1

An iterative thresholding algorithm for jointdeconvolution and separation of multichannel data

Y. Moudden, J. Bobin and J.-L. StarckCEA, IRFU, SEDI, Centre de Saclay, F-91191 Gif-sur-Yvette, France

Abstract—We report here on extending GMCA to the case ofconvolutive mixtures. GMCA is a recent source separation algo-rithm which takes advantage of the highly sparse representationof structured data in large overcomplete dictionaries to separatecomponents in multichannel data based on their morphology.Many successful applications of GMCA to multichannel dataprocessing have been reported : blind source separation, datarestoration, color image denoising and inpainting, etc. However,the data model in GMCA assumes an ideal instrument, notaccounting for instrumental impulse response functions whichcan be crucial in some important applications such as CMBdata analysis from the Planck experiment. We show here howto build on GMCA for that purpose and describe an iterativethresholding algorithm for joint multichannel deconvolution andcomponent separation. Preliminary numerical experiments giveencouraging results.

Index Terms—Multichannel data, Blind source separation,multichannel deconvolution, overcomplete representations, spar-sity, morphological component analysis, GMCA, morphologicaldiversity.

I. INTRODUCTION

In preparation for the challenging task of analyzing the full-sky CMB maps that will shortly be available from ESA’sPlanck1 experiment, dedicated multichannel data processingmethods are being actively developed. A precise analysis ofthe data is of huge scientific importance as most cosmologicalparameters in current models are constrained by the statistics(e.g. spatial power spectrum, Gaussianity) of the observedprimordial CMB field [1].

A first concern is that several distinct astrophysical pro-cesses contribute to the total observed radiation in the fre-quency range used for CMB observations [2]. The task ofunmixing the CMB, the Galactic dust and synchrotron emis-sions, the contribution of the Sunyaev-Zel’dovich (SZ) effectof galaxy clusters, to name a few, has attracted much attention.Several generic and dedicated source separation algorithmshave been applied and tailored to account for the specificprior knowledge of the physics at play ( [3]–[5] and referencestherein).

Among these methods, GMCA (Generalized MorphologicalComponent Analysis) has proven to be an efficient and flexiblesource separation algorithm. GMCA takes advantage of thesparse representation of structured data in large overcompletesignal dictionaries to separate sources based on their morphol-ogy. A full description of GMCA is given in [6], [7] and itsapplication to CMB data analysis is reported in [5].

1http://astro.estec.esa.nl/Planck

However, another issue is multichannel deconvolution : thedifferent channels of practical instruments have different pointspread functions. Hence, a correct fusion of the multichannelobservations for the purpose of estimating the different astro-physical component maps, requires a simultaneous inversionof the spectral mixing (i.e. each component contributes toeach channel according to its emission law) and of thespatial mixing (i.e. convolution by the instruments psf ineach channel). Our purpose here is to extend the GMCAframework and propose and algorithm that can achieve sucha joint deconvolution and separation. The devised methodrelies on the a priori sparsity of the initial components ingiven overcomplete representations (e.g. Fourier, wavelets,curvelets). It is an iterative thresholding algorithm with aprogressively decreasing threshold, leading to a salient-to-fineestimation process which has proven successful in related algo-rithms ( [8]–[10] and references therein). Results of numericalsimulations with synthetic 1D and 2D data are presented, inthe specific case where the mixing parameters are fully knowna priori. These preliminary toy experiments motivate currentefforts to extend the range of applications to the more difficultcase of blind separation where the mixing matrix and possiblythe beams are unknown. We are also working on applying theproposed method to multichannel CMB data.

