Digital Signal Processing Reference
In-Depth Information
remain 2. Other adaptive geometric representations, such as bandlets, are specifi-
cally designed to reach the optimal decay rate
O
(
m
−
κ
) (Le Pennec and Mallat 2005;
Peyr e and Mallat 2007).
5.5 CURVELETS AND CONTRAST ENHANCEMENT
Because some features are hardly detectable by eye in an image, we often transform
it before display. Histogram equalization is one of the most well known methods
for contrast enhancement. Such an approach is generally useful for images with a
poor intensity distribution. Because edges play a fundamental role in image under-
standing, a way to enhance the contrast is to enhance the edges. The very common
approach is to add to the image a fraction of its Laplacian: the so-called sharpening
method in image processing. However, with such a procedure, only features at the
finest scale are enhanced (linearly), and noise may be unreasonably amplified.
Since the curvelet transform is well adapted to represent images containing
edges, it is a good candidate for edge enhancement (Starck et al. 2003b). To enhance
edges in an image, each of its curvelet coefficients can be modified by multiplying
it by an appropriately chosen continuous enhancement function
. Such a function
must be even-symmetric to enhance the coefficients independently of their sign. In
Starck et al. (2003b), the following choice was proposed:
E
⎧
⎨
1
if
t
<τσ
τσ
γ
t
−
τσ
τσ
2
τσ
−
t
+
if
t
<
2
τσ
τσ
E
(
t
;
σ
):
=
t
γ
∀
t
≥
0
,
⎩
if 2
τσ
≤
t
<μ
t
if
t
≥
μ
where
σ
is the noise standard deviation. Four parameters are involved:
γ
,
,
μ
, and
τ
determines the degree of nonlinearity in the nonlinear rescaling of
the luminance and must be in [0
. Variable
γ
,
1];
≥
0 introduces dynamic range compression.
Using a nonzero
will enhance the faintest edges and soften the strongest edges at
the same time. Variable
τ
is a normalization parameter, and a value of
τ
typically
larger than 3 ensures that a coefficient with amplitude less than 3
σ
will be left un-
touched; that is, the noise will not be amplified. The parameter
is the value under
which the coefficient is enhanced. It appears natural, then, to make the value of this
parameter depend on the curvelet subband at scale
j
and orientation
μ
. Two options
to automatically set this parameter are advocated:
Variable
μ
can be derived from the noise standard deviation
σ
as
μ
=
τ
μ
σ
using
τ
μ
has an intuitive meaning and is
independent of the curvelet coefficient values and is therefore easier to set. For
instance, using
an extra parameter
τ
μ
. The advantage is that
τ
=
τ
μ
=
3 and
10 amplifies all coefficients with a SNR between
3 and 10.
μ
Variable
can also be derived from the maximum curvelet coefficient at sub-
max
j
max
j
band (
j
,
)
α
,
=
max
k
|
α
j
,,
k
|
, that is,
μ
=
ρα
,with0
≤
ρ<
1. Choosing, for
,
instance,
τ
=
3 and
ρ
=
0
.
5 will enhance all coefficients with an absolute value
between 3
σ
and half the maximum absolute value of the subband.
