This question is inspired from this reading here
Let's take a simple example, take a kernel matrix of any size that holds any values. Take a zero matrix of any size larger than the kernel, and put a one somewhere in it. Convolution with the kernel and the matrix will simply reproduce the kernel surrounding where the 1 was in the matrix, a feature known as transitional equivariance (?).
While I can understand this conceptually and write out examples, I'm having a bit of trouble conceptualizing the written proof for the above case that shows that the convolution of a kernel with a zero-with-one-1 matrix. I was hoping someone could lead me in the right direction in coming up with a place to start.
Thanks.
I would start with the nice formula that your article gives you:
$$F*I(x,y)=\sum_{j=-N}^N\sum_{i=-N}^NF(i,j)I(x-i,y-j)$$
Now if $F(i,j)=\begin{cases}1&\text{if}\ i=i_0,j=j_0\ \text{for some specific $i_0,j_0$,}\\0&\text{else,}\end{cases}$ then we get $$F*I(x,y)=\sum_{j=-N}^N\sum_{i=-N}^NF(i,j)I(x-i,y-j) = F(i_0,j_0)I(x-i_0,y-j_0)+0=I(x-i_0,y-j_0)$$ because all terms where $i\neq i_0$ or $j\neq j_0$ are $0$.
Indeed, $I(x-i_0,y-j_0)$ is just the matrix $I$, but translated by $i_0,j_0$.