II. LINEAR MIXING AND CHANNEL-WISE CONVOLUTION

The simplest model encountered in the field of componentseparation states that the observations are a noisy linearinstantaneous mixture of the initial components. Such a modelhas, for instance, often been used to describe multichannelCMB observations : the total sky emission observed in pixelj with detector i sums the contributions of n components

xji =

n∑k=1

aki sj

k + nji (1)

where sjk is the emission template for the kth astrophysical

process, herein referred to as a source or a component. Thecoefficient ak

i reflects the emission law of the kth source inthe frequency band of the ithsensor, and nj

i stands for thenoise contributions to the observations. When t observationsare obtained from m detectors, this equation can be put in amore convenient vector-matrix form:

X = AS + N (2)

where X and N are m × t matrices, S is n × t, and A isthe m×n mixing matrix. For simplicity, the noise is assumed

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uncorrelated intra- and inter- channel, with variance σ2k in the

kth channel.Commonly, the different channels of typical multichannel

instruments do not have the same, what is more high reso-lution, impulse response or point spread function (psf ). Theabove model is then not capable of correctly rendering theobservations : assuming the psf ’s are known, fitting that modelto the observations is not making the most of the data andwill not produce estimates of the component maps at thebest possible resolution. A notable exception is when themultichannel observations offer multiple views of a singleprocess in which case there is no separation to be performedand the analysis can be reduced from a multichannel to asingle channel deconvolution problem without loss of statis-tical efficiency [11]. As mentioned above, our interest hereis with analyzing multiple convolutive mixtures of multiplesources, when the different channels have different pointspread functions :

X = H(AS) + N (3)

denoting H the multichannel convolution matrix-operator, act-ing on each channel separately. The data observed in channeli is

xi = aiSHi + ni (4)

where Hi is the t × t matrix of the convolution operator inthis channel, xi, ai and ni are respectively the ith lines of X ,A and N .

In practice, both the sources S and their emission lawsA may be unknown, a case referred to as blind, or onlypartly known. Although blind deconvolution would be anadded excitement, we consider here that the instrumental psf ’sare fully known a priori. The purpose of source separationalgorithms is then to invert the mixing process and provideestimates of the unknown parameters. This is most often an ill-posed problem, requiring additional prior information in orderto be solved : one needs to know what in essence makes thesources different.

For instance, they may have a priori disjoint supports in agiven representation such as time and/or frequency leading toclustering or segmentation algorithms. In some cases, knowinga priori that the mixed sources are statistically independentprocesses will enable their recovery : this is the frameworkof Independent Component Analysis (ICA), a growing setof multichannel data analysis techniques, which have provensuccessful in a wide range of applications [12]. Indeed, al-though statistical independence is a strong assumption, it isin many cases physically plausible. ICA algorithms for blindcomponent separation and mixing matrix estimation dependon the a priori model used for the probability distributions ofthe sources [12]–[15].

An especially important case is when the mixed sources arehighly sparse, meaning that each source is only rarely activeand mostly nearly zero. The independence assumption thenensures that the probability for two sources to be significantsimultaneously is extremely low so that the sources may againbe treated as having nearly disjoint supports. It is shownin [16] that first moving the data into a representation in

which the sources are sparse greatly enhances the quality ofthe separation.

Working with combinations of several bases or with very re-dundant dictionaries such as the undecimated wavelet frames,ridgelets or curvelets [17] leads to even sparser representations: this is exploited by GMCA. Sparsity is now generallyrecognized as a valuable property for blind source separation.At the same time, the benefits of sparsity for single channeldeconvolution problems in signal and image processing havebeen experimented in [18]–[20]. Hence, the joint multichanneldeconvolution and source separation problem presumably hasmuch to gain from sparse representations in large dictionaries,with a GMCA-like recovery process. Following the assump-tions in GMCA which we are extending here to the case ofconvolutive mixtures, we assume that the source processeshave sparse representations in a priori known dictionaries. Forthe sake of simplicity, we consider that the components allhave a sparse representation in the same orthonormal basisΦ. Recent analyses in [21]–[23] give theoretical grounds foran extension to frames and redundant dictionaries. We alsoassume that the source processes are generated from sparsecoefficient vectors αi sampled identically and independentlyfrom Laplacian distributions

