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).e^{i.φ(h,k)}.
What will be the image formed using an inverse Fourier transformation combining moduli F_{1}(h,k) (of the Fourier transform) of image I_{1}(n,m) with phases φ_{2}(h,k) (of the Fourier transform) of a second image I_{2}(n,m)
and vice versa?
Image
1
Image 2
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.
