a new look to multichannel blind image deconvolution

14
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 7, JULY 2009 1487 A New Look to Multichannel Blind Image Deconvolution Wided Souidene, Member, IEEE, Karim Abed-Meraim, Senior Member, IEEE, and Azeddine Beghdadi, Senior Member, IEEE Abstract—The aim of this paper is to propose a new look to MBID, examine some known approaches, and provide a new MC method for restoring blurred and noisy images. First, the direct image restoration problem is briefly revisited. Then a new method based on inverse filtering for perfect image restoration in the noiseless case is proposed. The noisy case is addressed by introducing a regularization term into the objective function in order to avoid noise amplification. Second, the filter identification problem is considered in the MC context. A new robust solution to estimate the degradation matrix filter is then derived and used in conjunction with a total variation approach to restore the original image. Simulation results and performance evaluations using re- cent image quality metrics are provided to assess the effectiveness of the proposed methods. I. INTRODUCTION M ULTICHANNEL (MC) image processing is nowadays a relatively active field of research. Preliminary results of multichannel deconvolution were first found in the signal case then extended to the image. This development is due to the in- creasing number of applications where several versions of the captured image are available. In multichannel framework, sev- eral images are observed from a single scene that passes through different channels. Applications where multichannel techniques could be used include, among others, polarimetric [1], satellite [2], astronomical [3], [4], and microscopic [5] imagery. When channels are frequency bands, we refer to it as multispectral im- ages. If the same scene is captured at different time slots, we talk about image sequences. If different representations of the same image are provided at different resolutions, we also can treat this as a multichannel representation. The main advantage we can draw from MC processing for the deconvolution is to exploit the diversity and redundancy of information in the dif- ferent acquisitions. Hence, it is worth to note that the set of the observed images is considered as one entity. Manuscript received July 07, 2008; revised January 17, 2008. First published May 12, 2009; current version published June 12, 2009. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Ljubisa Stankovic. W. Souidene is with the ESIGETEL, 77215 Avon-Fontainebleau, and also with the L2TI, Université Paris 13, 93430, Villetaneuse (e-mail: [email protected]). A. Beghdadi is with the L2TI, Université Paris 13, 93430, Villetaneuse (e-mail: [email protected]). K. Abed-Meraim is with the Telecom ParisTech, 75013, Paris, and also with the ECE Department, College of Engineering, University of Sharjah, UAE (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIP.2009.2018566 Image deconvolution/restoration solutions can be divided into two classes: stochastic and deterministic. Stochastic methods consider observed images as random fields and esti- mate the original image as the most probable realization of a certain random process. These methods are, mainly, the linear minimum mean squares error (LMMSE) [6], the maximum likelihood (ML) [7], and the maximum a posteriori (MAP) [8]. These methods have two major drawbacks: i) they are very sensitive to perturbations and modeling errors, ii) strong statistical hypothesis are made on the image and the noise which are considered as uncorrelated homogeneous random processes. On the other hand, deterministic methods do not rely on such hypothesis and estimate the original image by minimizing a norm of a certain residuum. These methods include among others: constrained least square (CLS) methods which incorporate a regularization term [9], the iterative blind deconvolution technique [10] and the non-negativity and support constraint—recursive inverse filtering (NAS-RIF) algorithm [11]. The latter methods are based on minimizing a certain criterion under some constraints like non-negativity of the original image, finite support of the convolution masks, smoothness of the estimate, etc. Blind MC image deconvolution can be performed in two ways. One can first identify the point spread function (PSF) of the degradation also called blur function and then restore the image using this knowledge. This approach belongs to the class of identification techniques [12]. When the original image is directly restored, we refer to the solution as an equalization or inverse filtering technique. In this article, we deal with an equalization technique called the mutually referenced equal- izers (MRE) where a regularization term is incorporated is order to prevent from noise amplification. Then, we consider another approach which belongs to the class of channel identification techniques. We carry out a comparative study in order to choose the most efficient blind channel identification algorithm. We compared these techniques in terms of estimation accuracy (restored image quality), and identification accuracy (distance between the real and the identified filters). Finally, we develop a joint identification/restoration technique using the total varia- tion (TV) aiming at the restoration of the original image. More precisely, our contributions consist of i) a new blind restoration method using the mutually referenced equalizers technique in conjunction with a regularization method that trun- cates the greatest singular values of the inverse filter matrix; ii) a new multichannel identification technique that generalizes the least squares smoothing (LSS) method [13] from the 1-D to the 2-D case. This method is then compared with other 1057-7149/$25.00 © 2009 IEEE

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Page 1: A New Look to Multichannel Blind Image Deconvolution

