Image Processing Reference
In-Depth Information
Conversely, the Hartley transform can be calculated from the Fourier transform by:
H
(
u
) = Re[
F
(
u
)] - Im[
F
(
u
)] (2.43)
where Re[ ] and Im[ ] denote the real and the imaginary parts, respectively. This emphasises
the natural relationship between the Fourier and the Hartley transform. The image of
Figure
2.21
(c) has been calculated via the 2D FFT using Equation 2.43. Note that the
transform in Figure
2.21
(c) is the complete transform whereas the Fourier transform in
Figure
2.21
(a) shows magnitude only. Naturally, as with the DCT, the properties of the
Hartley transform mirror those of the Fourier transform. Unfortunately, the Hartley transform
does not have shift invariance but there are ways to handle this. Also, convolution requires
manipulation of the odd and even parts.
2.7.3
Introductory wavelets; the Gabor wavelet
Wavelets
are a comparatively recent approach to signal processing, being introduced only
in the last decade (Daubechies, 1990). Their main advantage is that they allow multi-
resolution analysis (analysis at different scales, or resolution). Furthermore, wavelets allow
decimation in space and frequency
simultaneously
. Earlier transforms actually allow
decimation in frequency, in the forward transform, and in time (or position) in the inverse.
In this way, the Fourier transform gives a measure of the frequency content of the whole
image: the contribution of the image to a particular frequency component. Simultaneous
decimation allows us to describe an image in terms of frequency which occurs at a position,
as opposed to an ability to measure frequency content across the whole image. Clearly this
gives us a greater descriptional power, which can be used to good effect.
First though we need a basis function, so that we can decompose a signal. The basis
functions in the Fourier transform are sinusoidal waveforms at different frequencies. The
function of the Fourier transform is to convolve these sinusoids with a signal to determine
how much of each is present. The
Gabor wavelet
is well suited to introductory purposes,
since it is essentially a sinewave modulated by a Gaussian envelope. The Gabor wavelet
gw
is given by
2
t
-
t
0
-
-
jf t
a
gw t
() =
e
e
(2.44)
0
where
f
0
is the modulating frequency,
t
0
dictates position and
a
controls the width of the
Gaussian envelope which embraces the oscillating signal. An example Gabor wavelet is
shown in Figure
2.22
which shows the real and the imaginary parts (the modulus is the
Gaussian envelope). Increasing the value of
f
0
increases the frequency content within the
envelope whereas increasing the value of
a
spreads the envelope without affecting the
frequency. So why does this allow simultaneous analysis of time and frequency? Given that
this function is the one convolved with the test data, then we can compare it with the
Fourier transform. In fact, if we remove the term on the right-hand side of Equation 2.44
then we return to the sinusoidal basis function of the Fourier transform, the exponential in
Equation 2.1. Accordingly, we can return to the Fourier transform by setting
a
to be very
large. Alternatively, setting
f
0
to zero removes frequency information. Since we operate in
between these extremes, we obtain position and frequency information simultaneously.
Actually, an infinite class of wavelets exists which can be used as an expansion basis in
signal decimation. One approach (Daugman, 1988) has generalised the Gabor function to
