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.

 

Fig. 12 Fourier transform of a top hat function

2D Case

If f(x,y) is a function, for example the brightness in an image, its Fourier transform is given by

and the inverse transform, as might be expected, is

Images are digitised !!

Thus, we need a discrete formulation of the Fourier transform, which takes such regularly spaced data values, and returns the value of the Fourier transform for a set of values in frequency space which are equally spaced.

This is done quite naturally by replacing the integral by a summation, to give the discrete Fourier transform or DFT for short.

In 1D it is convenient now to assume that x goes up in steps of 1, and that there are N samples, at values of x from 0 to N-1.

So the DFT takes the form

while the inverse DFT is

NOTE: Minor changes from the continuous case are a factor of 1/N in the exponential terms, and also the factor 1/N in front of the forward transform which does not appear in the inverse transform.

The 2D DFT works is similar. So for an grid in x and y we have

and

Often N=M, and it is then it is more convenient to redefine F(u,v) by multiplying it by a factor of N, so that the forward and inverse transforms are more symmetrical:

and