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 7, JULY 2009 1487

A New Look to Multichannel BlindImage Deconvolution

Wided Souidene, Member, IEEE, Karim Abed-Meraim, Senior Member, IEEE, andAzeddine Beghdadi, Senior Member, IEEE

Abstract—The aim of this paper is to propose a new look toMBID, examine some known approaches, and provide a newMC method for restoring blurred and noisy images. First, thedirect image restoration problem is briefly revisited. Then a newmethod based on inverse filtering for perfect image restorationin the noiseless case is proposed. The noisy case is addressed byintroducing a regularization term into the objective function inorder to avoid noise amplification. Second, the filter identificationproblem is considered in the MC context. A new robust solution toestimate the degradation matrix filter is then derived and used inconjunction with a total variation approach to restore the originalimage. Simulation results and performance evaluations using re-cent image quality metrics are provided to assess the effectivenessof the proposed methods.

I. INTRODUCTION

M ULTICHANNEL (MC) image processing is nowadaysa relatively active field of research. Preliminary results

of multichannel deconvolution were first found in the signal casethen extended to the image. This development is due to the in-creasing number of applications where several versions of thecaptured image are available. In multichannel framework, sev-eral images are observed from a single scene that passes throughdifferent channels. Applications where multichannel techniquescould be used include, among others, polarimetric [1], satellite[2], astronomical [3], [4], and microscopic [5] imagery. Whenchannels are frequency bands, we refer to it as multispectral im-ages. If the same scene is captured at different time slots, wetalk about image sequences. If different representations of thesame image are provided at different resolutions, we also cantreat this as a multichannel representation. The main advantagewe can draw from MC processing for the deconvolution is toexploit the diversity and redundancy of information in the dif-ferent acquisitions. Hence, it is worth to note that the set of theobserved images is considered as one entity.

Manuscript received July 07, 2008; revised January 17, 2008. First publishedMay 12, 2009; current version published June 12, 2009. The associate editorcoordinating the review of this manuscript and approving it for publication wasProf. Ljubisa Stankovic.

W. Souidene is with the ESIGETEL, 77215 Avon-Fontainebleau, andalso with the L2TI, Université Paris 13, 93430, Villetaneuse (e-mail:[email protected]).

A. Beghdadi is with the L2TI, Université Paris 13, 93430, Villetaneuse(e-mail: [email protected]).

K. Abed-Meraim is with the Telecom ParisTech, 75013, Paris, and also withthe ECE Department, College of Engineering, University of Sharjah, UAE(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIP.2009.2018566

Image deconvolution/restoration solutions can be dividedinto two classes: stochastic and deterministic. Stochasticmethods consider observed images as random fields and esti-mate the original image as the most probable realization of acertain random process. These methods are, mainly, the linearminimum mean squares error (LMMSE) [6], the maximumlikelihood (ML) [7], and the maximum a posteriori (MAP)[8]. These methods have two major drawbacks: i) they arevery sensitive to perturbations and modeling errors, ii) strongstatistical hypothesis are made on the image and the noisewhich are considered as uncorrelated homogeneous randomprocesses. On the other hand, deterministic methods do notrely on such hypothesis and estimate the original image byminimizing a norm of a certain residuum. These methodsinclude among others: constrained least square (CLS) methodswhich incorporate a regularization term [9], the iterativeblind deconvolution technique [10] and the non-negativityand support constraint—recursive inverse filtering (NAS-RIF)algorithm [11]. The latter methods are based on minimizinga certain criterion under some constraints like non-negativityof the original image, finite support of the convolution masks,smoothness of the estimate, etc.

Blind MC image deconvolution can be performed in twoways. One can first identify the point spread function (PSF)of the degradation also called blur function and then restorethe image using this knowledge. This approach belongs to theclass of identification techniques [12]. When the original imageis directly restored, we refer to the solution as an equalizationor inverse filtering technique. In this article, we deal with anequalization technique called the mutually referenced equal-izers (MRE) where a regularization term is incorporated is orderto prevent from noise amplification. Then, we consider anotherapproach which belongs to the class of channel identificationtechniques. We carry out a comparative study in order to choosethe most efficient blind channel identification algorithm. Wecompared these techniques in terms of estimation accuracy(restored image quality), and identification accuracy (distancebetween the real and the identified filters). Finally, we developa joint identification/restoration technique using the total varia-tion (TV) aiming at the restoration of the original image.

