Digital Signal Processing Reference
In-Depth Information
In summary, the findings of this experiment allow us to conclude the following
safely:
Sparsity brings better results. Among the methods we used, only JADE is not
a sparsity-based separation algorithm. Whatever the method, separating in a
sparse representation domain enhances the separation quality: RNA, EFICA,
and GMCA clearly outperform JADE.
GMCA takes better advantage of overcompleteness and morphological diver-
sity. GMCA takes better advantage of overcomplete sparse representations than
RNA and EFICA.
9.5.3 Multichannel Image Inpainting
Similarly to the MCA (see Section (8.7.2)), GMCA can be readily extended to han-
dle multichannel missing data. Although the rest of the section holds for any overde-
termined multichannel data, without loss of generality, we consider color images
in which the observed data
Y
consist of
N
c
=
3 observed channels corresponding
to each color layer (e.g., red, green, and blue), and the number of sources is also
N
s
=
3.
GMCA inpainting seeks an unmixing scheme, through the estimation of
A
,
which leads to the sparsest sources
S
in the dictionary
, taking into account the
missing data mask
M
j
(the main diagonal of
M
j
encodes the pixel status in chan-
nel
j
; see Section 8.7.2 for more details). The resulting optimization problem to be
solved is then
N
s
2
2
+
λ
N
c
N
s
K
α
i
,
k
1
2
p
p
min
y
j
−
M
j
A
[
j
,
i
]
α
i
A
,α
1
,
1
,...,α
N
s
,
K
j
=
1
i
=
1
i
=
1
k
=
1
s
.
t
.
a
i
2
=
1
∀
i
∈{
1
,...,
N
s
}
.
(9.34)
If
M
j
=
M
for all channels, equation (9.34) becomes
N
s
K
2
Y
T
M
α
i
,
k
1
2
F
+
λ
p
p
min
−
A
α
A
,α
1
,
1
,...,α
N
s
,
K
i
=
1
k
=
1
s
.
t
.
a
i
2
=
1
∀
i
∈{
1
,...,
N
s
}
.
(9.35)
The GMCA inpainting algorithm is similar to Algorithm 34, except that the up-
date of the residual
R
(
t
)
i
k
is modified to
,
Y
M
x
(
t
−
1)
T
i
,
a
(
t
−
1)
i
R
(
t
)
i
=
−
.
,
k
k
(
i
,
k
)
=
(
i
,
k
)
Figure 9.8 (top) shows the original “Barbara” color image (in RGB space) and a
zoom. (middle) Masked color images where 90 percent of the color pixels were miss-
ing. (bottom) Recovered images with the color space-adaptive GMCA algorithm,
where
A
was estimated along with the inpainted sources. It was shown by Bobin
et al. (2009) that the adaptive color space GMCA inpainting performs much better
than inpainting each color channel separately using the algorithms of Section 8.7.
