Solving Image Processing Problem: Fourier Transform and Image Dimensions

[mr_k]

hi!
i need help with problem:
Let F(u,v) be the Fourier transform of an M x N image f(x,y). Let g(x,y) be an image of dimensions (2 M) x (2 N) whose Fourier transform G(u,v) is defined as follows:

What does the image g(x,y) look like in terms of f(x,y)?

my solution : g(x,y)=0.25f(x/2,y/2).

Is that true?

thanks :)

 

[marcusl]

No, you're on the wrong track. You have an image that has looks basically the same but has twice (in this case) as many points. These extra points are interpolated between the points of the original f. This is known as zero-padding. See if you can work it out from the DFT expressions.

 

[AlephZero]

You are not completely on the wrong track, but your math solution is wrong, because the functions f(x,y) etc are only defined when x and y are integers. You can't talk about "half a pixel", so f(x/2, y/2) doesn't mean anything if x or y are odd numbers.

On the other hand if you say what your formula means "in words" rather than as a math formula, it does describe what the image "looks like".

 

[mr_k]

hi,my solution:
http://img713.imageshack.us/img713/3815/lllgu.png



What's the problem with the answer?

thanks.

 

[I like Serena]

hi,my solution:

What's the problem with the answer?

thanks.

Your answer is fine for even x and y.
But what happens if x or y is odd?

 

[mr_k]

ok,So what's the right answer?

 

[I like Serena]

Your answer is right, but you should add that if x or y is odd, that the value is effectively interpolated.

 

[mr_k]

thanks!