Biomedical Engineering Reference
In-Depth Information
L
2
f
h
[
-n
]
↓
2
L
1
f
↓
2
h
[
-n
]
f(n
)
W
2
f
g
[
-n
]
↓
2
g
[
-n
]
↓
2
W
1
f
↓
2
downsampling by 2
Figure 6.2: Illustration of orthogonal wavelet transform of a discrete signal
f
(
n
) with CMF. A two-level expansion is shown.
Wavelet coefficients
W
j
f
at scale
s
=
2
j
have a length of
N
/
2
j
and the largest
decomposition depth
J
is bounded by the signal length
N
as (sup(
J
)
=
log
2
N
).
For fast implementation (such as filter bank algorithms), a pair of conjugate
mirror filters (CMF)
h
and
g
can be constructed from the scaling function
φ
and
wavelet function
ψ
as follows:
1
√
2
φ
t
2
,φ
(
t
−
n
)
1
√
2
ψ
t
2
,φ
(
t
−
n
)
h
[
n
]
=
and
g
[
n
]
=
(6.12)
.
A conjugate mirror filter
k
satisfies the following relation:
k
(
ω
)
k
(
ω
+
π
)
2
2
k
(0)
=
2
.
+
=
2
and
(6.13)
It can be proven that
h
is a low-pass filter and
g
is a high-pass filter. The discrete
orthogonal wavelet decomposition in Eq. (6.11) can be computed by applying
these two filters to the input signal and recursively decomposing the low-pass
band, as illustrated in Fig. 6.2. A detailed proof can be found in [15].
For orthogonal basis, the input signal can be reconstructed from wavelet
coefficients computed in Eq. (6.11) using the same pair of filters, as illustrated
in Fig. 6.3.
↑
2
h
↑
2
h
f(n)
L
2
f
L
1
f
↑
2
g
↑
2
g
W
2
f
W
1
f
↑
2
upsampling by 2
Figure 6.3:
Illustration of inverse wavelet transform implemented with CMF. A
two-level expansion is shown.