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I have an image and its fourier transform. When I rotate it, its fourier transform rotates too, but I can't figure it out. Why does this happen? On the other hand, when I shift the image, its fourier transform doesn't change. As I know, time shifting means frequency shifting. Am Iwrong?

enter image description here

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Time shift corresponds to a phase shift in frequency domain, not a frequency shift. Since you display only the magnitude of the 2D FFT before and after the shift, you do not observe the phase shift.

$$f(x) \Rightarrow F(w) \\ f(x-a) \Rightarrow e^{-j2\pi wa} F(w) $$

Rotation in space corresponds to a rotation in frequency domain as you observe.

For proofs, see these notes on Some Properties of Fourier Transform

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