Uncorrelated and zero mean image noise

Question

Consider a noisy image g(x,y) formed by the addition of noise n(x,y) to an original image f(x,y); that is

g(x,y) = f(x,y) + n(x,y)

where the assumption is that at every pair of coordinates (x,y) the noise is uncorrelated and has zero average value.

What does it mean for noise to be uncorrelated and have zero average value. I know the term uncorrelated for random variables, and mean of probability distribution function, but I can't understand these concept for images.

How can noise have zero expected value at every pair of coordinates (x,y), or be uncorrelated ?

Answer 1

Let us consider two random variables $X$ and $Y$. You are aware of the following terminologies with respect to (wrt) the random variables:

(Zero Mean) : $\mathbb{E}(X) = 0$ and $\mathbb{E}(Y) = 0$,

(Uncorrelated) : $\mathbb{E}(X Y) = \mathbb{E}(X) \mathbb{E}(Y)$. It is interesting to figure out how it is different from independence.

Now, an image specified, say $g(x,y) = f(x,y) + n(x,y)$ is a discrete random random process. The notion of zero mean and uncorrelated are not very different here, as for any two points $(x,y) = (x_1,y_1)$ and $(x,y) = (x_2,y_2)$, the image intensity at these pixel locations are random variables.

Answer 2

So zero average value means that: $$ E[n(x,y)] = 0 $$ i.e. the expected value of the noise is zero.

Uncorrelated means that: $$ E[n(x_1,y_1)n(x_2,y_2)] = 0 $$ where $x_1 \not = x_2$ and $y_1 \not = y_2$.