Digital Signal Processing Reference
In-Depth Information
N
×
2
where div is the discrete divergence operator (Chambolle 2004) and
G
:
R
→
p
−
2
for
R
N
×
2
is a m
ultiv
alued function whose
i
th entry is the vector field
p
ζ
|
ζ
|
ζ
∈ R
2
and
.
A solution to equation (8.13) is computed via projected gradient descent:
|
ζ
|=
ζ
T
ζ
x
(
t
)
K
F
(
x
(
t
)
)
x
(
t
+
1)
H
∗
k
M
k
M
k
T
k
(
y
H
x
(
t
)
)
= P
+
+
μ
−
−
λ
∇
,
(8.16)
k
=
1
where
0 is a descent step-size parameter, chosen either by a line search or
as a fixed step size of moderate value to ensure convergence.
3
μ>
In fact, with the
/
|||
λ
as
p
2
k
|||
k
|||
1
+
, we observed conver-
gence. Note that because the noise is controlled by the multiresolution supports, the
regularization parameter
2
upper-bound estimate 2
H
|||
+
8
→
does not have the same importance as in standard decon-
volution methods. A much lower value is enough to remove the artifacts produced
by the wavelets and the curvelets.
λ
8.4.2.1 Example
Figure 8.4 (top) shows the Shepp-Logan phantom and the degraded image; that
is, the original image is blurred with a Gaussian PSF (full width at half maximum
(FWHM)
3.2 pixels) and then corrupted with Poisson noise. The multiresolution
supports were obtained from the Anscombe stabilized image. Figure 8.4 (bottom)
shows the deconvolution with (left) only the wavelet transform (no penalization
term) and (right) the combined deconvolution method (
=
λ
=
0
.
4).
8.5 MORPHOLOGICAL COMPONENT ANALYSIS
8.5.1 Signal and Image Decomposition
Although mathematics has its million-dollar problems, in the form of Clay Math
Prizes, there are several billion-dollar problems in signal and image processing. Fa-
mous ones include the cocktail party problem. These signal processing problems
seem to be intractable according to orthodox arguments based on rigorous mathe-
matics, and yet they keep cropping up in problem after problem.
One such fundamental problem involves decomposing a signal or image into su-
perposed contributions from different sources. Think of symphonic music, which
may involve superpositions of acoustic signals generated by many different instru-
ments - and imagine the problem of separating these contributions. More abstractly,
we can see many forms of media content that are superpositions of contributions
from different content types (i.e., the morphological components), and we can imag-
ine wanting to separate out the contributions from each. We again see that there is
a fundamental obstacle to this: for an
N
-sample signal or image
x
created by super-
posing
K
different components, we have as many as
N
K
unknowns (the contri-
bution of each content type to each pixel) but only
N
equations. But as we argued
in Section 8.1.1, if we have information about the underlying components - using
·
3
The objective (8.14) is not Lipschitz differentiable, but in practice, the iteration still works for appro-
priately chosen
μ
.
