Digital Signal Processing Reference
In-Depth Information
Algorithm 37
The Undecimated Wavelet Transform on the Sphere
Task:
Compute the UWTS of a discrete
X
.
Parameters:
Data samples
X
and number of wavelet scales
J
.
Initialization:
c
0
=
X
.
,
H
, and
G
numerically.
Compute the corresponding spherical harmonics transform of
c
0
.
Compute the B
3
-spline scaling function and derive
ˆ
ψ
for
j
=
0
to
J
−
1
do
1. Compute the spherical harmonics transform of the scaling coefficients:
c
j
+
1
=
c
j
H
j
.
2. Compute the inverse spherical harmonics transform of
c
j
+
1
to get
c
j
+
1
.
3. Compute the spherical harmonics transform of the wavelet coefficients:
ˆ
c
j
G
j
.
4. Compute the inverse spherical harmonics transform of ˆ
w
=
j
+
1
w
j
+
1
to get
w
j
+
1
.
Output:
W ={
w
1
,w
2
,...,w
J
,
c
J
}
, the UWTS of
X
.
10.5.1.4 Inverse Transform
If the wavelet is the difference between two resolutions, a straightforward recon-
struction of an image from its wavelet coefficients
W ={
w
1
,...,w
J
,
c
J
}
is
J
c
0
(
θ,ϑ
)
=
c
J
(
θ,ϑ
)
+
1
w
j
(
θ,ϑ
)
.
(10.27)
j
=
This reconstruction formula is the same as with the starlet algorithm.
But since the transform is redundant, there is actually no unique way to re-
construct an image from its coefficients (the filter bank design framework of Sec-
tion 3.6). Indeed, using the relations
=
H
j
[
l
c
j
+
1
[
l
,
m
]
,
m
]
c
j
[
l
,
m
]
=
G
j
[
l
w
1
[
l
,
m
]
,
m
]
c
j
[
l
,
m
]
,
(10.28)
ˆ
j
+
a least squares estimate of
c
j
from
c
j
+
1
and
w
j
+
1
gives
c
j
+
1
H
j
+
w
j
+
1
G
j
,
c
j
=
ˆ
(10.29)
where the dual filters
h
and
g
satisfy
4
H
j
2
G
j
π
H
j
h
j
2
=
H
∗
j
/
=
+
2
l
+
1
4
H
j
2
+
G
j
π
G
j
=
2
g
j
=
G
∗
j
/
.
(10.30)
2
l
+
1
