Digital Signal Processing Reference
In-Depth Information
(c)
(a)
(b)
(d)
Figure 5.13. (a) Continuous curvelet frequency tiling. The gray area represents a wedge
obtained as the product of the radial window (annulus shown in lighter color) and
the angular window (darker color). (b) The Cartesian grid in space associated with the
construction in (a) whose spacing also obeys the parabolic scaling by duality. (c) Discrete
curvelet frequency tiling. The window
u
j,
isolates the frequency near the trapezoidal
wedge such as the ones shown in gray. (d) The wrapping transformation. The dashed
line shows the same trapezoidal wedge as in (b). The parallelogram contains this wedge
and hence the support of the curvelet. After periodization, the wrapped Fourier samples
can be collected in the rectangle centered at the origin. (
See color plates.
)
, the CurveletG2 coefficients of the 2-D function
f
(
t
)
are defined as the inner product
In continuous frequency
ν
=
f
,ϕ
j
,,
k
=
)
e
i
t
j
,
f
(
·
ν
d
α
j
,,
k
:
ν
)ˆ
ϕ
j
(
R
θ
ν
ν
.
(5.8)
k
2
R
This construction implies a few properties:
2
).
2. The effective length and width of these curvelets obey the parabolic scaling
relation (2
−
j
1. The CurveletG2 defines a tight frame of
L
2
(
R
2
−
j
/
2
)
2
.
3. The curvelets exhibit an oscillating behavior in the direction perpendicular to
their orientation.
=
width)
=
(length
=
