# What do you mean by noise vairance and mean in an image?

I know the sudden variation in the intensity level of an image is termed as noise. But I couldn't understand what do mean and variance of noise in an image refer to? What is the role of noise variance? As far as I have observed till now, most of the noise reduction algorithm's results vary depending on noise variance. Can someone help me in understanding the concept of noise mean and variance?

As I am very new to image processing, it would be better if someone could give a more elementary explanation. Thank you in advance!

Noise is not just a sudden variation in intensity; that could also be an edge of an object, a texture pattern etc. DC bias variations could also be considered as noise.

Noise is mostly an unrelated (independent of the signal) and random signal, most typically, imposed on the true signal of interest. Note that distortion (which depends on the signal) is something else.

For practical reasons, noise is mostly assumed to be white, and uncorrelated for simplicty but there are various different noise types as well, such as bandlimited (colored) noise, correlated noise etc.

For a WSS, zero mean, noise source, the variance gives its power. Stated mathematically: The variance of a WSS random process (noise) $$v[n]$$ is $$\sigma_v^2 = \mathcal{E} \{ (v[n]-\mu_x[n])^2 \} = \mathcal{E} \{ v[n]^2 \} - \mu_x^2$$

When the process is zero mean; i.e. $$\mathcal{E} \{ \mu_x[n]\} = 0$$, then $$\sigma_v^2 = \mathcal{E} \{ v[n]^2 \}$$

Which is the definition of power of the WSS random process (noise) $$v[n]$$

Mean of ergodic noise is, its time (sample) average in the most practical sense. Or it's exptected value as ensemble average, in a more theroretical sense.

It's difficult to separate noise from true signal. A class of techniques to reduce noise, relies on statistical methods to characterize and differentiate noise and true signal, and adjust their filter characteristics according to noise variance (noise power) versus signal variance (signal power).

Optimal Wiener filters exemplify a class of linear (LTI) filters that compute minimum MSE expectations of true signals in noise, for given signall and noise statistical characteristics.

• Thank you that's a nice explanation. But what do you mean by power of a noise or true signal? – shwetha Dec 15 '18 at 16:19
• @shwetha edited... – Fat32 Dec 15 '18 at 16:33