Digital image warping

The Fourier transform of a signal is often plotted as magnitude versus frequency,

ignoring phase angle. This form of display has become conventional because the bulk of

the information relayed about a signal is embedded in its frequency content, as given by

the magnitude spectrum IF (u)[. For example, Figs. 2.5a and 2.5b show a square wave

and its spectrum, respectively. In this example, it just so happens that the phase function

(u) is zero for all u, with the spectrum being defined in terms of the following infinite

1  1 . 1 _

F(u) = cosu - cos.u + -cos u - cosu + ... (2.1.18)

1 1 sin7u+ '"

= sinu + -sin3u + sin5u +

Consequently, the spectrum is real and we display its values directly. Note that both

positive and negative frequencies are displayed, and the amplitudes have been halved

accordingly. An application of Fourier synthesis is shown in Fig. 2.5c, where the first

five nonzero components ofF(u) are added together. With each additional component,

the reconst'ucted function increasingly takes on the appearance of the original square

wave. The ripples that persist are a consequence of the oscillatory behavior of the

sinusoidal basis functions. They remain in the reconstruction unless all frequency com-

ponents are considered in the reconstruction. This artifact is known as Gibbs

phenomenon which predicts an overshoot/undershoot of about 22% near edges that are

not fully reconstructed [Antoniou 79].

A second example is given in Fig. 2.6. There, an arbitrary waveform undergoes

Fourier analysis and synthesis. In this case, F(u) is complex and so only the magnitude

spectrum IF(u) l is shown in Fig. 2.6b. Since the spectrum is defined over infinite fre-

quencies, only a small segment of it is shown in the figure. The results of Fourier syn-

thesis with the first ten frequency components are shown in Fig. 2.6c. As before, incor-

porating the higher frequency components adds finer detail to the reconstructed function.

The two examples given above highlight an important property of Fourier

transforms that relate to periodic and aperiodic functions. Periodic signals, such as the

square wave shown in Fig. 2.5a, can be represented as the sum of phase-shifted sine

waves whose frequencies are integral multiples of the signal's lowest nonzero frequency

component. In other ._..,r_d a periodic signal contains all the frequencies that are har-

monics of the fundamental frequency. We normally associate the analysis of periodic

signals with Fourier series rather than Fourier transforms. The Fourier series can be

expressed as the following summation

f (x) = a c(nuo)ei2nux (2.1.19)

where c (nuo) is the nth Fourier coefficient

c(nuo) = f f (x)e-i2nnuX dx (2.1.20)

2.1 FUNDAMENTALS

f (x) F (u)

(a) (b)

sin(x)

sin(5x)

(c)

23

(2.1.20) that is used to compute the Fourier coefficients must only integrate over period

X0.

Aperiodic signals do not enjoy the same compact representation as their periodic

components that are integer multiples of some fundamental frequency, an aperiodic sig-

nal must necessarily be represented as an integral over a continuum of frequencies, as in

Eq. (2.1.13). This is reflected in the spectra of Figs. 2.5b and Fig. 2.6b. Notice that the

square wave spectrum consists of a discrete set of impulses in the frequency domain,

while the spectrum of the aperiodic signal in Fig. 2.6 is defined over all frequencies. For

this reason, we distinguish Eq. (2.1.13) as the Fourier integral. It can be shown that the

Fourier series is a special case of the Fourier integral. In summary, periodic signals have

discrete Fourier components and are described by a Fourier series. Aperiodic signals

24