Digital Signal Processing Reference
In-Depth Information
k
1
= B
/
2
+
k
mod
B
/
2,
l
1
= B
/
2
+
l
mod
B
/
2,
k
2
=
k
1
−B
/
2, and
l
2
=
l
1
−B
/
2.
We compute a pixel value,
f
[
k
,
l
], in the following way:
f
1
=
(2
k
2
/
B
)
B
b
1
,
b
2
[
k
1
,
l
1
]
+
(1
−
2
k
2
/
B
)
B
b
1
+
1
,
b
2
[
k
2
,
l
1
]
f
2
=
(2
k
2
/
B
]
B
b
1
,
b
2
+
1
[
k
1
,
l
2
]
+
(1
−
2
k
2
/
B
)
B
b
1
+
1
,
b
2
+
1
[
k
2
,
l
2
]
f
[
k
,
l
]
=
(2
l
2
/
B
)
f
1
+
(1
−
2
l
2
/
B
)
f
2
,
(5.5)
cos
2
(
where
2) is the window. Of course, one might select any other
smooth, nonincreasing function satisfying
(
t
)
=
π
t
/
(0)
=
1,
(1)
=
0, with a derivative
(0)
=
0 and obeying the symmetry property
(
t
)
+
(1
−
t
)
=
1.
5.3.6 Sparse Representation by Ridgelets
The continuous ridgelet transform provides optimally sparse representation of
smooth functions that may exhibit linear singularities. As a prototype function, con-
sider the mutilated function
f
(
t
1
,
t
2
)
=
1
{
t
1
cos
θ
+
t
2
sin
θ
−
b
>
0
}
g
(
t
1
,
t
2
)
,
where
g
is compactly supported, belonging to the Sobolev space
W
2
,
0.
1
Vari-
s
>
able
f
is a smooth function away from singularity along the line
t
1
cos
θ
+
t
2
sin
θ
−
b
0. It was proved by Candes (2001) that the ridgelet coefficient sequence of
f
is as sparse as if it were without singularity. For instance, the
m
-term approxima-
tion - that is, the nonlinear approximation of
f
by keeping only the
m
largest co-
efficients in magnitude in the ridgelet series - achieves the rate of order
m
−
s
in
squared
L
2
error (Candes 2001; Cand es and Donoho 1999). This is much better
than the wavelet representation, which has only a rate
O
(
m
−
1
)
=
s
. In summary, the
ridgelet system provides sparse representation for piecewise smooth images away
from global straight edges.
We have seen previously that there are various DRTs, that is, expansions with
a countable discrete collection of generating elements. The ones we discussed here
correspond to frames. The preceding sparsity of the continuous ridgelets suggests
that a DRT can be made to give a sparse representation of discrete images with
singularities along (discrete) lines.
,
∀
5.4 CURVELETS
5.4.1 The First-Generation Curvelet Transform
In image processing, edges are curved, rather than straight lines, and ridgelets are
not able to effectively represent such images. However, one can still deploy the
ridgelet machinery in a localized way, at fine scales, where curved edges are al-
most straight lines (as illustrated in Fig. 5.10). This is the idea underlying the first-
generation curvelets (termed here
CurveletG1
) (Candes and Donoho 2002).
1
W
2
is the Sobolev space of square integrable functions whose
s
th derivative is also square integrable.
