Effect of Translation & Rotation on the Fourier Transform

The effect of translation and rotation on the fourier transform was studied in the following manner -

Effect of translation -

1. An image of a white rectangle on a black background was taken and its fourier spectra was obtained after shifting
   the origin of the frequency plane to the center of the image.
2. Then this image of the white rectangle was translated to various positions as shown below and again the spectra was
   obtained using the above process.

It can be seen that the process of  shifting the origin of the frequency plane make the fourier transform translation invariant,
i.e. no matter where we may translate the image, the origin of the spectra can always be centered on the image. If no centering
is done then the results obtained are shown as well.

Note that the origin of the frequency plane can be translated to any position (x,y) in the image and not just
 
 
 

Image	Fourier Spectra without shifting	Fourier Spectra after shifting origin of frequency plane  to center of image

 

Effect of ratation -

1. Again an image of a white rectangle on a black background was taken and its Fourier Spectra was obtained.
2. Then the rectangle was rotated by 40 and 90 degrees and the corresponding spectra were obtained again.

It can bee seen clearly that ratating the image by any amount rotates the spectra by the same angle. So similar to translation the
Fourier transform can be made rotation invariant as well.
 
 

Image	Fourier Spectra

It may be noted that to test these no seperate program is necessary, since these properties are inherent in any Fourier Transform
so can be tested very easily using any implimentation of the Fourier Transform.

Page last updated on 28 January, 2004.	AT cse.iitd.ac.in	© Parag Chaudhuri , 2009