Back to the Chinese Musician: physical meaning of the Fourier coefficients



The Fourier Transformation can then be applied to our reference digital photograph:





                                  
Moduli map (log scale) of the Fourier coefficients of the original image                     Phases map of the Fourier coefficients of the original image

Now, one may wonder of what will be the inverse Fourier transform of these moduli and phases maps if one erases part of them ?
For example, one can set to zero all Fourier coefficients F(h,k) that are beyond a given threshold from the origin of the Fourier space and apply an inverse Fourier transform:

                                
Only the Fourier coefficients close to the origin (F(0,0)) are kept             Inverse Fourier Transform of the modified Fourier coefficients


One can also do the contrary, eliminating the Fourier coefficients close to (not far from) the origin:

                               
              Fourier coefficients close to F(0,0) are set to zero                            Inverse Fourier Transform of these modified Fourier coefficients


The comparison of these modified images with the original photograph leads to the following statements:

        • The image built using only the Fourier coefficients close to the origin is blurred, without the details of the original image but displays the relative gray levels between the different areas (white statue in the foreground and black forest in the background).

       • On the contrary, the image built using only the Fourier coefficents far from the origin presents much more details, but on the other side the information about the relative gray levels througout the image is lost: the image is characterized by an uniform gray level, strewed with details lines.


From these observations one can conclude:

       • The Fourier coefficients close to the origin (h=0;k=0) bear information about slowly varying charateristics accros the image (low frequency charateristics) and on the relative gray levels between different areas.

      • The Fourier coefficients far from the origin code for details of the image, such as the outline of the statue, the flower bed.

    If more and more Fourier coefficients are included in the inverse Fourier Transformation, the resolution of the reconstructed image will be increased (the image will be less blurred): the resolution is thought as the hability to distinguish details on the image. Then, the Fourier coefficients close to the origin are qualified of low resolution, whereas those far from the origin are labelled as high resolution.