Digital Signal Processing Reference
In-Depth Information
Finally, the reconstruction algorithm is summarized as follows:
Algorithm 9
2-D Pyramidal Wavelet Transform Reconstruction Algorithm in
Fourier Domain
Task:
Compute the inverse 2-D pyramidal wavelet transform in Fourier domain
of
,...,w
J
,
c
J
}
×
W ={
w
, the pyramidal wavelet transform of an
N
N
image
X
.
1
Parameters:
ˆ
B
3
-spline.
Initialization:
From the scaling function
φ
=
h
, and
g
numerically;
J
φ
, compute
ψ
,
=
log
2
N
.
Compute the 2-D FFT of
c
J
; name the resulting complex image
c
J
.
for
j
=
−
J
to 1, with step=
1
do
1. Compute the 2-D FFT of the wavelet coefficients at scale
j
,ˆ
w
j
.
ν
1
,ν
2
)by
g
(2
j
2
j
2. Multiply the wavelet coefficients ˆ
w
j
(
ν
1
,
ν
2
).
ν
1
,ν
2
)by
h
(2
j
3. Multiply the scaling coefficients
c
j
(
ν
1
,
2
j
ν
2
).
h
.
g
4. Compute
c
j
−
1
=
w
j
ˆ
+
c
j
Take the 2-D inverse FFT of
c
0
to get
c
0
.
Output:
c
0
=
W
X
, the reconstruction from
.
The use of a scaling function with a cutoff frequency allows a reduction of sam-
pling at each scale and limits the computational load and the memory storage.
3.7.3 Other Pyramidal Wavelet Constructions
There are other pyramidal wavelet constructions. For instance, the steerable
wavelet transform (Simoncelli et al. 1992) allows us to choose the number of di-
rections in the multiscale decomposition, and the redundancy is proportional to this
number. Do and Vetterli (2003a) studied the Laplacian pyramid using frame theory
and showed that the Laplacian pyramid with orthogonal filters is a tight frame, and
thus the reconstruction using the dual-synthesis filters is optimal and has a simple
structure that is symmetric with respect to the forward transform. We note also the
work of Unser and his collaborators on pyramidal wavelets using splines (Unser
et al. 1993; Unser 1999; Brigger et al. 1999).
3.8 GUIDED NUMERICAL EXPERIMENTS
3.8.1 Denoising by Undecimated Wavelet Transform
One of the main applications of redundant transforms is restoration and, in particu-
lar, denoising. There are numerous methods for the removal of additive noise from
an image, and the wavelet-based methods draw special interest because of their the-
oretical underpinning, their success in practice, and their fast implementation. We
here anticipate Chapter 6, devoted to denoising, and describe a few ingredients nec-
essary to carry out this experiment. Hard thresholding consists of killing (setting
to 0) all wavelet coefficients having a negligible value compared to noise and, in this
