Joint Multichannel Deconvolution and Blind Source Separation

Abstract

Blind Source Separation (BSS) is a challenging matrix factorization problem that plays a central role in multichannel imaging science. In a large number of applications, such as astrophysics, current unmixing methods are limited since real-world mixtures are generally affected by extra instrumental effects like blurring. Therefore, BSS has to be solved jointly with a deconvolution problem, which requires tackling a new inverse problem: deconvolution BSS (DBSS). In this article, we introduce an innovative DBSS approach, called DecGMCA, based on sparse signal modeling and an efficient alternative projected least square algorithm. Numerical results demonstrate that the DecGMCA algorithm performs very well on simulations. It further highlights the importance of jointly solving BSS and deconvolution instead of considering these two problems independently. Furthermore, the performance of the proposed DecGMCA algorithm is demonstrated on simulated radio-interferometric data.

Full text

Joint Multichannel Deconvolution and Blind Source Separation
Ming Jiang∗, J´erˆome Bobin∗,and Jean-Luc Starck∗
Abstract. Blind Source Separation (BSS) is a challenging matrix factorization problem that plays a central role
in multichannel imaging science. In a large number of applications, such as astrophysics, current
unmixing methods are limited since real-world mixtures are generally affected by extra instrumental
effects like blurring. Therefore, BSS has to be solved jointly with a deconvolution problem, which
requires tackling a new inverse problem: deconvolution BSS (DBSS). In this article, we introduce
an innovative DBSS approach, called DecGMCA, based on sparse signal modeling and an efficient
alternative projected least square algorithm. Numerical results demonstrate that the DecGMCA al-
gorithm performs very well on simulations. It further highlights the importance of jointly solving BSS
and deconvolution instead of considering these two problems independently. Furthermore, the per-
formance of the proposed DecGMCA algorithm is demonstrated on simulated radio-interferometric
data.
Key words. Multichannel restoration, Blind Source Separation, Deconvolution, Sparsity
AMS subject classifications. 68U10
1. Introduction. In many imaging applications, such as astrophysics, the advent of multi-
wavelength instruments mandates the development of advanced signal/image processing tools
to extract relevant information. It is especially difficult to identify and extract the character-
istic spectra of sources of interest when they are blindly mixed in the observations.
For this reason, multichannel data has generated interest in study of the Blind Source
Separation (BSS) problem. In the BSS model, suppose we have Ncchannels, each channel
{xi}1≤i≤Ncdelivers an observation which is the linear combination of Nssources and contam-
inated by the noise {ni}1≤i≤Nc. More precisely,
(1) ∀i∈ {1,2,· · · ,Nc},xi=
Ns
X
j=1
Aij sj+ni,
where the matrix Ais the mixing matrix. BSS problems appear in a variety of applications,
such as in astronomy [4], neuroscience [15,21], or medical imaging [16]. Various BSS meth-
ods have been proposed in the literature. Depending on the way the sources are separated,
most BSS methods can be divided into two main families. The first family is based on a sta-
tistical approach, called Independent Component Analysis (ICA), such as FastICA [11] and
its derivatives [14]. These approaches assume that the sources are statistically independent
and build upon statistical independence as a discriminative measure of diversity between the
sources. However, they are generally not robust at low Signal to Noise Ratio (SNR). In the
last decade, a second family of approaches, based on morphological diversity and sparsity,
has blossomed. The pioneering work of Zibulevsky et al. [26] introduced a sparsity prior to
distinguish sources. The sparsity constraint can be imposed directly on the image pixel values
or on its coefficients in a given sparse representation in a different space, the GMCA method
∗Service d’Astrophysique, CEA Saclay, France (ming.jiang@cea.fr,jerome.bobin@cea.fr,jean-luc.starck@cea.fr).
1
arXiv:1703.02650v2 [stat.AP] 14 May 2017

2 MING JIANG, J´
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proposed by Bobin et al. [3] and its derivative AMCA [2] enforce the sparsity constraint in
the sparse domain and employ an adaptive threshold strategy to extract sparse sources. It
has been shown that these methods are more tolerant to non-independent sources and more
robust to noisy data than ICA based approaches.
BSS alone is a complex inverse problem since it is a non-convex problem matrix factor-
ization problem. This problem becomes even more challenging when the data are not fully
sampled or blurred due to the Point Spread Function (PSF). For instance, in radio-astronomy
instruments such as the future Square Kilometre Array (SKA [8]) provide an incomplete
sampling in Fourier space, leading to an ill-conditioned system. Furthermore, the PSF is not
identical for the different channels of the instrument, which degrades the observed image chan-
nels differently. Therefore, for such instruments, apart from the source separation problem,
we also need to restore images from masking or blurring effects. Mathematically, the masking
or the blurring can be modeled as a linear operator. Hence, we have to jointly solve both a
deconvolution and a BSS problem, yielding a Deconvolution Blind Source Separation
(DBSS) problem.
Deconvolution BSS problems have not been extensively studied in the literature. To the
best of our knowledge, solving BSS problems from incomplete measurements has only been
investigated in the framework of Compressed Sensing (CS) in work by Kleinsteuber [12].
The CS-BSS problem can be considered as a specific case of DBSS with the linear operation
specialized in masking. However, the proposed approach only applies to Compressed Sensing
measurements, which is a very specific case of the DBSS problem we investigate in this paper.
In CS framework as well, the Compressive Source Separation (CSS) method proposed in [10]
processes the source separation of hyperspectral data, but under the assumption that the
mixing matrix is known. In the framework of ICA, the DBSS algorithm can be recast as a
special case of BSS from convolutive mixture models [9,13,22]. However, the methods that
have been introduced to unmix convolutive mixtures provide an estimate of the mixing matrix
but not a joint estimation with the sources. These methods are limited to well-conditioned
convolution operators, which excludes the ill-posed convolution operators we consider in this
paper.
Contribution. In this paper, we introduce a novel sparsity-based BSS algorithm that
jointly addresses blind source separation and deconvolution. The proposed algorithm, coined
DecGMCA, allows tackling blind separation problems that involve linear, potentially ill-
conditioned or ill-posed convolution operators. This includes incomplete measurements or
blurring effect.
This paper is organized as follows: we will present our DBSS model in section 2. In
section 3 we will introduce our deconvolution BSS method, called DecGMCA. Numerical
experiments follow in section 4 and we will demonstrate the performance of our method. Then
in section 5 we will apply our method to realistic interferometric data to give an illustration
of the technique.
Notation. Before moving to the problem setup, we will introduce the notation which
will be used hereafter. Matrices and vectors are denoted in boldface uppercase letters and
lowercase letters such as Xand xrespectively. The individual entry of a matrix or a vector

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 3
is denoted in normal font such as Xi,j or xi.||x|| denotes the length or `2norm of a vector
x.Xtdenotes nonconjugate transpose of X, while X∗denotes the conjugate transpose of X.
The Fourier transform of Xis defined as ˆ
X. For any matrix X,||X||pdenotes the p-norm
of the matrix. Specifically, ||X||Fis called the Frobenius norm of the matrix. The identity
matrix of size nis denoted by In. In addition, denotes the Hadamard piecewise product
and ∗denotes convolution. Finally, as for the proximal operation, Y= Thλ(X) returns the
thresholding of the input Xwith threshold λ(soft or hard thresholding is not specified, it will
be clarified if needed).
More precisely, in terms of our DBSS problem, given Nssources of length Np, sources are
written as a stack of row vectors (two dimensional source will be aligned into row vector),
denoted as S= (Si,j )1≤i≤Ns
1≤j≤Np
= [st
1,st
2,· · · ,st
Np]t, therefore {si}1≤i≤Npdenotes the i-th source.
In order to simplify the presentation of the model hereafter, the source matrix will also be
written as concatenated column vectors such as S= [s1,s2,· · · ,sNp], where {sj}1≤j≤Npis a
column vector of all sources at position j. Assuming we have Ncchannels, the mixing matrix
is written as a stack of row vectors such as A= (Ai,j )1≤i≤Nc
1≤j≤Ns
= [at
1,at
2,· · · ,at
Nc]t, where
{ai}1≤i≤Ncis a row vector of the contribution of all sources at channel index i. The kernel
H= (Hi,j )1≤i≤Nc
1≤j≤Np
, which takes into account the masking or the blurring effect due to the
PSF, is written as H= [ht
1,ht
2,· · · ,ht
Nc]t, where {hi}1≤i≤Ncis a row vector of the PSF at
channel index i. Finally, the observation is denoted as Y= (Yν,k )1≤ν≤Nc
1≤k≤Np
= [yt
1,yt
2,· · · ,yt
Nc]t,
where {yi}1≤i≤Ncis a row vector giving the observation at channel index i.
2. The DBSS Problem. As shown in Equation (1), we have Ncchannels available for
the observation and each observation channel is assumed to be a mixture of Nssources (each
source is of length Np). The columns of mixing matrix Adefine the contribution of the
sources in the mixture and are regarded as spectral signatures of the corresponding sources.
We assume herein that the number of channels is greater than or equal to the number of
sources: Nc≥Nsand Ais a full-rank matrix. Besides the mixing stage, the observations are
degraded by a linear operator H:
- On the one hand, the data may be sub-sampled and this issue is related to the com-
pressed sensing data. Hcan be therefore interpreted as a sub-sampling matrix or a
mask.
- On the other hand, the data may be blurred by a PSF and His a convolution operator.
Moreover, the observed data are contaminated with the additive noise N. Hence, the proposed
imaging model can be summarized as follows:
(2) ∀ν∈ {1,2,· · · ,Nc},yν=hν∗xν=hν∗

