How to find the Fourier transfer function of a Laplacian operation?

My practice exam question is:

A discrete Laplacian operation on an image $f(x,y)$ results in the following output image: $$g(x,y) = 4f(x,y) - f(x+1,y)-f(x-1,y)-f(x,y+1)-f(x,y-1)$$

Find the 2 dimensional discrete Fourier transfer function of the Laplacian operation and show that the laplacian operation is a high pass filter.

I have the answer sheet, but I really don't understand it so am asking for a bit more of an explanation on the steps to solve it.

• Can you be more precise in what you don't understand? – MBaz Nov 10 '16 at 2:55
• @MBaz because there's no explanation the first step in the solutions is just a bunch of e to the power of 2pi*i terms. I don't get what has been done or how. I have no clue where to start on this question – Aequitas Nov 10 '16 at 3:00
• Can you add the solution to the post so someone will be able to give explanation in the step you are confused? – Navin Prashath Nov 10 '16 at 8:21
• @NavinPrashath I think I have worked out the solution, I will post an answer with it sometime tomorrow. – Aequitas Nov 10 '16 at 8:47
• As a general suggestion, when you can't solve a homework problem, your first resource should be your professor's office hours. – MBaz Nov 10 '16 at 13:44

$$H(x,y)=4\delta(x,y)-\delta(x+1,y)-\delta(x-1,y)-\delta(x,y+1)-\delta(x,y-1)$$
$$H_{xy}=\begin{bmatrix} 0 & -1 & 0\\ -1 & 4 & -1\\ 0 & -1 & 0 \end{bmatrix}$$
This is a representation of the 2D derivative operator, which is known to have a "high filtering" shape both in $x$ and $y$ axis