Moduli and phases of the Fourier coefficients


    The Fourier tranform of the image I(n,m) is an array of complex numbers which can be represented on polar form |F(h,k)|.ei.φ(h,k). What will be the image formed using an inverse Fourier transformation combining moduli |F1(h,k)| (of the Fourier transform) of image I1(n,m) with phases φ2(h,k) (of the Fourier transform) of a second image I2(n,m) and vice versa?

                     
Image 1                                                                                                           Image 2


Image obtained by combining the moduli of the FT of image 1 with phases of the FT of image 2 Image obtained by combining the moduli of the FT of image 2 with phases of the FT of image 1


Experiment shows us that it is mainly the image from which one takes the phases that blows up!

This property can be shown from the definition of the inverse Fourier transform:


Each pixel of an image reconstructed from Fourier coefficients results from the interference (addition in the complex plane) of moduli & phases; this can be represented as an Argand scheme:


The intensity of a pixel is proportional to the length of the dashed line arrow, resulting from the interference of all the complex Fourier coefficients.


    This shows that if phases of the Fourier coefficients (i.e. angles between arrows and the horizontal direction) are modified, then the total amplitude will be dramatically changed. This behavior is also to be linked to the similarity between the Fourier coefficients moduli of objects of same nature, as it will be shown later.