Ns
X
j=1
Aν,j sj
+nν=hν∗(aνS) + nν.
If we apply a Fourier transform on both sides of the above equation, our model can be
more conveniently described in the Fourier domain. We denote ˆ
Si,·the Fourier transform of
the i-th source (for a two dimensional source, a two dimensional Fourier transform is applied
and the Fourier coefficients are aligned as a row vector). For simplicity, ˆ
Sis defined as a stack

4 MING JIANG, J´
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of row vectors of the Fourier coefficients of all sources. Using the same convention, the Fourier
transform of the observation, the kernel and the noise can be defined respectively as ˆ
Y,ˆ
Hand
ˆ
N. The matrix ˆ
Swill be written as concatenated column vectors such as ˆ
S= [ˆ
s1,ˆ
s2,· · · ,ˆ
sNp],
or a stack of row vectors [ˆ
st
1,ˆ
st
2,· · · ,ˆ
st
Ns]t, while ˆ
Y= [ˆ
yt
1,ˆ
yt
2,· · · ,ˆ
yt
Nc]t,ˆ
H= [ˆ
ht
1,ˆ
ht
2,· · · ,ˆ
ht
Nc]t
and ˆ
N= [ˆ
nt
1,ˆ
nt
2,· · · ,ˆ
nt
Nc]tare written as stacks of row vectors. Thus, in the Fourier domain,
our model can be recast as follows:
(3) ∀ν∈ {1,2,· · · ,Nc},ˆ
yν=ˆ
hνˆ
xν=ˆ
hν

Ns
X
j=1
Aν,jˆ
sj
+ˆ
nν=ˆ
hνaνˆ
S+ˆ
nν,
where denotes the Hadamard product. More precisely, at frequency kof channel ν, the
entity of ˆ
Ysatisfies:
(4) ˆ
Yν,k =ˆ
Hν,kaνˆ
sk+ˆ
Nν,k.
This forward model applies to a large number of applications. For instance, in radioastron-
omy or in medicine, instruments such as a radio interferometer or a Magnetic Resonance Imag-
ing (MRI) scanner actually measure Fourier components. The observations are sub-sampled
or blurred during the data acquisition and sources of interest are mixed blindly. Therefore,
blind source separation from degraded data has generated interest in both domains.
Sparsity has been shown to highly improve the separation of sources [3]. We want to
utilize this concept to facilitate the source separation. To solve the DBSS problem, we assume
the Nssources forming the source matrix Sare sparse in the dictionary Φ. Namely,
(5) ∀i∈ {1,2,· · · ,Ns};si=αiΦ,
where Φis also called the synthesis operator, which reconstructs siby assembling coefficients
αi. Conversely, the analysis operator Φtdecomposes the source by a series of coefficients αi
attached to atoms: αi=siΦt. In addition, the dictionary Φis supposed to be (bi-)orthogonal
in the above equation and the algorithm hereafter. If a dictionary is (bi-)orthogonal, it sat-
isfies ΦΦt=ΦtΦ=I. However, an overcomplete or a redundant dictionary allows for more
degrees of freedom which helps signal/image restoration. Such a dictionary satisfies the exact
reconstruction property ΦtΦ=I, but ΦΦtis not guaranteed to be an identity matrix. For-
tunately, for most redundant dictionaries, such as curvelets [19], ΦΦtis diagonally dominant
and such dictionaries can be considered as a good approximation of a tight frame. Although
the demonstration and the algorithm hereafter are based on the (bi-)orthogonal dictionary,
they are good approximations when tight frame dictionaries are used.
Therefore, under the sparsity constraint, our problem can be written in Lagrangian form
as follows:
(6) min
S,A
1
2
Nc
X
ν
Np
X
k
|| ˆ
Yν,k −ˆ
Hν,kaνˆ
sk||2
2+
Ns
X
i
λi||siΦt||p,
where the `pnorm, which can be replaced by the `0norm or the `1norm, enforces the sparsity
constraint in the dictionary Φ, while the quadratic term guarantees the data fidelity. Our goal

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 5
is to recover the sources Sand the mixing matrix Aby jointly solving a deconvolution and a
BSS problem. However, such problems are challenging, as BSS integrates deconvolution for
multichannel data. First of all, the DBSS problem involves non-convex minimization, hence
only a critical point can be expected. Then, the convolution kernel ˆ
Hcan be ill-conditioned
or even rank deficient. As a consequence, the deconvolution can be unstable if not well
regularized.
3. DecGMCA: a sparse DBSS method. The GMCA framework proposed by Bobin et
al. [3] is an efficient BSS method taking advantage of morphological diversity and sparsity
in a transformed space. Compared to ICA-based methods, it has also been demonstrated to
be more robust to noisy data. However, GMCA does not take deconvolution into account,
which is limited in practical applications. Therefore, a more rigorous BSS method should be
conceived for the DBSS problem.
In this section, we will firstly present several ingredients of our method before moving
onto the whole algorithm. Then, we will discuss the initialization of the algorithm and the
choice of parameters. We will discuss the convergence at the end of this section.
3.1. Two-stage estimate. As the original problem (6) is non-convex due to indeterminacy
of the product Aˆ
S, reaching the global optimum can never be guaranteed. In the spirit of
BCR [23], the product Aˆ
Scan be split into two variables Aand ˆ
S, which allows the original
problem to be split into two alternating solvable convex sub-problems: estimate of Sknowing
A
(7) min
S
1
2
Nc
X
ν
Np
X
k
|| ˆ
Yν,k −ˆ
Hν,kaνˆ
sk||2
2+
Ns
X
i
λi||siΦt||p,
and estimate of Aknowing S
(8) min
A
1
2
Nc
X
ν
Np
X
k
|| ˆ
Yν,k −ˆ
Hν,kaνˆ
sk||2
2.
3.1.1. Estimate of S. Problem (7) is convex but does not generally admit an explicit
solution. To compute its minimizer requires resorting to iterative algorithms such as proximal
algorithms (e.g. FISTA [1], Condat-Vu splitting method ([7,25]) to only name two). However,
these methods are very computationally demanding. In most cases the least squares method
is sufficient to have a computationnally cheap rough estimate of the sources. Therefore, in the
spirit of the GMCA algorithm, we will employ a projected least-squares estimation strategy.
Assuming faν,ˆ
sk=1
2
Nc
P
ν
Np
P
k
|| ˆ
Yν,k −ˆ
Hν,kaνˆ
sk||2
2. In order to estimate ˆ
Swith respect to A,
we should let the deviation of faν,ˆ
skof ˆ
skvanish: ∂f(aν,ˆ
sk)
∂ˆ
sk= 0. In other words,
∂f aν,ˆ
sk
∂ˆ
sk=
Nc
X
ν
Np
X
kˆ
Hν,kaνtˆ
Yν,k −ˆ
Hν,kaνˆ
sk
(9a)
=
Nc
X
ν
Np
X
k
ˆ
Hν,k ˆ
Yν,kat
ν−
Nc
X
ν
Np
X
kˆ
Hν,kaνtˆ
Hν,kaνˆ
sk= 0.(9b)

