image processing - What does zero-mean noise mean?

I don't know where to start from. and this question might be very silly,

I have found such as comment from someone at here,

"I think if you noise is zero-mean it doesn't matter, if not it does (you should de-noise first). I'm not entirely sure though so I won't put it in an answer yet."

Q1. Actually, I want to know what does zero-mean noise mean? and what does it not matter means?

Q2. Why we presume that the noise in Gaussian filter have sigma 1 and mean 0 ?

Answer

It is an easy concept but a sometimes it is hard to express it compactly. The difficulty arises because of additional concepts to be known for any formal description.

As you may know, mathematical analysis of noise variables and signals are handled via the tools of probability theory, in particular by the stochastic (random) processes.

The usual progress is such that first properties of random variables are described and then properties of random signals are derived based on it.

The mean for a random variable, X, is defined to be its Expected Value E(X) and computed based on its Probability Density Function $f_X(x)$ such as $\mu_X = \int {xf_X(x) dx}$ with the limits from minus to plus infinity for a continuous random variable.

So a zero mean random variable is that one for which the above integral is zero. (for a discrete random variable the integral is replaced by a sum and pdf is replaced by a pmf). This is also called as the average value.

Random signals are considered to be made up of an infinite set of random variables each making a single value at a single instance of observation of the instance being created. Hence there is the concept of ensemble and time averages.

Now coming to random signals X(t) (noise) expected value of a random signal is also expressed as E(X(t)) which for a Stationary (or at east Weakly Stationary up to second order) Process is a fixed value.

Your answer is that a zero mean noise is that one for which E(X(t)) is zero for all t.

However above is a theoretical description of Mean. In practice to compute mean and other theoretical parameters, you will refer to statistical computation which deal with practically observed data and computations based on that.

The key which connects statistically made (practical) computations to theoretical computations is the Ergodicity Theorem which states that for an ergodic random process theoretical computations are equal to statistical ones.