More precisely, our contributions consist of i) a new blindrestoration method using the mutually referenced equalizerstechnique in conjunction with a regularization method that trun-cates the greatest singular values of the inverse filter matrix;ii) a new multichannel identification technique that generalizesthe least squares smoothing (LSS) method [13] from the 1-Dto the 2-D case. This method is then compared with other

1057-7149/$25.00 © 2009 IEEE

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Fig. 1. (a) Set of degraded images used for MC deconvolution and altered byPSFs of size 7� 7; (b)–(c) restored image using a mono-channel deconvolutiontechnique; (d) perfectly restored image in the noiseless case using the multi-channel deconvolution technique described in this article.

existing identification methods with respect to the estimationaccuracy and computational cost; iii) a new robust multichannelrestoration method based on a total variation technique; andiv) a performance evaluation and comparative study of thedifferent blind restoration methods using some objective imagequality metrics.

The rest of the paper is organized as follows. Section II high-lights the advantages of the multichannel processing approach.Section III introduces the image acquisition model and statesthe objectives of this paper. In Section IV, the regularized MREmethod is developed. Section V presents the generalized LSSmethod and briefly reviews some other existing channel identi-fication algorithms for comparison purposes. In Section VI, weintroduce our image restoration technique using the total vari-ation approach. Simulation results and performance evaluationare provided in Section VII. The last section is for the conclu-sion and final remarks.

II. MOTIVATIONS FOR MC PROCESSING

The motivations behind the use of MC framework as com-pared to the standard mono-channel one are first the increasingnumber of applications in this field, but more importantly, thepotential diversity gain that may lead to significant improve-ments in the restored image quality. Indeed, diversity combiningtechniques have been recently the focus of an intensive researcheffort in different application fields including radar processing[14], wireless communications [15], and image processing [16].It is shown in particular that, in the noiseless case, the diversitycombining techniques allow a perfect image restoration [17].This is illustrated by the example in Fig. 1, where we com-pare a blind mono-channel restoration technique developed in[18], and based on the iterative Richardson–Lucy scheme, withthe multichannel one developed in Section IV of this article.Clearly, this example highlights the performance gain that canbe obtained by MC processing. This gain is due to the inherentdiversity of multichannel systems where multiple replica of thesame signal/image are observed through different (independent)channels. In that case, if a part of the original information is lost(degraded) in one of the observed images, it can be retrievedfrom the other observed images upon certain diversity condi-tions. More precisely, this is possible when the same image dis-

tortion does not occur on all the observed images simultane-ously. Mathematically, this is expressed in the condition that thespectral transforms of the PSFs do not share common zeros.1 In-deed, a PSF’s zero represents roughly a “fading” around a par-ticular frequency point, and, hence, common zeros represent thesituation where the same fading occurs on all channels simulta-neously. In the presence of noise, perfect reconstruction is notpossible but the diversity gain provides significant improvementin the restored image quality. This is illustrated by Fig. 2, wherewe plot the restored image quality evaluated by the objectivemeasure proposed in [19] versus the number of observed im-ages (channels).

In the mono-channel case, the restoration method used is theone introduced in [18]. For the MC case, we use the restora-tion method with the total variation approach (Section VI). Onecan observe that the gain increases in terms of image restorationquality when using the MC processing. Furthermore, this gainis obtained with a relatively small number of independent PSFs(3 to 4, depending on the SNR level) which limits the extra com-putational burden of MC processing.

III. PROBLEM STATEMENT

A. Notations

It is assumed that a single image passes through inde-pendent channels and, hence, different noisy blurred imagesare observed. Each channel corresponds to a degrading filter.The notations used are as follows.

— the original image of size .— the output images each of size .— the point spread functions (PSFs) each of

size and their row-wise vectorizedversions. We denote by the vector ofall PSF parameters. To satisfy the diversity condition, thePSF functions are assumed to have no common zeros, i.e.,the polynomials

are strongly co-prime.the additive noise in each channel.

For the seek of notational simplicity, we adopt here a “causal”representation of the considered filters so that the system modelcan be written as: for

In the sequel, the images and impulse responses will be pro-cessed in a vectorized, windowed form of size .Hence, we denote by the data vector corresponding

1Note that this condition can be met only if� � � different PSFs (channels)are available.

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SOUIDENE et al.: NEW LOOK TO MULTICHANNEL BLIND IMAGE DECONVOLUTION 1489

Fig. 2. Evolution of the restored image quality (using the PSF identificationmethod followed by TV-based restoration) versus the number of observed im-ages (PSFs) for different SNR values: Cameraman image.

to a window of the th image where the last pixel is indexed by, i.e., the right bottom pixel

(1)

In order to exploit the diversity and the redundancy offeredby the multiple observations of , we deal simultaneously withall observed images by merging them into a single observationvector

(2)

where is an vectorgiven by

and being the filter matrix associatedto given by

. . .. . . (3)

with blocks in the row direction and blocksin the column direction and [see (4), shown at the bottom of thepage] of size .