6 MING JIANG, J´
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For each position k, noticing that ˆ
Hν,k is a scalar, we have
Nc
X
ν
ˆ
Hν,k ˆ
Yν,kat
ν− Nc
X
νˆ
Hν,kaνtˆ
Hν,kaν!ˆ
sk= 0(10a)
⇒ˆ
sk= Nc
X
ν
(ˆ
Hν,kaν)t(ˆ
Hν,kaν)!−1Nc
X
ν
ˆ
Hν,k ˆ
Yν,kat
ν.(10b)
In this article, the convolution kernels ˆ
Hcan be ill-conditioned or rank deficient. In
this setting, the least-square estimate is either not defined if the inverse of the kernel is
unbounded or highly unstable with an amplified level of noise. Therefore, we propose resorting
to a Tikhonov regularization of the least-square estimate in Fourier space to stabilize the
multichannel deconvolution step:
(11) ˆ
sk= Nc
X
νˆ
Hν,kaνtˆ
Hν,kaν+0INs!−1Nc
X
ν
ˆ
Hν,k ˆ
Yν,kat
ν,
where INsis an identity matrix of size Nswith Nssources. 0INsis a regularization term that
controls the condition number of the system. Since the condition number is dependent on the
Fourier frequency k(as our working space is Fourier space, kcorresponds to the frequency in
Fourier space), denoting P(k) =
Nc
P
νˆ
Hν,kaνtˆ
Hν,kaν, we choose 0to be proportional to
the spectral norm of matrix Psuch that 0(k) = ||P(k)||2with the regularization parameter
to be discussed in subsection 3.2.2.
Unfortunately, the noise is not cleanly removed and artifacts are present after the above
procedure. The next step consists in enforcing the sparsity of the sources in the wavelets
space, which yields the following estimate of the sources:
(12) ∀i∈ {1,2,· · · ,Ns};si=ThλisiΦtΦ,
where Thλi(·) denotes the thresholding operation that will be discussed in subsection 3.2.3.
Besides, as mentioned before, Φis a (bi-)orthogonal dictionary during the demonstration.
In summary, equipped with the wavelet shrinkage, the multichannel hybrid Fourier-wavelet
regularized deconvolution presented above performs regularization in both the Fourier and
wavelet spaces and it can be interpreted as a multichannel extension of the ForWaRD decon-
volution method [17].
Using such a projected regularized least-square source estimator is motivated by its lower
computational cost. If this procedure provides a more robust separation process, it does not
provide an optimal estimate of the sources. Consequently, in the last iteration, the problem
is properly solved so as to provide a very clean estimate of S. Solving (7) is then carried out
with a minimization method based on the Condat-Vu splitting method ([7,25]).
3.1.2. Estimate of A. Similarly, we derive the mixing matrix Afrom ˆ
Sby vanishing the
deviation of faν,ˆ
skof aν:∂f (aν,ˆ
sk)
∂aν= 0. Having noticed that ˆ
skis complex valued, we

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 7
obtain:
∂f aν,ˆ
sk
∂aν
=
Nc
X
ν
Np
X
kˆ
Yν,k −ˆ
Hν,kaνˆ
skˆ
Hν,kˆ
sk∗
(13a)
=
Nc
X
ν
Np
X
k
ˆ
Hν,k ˆ
Yν,k ˆ
sk∗−
Nc
X
ν
Np
X
k
aνˆ
Hν,kˆ
skˆ
Hν,kˆ
sk∗= 0.(13b)
For each frequency channel ν, noticing that ˆ
Hν,k is a scalar, the final expression is given
by
Np
X
k
ˆ
Hν,k ˆ
Yν,k ˆ
sk∗−aν

Np
X
kˆ
Hν,kˆ
skˆ
Hν,kˆ
sk∗
= 0(14a)
⇒aν=

Np
X
k
ˆ
Hν,k ˆ
Yν,k ˆ
sk∗


Np
X
kˆ
Hν,kˆ
skˆ
Hν,kˆ
sk∗

−1
.(14b)
Since NpNc, the least-square term of update A, which involves summations over all
the Npsamples at channel ν, is not rank deficient and robust to be inverted. The estimate of
Adoes not require an extra regularization parameter. As the `p-norm constraint imposes a
minimal norm of S, the global optimization problem may diverge to an unexpected solution
as S=0and A=∞. Therefore, it is necessary to renormalize the columns of Aas unit
vectors before updating next S:
(15) ∀j∈ {1,2,· · · ,Ns};¯
aj=aj
||aj||.
3.2. DecGMCA algorithm. Assembling the two-stage estimates, we summarize our Deconvolved-
GMCA (DecGMCA) algorithm presented in Algorithm 1:
3.2.1. Initialization. For the initialization of A, we can simply take a random value as
the first guess. Apart from random initialization, we can also utilize different strategies for
the initialization following the specific form of the data:
- If the data are not sub-sampled, we can utilize SVD decomposition to help the ini-
tialization. Due to the size of ˆ
Y(NcNp), we perform a thin SVD decomposition
such that ˆ
Y=UNcΣNcV∗, the matrix Uis thus of size Nc×Nc,ΣNcis Nc×Nc
diagonal matrix and Vis of size Nc×Np. Then, the first guess of Ais set to the first
Nsnormalized columns of U.
- If the data are sub-sampled, the discontinuity effect of the data affects the SVD ini-
tialization. In order to reduce such discontinuity, we perform a matrix completion
scheme using the SVT algorithm [5] before the initialization:
(16) min || ˆ
X||∗s.t. || ˆ
Y−Hˆ
X||2
F< err,
where || · ||∗denotes the nuclear norm.

8 MING JIANG, J´
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Algorithm 1 Deconvolved-GMCA (DecGMCA)
1: Input: Observation ˆ
Y, operator ˆ
H, maximum iterations Ni,(0)
2: Initialize A(0)
3: for i= 1,...,Nido
4: ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗Estimating S∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
5: for k= 1,...,Npdo
6: •Compute the current ˆ
skwith respect to the current estimate of A(i):
7: (ˆ
sk)(i)=Nc
P
νˆ
Hν,ka(i)
νtˆ
Hν,ka(i)
ν+0IN−1Nc
P
ν
ˆ
Hν,k ˆ
Yν,k(a(i)
ν)t
8: end for
9: •Obtain sources in image space by inversing FFT:
10: S(i)= Re(FT−1(ˆ
S(i)))
11: ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗Sparse thresholding ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
12: for j= 1,...,Nsdo
13: •Apply sparsity prior in wavelet space and estimate the current coefficients by thresh-
olding:
14: αj= Thλ(i)
j
(s(i)
jΦt)
15: •Obtain the new estimate of Sby reconstructing treated coefficients
16: s(i)
j=αjΦ
17: end for
18: •Obtain sources in Fourier space by FFT:
19: ˆ
S(i)= FT(S(i))
20: ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗Estimating A∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
21: for ν= 1,...,Ncdo
22: •Compute the current aνwith respect to the current estimate of S(i):
23: a(i)
ν= Np
P
k
ˆ
Hν,k ˆ
Yν,k (ˆ
sk)(i)∗!Nc
P
kˆ
Hν,k(ˆ
sk)(i)ˆ
Hν,k(ˆ
sk)(i)∗−1
24: end for
25: •Update the threshold λand 
26: end for
27: ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗Ameliorating S∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
28: Solve Equation (7) with respect to Ausing proximal methods
29: return A,S
3.2.2. Regularization parameters. As shown in Equation (11), the Tikhonov term is of
great importance to regularize the deconvolution procedure. Therefore, this section con-
cerns the choice of the regularization parameter . Recalling that 0=||P(k)||, where
P(k) =
Nc
P
νˆ
Hν,kaνtˆ
Hν,kaν, the Tikhonov term 0INswill control the condition num-
ber. Intuitively, a larger makes the system more regularized, but the detailed information
will be smooth, yielding a loss of precision. In contrast, a smaller can conserve more details,
but the system will not be sufficiently regularized and the deconvolution will be unstable.

