Biomedical Engineering Reference
In-Depth Information
brushlet basis functions:
f
=
f
n
,
j
u
n
,
j
,
(6.31)
n
j
f
n
,
j
being the brushlet coeffi-
with
u
n
,
j
being the brushlet basis functions and
cients.
The original signal
f
can then be reconstructed by:
f
n
,
j
w
n
,
j
,
f
=
(6.32)
n
j
where
w
n
,
j
is the inverse Fourier transform of
u
n
,
j
, which is expressed as:
l
n
e
2
i
π
a
n
x
e
i
π
l
n
x
(
−
1)
j
b
n
(
l
n
x
−
j
)
−
2
i
sin(
π
l
n
x
)
v
(
l
n
x
+
j
)
(6.33)
with
b
n
and
v
being the Fourier transforms of the window functions
b
n
and
v
.
Since the Fourier operator is a unitary operator, the family of functions
w
n
,
j
is also an orthogonal basis of the real axis. We observe here the wavelet-like
structure of the
w
n
,
j
functions with scaling factor
l
n
and translation factor
j
.An
illustration of the brushlet analysis and synthesis functions is provided in Fig. 6.8.
Projection on the analysis functions
u
n
,
j
can be implemented efficiently by a
folding operator and Fourier transform. The folding technique was introduced
by Malvar [31] and is described for multidimensional implementation by Wick-
erhauser in [21]. These brushlet functions share many common properties with
Gabor wavelets and wavelet packets regarding the orientation and frequency
selection of the analysis but only brushlet can offer an orthogonal framework
w
n
,
j
(
x
)
=
10
−1
10
−2
2
5
0
0
time
frequency
−
2
−
5
a
-
e
a
+
+
e
-
j
j
1
l
n
n
(a)
(b)
Figure 6.8: (a) Real part of analysis brushlet function
u
n
,
j
. (b) Real part of
synthesis brushlet function
w
n
,
j
.
