What is an arithmetic mean filter? Defined as:
  
  $$\hat{f}(x,y)=\frac{1}{mn}\sum_{(s,t)\in S_{xy}}g(s,t) $$


As far as I understand:


$g(x, y)$ is the original image
$g(s, t)$ is a sub-image of $g(x, y)$ with a dimension of mxn
$\hat f(x, y)$ is the filtered image
Sub-images are summed up and then multiplied by ${ \frac {1}{mn}}$.


Am I correct?

If yes, then, I have several questions:


Is $g(s,t)$ a pixel in the neighborhood? What does it mean by that?
What is $S_{xy}$ ?
What are the meanings of $s$ and $t$?
What is the meaning of $(s,t) \in S_{xy}$?
In one picture: here, the mask is of size $m=3\times n=5$, filled with value $\frac{1}{3\times 5}$, the light gray shadow on the input image defines the neighborhood of the dark gray pixel: a set of pixels (all your $g(s,t)$) close to the center one (it moves with the pixel). Let us suppose the dark gray pixel has coordinates $x=5$ and $y=6$. The shadow defines $S_{5,6}$ having $s\in[5-(m-1),\dots,5+(m-1)]$ and $t\in[6-(n-1),\dots,6+(n-1)]$. $(s,t)\in S_{5,6} $ means for instance that the light gray pixel on the bottom left of the dark gray one, with coordinates $(6,5)$ belongs to the shadow.