Digital Signal Processing Reference
In-Depth Information
Figure 5.14. An example of a second-generation real curvelet. (left) Curvelet in spatial
domain. (right) Its Fourier transform.
digital CurveletG2 construction suggests Cartesian curvelets that are translated and
sheared versions of a mother Cartesian curvelet ˆ
j
(
ϕ
ν
)
=
u
j
,
0
(
ν
), where
j
S
T
m
D
j
2
3
j
/
4
D
ϕ
k
(
t
)
=
ϕ
t
−
(5.12)
,,
θ
2
−
j
/
2
l
).
Figure 5.14 left shows a curvelet at a given scale and orientation and its Fourier
transform on the right.
(2
−
j
k
and
m
=
,
5.4.2.2 Digital Implementation
The goal here is to find a digital implementation of the DCTG2, whose coefficients
are now given by
=
f
k
=
)
e
iS
−
T
f
(
m
·
ν
d
j
j
(
S
−
1
α
j
,,
k
:
,ϕ
ν
)ˆ
ϕ
θ
ν
ν
.
(5.13)
θ
,,
R
2
This suggests the following steps to evaluate this formula with discrete data:
1. Compute the 2-D FFT to get
f
.
2. Form the windowed frequency data
f u
j
,
.
3. Apply the inverse Fourier transform.
But the last step necessitates evaluating the FFT at the sheared grid
S
−
T
θ
m
,forwhich
the classical FFT algorithm is not valid. Two implementations were then proposed
(Candes et al. 2006a), essentially differing in their way of handling the grid:
ν
A tilted grid mostly aligned with the axes of
u
j
,
(
), which leads to the unequi-
spaced FFT (USFFT)-based DCTG2. This implementation uses a nonstandard
interpolation. Furthermore, the inverse transform uses conjugate gradient iter-
ation to invert the interpolation step. This will have the drawback of a higher
