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Transform inverse result

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Problem Detail: 

let us assume that:
f is a grayscale image of size NxN, and F is the Fourier transform of f.
G is a 2Nx2N transform obtained by inserting 0 between every value in F: enter image description here

What is the inverse transform of G?

solution: so i have programmed it, and the picture will be 4 times bigger with the picture 4 times in it. g= [f f;f f] can you please explain why this is the answer?

Asked By : Gilad
Answered By : boshnak

Think of it the other way around.
If you duplicate an image then you get zeros in between the transform ( This can intuitively be explained:
duplicating an image is like convolving with 2 deltas (comb) of wavelength 1/2 the new image size.
so is equivalent under convolution theorem to multiplying with comb of freq size of image / 2 which gives a comb with delta every other pixel -> every other pixel is zero.

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