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 9
Therefore, the parameter is fixed so as to provide a trade-off between precision and
stability. During the first iterations, the source Sis not well estimated and the estimation
is vulnerable to the amplified noise. Thus, we apply a large to mainly regularize the ill-
conditioned system and ensure that the solution will not get stuck in a local optimum. As the
algorithm goes on, the sources Stend to converge towards a more stable solution. Then, we
decrease to improve the estimate. However, can never be decreased to zero as zero regular-
ization will make the estimate unstable again. In practice, decays linearly or exponentially
from 10−1to a very small non-zero value, for example 10−3. Besides, since the choice of final 
is dependent on the global condition number of the system and the tolerance of the precision,
it should be adapted to the specific case in practice.
3.2.3. Thresholding strategy. We didn’t specify the `pnorm in the optimization prob-
lem (6). Indeed, the `pnorm can be either `0or `1. The `0norm problem using hard-
thresholding gives an exact sparse solution, while the `1norm problem using soft-thresholding,
leading to a convex sub-problem, can be regarded as a relaxation of the `0norm. Nevertheless,
hard-thresholding often converges to a better result in practice as it does not produce bias.
Therefore, the sparsity constraint is written as an `0norm regularizer instead of an `1norm.
The sparsity parameters {λi}1≤i≤Nscan be implicitly interpreted as thresholds in Equa-
tion (12). In addition, the choice of thresholds {λi}1≤i≤Nsis a vital point in the source
separation process. The DecGMCA algorithm utilizes an adapted thresholding strategy. The
initial thresholds are set to high values so that the most significant features of the sources
can be extracted to facilitate source separation. In addition, the high thresholds prevent the
algorithm from being trapped on local optima. When the most discriminant features of the
sources are extracted following the high thresholds, the sources are separated with high prob-
ability. Then, to retrieve more detailed information about the sources, the thresholds decrease
towards the final values. The final thresholds can be chosen as τσiwith σithe standard devi-
ation of noise of the i-th source. In practice, Median Absolute Deviation (MAD) is a robust
empirical estimator for Gaussian noise. The value τranges between 2 ∼4. In practice, there
are many ways to chose the decreasing function of the threshold. We present our strategy
of decreasing threshold called “percentage decreasing threshold”, which is the most robust
according to our tests. Assuming at iteration i, as for an ordered absolute wavelet coefficient
set of the j-th source |αj|, the current threshold is selected as the p(i)
j-th element in |αj|such
as λ(i)
j=|αj|hp(i)
ji, where p(i)
jsatisfies:
(17) ∀j∈ {1,2,· · · ,Ns};p(i)
j= (1 −p(0)
j)(i−1)
Ni−1+p(0)
j!card(|αj| ≥ τσj),
with p(0)
jthe initial percentage (for example 5%). Hence, p(i)
jincreases linearly from p(0)
jto
100%, or the thresholds decay until τσj.
3.2.4. Convergence analysis. It is well-known that BSS problems are non-convex matrix
factorization problems. Therefore, one can only expect to reach a critical point of the problem
(6). Let us recall that the DecGMCA algorithm is built upon the Block Coordinate Relaxation
minimization procedure where the two blocks are defined by the source matrix Sand the

10 MING JIANG, J´
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OME BOBIN, JEAN-LUC STARCK
mixing matrix A. More generally, the DBSS problem can be described by the following
generic formulation:
(18) min
A,Sf(A) + g(S) + h(AS)
where fstands for the `2ball constraint on the columns of the mixing matrix, gfor the `1-norm
penalization and hfor the data fidelity term. The convergence of BCR for this type of matrix
factorization problem has been investigated by Paul Tseng in [23]. In this article, Tseng
introduces conditions on the minimization problem that guarantees that the BCR alternate
minimization procedure converges to a critical point of (6). We previously emphasized in
[18] that these convergence conditions apply to algorithms based on GMCA as long as the
regularization parameters are kept fixed (i.e. the thresholds {λj}and ). In the DecGMCA
algorithms, these parameters evolve; this evolution is key to increase the robustness of the
algorithm with respect to local stationary points. However, it has been noticed in [2] that
these parameters tend to stabilize at the end of GMCA-like algorithms. In that case, the
algorithm tends to be close to a regime where convergence is guaranteed by Tseng’s paper.
The same argument applies to the proposed DecGMCA algorithm.
4. Numerical results on simulations. In radioastronomy, the recent advent of giant
ground-based radio interferometers brought improved angular, spectral and temporal reso-
lutions. However, the interferometric data are sub-sampled and blurred in Fourier space,
and the sources of interest are often mixed in multichannel interferometry imaging. Radio-
interferometric data are the perfect candidate where a joint deconvolution and blind source
separation problem needs to be solved. In the following, we will investigate the two following
cases for the linear operator ˆ
H:
- the data are sub-sampled because of a limited number of antennas of the interferom-
eter. As for the compressed sensing data, the operator ˆ
H, which is associated to the
sub-sampling effect, can be regarded as a mask with value 1 for active data and 0 for
inactive data;
- furthermore, the angular resolution of the interferometer is limited by its beamforming.
In practice, the PSF of the interferometer is determined by its beam. Therefore, ˆ
H, in
more general case, can be considered as a PSF kernel, which can take any real value.
Hence, depending on the form of the operator ˆ
H, we will apply the DecGMCA algorithm
to simulations corresponding to each case, namely a simulation on multichannel compressed
sensing (i.e. incomplete measurements) and a simulation on multichannel deconvolution.
Firstly, we generate simulated but rather complex data so that we can easily launch
Monte-Carlo tests with different parameter settings. The sources are generated as follows:
1. The mono-dimensional sources are K-sparse signals. The distribution of the active
entries satisfy a Bernoulli process πwith parameter ρsuch that:
(19) P[π= 1] = ρ, P[π= 0] = 1 −ρ.
2. Then the K-sparse sources are convolved with a Laplacian kernel with FWHM (Full
Width at Half Maximum) equal to 20.
Each source contains Np= 4096 samples and K is equal to 50. As mentioned in section 2,
an overcomplete dictionary outperforms (bi-)orthogonal dictionary in terms of signal/image

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 11
restoration. According to the form of the simulated sources or even the astrophysical sources
in more general cases which are isotropic, the starlet is optimal to sparsely represent such
sources [20].
Before moving to our numerical experiments, we firstly define the criteria that will be used
to evaluate the performance of the algorithms:
- the criterion for the mixing matrix is defined as: ∆A=−log10
||A†
estAref −INs||1
N2
s, which
is independent of the number of sources Ns. Intuitively, when the mixing matrix is
perfectly estimated, ∆A= +∞. Thus, the larger ∆Ais, the better the estimate of A
will be.
- according to [24], the estimated sources can be evaluated by a global criterion Source
to Distortion Ratio (SDR) in dB: SDR = 10 log10
||Sref ||2
||Sest−Sref ||2. The higher the SDR
is, the better the estimate of Swill be.
The rest of this section is organized as follows: for each case of simulations, we firstly
study the performance of DecGMCA in terms of the condition of ˆ
H, the number of sources
and the SNR, then we compare DecGMCA with other methods. If not specified, all criteria
will be chosen as the median of 50 independent Mont-Carlo tests.
(a) Example of a real masked observation compared with the complete data. Remark: The raw
data are in Fourier space, but they are transformed to pixel space for better visualization. The
percentage of active data is 50% and the SNR is 60 dB.
(b) Example of a recovered source from the above masked observations through DecGMCA super-
posed with the ground-truth.
Figure 1. Examples of the data and recovered sources superposed with ground-truth.

