Images, sampling and frequency domain processing - Feature Extraction and Image Processing - page 47

p x

Fp u

x
(a) Sampled signal

u
(b) Transform of sampled signal

Figure 2.12 A sampled signal and its discrete transform

The Fourier transform in Figure 2.12 (b) can be used to reconstruct the original signal in
Figure 2.12 (a), as illustrated in Figure 2.13 . Essentially, the coefficients of the Fourier
transform tell us how much there is of each of a set of sinewaves (at different frequencies),
in the original signal. The lowest frequency component Fp 0 , for zero frequency, is called
the d.c. component (it is constant and equivalent to a sinewave with no frequency) and it
represents the average value of the samples. Adding the contribution of the first coefficient
Fp 0 , Figure 2.13 (b), to the contribution of the second coefficient Fp 1 , Figure 2.13 (c), is
shown in Figure 2.13 (d). This shows how addition of the first two frequency components
approaches the original sampled pulse. The approximation improves when the contribution
due to the fourth component, Fp 3 , is included, as shown in Figure 2.13 (e). Finally, adding
up all six frequency components gives a close approximation to the original signal, as
shown in Figure 2.13 (f).
This process is, of course, the inverse DFT . This can be used to reconstruct a sampled
signal from its frequency components by:

2

N -1

j ux

N

p =

=0

Fp e (2.21)

x u

u

Note that there are several assumptions made prior to application of the DFT. The first is
that the sampling criterion has been satisfied. The second is that the sampled function
replicates to infinity. When generating the transform of a pulse, Fourier theory assumes
that the pulse repeats outside the window of interest. (There are window operators that are
designed specifically to handle difficulty at the ends of the sampling window.) Finally, the
maximum frequency corresponds to half the sampling period. This is consistent with the
assumption that the sampling criterion has not been violated, otherwise the high frequency
spectral estimates will be corrupt.

2.5.2 Two-dimensional transform

Equation 2.15 gives the DFT of a one-dimensional signal. We need to generate Fourier
transforms of images so we need a two - dimensional discrete Fourier transform . This is a
transform of pixels (sampled picture points) with a two-dimensional spatial location indexed
by co-ordinates x and y . This implies that we have two dimensions of frequency, u and v ,