P (αi) ∝ exp (−λi‖αi‖1) (5)

with inverse scale parameter λi, and si = αiΦ.The linear operator considered here, linear mixing followed

by convolution in each channel, is somewhat cumbersome tohandle : mixing acts on each column of S separately whileconvolution acts on the each line of AS separately. The fulloperator does not factorize very nicely. The order in which themixing and the convolution matrices apply obviously matters.In the case where A and H are invertible, inverting the fulloperator consists in first applying the inverse of H and then theinverse of A. When additive noise corrupts the observed data,or when the channel-wise convolution operator or the mixingmatrix are underdetermined, inversion asks for some sort ofregularization. We consider here inverting the joint mixtureand convolution operator in a possibly underdetermined andnoisy context thanks to regularization induced by the a priorisparsity of the component signals to be recovered. Makingthe most of the noisy data to estimate the sources neitherreduces to applying the inverse of A and deconvolving eachsource separately, nor deconvolving each channel separatelyand applying the inverse of A. This is also true of estimatingA with S fixed. A global necessarily intricate multichanneldata processing is necessary.

III. FULL OBJECTIVE FUNCTION

With the assumptions in the previous section, the jointdeconvolution and separation task is readily expressed as aminimization problem, in all respects equivalent to a maximuma posteriori estimation of A and S assuming an improperuniform prior on the entries of A :

minA,S

12‖X −H( AS )‖2Σ +

∑i

λi‖siΦt‖1 (6)

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where the subscript Σ reminds us of the noise covariancematrix, and Φt is the transpose of Φ. Equivalently, one has :

minA,{αi}

12‖X −H(

∑i

aiαiΦ )‖2Σ +∑

i

λi‖αi‖1 (7)

The above objective balances between a quadratic measureof fit and an `1 sparsity measure of the solution. The `1is used here in place of the combinatorial `0 pseudo normwhich is the obvious sparsity constraint, however much lessconveniently handled [24]. Still, the above problem is nonconvex. Nevertheless, starting form an initial guess, a solutionof reduced cost can be reached using and iterative blockcoordinate relaxation approach [25], [26], where the pairsai, si (ith column of A, ith line of S) are updated one at atime. Assuming without much loss of generality that Φ isthe identity matrix, we have, alternating over i, the followingminimization problems :

minai,si

12‖Ri −H( aisi )‖2Σ + λi‖si‖1 (8)

whereRi = X −H(

∑j 6=i

ajsj ) (9)

Rewriting the quadratic matrix norm with an explicit sum overlines leads to :

minai,si

∑k

12σ2

k

‖Ri,k − aiksiHk‖2 + λi‖si‖1 (10)

where Ri,k is the kth line of Ri. Keeping si fixed, zeroing thegradient of the above with respect to ai

k leads to the followingupdate rule :

aik =

(siHkHt

ksti

)−1siHkRt

i,k (11)

Conversely, keeping ai fixed, zeroing the gradient2 of theabove with respect to si leads to :

si =

Xk

aik

σ2k

Ri,kHtk − λisign(si)

! Xk

“ aik

σk

”2HkHt

k

!−1

(12)

provided the rightmost term exists. Save for the sign func-tion, this is the least squares solution to the multichanneldeconvolution in the single-input multiple-output (SIMO) case.An important feature here is the effective beam used for thedeconvolution : the effective beam may be invertible evenif the individual beams are not. Following the discussionin [11] any robust 1D deconvolution algorithm could beapplied to the 1D-projected data. Here, as hinted to by thesign function, robustness will come from shrinkage. However,equation (12) is not easily inverted and probably has no closedform solution owing to the coupling matrix on the right whichwould commonly not be diagonal. It would have to be solvednumerically in an iterative scheme, in the general case. Whenthis factor is diagonal (or approximately diagonal), the closedform solution is known as soft-thresholding. In all cases, thefull set of update rules leads to a slow algorithm which is likely

2We stick to loose definitions, knowing that rigorous statements can bemade with no impact on the result.

to reach and stay stuck by the local minimum closest to thestarting point. In the following, we focus on approximationsfor faster and more robust algorithms and propose a practicalrule for updating the components S assuming the mixingmatrix A is known and fixed. Future work will combine thiswith a practical learning step on A to complete the algorithm.

