Digital Signal Processing Reference
In-Depth Information
Figure 5.2. A few examples of ridgelets. The second, third, and fourth graphs are obtained
after simple geometric manipulations of the first ridgelet, namely, rotation, rescaling, and
shifting.
sufficient decay and satisfying the admissibility condition
ˆ
2
2
d
|
ψ
(
ν
)
|
/
|
ν
|
ν<
∞
,
(5.1)
has a vanishing mean
ψ
which holds if, say,
ψ
(
t
)
dt
=
0. We will suppose a special
so that
0
|
ˆ
ν
−
2
d
normalization about
ψ
ψ
(
ν
)
|
2
ν
=
1.
For each scale
a
>
0, each position
b
∈ R
, and each orientation
θ
∈
[0
,
2
π
), we
define the bivariate
ridgelet
ψ
:
R
2
→ R
by
a
,
b
,θ
a
−
1
/
2
ψ
a
,
b
,θ
(
t
)
=
ψ
a
,
b
,θ
(
t
1
,
t
2
)
=
·
ψ
((
t
1
cos
θ
+
t
2
sin
θ
−
b
)
/
a
)
,
(5.2)
where
t
const.
Transverse to these ridges, it is a wavelet. Figure 5.2 depicts a few examples of
=
(
t
1
,
t
2
)
∈ R
2
. A ridgelet is constant along lines
t
1
cos
θ
+
t
2
sin
θ
=
