So far we have been processing images by looking at the grey level at each point in the image. These methods are known as spatial methods.

However, there are many ways of transforming image data into
alternative representations that are more amenable for certain
types of analysis. The most common image transform takes
spatial data and transforms it into frequency data. This is done
using the *Fourier transform*.

The Fourier transform is simply a method of expressing a function
(which is a point in some infinite dimensional vector space of
functions) in terms of the sum of its projections onto a set of basis
functions. Since an image is only defined on a closed and bounded
domain (the image window), we can assume that the image is defined as
being zero outside this window. In other words, we can assume that
the image function is *integrable* over the real line.

To see how the Fourier transform works, we will begin with a one-dimensional signal and consider a simple step function. This is equivalent to taking a horizontal slice through an image that is black on its left half and white on its right half, as shown in figure 1.

Now, a step function (or a square wave form) can be represented as a sum of sine waves of frequency , where is the frequency of the square wave, and we recall that frequency = 1/wavelength. Normally, frequency refers to the rate of repetitions per unit time, that is, the number of cycles per second (Hertz). In images we are concerned with spatial frequency, that is, the rate at which brightness in the image varies across the image, or varies with viewing angle. Figure 2 shows the sum of the first few terms in a sine wave decomposition of a square wave. This sum converges to the square wave as the number of terms tends to infinity.

From the decomposition of the signal into varying sinusoidal components we can construct a diagram displaying the amplitudes of all the sinusoids for all the frequencies. A graph of such a diagram is given in figure 3 below for the square wave.

Note that we have to consider negative frequencies (whatever that
might actually mean) so the sinusoidal component of frequency *f* and
amplitude *A _{1}* has to be split into two components of amplitude

For example, a vector **v** in 3-space is described in terms of
3 orthogonal unit vectors **i**, **j** and **k**, and we
can write **v** as the sum of its projections onto these 3 basis
vectors:

A similar process is used to calculate the Fourier transform of a function. The function is just, conceptually, a point in some vector space (although now the vector space is infinitely dimensional). Given our orthogonal basis functions, we calculate the component of our given function in each of the basis functions by calculating the inner product between the two. The standard basis functions used for Fourier transform are or, equivalently . It is the frequency that varies over the set of all real numbers to give us an infinite collection of basis functions. Since

we see that the Fourier transform has real and imaginary components. Moreover, the exponential form of basis function allows us to represent both real and complex valued functions by their Fourier transform.We can show that any two basis functions of different frequencies are orthogonal by calculating their inner product and showing that it is 0. For example, for the real case and considering only the cosine terms,

for , because the function being integrated is actually a cosine function itself () and so it has equal areas above and below the
Thus, we project our given function *f* onto our basis functions
to get the Fourier amplitudes for each frequency :

We often express *F* in polar form though:

The norm of the amplitude, is called the *
Fourier spectrum* of *f*, and the exponent is called
the *phase angle*. The square of the amplitude is just and is called the *power spectrum*
of *f*.

In many applications only the amplitude information is needed and the phase information is discarded. However, despite this common practice, phase information should not be ignored. In images, as in sound signals, phase carries considerable information [3]. Oppenheim and Lim have shown that if we construct synthetic images made from the amplitude information of one image and the phase information of another, it is the image corresponding to the phase data that we perceive, if somewhat degraded.

**Example** Consider again the square wave form shown in the figure
below.

This is a complex-values quantity, and the Fourier spectrum is given by its modulus, .

Suppose we are given two functions *f* and *g*, with Fourier
transforms *F* and *G*, and suppose that *a* and *b* are constants. Then

- The Fourier transform is linear, that is,
- Changing spatial scale inversely affects frequency and amplitude, that is,
- Shifting the function only changes the phase of the spectrum, that is,

- if
*f*(*x*) is real, then - if
*f*(*x*) is imaginary, then - if
*f*(*x*) is even, then - if
*f*(*x*) is odd, then .

The *Convolution Theorem* tells us that convolution in the spatial
domain corresponds to multiplication in the frequency domain, and vice versa.
That is,

Now an image is thought of as a two dimensional function and so the
Fourier transform of an image is a two dimensional object. Thus,
if *f* is an image, then