12 MING JIANG, J´
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OME BOBIN, JEAN-LUC STARCK
4.1. Multichannel compressed sensing and BSS simulation. First of all, to give a general
idea of simulated data, Figure 1(a) displays 1 mixture out of 20 of masked data (red dashed
curve) compared with that of complete data (blue dashed curve). The percentage of active
data is 50% and the SNR is 60 dB. Since the data are masked in Fourier space, we perform an
inverse Fourier transform to better visualize the data. To show the performance of our method
DecGMCA, Figure 1(b) displays 1 recovered source (red dashed curve) out of 2 superposed
with the simulated ground-truth source (blue solid curve). We can observe that DecGMCA
is able to recover all peaks and the estimated source is not biased. Besides, the noise is well
removed in the estimated source.
In this part we will discuss precisely the performance of DecGMCA on the multichannel
compressed sensing data in terms of the sub-sampling effect, the number of sources and the
SNR. The initial regularization parameter (0) is set to be 1 and decreases to 10−3for all
the experiments in this section.
4.1.1. On the study of DecGMCA.
-Sub-sampling effect: To study the impact of the sub-sampling effect, the mask in
Fourier space is varied from 10% to 90% where the value denotes the percentage of
active data. The number of sources Nsis fixed to 5 and the SNR is fixed to 60 dB. The
number of observation channels Ncis set to 5, 10 and 20. We applied our DecGMCA
method and Figure 2(a) shows an error bar plot of the criterion of Aand SDR in
terms of the mask. The blue, green and red curves represent results corresponding
Nc= 5, 10 and 20 respectively. In the figure, the first conclusion we can draw is
that when the percentage of active data is below 20%, DecGMCA performs badly in
the sense of ∆Aand SDR. This is due to the fact that when the mask is very ill-
conditioned, we almost have no observation in the dataset so that we cannot correctly
recover the mixing matrix and the sources. As the mask becomes better conditioned
(the percentage of active data increases), we have more and more observations and
we are expected to have better performance. It is interesting to notice that when the
Nc= 5, no matter which mask is used, the performance of DecGMCA is not good.
The lack of performance is caused by the underdetermination of the system in the
presence of mask when Nc= Ns. Besides, we can also observe that when Nc= 10,
DecGMCA performs well when the percentage of active data is above 50%. It could
be argued that though each of 10 channels is sub-sampled, statistically, the loss of
observation can be fully compensated when the percentage of active data is 50%, or in
other words, we have on average 5 fully sampled channels. Considering Nc= 20, we
can see all criteria are stabilized when the percentage of active data is over 50%. This
is due to the fact that the DecGMCA reaches peak performance in this test scenario
when the system is more and more well-conditioned.

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 13
10% 20% 30% 40% 50% 60% 70% 80% 90%
Mask
0.0
0.5
1.0
1.5
2.0
2.5
Criterior of A
10% 20% 30% 40% 50% 60% 70% 80% 90%
Mask
20
10
0
10
20
30
40
50
SDR(dB)
Ch num=5 Ch num=10 Ch num=20
(a) Performance of DecGMCA in terms of the percentage of active data. The number
of sources is 5 and the SNR is 60 dB. Abscissa: percentage of active data. Ordinate:
criterion of mixing matrix for left figure and SDR for right figure.
0 2 4 6 8 10 12 14 16
Number of sources
0.5
1.0
1.5
2.0
2.5
3.0
Criterior of A
0 2 4 6 8 10 12 14 16
Number of sources
10
0
10
20
30
40
50
SDR(dB)
Ch num=10 Ch num=20 Ch num=30
(b) Performance of DecGMCA in terms of the number of sources. The percentage of
active data is 50% and the SNR is 60 dB. Abscissa: number of sources. Ordinate:
criterion of mixing matrix for left figure and SDR for right figure.
10 20 30 40 50 60
SNR(dB)
0.0
0.5
1.0
1.5
2.0
2.5
Criterior of A
10 20 30 40 50 60
SNR(dB)
10
0
10
20
30
40
SDR(dB)
Ch num=5 Ch num=10 Ch num=20
(c) Performance of DecGMCA in terms of SNR. The percentage of active data is 50%
and the number of sources is 5. Abscissa: SNR. Ordinate: criterion of mixing matrix for
left figure and SDR for right figure.
Figure 2. Multichannel compressed sensing simulation (1): study of DecGMCA. The parameters are the
percentage of active data, the number of sources and the SNR from top to bottom. The curves are the medians
of 50 realizations and the errorbars corresponds to 60% of ordered data around the median. Blue curves, green
curves, red curves and cyan curves represent the number of channels=5, 10, 20 and 30 respectively.

14 MING JIANG, J´
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OME BOBIN, JEAN-LUC STARCK
-Number of sources: The number of sources is also an important factor for the perfor-
mance of DecGMCA. In this paragraph, Nsis set to 2, 5, 10 and 15. The percentage
of active data is fixed to 50% and the SNR is fixed to 60 dB. Ncis set to 10, 20 and
30. In Figure 2(b), when Ns= 2, we can observe that the sub-sampling effect does
not affect the performance. This is due to the fact that when the number of chan-
nels is sufficiently large compared to the number of sources, the loss of observation
in one channel can be completely compensated by observations from other channels,
which makes the matrix inversion easier and more accurate in the source estimate step
(see Equation (11)). Conversely, when Ns= 15, noticing that the mask is 50%, it is
impossible to have good results when the number of channels is below 30. Thus, we
can see in the figure that sources are not well recovered (SDR <20 dB) when Ncis
20 or 30. Besides, given a fixed number of channels, the performance of DecGMCA
decays as the number of sources increases, which is consistent with our expectation.
Indeed, when the number of sources increases, the number of columns of the mixing
matrix increases, which makes it more difficult to jointly recover the mixing matrix
and reduce the masking effect.
-SNR: The third parameter which affects the performance of DecGMCA is the SNR.
In this paragraph, the SNR is varied from 10 dB to 55 dB. The percentage of active
data is fixed to 50% as well and Nsis fixed to 5. Ncis set to 5, 10 and 20. We can
observe in Figure 2(c) that as expected the performance gets worse as SNR decreases.
Particularly, when Nc= 5, irrespective of the SNR, DecGMCA performs poorly. This
is owing to the fact when Nc= 5 and the mask is 50%, we are not able to successfully
recover 5 sources (SDR is around 0 dB). Therefore, in this case, the number of channels
instead of SNR is the bottleneck of the algorithm. When the SNR is below 25 dB for
Nc= 10 and 45 dB for Nc= 20, we can see increasing the SNR significantly helps to
improve the performance of DecGMCA. This can be argued that when the number of
channels is no longer the restriction of the algorithm, the contamination of noise in
the data becomes the dominant restriction of the performance of DecGMCA. When
the SNR is high, it is easier to extract useful information to estimate sources. Thus,
the sources are estimated with high precision. However, one should notice that the
performance of DecGMCA cannot eternally grow along with the SNR. The reason for
this is that we can already successfully extract information to estimate sources and
an even higher SNR will not help us significantly. In this case, it is the number of
channels that becomes the limiting factor for better estimating sources. Therefore, we
can observe in the figure that the saturation points of the criteria of Nc= 20 appears
later than those of Nc= 10.

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 15
10% 20% 30% 40% 50% 60% 70% 80% 90%
Mask
0.0
0.5
1.0
1.5
2.0
2.5
Criterior of A
10% 20% 30% 40% 50% 60% 70% 80% 90%
Mask
20
10
0
10
20
30
40
SDR(dB)
DecGMCA,Ch num=5
MC+GMCA,Ch num=5
DecGMCA,Ch num=10
MC+GMCA,Ch num=10
DecGMCA,Ch num=20
MC+GMCA,Ch num=20
(a) Comparison between DecGMCA and MC+GMCA in terms of the per-
centage of active data. The number of sources is 5 and the SNR is 60 dB.
Abscissa: percentage of active data. Ordinate: criterion of mixing matrix for
left figure and SDR for right figure.
0 2 4 6 8 10 12 14 16
Number of sources
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Criterior of A
0 2 4 6 8 10 12 14 16
Number of sources
10
0
10
20
30
40
50
SDR(dB)
DecGMCA,Ch num=10
MC+GMCA,Ch num=10
DecGMCA,Ch num=20
MC+GMCA,Ch num=20
DecGMCA,Ch num=30
MC+GMCA,Ch num=30
(b) Comparison between DecGMCA and MC+GMCA in terms of the number
of sources. The percentage of active data is 50% and the SNR is 60 dB.
Abscissa: number of sources. Ordinate: criterion of mixing matrix for left
figure and SDR for right figure.
10 20 30 40 50 60
SNR(dB)
0.0
0.5
1.0
1.5
2.0
2.5
Criterior of A
10 20 30 40 50 60
SNR(dB)
10
0
10
20
30
40
SDR(dB)
DecGMCA,Bd num=5
MC+GMCA,Bd num=5
DecGMCA,Bd num=10
MC+GMCA,Bd num=10
DecGMCA,Bd num=20
MC+GMCA,Bd num=20
(c) Comparison between DecGMCA and MC+GMCA in terms of SNR. The
percentage of active data is 50% and the number of sources is 5. Abscissa:
SNR. Ordinate: criterion of mixing matrix for left figure and SDR for right
figure.
Figure 3. Multichannel compressed sensing simulation (2): comparison between DecGMCA (joint decon-
volution and BSS) and MC+GMCA (matrix completion followed by BSS). The parameters are the percentage
of active data, the number of sources and the SNR from top to bottom. The curves are the medians of 50
realizations. Blue curves, green curves, red curves and cyan curves represent the number of channels=5, 10,
20 and 30 respectively.

