Digital Signal Processing Reference
In-Depth Information
2
The Wavelet Transform
2.1 INTRODUCTION
In this chapter, we start with the continuous wavelet transform followed by imple-
mentations of it using the Morlet and Mexican hat wavelets. The Haar wavelet
is then discussed. With the continuous transform, we seek all possible, practical
resolution-related information. The Morlet wavelet, with many oscillations, is good
for oscillatory data. When the data are not very oscillatory, then the less oscillatory
Mexican hat wavelet is more appropriate.
We then move to the discrete wavelet transform. Multiresolution analysis ex-
presses well how we need to consider the direct, original (sample) domain and
the frequency, or Fourier, domain. Pyramidal data structures are used for practi-
cal reasons, including computational and storage size reasons. An example is the
biorthogonal wavelet transform, used in the JPEG-2000 image storage and com-
pression standard. The Feauveau wavelet transform is another computationally and
storage-efficient scheme that uses a different, nondyadic decomposition. We next
look at the lifting scheme, which is a very versatile algorithmic framework. Fi-
nally, wavelet packets are described. A section on guided numerical experiments in
MATLAB ends the chapter.
2.2 THE CONTINUOUS WAVELET TRANSFORM
2.2.1 Definition
The continuous wavelet transform uses a single function
(
t
) and all its dilated
and shifted versions to analyze functions. The Morlet-Grossmann definition (Gross-
mann et al. 1989) of the continuous wavelet transform (CWT) for a one-dimensional
(1-D) real-valued function
f
(
t
)
ψ
∈
L
2
(
R
), the space of all square-integrable func-
tions, is
ψ
∗
t
dt
+∞
1
√
a
−
b
¯
W
(
a
,
b
)
=
f
(
t
)
=
f
∗
ψ
a
(
b
)
,
(2.1)
a
−∞
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