Biomedical Engineering Reference
In-Depth Information
where
l
n
=
a
n
+
1
−
a
n
and
c
n
=
l
n
/
2. The two window functions
b
n
and
are
v
derived from the ramp function
r
:
0 f
t
≤−
1
1 f
t
≥
1
r
(
t
)
=
(6.26)
and
r
2
(
t
)
+
r
2
(
−
t
)
=
1
,
∀
t
∈ R
.
(6.27)
The bump function
v
is defined as:
v
(
t
)
=
r
t
ε
r
−
t
ε
t
∈
[
ε, ε
]
.
(6.28)
,
The bell function
b
n
is defined by:
r
2
t
+
l
n
/
2
ε
⎧
⎨
⎩
if
t
∈
[
−
l
n
/
2
−
ε,
−
l
n
/
2
+
ε
]
b
n
(
t
)
=
(6.29)
1
if
t
∈
[
−
l
n
/
2
+
ε,
l
n
/
2
−
ε
]
.
r
2
l
n
/
2
−
t
ε
if
t
∈
[
l
n
/
2
−
ε,
l
n
/
2
+
ε
]
An illustration of the windowing functions is provided in Fig. 6.7.
Finally, the complex-valued exponentials
e
j
,
n
are defined as:
1
√
l
n
e
−
2
i
π
j
(
x
−
a
n
)
e
j
,
n
(
x
)
=
(6.30)
.
l
n
In order to decompose a given signal
f
along directional texture components,
the Fourier transform
f
of the signal and not the signal itself is projected on the
2
e
l
n
2
e
1
0.5
b
n
(x )
v(x)
a
n+1
a
n+1
a
n
a
n
+
e
a
n
−
e
a
n+1
−
e
v(x)
Figure 6.7:
Windowing functions
b
n
and bump functions
ν
defined on the inter-
val [
a
n
−
ε,
a
n
+
1
+
ε
].