16 MING JIANG, J´
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OME BOBIN, JEAN-LUC STARCK
4.1.2. Comparison with other methods. Since the data are not completely available,
another concept is to utilize matrix completion method to interpolate the data and then
apply a BSS method on the interpolated data. Therefore, in this subsection, we will compare
DecGMCA with the matrix completion followed by a BSS method. Herein, we use a classical
SVT algorithm [5] to solve the Matrix Completion (MC) problem and GMCA to solve the
BSS problem on the interpolated data. In the rest of this subsection, we repeat the same
simulation of DecGMCA but utilize MC+GMCA and we compare their performances. For all
the figures in this subsection, solid curves represent medians of criteria by applying DecGMCA
while dashed curves represent medians of criteria by applying MC+GMCA.
-Sub-sampling effect: Similarly, we study firstly the sub-sampling effect. In Figure 3(a),
we can see the performance of MC+GMCA decreases dramatically when the mask
degrades. This might be likely when the loss of information becomes severe, the
low-rank hypothesis is no longer valid. As a result, the matrix completion cannot
correctly interpolate the data, which means that the following source separation pro-
cedure performs badly. Comparing the performance of DecGMCA with MC+GMCA
at its turning points (30% and 60% for the number of channels 20 and 10 respec-
tively), we can see that DecGMCA conserves well the continuity of both criteria and
outperforms MC+GMCA even when the mask is very bad-conditioned. One should
notice that when mask is relatively good, DecGMCA still outperforms MC+GMCA.
This is due to the fact that DecGMCA takes all of the data into account and si-
multaneously processes source separation and sub-sampling effect, while MC+GMCA
considers them separately. Consequently, the blind source separation in MC+GMCA
relies on the quality of matrix completion, which in fact approximates the data inter-
polation and produces a negligible bias. Interestingly, the separation performances of
the DecGMCA seem to degrade when the average number of available measurements
per frequency in the Fourier domain (i.e. the product of the sub-sampling ratio and
the total number of observations) is roughly of the order of the number of sources. In
that case, the resulting problem is close to an under-determined BSS problem. In that
case the identifiability of the sources is not guaranteed unless additional assumptions
about the sources are made. In this setting, it is customary to assume that the sources
have disjoint supports in the sparse domain, which is not a valid assumption in the
present paper. Additionally, radio-interferometric measurements are generally com-
posed of a large amount of observations for few sources to be retrieved. Furthermore,
in contrast to the fully random masks we considered in these experiments, real inter-
ferometric masks exhibit a denser amount of data at low frequency and their evolution
across channels is mainly a dilation of the sampling mask in the Fourier domain. This
entails that the sampling process across wavelegengths is highly correlated, which is a
more favorable setting for blind source separation. Altogether, these different points
highly mitigate the limitations of the DecGMCA algorithm due to sub-sampling in a
realistic inferometric imaging setting.

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 17
0 1 2 3 4 5 6 7 8 9
FWHMmax/FWHMmin
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
Criterior of A
0 1 2 3 4 5 6 7 8 9
FWHMmax/FWHMmin
25
30
35
40
45
SDR(dB)
Ch num=10 Ch num=20
(a) Performance of DecGMCA in terms of resolution ratio. The number of sources is 5
and the SNR is 60 dB. Abscissa: resolution ratio between best resolved PSF and worst
resolved PSF. Ordinate: criterion of mixing matrix for left figure and SDR for right
figure.
0 2 4 6 8 10 12 14 16
Number of sources
0.5
1.0
1.5
2.0
2.5
3.0
Criterior of A
0 2 4 6 8 10 12 14 16
Number of sources
0
10
20
30
40
50
60
SDR(dB)
Ch num=10 Ch num=20 Ch num=30
(b) Performance of DecGMCA in terms of the number of sources. The resolution ratio
is 3 and the SNR is 60 dB. Abscissa: number of sources. Ordinate: criterion of mixing
matrix for left figure and SDR for right figure.
10 20 30 40 50 60
SNR(dB)
0.0
0.5
1.0
1.5
2.0
2.5
Criterior of A
10 20 30 40 50 60
SNR(dB)
10
0
10
20
30
40
50
SDR(dB)
Ch num=5 Ch num=10 Ch num=20
(c) Performance of DecGMCA in terms of SNR. The resolution ratio is 3 and the number
of sources is 5. Abscissa: SNR. Ordinate: criterion of mixing matrix for left figure and
SDR for right figure.
Figure 4. Multichannel deconvolution BSS simulation (1): study of DecGMCA. The parameters are the
resolution ratio, the number of sources and the SNR from top to bottom. The curves are the medians of 50
realizations and the errorbars corresponds to 60% of ordered data around the median. Blue curves, green curves,
red curves and cyan curves represent the number of channels=5, 10, 20 and 30 respectively.

18 MING JIANG, J´
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OME BOBIN, JEAN-LUC STARCK
-Number of sources: Considering the number of sources, in Figure 3(b), we can observe
that the performance of MC+GMCA decreases more rapidly than DecGMCA. It can
be explained by the fact that a large number of sources complicates the matrix com-
pletion (MC) step. Thus, the interpolated data is biased, which affects the following
source separation (GMCA). Conversely, as for DecGMCA, jointly solving the data
interpolation and the source separation avoids the bias and gives better results.
-SNR: In terms of SNR, in Figure 3(c), both DecGMCA and MC+GMCA are sta-
ble. However, DecGMCA still outperforms MC+GMCA. One should remark that in
MC+GMCA the noise affects both the MC and GMCA steps, which makes the results
less accurate. In DecGMCA, as we integrate the demasking in the source separation,
we reduce the bias produced by the noise. Hence, DecGMCA has more tolerance to
noisy data. The DecGMCA tends to perform correctly for large values of the SNR (i.e.
typically larger than 40 dB) but rapidly fails at low SNR. As we mentionned before,
the experimental setting we considered in the present paper is more conservative than
realistic inferometric settings. Indeed, radio-interferometric data are generally made of
hundreds of measurements, which will dramatically help improving the performances
of the DecGMCA algorithm in the low SNR regime..
4.2. Multichannel deconvolution and BSS simulation. Then, as for the general form
of the operator ˆ
H, we will extend the above experiments to a more general multichannel
deconvolution case. In practice, because of instrumental limits, the resolution of the PSF
of multichannel sensors is dependent on the channel. We assume that the resolution of the
Gaussian-like PSF increases linearly along the channel index. The PSF function in channel
νis defined by: Fν(x) = exp( x2
2σ2
ν), where the coordinate in Fourier space x∈[−2047,2048].
As FWHM (Full Width at Half Maximum, which is conceptually considered as a standard
deviation) is commonly used to define the resolution of the PSF, we define a resolution ratio
based on FWHM to quantify the variation of the PSF hereafter: ratio = FWHMmax
FWHMmin . In this
section, the best resolved Gaussian-like PSF is always fixed with σmax = 1800. We will
utilize our DecGMCA method to realize the multichannel deconvolution BSS. We will study
the performance in three parts as well: resolution ratio, the number of sources and SNR.
The initial regularization parameter (0) is set to be 1 and decreases to 10−5for all the
experiments in this section.
4.2.1. On the study of DecGMCA.
-Resolution ratio: The first parameter is the resolution ratio. In this paragraph, we
define different resolution ratios to study the performance of DecGMCA. The reso-
lution ratio is varied from 1 to 8. Therefore, ratio=1 means all channels have the
same resolution, while ratio=8 means the largest difference between the best resolu-
tion and the worst resolution. Nsis fixed to 5 and the SNR is fixed to 60 dB. Ncis
set to 10 and 20. In Figure 5(a), we can observe that if Ns= 10, the performance
of DecGMCA becomes unstable as resolution ratio increases. This means that when
the resolution ratio becomes large, the system becomes ill-conditioned and the noise is
likely to explode. However, if we increase the number of channels to 20, both criteria
become more stable. Indeed, as the number of channels increases, we have more in-
formation to help to estimate Aand S, yielding more accurate estimates. One might

