Digital Signal Processing Reference
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assumed to be sparsely represented in a dictionary
:
s
i
=
α
i
∀
i
=
1
,...,
N
s
,
(9.10)
where the coefficients were assumed independent with a sharply peaked (i.e., lep-
tokurtic) and heavy-tailed pdf:
e
−
λ
i
ψ
(
α
i
[
l
])
pdf
(
α
1
,...,α
N
s
)
∝
,
(9.11)
α
i
,
l
where
p
norm cor-
responding to a GGD prior. Zibulevsky and Pearlmutter (2001) used a convex
smooth approximation of the
ψ
(
α
i
[
l
]) is a sparsity-promoting penalty, for example, the
1
norm (Laplacian prior) and proposed to estimate
A
and
S
via a maximum a posteriori (MAP) estimator. The resulting optimiza-
tion problem was solved with a relative Newton algorithm (RNA) (Zibulevski
2003). This work paved the way for the use of sparsity in BSS. Note that several
other works emphasized the use of sparsity in a parametric Bayesian approach; see
Ichir and Djafari (2006), and references therein. Recently, sparsity has emerged as
an effective tool for solving underdetermined source separation problems; see Li
et al. (2006), Georgiev et al. (2005), Bronstein et al. (2005), and Vincent (2007), and
references therein.
9.3 SPARSITY AND MULTICHANNEL DATA
In this section, will see how the story of the monochannel sparse decomposition
problem described and characterized in Section 8.1.1 can be told in the language
of multichannel data. This will be a consequence of a key observation dictated by
equation (9.3).
9.3.1 Morphospectral Diversity
Extending the redundant representation framework to the multichannel case re-
quires defining what a multichannel overcomplete representation is. Let us assume
in this section that
A
N
c
×
N
s
is a
known spectral
dictionary and
=
[
ϕ
ν,
1
,...,ϕ
ν,
N
c
]
∈ R
T
is a
spatial
or
temporal
dictionary.
1
We assume that
each source
s
i
can be represented as a (sparse) linear combination of atoms in
N
×
that
=
[
ϕ
1
,...,ϕ
T
]
∈ R
;
i
.
From equation (9.3), the multichannel noiseless data
Y
can be written as
s
i
=
α
i
.Let
α
be the
N
s
×
T
matrix whose rows are
α
N
s
T
ϕ
ν,
i
ϕ
j
α
i
[
j
]
T
T
=
α
=
.
Y
A
(9.12)
=
=
i
1
j
1
Consequently, each column of
Y
reads
Y
[
.,
l
]
=
(
A
⊗
[
l
,.
]) vect(
α
)
,
∀
l
=
1
,...,
N
,
(9.13)
1
The adjectives
spectral
and
spatial
that characterize the dictionaries are not formal. Owing to the sym-
metry of the multichannel sparse decomposition problems,
A
and
have no formal difference. In
practice, and more particularly, in multi/hyperspectral imaging,
A
will refer to the dictionary of physi-
cal spectra and
to the dictionary of image/signal waveforms. In the BSS problem,
A
is unknown.
