Digital Signal Processing Reference
In-Depth Information
2.
Predict:
Calculate the wavelet coefficient
w
j
+
1
[
l
] as the prediction error of
c
j
[
l
] from
c
j
[
l
] using the prediction operator
P
:
c
j
[
l
]
(
c
j
)[
l
]
w
j
+
1
[
l
]
=
−P
.
(2.31)
3.
Update:
The coarse approximation
c
j
+
1
of the signal is obtained by using
c
j
[
l
]
and
w
j
+
1
[
l
] and the update operator
U
:
c
j
[
l
]
c
j
+
1
[
l
]
=
+U
(
w
j
+
1
)[
l
]
.
(2.32)
The lifting steps are easily inverted by
c
j
[
l
]
c
j
[2
l
]
=
=
c
j
+
1
[
l
]
−U
(
w
j
+
1
)[
l
]
c
j
[
l
]
(
c
j
)[
l
]
c
j
[2
l
+
1]
=
=
w
j
+
1
[
l
]
+P
.
(2.33)
2.7.1 Examples of Wavelet Transforms via Lifting
We here provide some examples of wavelet transforms via the lifting scheme.
2.7.1.1 Haar Wavelet via Lifting
The Haar wavelet transform can be implemented via the lifting scheme by taking
the prediction operator as the identity and an update operator that halves the dif-
ference. The transform becomes
c
j
[
l
]
c
j
[
l
]
w
1
[
l
]
=
−
j
+
+
w
j
+
1
[
l
]
2
c
j
[
l
]
c
j
+
1
[
l
]
=
.
All computations can be done in place.
2.7.1.2 Linear Wavelets via Lifting
The identity predictor used earlier is appropriate when the signal is (piecewise) con-
stant. In the same vein, one can use a linear predictor, which is effective when the
signal is linear or piecewise linear. The predictor and update operators now become
2
c
j
[
l
]
1]
1
(
c
j
)[
l
]
c
j
[
l
=
+
+
P
1
4
(
U
(
w
j
+
1
)[
l
]
=
w
j
+
1
[
l
−
1]
+
w
j
+
1
[
l
])
.
It is easy to verify that
1
8
c
j
[2
l
1
4
c
j
[2
l
3
4
c
j
[2
l
]
c
j
+
1
[
l
]
=−
−
2]
+
−
1]
+
1
4
c
j
[2
l
1
8
c
j
[2
l
+
+
1]
−
+
2]
,
which is the biorthogonal Cohen-Daubechies-Feauveau (Cohen et al. 1992) wavelet
transform.
