationary) Noise process $$\mathbf w$$. Wide Sense Stationarity of a random process means that we can assume certain simplification regarding how is the random process changing with time. There are 2 properties:

1. $$\mathbf E w[n] = c$$, meaning the process's mean remains constant no matter when it is observed. Expectation is not a function of time.
2. $$\mathbf Ew[n]w[n-l] = \mathcal R[l]$$, meaning auto-correlation is a function of only the lag $$l$$, and not when the auto-correlation is observed.

Now, if the Noise Process is a 0 mean WSS Process and if you can also model it as Uncorrelated, then you get a very simple auto-correlation function as follows:

$$\mathcal R[l] = \sigma^2 \delta[l]$$ Why? Because $$w[n]$$ and $$w[n-l]$$ are two random variables sampled from the noise process, and if we have assumed uncorrelatedness, then $$\mathbf Ew[n]w[n-l]$$ will be $$0$$ except when $$l=0$$. When $$l=0$$, we are correlating the same sample with itself which will result is variance : $$\mathbf Ew[n]w[n] = \sigma^2$$.

This means that the spectrum of noise is constant for all frequencies and equal to its variance $$\sigma^2$$, because fourier transform of $$\mathscr F\{\delta[l]\} = 1$$. Because Power spectral Density (PSD) of a random process is defined as Fourier transform of the auto-correlation function.

On the other side, if you cannot make this assumption that Noise Process is Uncorrelated, then you cannot model it as White Noise. And, as a result your image spectrum will be impacted by this noise differently at different frequencies.