Digital Signal Processing Reference
In-Depth Information
to have a wavelet transform that maps integer values to integers. We can build an
integer version of every wavelet transform (Calderbank et al. 1998). For instance,
denoting
as the largest integer not exceeding
t
, the integer Haar transform (also
called the
S transform
) can be calculated by
t
c
j
[
l
]
c
j
[
l
]
w
j
+
1
[
l
]
=
−
w
j
+
1
[
l
]
2
(4.1)
c
j
[
l
]
c
j
+
1
[
l
]
=
+
,
where, as in the previous chapter, superscripts
o
and
e
represent odd and even in-
dexed values. The reconstruction is trivially given by
w
j
+
1
[
l
]
2
c
j
[
l
]
=
c
j
[2
l
]
=
c
j
+
1
[
l
]
−
(4.2)
c
j
[
l
]
=
c
j
[2
l
+
1]
=
w
j
+
1
[
l
]
+
c
j
[2
l
]
.
More generally, the lifting operators for an integer version of the wavelet trans-
form are
1
2
(
c
j
[
l
])
p
[
k
]
c
j
[
l
P
=
−
k
]
+
k
(4.3)
1
2
U
(
w
j
+
1
[
l
])
=
u
[
k
]
w
j
+
1
[
l
−
k
]
+
,
k
where
p
and
u
are appropriate filters associated with primal and dual-lifting steps.
For instance, the linear integer wavelet transform
1
is given by
1
2
c
j
[
l
]
1]
+
1
2
c
j
[
l
]
c
j
[
l
w
1
[
l
]
=
−
+
+
j
+
(4.4)
1
4
(
1
2
c
j
[
l
]
=
+
−
+
w
j
+
1
[
l
])
+
c
j
+
1
[
l
]
w
j
+
1
[
l
1]
.
More filters can be found in the work of Calderbank et al. (1998). In lossless
compression of integer-valued digital images, even if there is no filter that consis-
tently performs better than all other filters on all images, it has been observed that
the linear integer wavelet transform generally performs better than integer wavelet
transforms using other filters (Calderbank et al. 1998).
4.2.2 Wavelet Transform on Irregular Grid
A wavelet transform on irregularly sampled data can be performed by introduc-
ing weighted inner products (Sweldens and Schr oder 1996; Daubechies et al. 1999),
1
This integer wavelet transform is based on a symmetric, biorthogonal wavelet transform built from the
interpolating Deslauriers-Dubuc scaling function, where both the high-pass filter and its dual have two
vanishing moments (Mallat 2008).
