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