Biomedical Engineering Reference
In-Depth Information
Figure 6.6: (a) Dyadic wavelet decomposition tree. (b) Wavelet packets decom-
position tree. (c) An example of an orthogonal basis tree with wavelet packets
decomposition.
the enhancement and denoising task of a noisy signal if the wavelet packets
basis are selected properly [30]. In practical applications for various medical
imaging modalities and applications, features of interest and noise properties
have significantly different characteristics that can be efficiently characterized
separately with this framework.
A fast algorithm for wavelet-packets best basis selection was introduced by
Coifman and Wickerhauser in [30]. This algorithm identifies the “best” basis for
a specific problem inside the wavelet packets dictionary according to a criterion
(referred to as a cost function) that is minimized. This cost function typically
reflects the entropy of the coefficients or the energy of the coefficients inside
each subband and the optimal choice minimizes the cost function comparing
values at a node and its children. The complexity of the algorithm is
O
(
N
log
N
)
for a signal of
N
samples.
6.2.3.3 Brushlets
Brushlet functions were introduced to build an orthogonal basis of transient
functions with good time-frequency localization. For this purpose, lapped or-
thogonal transforms with windowed complex exponential functions, such as
Gabor functions, have been used for many years in the context of sine-cosine
transforms [31].
Brushlet functions are defined with true complex exponential functions on
subintervals of the real axis as:
u
j
,
n
(
x
)
=
b
n
(
x
−
c
n
)
e
j
,
n
(
x
)
+
v
(
x
−
a
n
)
e
j
,
n
(2
a
n
−
x
)
−
v
(
x
−
a
n
+
1
)
e
j
,
n
(2
a
n
+
1
−
x
)
,
(6.25)
