Update:
So, according to this article.
For a 2D mean filter,
$$
H[f(x,y)]=\frac{1}{NM}\sum_{k=0}^{N-1}\sum_{p=0}^{M-1}f(x-k,y-p)=g(x,y)\tag{1}
$$
$$
g(x+x_0,y+y_0)=\frac{1}{NM}\sum_{k=0}^{N-1}\sum_{p=0}^{M-1}f(x+x_0-k,y+y_0-p)\tag{2}
$$
$$
H[f(x+x_0,y+y_0)]=\frac{1}{NM}\sum_{k=0}^{N-1}\sum_{p=0}^{M-1}f(x+x_0-k,y+y_0-p)\tag{3}
$$
because (2) equals to (3), a 2D mean filter is shift-invariant.
Is the proof right?
BTW, if f(x,y) is a 2D image, such as
1 2 1
2 3 2
1 2 1
and g(x,y) is (using zero padding)
8/9 11/9 8/9
11/9 15/9 11/9
8/9 11/9 8/9
What does these
$$
H[f(x+x_0,y+y_0)]\tag{4}
$$
and
$$
g(x+x_0,y+y_0)\tag{5}
$$
mean?
(I want to prove it by using a image.)
Origin:
I read this article
and this Wikipedia page.
However, I still don't know what it means for (or how to apply to) a mean filter.
Can anyone give me an image example?
(for example, https://dsp.stackexchange.com/a/14435 is easier to understand)