1
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ Can you be more precise in what you don't understand? $\endgroup$ – MBaz Nov 10 '16 at 2:55
  • $\begingroup$ @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 $\endgroup$ – Aequitas Nov 10 '16 at 3:00
  • $\begingroup$ Can you add the solution to the post so someone will be able to give explanation in the step you are confused? $\endgroup$ – Navin Prashath Nov 10 '16 at 8:21
  • 1
    $\begingroup$ @NavinPrashath I think I have worked out the solution, I will post an answer with it sometime tomorrow. $\endgroup$ – Aequitas Nov 10 '16 at 8:47
  • $\begingroup$ As a general suggestion, when you can't solve a homework problem, your first resource should be your professor's office hours. $\endgroup$ – MBaz Nov 10 '16 at 13:44
1
$\begingroup$

The answer is trivial:

$$H(x,y)=4\delta(x,y)-\delta(x+1,y)-\delta(x-1,y)-\delta(x,y+1)-\delta(x,y-1)$$

Or in a coefficient representation:

$$H_{xy}=\begin{bmatrix} 0 & -1 & 0\\ -1 & 4 & -1\\ 0 & -1 & 0 \end{bmatrix}$$

This is a representation of the 2D derivative operator, which is known to have a "high filtering" shape both in $x$ and $y$ axis

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.