Fourier Theory

The tool which converts a spatial (real space) description of an image into one in terms of its frequency components is called the Fourier transform
The new version is usually referred to as the Fourier space description of the image.
The corresponding inverse transformation which turns a Fourier space description back into a real space one is called the inverse Fourier transform.
1D Case
Considering a continuous function f(x) of a single variable x representing distance.
The Fourier transform of that function is denoted F(u), where u represents spatial frequency is defined by


Note: In general F(u) will be a complex quantity even though the original data is purely real.
The meaning of this is that not only is the magnitude of each frequency present important, but that its phase relationship is too.
The inverse Fourier transform for regenerating f(x) from F(u) is given by

which is rather similar, except that the exponential term has the opposite sign.
Let's see how we compute a Fourier Transform: consider a particular function f(x) defined as

shown in Fig. 11.
 
Fig. 11 A top hat function
So its Fourier transform is: (refer formulae sheet)


In this case F(u) is purely real, which is a consequence of the original data being symmetric in x and -x. A graph of F(u) is shown in Fig. 12. This function is often referred to as the Sinc function.