Digital Signal Processing Reference
In-Depth Information
Curvelets as just constructed are complex-valued. It is easy to obtain real-valued
curvelets by working on the symmetrized version ˆ
ϕ
j
(
r
,θ
)
+
ϕ
j
(
r
ˆ
,θ
+
π
).
Discrete Coronization
The discrete transform takes as input data defined on a Cartesian grid and outputs
a collection of coefficients. The continuous-space definition of the CurveletG2 uses
coronae and rotations that are not especially adapted to Cartesian arrays. It is then
convenient to replace these concepts by their Cartesian counterparts, that is, con-
centric squares (instead of concentric circles) and shears (instead of rotations). See
Fig. 5.13(c).
The Cartesian equivalent to the radial window ˆ
(2
−
j
) would be a
bandpass frequency-localized window, which can be derived from the difference
low-pass windows:
j
(
ν
)
=
ˆ
ν
h
j
+
1
(
h
j
(
h
(
ν
1
)
h
(
j
(
ˆ
ν
)
=
ν
)
−
ν
)
,
∀
j
≥
0
,
and ˆ
0
(
ν
)
=
ν
2
)
,
(5.9)
where
h
j
is separable,
h
j
(
h
1D
(2
−
j
1
)
h
1D
(2
−
j
ν
=
ν
ν
,
)
2
)
and
h
1D
is a 1-D low-pass filter. Another possible choice is to select these windows
inspired by the construction of Meyer wavelets (Meyer 1993; Cand es and Donoho
2004). See Cand es et al. (2006a) for more details of the construction of the Cartesian
version of ˆ
j
.
Let us now examine the angular localization. Each Cartesian corona has four
quadrants: east, north, west, and south. Each quadrant is separated into 2
j
/
2
ori-
entations (wedges) with the same areas. Take, for example, the east quadrant
(
−
π/
4
≤
θ
<π/
4). For the west quadrant, we would proceed by symmetry around
the origin, and for the north and south quadrants, by exchanging the roles of
ν
1
and
ν
2
. Define the angular window for the
th direction as
2
j
/
2
ν
−
ν
1
tan
θ
2
v
ˆ
(
ν
)
=
v
ˆ
,
(5.10)
j
,
ν
1
2
−
j
/
2
with the sequence of equispaced slopes (and not angles) tan
θ
=
for
2
j
/
2
,...,
2
j
/
2
−
=−
1. We can now define the window that is the Cartesian ana-
ϕ
log of ˆ
j
above,
u
j
,
(
ν
)
=
j
(
ˆ
ν
)ˆ
v
j
,
(
ν
)
=
j
(
ˆ
ν
)ˆ
v
j
,
0
(
S
θ
ν
)
,
(5.11)
where
S
θ
is the shear matrix:
1
0
S
θ
=
.
−
tan
θ
1
From this definition, it can be seen that
u
j
,
is supported near the trapezoidal
2
)
2
j
{
ν
=
ν
,ν
≤
ν
≤
2
j
+
1
,
−
2
−
j
/
2
≤
ν
/ν
−
θ
≤
2
−
j
/
2
}
wedge
(
tan
. The collec-
1
1
2
1
tion of
u
j
,
gives rise to the frequency tiling shown in Fig. 5.13(c). From
u
j
,
(
ν
), the
