Digital Signal Processing Reference
In-Depth Information
Relative to other orthonormal wavelet transforms, the Haar wavelet lacks
smoothness, and although the Haar wavelet is compact in original space, it decays
slowly in Fourier space.
2.4 CONTINUOUS WAVELET TRANSFORM ALGORITHM
In practice, to compute the CWT of sampled signal
X
[
n
]
f
(
nT
s
)(
T
s
is the sam-
pling period), we need to discretize the scale space, and the CWT is computed
for scales between
a
min
and
a
max
, with a step
=
a
. Variable
a
min
must be chosen
large enough to discretize properly the wavelet function, and
a
max
is limited by the
number
N
of samples in the sampled signal
X
. For the example shown in Fig. 2.3
(i.e., Mexican hat wavelet transform),
a
min
was set to 0
66, and because the dilated
Mexican hat wavelet at scale
a
is approximately supported in [
.
−
4
a
,
4
a
], we choose
N
a
max
=
8
. The number of scales
J
is defined as the number of voices per octave
multiplied by the number of octaves. The number of octaves is the integral part
of log
2
(
a
max
/
a
min
). The number of voices per octave is generally chosen equal to
12, which guarantees a good resolution in scale and the possibility to reconstruct
the signal from its wavelet coefficients. We then have
J
=
12 log
2
(
a
max
/
a
min
) and
a
=
1).
The CWT algorithm follows.
(
a
max
−
a
min
)
/
(
J
−
Algorithm 1
Discretized CWT Algorithm
Task:
Compute CWT of a discrete finite-length signal
X
.
Parameters:
Wavelet
ψ
,
a
min
,
a
max
, and number of voices per octave. These values
depend on both
ψ
and the number of samples
N
.
Initialization:
J
=
voices
/
octave log
2
(
a
max
/
a
min
),
a
=
(
a
max
−
a
min
)
/
(
J
−
1)
=
for
a
a
min
to
a
max
with step
a
do
/
√
a
.
2. Convolve the data
X
with
¯
(
a
)
1. Compute
ψ
=
ψ
a
) (see equation (2.1)). The con-
volution product can be carried out either in the original domain or in the
Fourier domain.
3.
a
ψ
a
to get
W
(
a
,.
=
a
+
a
.
Output:
W
(
.,.
), the discretized CWT of
X
.
If discrete computations assume periodic boundary conditions, the discrete con-
volution
¯
ψ
a
X
can be performed in the Fourier domain:
¯
IFFT(FFT(
¯
ψ
a
X
=
ψ
a
)FFT(
X
))
,
where FFT and IFFT denote the fast Fourier transform and its inverse, respec-
tively. In this case, if the number of voices per octave is set to 12, it is easy to
check that the computation of the CWT requires
O
(12
N
log
2
N
)
2
operations
(Rioul and Duhamel 1992). If the convolution is done in the original (sample)