IV. KNOWN MIXING MATRIX

When A is known a priori, problem (6) reduces to a linearinverse problem with a sparsity constraint on the solution :

minS

12‖X −H( AS )‖2Σ +

∑i

λi‖si‖1 (13)

In the single channel case, deconvolution with an l1 penaltyhas been studied previously with a known mixing matrix [18]–[20]. More generally, solving linear inverse problems withsparse priors or l1 regularization has attracted much attentionlately [27]–[30]. If the combined mixture and convolutionlinear operator is underdetermined, the above cost functionis not strictly convex. This is where sparse decompositionmethods have proven especially profitable. When the operatoris overdetermined, the l1 sparsity constraint is a useful regu-larization when the data is corrupted by noise : the availablebudget will hopefully not be spent on representing smallnoise coefficients. Theoretical results have been establishedregarding the existence and uniqueness of a solution, andgeneral algorithms have been proposed to solve some of theseproblems ( [31] and references therein). In principle, thesemethods apply to the problem at hand. Among these, wefocus on iterative projected descent algorithms which havebeen proposed in [29], [30], [32], [33] for the inversion ofunderdetermined linear operators with sparsity constraints.

Consider again updating each source si alternately, assum-ing all the other sources fixed. The minimization problem (13)can be rewritten as follows :

minsi

f1(ai, si) + λif2(si) (14)

where

f1(ai, si) =X

k

1

2σ2k

‖Ri,k − aiksiHk‖2 and f2(si) = ‖si‖1 (15)

The gradient of f1 with respect to si assuming all the otherparameters are fixed is given by :

g =∑

k

−aik

σ2k

Ri,kHtk +

(aik

σk

)2

siHkHtk (16)

The Hessian of f1 is given by :

H =∑

k

(aik

σk

)2

HkHtk (17)

From here, several optimization schemes are possible all akinto a projected descent algorithm, the descent direction isgiven by the conjugate gradient. Typically, following recentdevelopments in optimization [29], [30], [32], [33]:

snewi = ∆λi

(sold

i − µgH−1)

(18)

where ∆λiis a soft or hard thresholding operator with

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threshold λi and µ is the usual step size parameter. It followsthat :

snewi = ∆λi

(1− µ)sold

i + µ

Xk

aik

σ2k

Ri,kHtk

!H−1

!(19)

Taking a single step of length µ = 1 at each iteration leadsto the following update rule :

si = ∆λi

0@ Xk

aik

σ2k

Ri,kHtk

! Xk

“ aik

σk

”2HkHt

k

!−11A (20)

which is simply thresholding the linear least squares so-lution. The numerical experiments in the next section wereconducted using this rule to update each source alternatelyand iteratively. Following a specific feature of MCA andGMCA-like algorithms, the threshold is initially set to a highenough value and is progressively decreased along iterations,leading to a robust salient to fine estimation process. Thestrategy by which the thresholds are progressively decreasedalong the iterations is an important issue discussed in [34].As mentioned earlier, the computation of the inverse Hessianshould be better behaved than inverting each single channeldeconvolution operator separately [11].

Other similar projected descent iterations with proper pre-conditioning were also experimented with but did not lead tobetter results. The following projected Landweber iteration,

Snew = ∆{λi}

“Sold + µAtHt

“X −H(AS)

””(21)

where µ is a small enough step size and Ht is the adjointof the channel-wise convolution operator, used in conjunctionwith a decreasing threshold, is slow but leads to very similarresults. Without the decreasing threshold, the latter iterationwas even slower to convergence and less robust in recoveringthe mixed sources. Other preconditioners could be used tolower the complexity of the iteration such as the inversediagonal or block-diagonal of the Hessian [23], or to handlecases where the Hessian is badly conditioned. Using a differentstep length for each scale, in a projected Landweber iterationto solve a wavelet-regularized deconvolution is an interestingpossibility discussed in [35]. Current work is also on otheriterations involving the inverse or pseudo inverse of A. Afterthe sources S have been estimated for a given mixing matrixA, the next step consists in updating the mixing matrix using asuitable learning step. Alternating source estimation steps andmixing matrix learning steps is current research to be reportedshortly.

V. NUMERICAL EXPERIMENTS

In a first set of experiments, with multichannel 1D data,source signals of length t = 128 were generated as Bernoullispike trains with parameter p. The spikes were given randomamplitudes uniformly distributed between −1 and 1. Randommixing matrices and beams were used. The latter are positivewith compact support. The beams and columns of the mixingmatrix were normalized to unit l2 norm. Noise was addedto the mixed and convolved sources, sampled from a normaldistribution with 0.1 standard deviation. The experiments wererepeated 500 times.