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 19
notice that the SDR does not change no matter which resolution ratio or number of
channels is used. This can be interpreted by the fact that even though the system
is ill-conditioned, DecGMCA can successfully regularize the ill-conditioned PSF. Al-
though the ∆Abecomes unstable when the resolution ratio becomes large, its median
is of good precision.
-Number of sources: The second parameter is the number of sources. In this paragraph,
Nsis set to 2, 5, 10 and 15. The resolution ratio is 3 and the SNR is 60 dB. Ncis
set to 10, 20 and 30. In Figure 4(b), we can observe that when Ncis 2 and 5, all
criteria are very good (∆A>2 and SDR >40 dB) and almost superposed. This is
due to the fact that the number of channels is always sufficiently large compared to
the number of sources and the ill-conditioned PSF is not difficult to be inverted. As
expected, when the number of sources increases, the system becomes more complicated
and both criteria decrease. In particular, considering the most difficult case in our test
(Nc= 10 for Ns= 10), DecGMCA does a poor job at regularizing the system and the
effect of ill-conditioned PSF significantly degrades both criteria.
-SNR: In the end, concerning the impact of noise on DecGMCA, the SNR is varied
from 10 dB to 55 dB. The resolution ratio is fixed to 3 and Nsis fixed to 5. Ncis set
to 5, 10 and 20. Figure 4(c) features the evolution of both criteria in terms of SNR.
Firstly, when the number of channels is 5, the performance of DecGMCA is always
poor. The reason is that recovering 5 sources from 5 channels is the limit of the BSS
problem, but the ill-conditioned PSF raises the difficulty. Even though DecGMCA can
regularize the system, its effectiveness is limited when SNR is very low. When Ncis
10 or 20, since more observations are available, both criteria grow rapidly along with
the SNR. When the SNR is low, the data is so noisy that even with regularization
DecGMCA cannot efficiently select useful information. Conversely, when SNR is high,
especially above 40 dB, DecGMCA is able to accurately estimate the mixing matrix
and the sources. One might notice that generally ∆Ais more unstable than SDR. It
means that the criterion of Ais more sensitive to the noise.
4.2.2. Comparison with other methods. DecGMCA considers BSS and deconvolution
simutaneously and naturally it gives a better result than considering them separately. In
order to validate, we compare DecGMCA with different approaches:
•Blind Source Separation only, without deconvolution. GMCA is used for BSS.
•a channel by channel deconvolution using ForWaRD followed by BSS (ForWaRD+GMCA)
In the rest of this subsection, we utilize GMCA and ForWaRD+GMCA to repeat our
DecGMCA simulation and compare their performances. For all the figures in this subsection,
solid curves represent medians of criteria by applying DecGMCA while pointed curves and
dashed curves represent medians of criteria by only GMCA and ForWaRD+GMCA respec-
tively.
-Resolution ratio: In terms of resolution ratio, Figure 5(a) displays the performance of
DecGMCA, GMCA and ForWaRD+GMCA. As GMCA does not consider the varied
PSFs and does not perform the deconvolution, GMCA provides the worst results.
Although ForWaRD+GMCA takes into account the deconvolution, it processes the
deconvolution on channel by channel instead of the whole dataset. Thus, it neglects

20 MING JIANG, J´
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the correlation between channels and gives a biased deconvolution, leading to a much
worse result than DecGMCA. It is also interesting to notice that when the resolution
ratio is small, the difference between the three methods is smaller than when the
resolution is large. This is because the PSFs are more varied when the resolution ratio
is larger. DecGMCA, which simultaneously considers deconvolution and BSS, is able
to capture the change of different PSFs and adapt to the complex data. Therefore,
DecGMCA outperforms the others when the PSFs are widely varied.
-Number of sources: Concerning the number of sources, from Figure 5(b), we can see
that as the number of sources increases, the system becomes more complicated and
the criteria of all of three methods degrade. Yet, DecGMCA always outperforms
ForWaRD+GMCA by at least 25 dB in SDR. This is because when considering simul-
taneous BSS and deconvolution, much more global information can be conserved and
the solution is less biased. Besides, among three methods, GMCA is again the worst
one as GMCA neglects the blurring effect caused by the PSF.
-SNR: Finally, in terms of SNR, in Figure 5(c), we can see the performance of GMCA is
very poor. This is not only the impact of the noise but also the neglect of the blurring
effect. As for ForWaRD+GMCA, both criteria grow slightly in terms of SNR (when
SNR <20 dB) as it takes the blurring effect into account and restores sources, but
ForWaRD+GMCA attains the saturation point rapidly as it cannot perform better
with channel by channel deconvolution. However, the performance of DecGMCA grows
rapidly as SNR increases because simultaneously considering deconvolution and BSS
benefits from the data for the recovery of the mixing matrix and sources.

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 21
0 1 2 3 4 5 6 7 8 9
FWHMmax/FWHMmin
0.0
0.5
1.0
1.5
2.0
2.5
Criterior of A
0 1 2 3 4 5 6 7 8 9
FWHMmax/FWHMmin
10
0
10
20
30
40
50
SDR(dB)
DecGMCA,Ch num=10
DecGMCA,Ch num=20
Fwd+GMCA,Ch num=10
Fwd+GMCA,Ch num=20
GMCA,Ch num=10
GMCA,Ch num=20
(a) Comparison among DecGMCA, GMCA and ForWaRD+GMCA in terms
of resolution ratio. The number of sources is 5 and the SNR is 60 dB. Abscissa:
ratio between the best resolved PSF and the worst resolved PSF. Ordinate:
criterion of mixing matrix for left figure and SDR for right figure.
0 2 4 6 8 10 12 14 16
Number of sources
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0
2.5
Criterior of A
0 2 4 6 8 10 12 14 16
Number of sources
20
10
0
10
20
30
40
50
SDR(dB)
DecGMCA,Ch num=10
DecGMCA,Ch num=20
DecGMCA,Ch num=30
Fwd+GMCA,Ch num=10
Fwd+GMCA,Ch num=20
Fwd+GMCA,Ch num=30
GMCA,Ch num=10
GMCA,Ch num=20
GMCA,Ch num=30
(b) Comparison among DecGMCA, GMCA and ForWaRD+GMCA in terms
of the number of sources. The resolution ratio is 3 and the SNR is 60 dB.
Abscissa: number of sources. Ordinate: criterion of mixing matrix for left
figure and SDR for right figure.
10 20 30 40 50 60
SNR(dB)
0.0
0.5
1.0
1.5
2.0
2.5
Criterior of A
10 20 30 40 50 60
SNR(dB)
10
0
10
20
30
40
50
SDR(dB)
DecGMCA,Ch num=10
DecGMCA,Ch num=20
Fwd+GMCA,Ch num=10
Fwd+GMCA,Ch num=20
GMCA,Ch num=10
GMCA,Ch num=20
(c) Comparison among DecGMCA, GMCA and ForWaRD+GMCA in terms
of SNR. The resolution ratio is 3 and the number of sources is 5. Abscissa:
SNR. Ordinate: criterion of mixing matrix for left figure and SDR for right
figure.
Figure 5. Multichannel deconvlution and BSS simulation (2): comparison among DecGMCA (joint decon-
volution and BSS), only GMCA (BSS) and ForWaRD+GMCA (channel by channel deconvolution followed by
BSS). The parameters are the resolution ratio, the number of sources and the SNR from top to bottom. The
curves are the medians of 50 realizations. Blue curves, green curves, red curves and cyan curves represent the
number of channels=5, 10, 20 and 30 respectively.

