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The Signal Processing SE post linked below shows how the Fourier Transform applied 4 times to a 1D function returns the original function, i.e. F{ F{ F{ F{ g(x) } } } } = g(x)

Link to 1D case: Fourier transform 4 times = original function (from Bracewell book)

Can someone prove that this does or doesn't apply to 2D or higher functions? In particular, if I use this method on a grayscale image, will I get back the original image? I know edge effects could be problematic, so lets say the image is a simple repeating pattern so that it is periodic. And yes, numerical/computational FFT issues might mean the image is not exactly like the original, but even if they are similar that would be pretty cool to see....

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The n-dimensional (discrete) Fourier transform is separable, that means the dimensions can be treated independently. So the property that the FT is a fourth root of the identity applies to n-dimensional transforms too. And the property is exact, so apart from a tiny numerical round off error you won't see any deviations.

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