0 20 40 60 80 100 120−0.5

0

0.5

0 20 40 60 80 100 120−0.2

0

0.2

0.4

0.6

amp

litu

de

0 20 40 60 80 100 120−1

−0.5

0

0.5

1

sample number

0 20 40 60 80 100 120−1

−0.5

0

0.5

1

sample number

amp

litu

de

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−1

−0.5

0

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sample number

amp

litu

de

Fig. 1. top : Synthetic mixtures in the three channels used in this experiment.bottom : The two original sources are in green. The source signals recoveredusing the proposed algorithm appear in red. The results obtained using asimple pseudo-inverse of the linear mixture and convolution operator appearin light blue.

Figure 1 shows typical synthetic observation in threechannels. The joint separation and deconvolution objectivewas approached using the proposed alternating iterativeprogressive thresholding scheme (20). The results werecompared to those obtained using a multichannel Wienerfilter and a simple pseudo-inverse of the full linear operator.Figure 2 display a comparison of these three methods interms of false postives to true postives curves, as the finalthreshold is decreased. The impact of an increasing numberof channels, or an increasing sparsity of the sources, oran increasing noise variance on the performance of thesealgorithms is shown to agree with what is expected. In allcases, the proposed iterative thresholding scheme performsmuch better than the other two filters.

In a second set of experiments, with multichannel 2D data,three natural images shown on the left of figure 5 were usedto generate six random mixtures. These were then degradedby 2D-convolution resulting in the blurred images shown onfigure 4, using the Laplacian beams shown on figure 3. Apply-ing the proposed algorithm in the curvelet domain, enables agood recovery of the input images from the multichannel dataas shown on figure 5 where the separated and deconvolvedimages are on the right.

5

0 0.05 0.1 0.15

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

false positives

tru

e p

osi

tive

s

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0

0.1

0.2

0.3

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tru

e p

osi

tive

s

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0

0.1

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0.5

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0.8

0.9

1

false positives

tru

e p

osi

tive

s

Fig. 2. Curves in red were obtained using the proposed alternate iterativeand progressive thresholding scheme. The pseudo-inverse gives the resultsin dark blue while the Multichannel Wiener Filter gives those in light blue.top : Lowering the Bernoulli parameter p from 0.4 to 0.1 gives sparser sourceprocesses for which the proposed scheme clearly performs better. middle :The standard noise deviation is lowered from 0.1 to 0.05, 0.025, 0.001 withthe expected impact on the algorithms performance. bottom : Increasing thenumber of channels from 2 to 3, 4, 6 also leads to better source estimation.

116 122 128 134 1400

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0.2

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116 122 128 134 1400

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116 122 128 134 1400

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116 122 128 134 1400

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116 122 128 134 1400

0.5

1channel 5

116 122 128 134 1400

0.5

1channel 6

Fig. 3. The six Laplacian beams used to obtain the blurred image mixturesshown on figure 4, by 2D convolution.

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Fig. 4. Simulated observations consisting of blurred linear mixtures of threeintial images shown on figure 5, in six channels with psf ’s shown on figure 3.

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Fig. 5. left : The three initial images. right : The three recovered imagesusing the proposed joint deconvolution and separation algorithm in the curveletrepresentation.

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VI. FUTURE WORK

The preliminary results reported here have demonstratedthe feasibility of a joint separation and deconvolution ofmultichannel data. We described an alternating progressive it-erative thresholding algorithm to obtain a robust inverse of themixing and convolution linear operator. Current work is withextending this method to the case where the mixing matrixis not known a priori, much in the line of GMCA for blindsource separation. The idea is to alternate updates of S and A,i.e. sparse coding and dictionary learning steps, in a salient-to-fine process with a threshold decreasing along iterations.An important application is CMB data analysis, which willbenefit from the versatility of GMCA-like algorithms whichcan easily account for prior knowledge of the emission lawsof the mixed components. Assuming unknown but parametricpsf ’s is yet another exciting perspective.

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