More precisely, let's say I apply a 45 degrees rotation to an image (in the spatial domain) say, in Matlab :

```
Ir=imrotate(myImage,45,'crop');
FT_I=fft2(I);

```

In the magnitude, i.e. abs(FT\_I) is it clear that the spectrum has be rotated too.

For example, trying to reconstruct/sort of "unrotate", wrt center (with shifted DC in center), in the frequency domaine like:

```
FT_Is=fftshift(FT_I);
I_rec_FT=abs(ifftshift(imrotate(FT_Is,45,'crop'))).*exp(-i*angle(ifftshift(imrotate(FT_Is,-45,'crop'))));

```

...does not make sense (ifft2()) does not show an image that seems at all sort of unrotated.

Therefore, what exactly happens with the ***phase*** part?

[![enter image description here](https://i.stack.imgur.com/0rFwD.jpg)](https://i.stack.imgur.com/0rFwD.jpg)

"unrotated" unsuccessfully, image.

[![enter image description here](https://i.stack.imgur.com/X9bjV.jpg)](https://i.stack.imgur.com/X9bjV.jpg)

From these spectra, one can clearly see that the magnitude spectrum has been rotated by the same amount. However, the phase has a random character, and one cannot visually see anything (from the phase image). I would like to know what mathematically and intuitively happens with the phase?

Moreover, it is mentioned in this lecture that rotation in spatial (or temporal) domain results in rotation in frequency domain, by the same angle, but unfortunately no precision are given w.r.t. the phase.

[![enter image description here](https://i.stack.imgur.com/ahA61.jpg)](https://i.stack.imgur.com/ahA61.jpg)

Moreover its stated in this lecture of Verona University and I heard it from image processing professors... 

<http://www.di.univr.it/documenti/OccorrenzaIns/matdid/matdid916567.pdf>

<https://www.slideshare.net/chinnannanperiasamy/fourier-transform-44374579>