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:

F ( u ) =<integrate from -∞ to ∞> f ( x ) exp(− j 2 π xu) dx

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:

f ( x ) =<integrate from -∞ to ∞> F ( u ) exp(j 2 π xu) du


Some references:
http://homepages.inf.ed.ac.uk/rbf/HIPR2/fourier.htm
https://www.cs.unm.edu/~brayer/vision/fourier.html
http://cns-alumni.bu.edu/~slehar/fourier/fourier.html