22 MING JIANG, J´
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5. Application to astrophysical data. In astrophysics, sources are often Gaussian-like.
The left column of Figure 8 displays three astrophysical sources. It has been shown that the
starlet dictionary [19] is the best representation for such isotropic sources. In spectroscopy,
the astrophysical source has a characteristic spectrum f(x)∝x−k, which generally respects
power law with a specific index. Through interferometers, we can capture these sources and
study their spectra. However, the problem of interferometry imaging is that the observation
is sub-sampled in Fourier space. Besides the sub-sampling effect, the PSF, or the angular
resolution of the interferometer, is limited by its beamforming and varies as a function of
wavelength. Therefore, we extend the numerical experiments in the previous section to the
case where the operator ˆ
Htakes not only the sub-sampling effect but also the blurring effect
into account.
(a) Example of the best resolved PSF
over total 20 channels.
(b) Example of the worst resolved PSF
over total 20 channels.
Figure 6. Illustration of masked PSFs (in Fourier space): the resolution ratio is 3 and the percentage of
active data is 50%).
Figure 7. Illustration of 1 out of 20 mixtures blurred by the masked PSFs and contaminated by the noise:
the resolution ration is 3, the percentage of active data is 50% and the SNR is 60 dB. Remark: The real data
are in Fourier space, but the data here is transformed to image space for better visualization.
For simplicity, we assume that the number of observation channels is 20. The resolution
ratio of the best resolved PSF and the worst resolved PSF is 3 and the percentage of active
data in Fourier space is 50%. In addition, the noise level is fixed to 60 dB. Figure 6 illustrates
two masked PSFs (the best resolved one and the worst resolved one) in Fourier space and
Figure 7 gives an example of 1 mixture out of 20. We can see that sources are drown in the

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 23
“dirty” image and mixed with each other. It seems to be very challenging to discriminate and
recover these sources from such an ill-conditioned PSF.
Figure 8. Illustration of DecGMCA applied on astrophysical images. The raw data is blurred by the masked
PSFs and contaminated by the noise: the resolution of PSF is linearly declined along 20 channels with resolution
ratio=3, besides, PSFs are masked with percentage of active data=50% and the SNR is 60 dB. Figure 7shows
an example of the raw data in image domain. We apply DecGMCA to separate and recover sources. Left
column: Ground-truth of three sources from top to bottom. Middle column: Estimate of three sources by using
DecGMCA from top to bottom. Right column: Estimate of three sources by using ForWaRD+GMCA from top
to bottom.
We set the wavelet scale to 5, final threshold level to 3σ(σstands for the standard deviation
of the noise) and the regularization parameter is initialized as 10−5and decreases to 10−6.
Having applied DecGMCA on such “dirty” data, we can see recovered sources presented in
the middle column of Figure 8. The sources are well deconvolved and separated. Visually,
compared to the references in the left column of Figure 8, the main structures of sources are
well positioned and even the detailed features of estimated sources are successfully recovered.
This means that the estimated sources by using DecGMCA have a very good agreement with
the references. However, if we first applied the ForWaRD method to perform the channel by
channel deconvolution and then applied GMCA method to separate sources, the results would
not be reliable. We can see in the right column of Figure 8 that the sources cannot be recovered
properly. The main structure of the first source is recovered but not well deconvolved and the
structure is biased; the second source cannot even be recovered with many artifacts present in
the background; the third source is successfully separated but the structures are biased and
not compact.
Furthermore, by computing residuals between estimated sources and ground-truth sources,
Figure 9 displays the error map of DecGMCA (left column) and ForWaRD+GMCA (right
column). We also compare their relative errors which are shown in Table 1. We can see

24 MING JIANG, J´
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OME BOBIN, JEAN-LUC STARCK
that DecGMCA is very accurate with the relative errors for three sources 0.14%, 0.27% and
0.36% respectively, which shows that our estimated sources have a good agreement with the
ground-truth. However, as for the ForWaRD+GMCA, since sources are not cleanly separated
and recovered, the residuals are significant. The relative errors are tremendous, respectively
54.74%, 1279.21% and 30.12% respectively.
(a) Residuals between estimated sources by using
DecGMCA and ground-truth sources, the relative
errors are 0.14%, 0.27% and 0.36% from top to
bottom.
(b) Residuals between estimated sources by using
ForWaRD+GMCA and ground-truth sources, the
relative errors are 54.74%, 1279.21% and 30.12%
from top to bottom.
Figure 9. Comparison between joint deconvolution and BSS using DecGMCA and channel by channel
deconvolution fol lowed by BSS using ForWaRD+GMCA. Left column: residuals between estimated sources
and ground-truth using DecGMCA. Right column: residuals between estimated sources and ground-truth using
ForWaRD+GMCA.
Table 1
Comparison of relative errors between DecGMCA and ForWaRD+GMCA
Sources DecGMCA ForWaRD+GMCA
1 0.14% 54.74%
2 0.27% 1279.21%
3 0.36% 30.12%
6. Software. In order to reproduce all the experiments, the codes used to generate the
plots presented in this paper will be available online at http://www.cosmostat.org/software/
gmcalab/

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 25
7. Conclusions. In this article, we investigated an innovative solution to the DBSS prob-
lems, where deconvolution and blind source separation need to be solved simultaneously. The
proposed algorithm, dubbed Deconvolution GMCA (DecGMCA) builds upon sparse signal
modeling and a novel projected least-squares algorithm to solve BSS problems from incom-
plete measurements and/or blurred data. Numerical experiments, in the multichannel com-
pressed sensing and multichannel deconvolution settings, have been carried out that show the
efficiency of the proposed DecGMCA algorithm. These results further emphasize the advan-
tage of the joint resolution of both the deconvolution and unmixing problems rather than an
independent processing. DecGMCA has been applied to astrophysical data, which illustrates
that it is a very good candidate for processing radioastronomy data. Future work will focus
on extending the proposed approach to account for more complex acquisition models.

26 MING JIANG, J´
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OME BOBIN, JEAN-LUC STARCK
Acknowledgments. This work is supported by the CEA DRF impulsion project COSMIC
and the European Community through the grants PHySIS (contract no. 60174), DEDALE
(contract no. 665044) and LENA (contract no. 678282) within the H2020 Framework Pro-
grame. The authors would like to thank Samuel Farrens for useful comments.
Appendix A. Proximal algorithms to solve Equation (7).Step 28 in Algorithm 1 consists
in using the Condat-V˜u [25,7] algorithm to improve the estimate of Swith respect to A. As
we formulate the sub-problem (7) in an analysis framework, the proximal operator || · Φt||p
(p= 0 or 1) is not explicit. The advantage of Condat-V˜u or other primal-dual algorithms is
that we do not need an inner loop to approach the proximal operator as done in the Forward-
Backward algorithm [6]. The detailed algorithm is presented in Algorithm 2.
Algorithm 2 Condat-V˜u algorithm
1: Input: Observation ˆ
Y, operator ˆ
H, mixing matrix A, maximum iterations Ni, threshold
λ,τ > 0,η > 0
2: Initialize S(0) as the last estimate by using ALS scheme,α(0) =SΦ
3: for i= 1,· · · ,Nido
4: •Obtain sources in Fourier space by FFT
5: ˆ
S(i)= FT(S(i))
6: •Compute the residual in Fourier space
7: R(i)=ˆ
Y−ˆ
HAˆ
S(i)
8: •Update S
9: S(i+1) =S(i)−τα(i)Φ+τRe FT−1Atˆ
H∗R(i)
10: •Introduce an intermediate variable
11: V(i+1) = 2S(i+1) −S(i)
12: for j= 1,· · · ,Nsdo
13: •Update αunder sparsity constraint
14: α(i+1)
j=I−Thλ(i)
jα(i)
j+ηV(i+1)
jΦt
15: end for
16: end for
17: return S(Ni)

JOINT MULTICHANNEL DECONVOLUTION AND BLIND SOURCE SEPARATION 27
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