# Uncorrelated and zero mean image noise

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 ?

• Correlation and distribution are two different concepts. Do you get it if we reduce to one dimensional signal $g(x) = f(x) + n(x)$ ? – AlexTP May 17 '17 at 20:27
• Correlation means that for any two samples of the noise $n(x_1,y_1)$ and $n(x_2,y_2)$ have a covariance of zero. The expected value of the entire noise signal is an integral that uses all of the values in question. Each sample's values may not be zero, but the average value of the noise is zero. – Envidia May 17 '17 at 20:32
• you may find this question useful dsp.stackexchange.com/questions/25719/zero-mean-noise-in-images/… – Balaji R May 18 '17 at 3:43

## 2 Answers

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.

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 $y_1 \not = y_2$.