More details about the Fourier coefficients



    In order to illustrate the meaning of the Fourier coefficients one can seek to which image corresponds a given limited number of Fourier moduli and phases. For example, what will be the image built by the inverse Fourier transform of an array of 16*16 all zero complex numbers excepted  F(-1,0)=-i & F(1,0)=i ?


k\h -8 -7 -6 -5 -4 -3 -2 -1
 0  1  2  3  4  5  6  7
-8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 -i 0 i 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Fourier trasnform of which one seeks the corresponding image (inverse Fourier transform)


Firstly, around (h,k)=(0,0) one have F(-h,-k)=F*(h,k): the resulting image will then be a real value image.
The Fourier coefficients F(-1,0) et F(1,0) are the first around the origin, in the horizontal direction: they will then create an horizontal sine wave of period equals to 16 pixels (it is the largest possible period on a 16 pixels wide image). This function varies between ±A (a constant), with a zero average since F(0,0) is null:


   
    
                                     (a)                                                                                   (b)                                                                                          (c)
Images and horizontal profiles obtained by inverse Fourier transform containing only the following Fourier coefficients: (a) F(-1,0)=-i & F(1,0)=i; (b) F(-2,0)=-i & F(2,0)=i; (c) F(-8,0)=1 : this case corresponds to the Nyquist frequency (highest possible frequency that can be displayed without aliasing (Moiré) effects), the sampling frequency of the image (periode 1 pixel) being the double of the created sine function (periode 2 pixels).

If only the Fourier coefficients F(-2,0) et F(2,0) are not zero (and complex conjugate to each other), the image obtained by inverse Fourier transform will be a sine wave of double frequency compared to the first example (images (a) & (b)). The highest horizontal frequency that could be contained in the created image corresponds to the Nyquist frequency, and is equal to 8 times the lowest frequency (images (a) & (c)).

A digital photograph such as the Chinese Musician results from the interference of the sine waves created from each Fourier coefficient couples, in each direction of the plan.