Digital Signal Processing Reference
In-Depth Information
Figure 3.8. Galaxy NGC 2997. The false colors correspond to an artificial
color map. (
See color plates.
)
bank using the “Astro” filter bank (
h
1
−
D
=
[1
,
4
,
6
,
4
,
1]
/
16,
g
1
−
D
=
δ
−
h
1
−
D
=
16 and
h
1
−
D
=
[
−
1
,
−
4
,
10
,
−
4
,
−
1]
/
g
1
−
D
=
δ
). In 2-D, this filter bank leads to a
w
1
j
,w
2
j
,w
3
wavelet transform with three orientations
j
at each scale
j
, but with the
same reconstruction property as for the IUWT; that is, the sum of all scales repro-
duces the original image:
J
3
q
j
[
k
c
0
[
k
,
l
]
=
c
J
[
k
,
l
]
+
1
w
,
l
]
.
(3.13)
j
=
1
d
=
Indeed, a straightforward calculation immediately shows that (Starck et al. 2007)
1
j
2
j
3
j
w
+
w
+
w
=
c
j
−
c
j
+
1
.
(3.14)
Therefore the sum of the three directions reproduces the IUWT detail subband at
scale
j
. Figure 3.9 shows the UWT of the galaxy NGC 2997 displayed in Fig. 3.8.
When we add the three directional wavelet bands at a given scale, we recover exactly
the starlet scale. When we add all bands, we recover exactly the original image. The
relation between the two undecimated transforms is clear.