B. Objectives

The ultimate goal is to restore the original image in a sat-isfactory way even in “severe” observation conditions. In prac-tice, the original image as well as the degrading filters are totallyunknown, so that our restoration procedure is totally blind. Totackle this problem two types of solutions are proposed. The di-rect image restoration technique [20] and the indirect one, thatfirst estimates the unknown PSFs and then restores the originalimage in a nonblind way (i.e., using the previously estimatedPSFs) [12].

1) Direct Restoration: Our objective here is to directly re-store the original image using only its degraded observed ver-sions. More precisely, we search for a unique equalizer or in-verse filter which, applied to the set of observations, allows usto restore the original image. We pay a particular attention to therobustness against additive noise by including in the estimationcriterion of the inverse filter an additional term that controls andlimits the noise effect.

2) Restoration via PSF Identification: Our objective here isto, first identify the PSF function and then inverse it in order torestore the original image. In the noisy case, the filter responseinversion leads to noise amplification, and, hence, we proposeto add a regularization term in order to reduce this undesiredeffect. For that, we use the total variation (TV) constraint of therestored image as a regularization criterion. In the following, wedeal with the direct restoration approach in Section IV and withthe indirect one in Section VI. The PSF estimation methods areconsidered in Section V.

IV. REGULARIZED MUTUALLY REFERENCED

EQUALIZERS (R-MRE)

In this section, we introduce our first image restorationmethod using the direct estimation of the inverse filters. Thesefilters are estimated by MRE method in [17]. We propose someimprovements in terms of computational cost and robustnessagainst additive noise as shown below.

A. MRE Method: Review and Improvements

We search, here, for the restoration filter denoted which,applied to the observed images, provides us with an esti-mate of the original image. This is a multichannel 2-D filterof size . This filter exists under the following as-sumptions: i) the PSFs have no common zeros (see [20] for more

. . .. . . (4)

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1490 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 7, JULY 2009

details) and ii) the filter matrix has full column-rank. More pre-cisely, it is proved in [17] that the channel matrix is left in-vertible if

(5)

When both conditions are satisfied, there exists a set of equal-izers each of them allowing us to restore the orig-inal image with a specific spatial shift . Note that thereexists an infinity of equalizers of different sizes (an infinity ofcouple ) that satisfy the condition in (5). Hence, when

is fixed, the original image is estimated up to a cer-tain constant factor and a certain spatial shift. The principle ofmutually referenced equalizers is as follows: suppose we havecomputed two equalizers and inducing the spatial shift

and , respectively

(6)

(7)

where denotes the 2-D convolution, is a given positive

scalar, and being the 2-D filter ofsize applied to the th observed image.

In the noiseless case, if we apply the equalizer toand the equalizer to , we

obtain exactly the same windowed area of the original image

(8)

In the original MRE method, it was shown that solving (8) forall equalizers inducing a spatial shift in the interval

leads to the computation of aset of equalizers which perfectly restore the original image inthe noiseless case. This solution presents a major drawback: itrequires the computation of a large number of equalizers (ex-actly equalizers), and, hence, it is com-putationally expensive. In [17] and [21], it is shown that wecan reduce the number of equalizers to be estimated and, con-sequently, the computational cost of this solution. More pre-cisely, it was proved that only 2 extremal equalizers cor-responding to the extremal spatial shifts and

are sufficient forperfect image restoration. In practice, these shifts do not ap-pear to perform good restoration quality due to the artefacts ap-pearing in the boundaries of the restored image. Therefore, wepropose to use a third equalizer corresponding to the medianshift where denotes the

integer part. Solving (8) for equalizers and , consistsin solving the following set of linear equations:

(9)

In practice, in order to take into account the additive noise,this set of equations is solved in the least squares sense leadingto a quadratic form that can be written as follows:

(10)

where and is therow-wise vectorized version of . The quadratic formis given by

whereand is the vector defined in (2) with. matrix is singular and its rank

depends on the equalizer size ; therefore, cri-terion must be minimized under unit-norm constraint.

B. Regularized MRE Method

In the noisy case, the MRE algorithm fails in restoring effi-ciently the image. This is due to the ill-conditioned filter matrixwhose inversion leads to noise amplification. In order to comethrough this difficulty, we propose, here, to combine the MREcriterion in (10) with a regularization technique inspired from[22] and adapted to the multichannel framework. In fact, thenoise amplification is due to the largest singular values of theInverse Filter Matrix (IFM). Therefore, our regularization tech-nique simply consists in the truncation of the largest singularvalues of the IFM. This truncation is realized through an adap-tive thresholding technique which is explained below. Now, toreduce the computational cost of the desired eigenvalues, we ex-ploit the Toeplitz structure and the large dimension of the inversefilter matrix in such a way to approximate it by a block circu-lant matrix [23] whose eigenvalues can be computed by meansof Fourier transform [24].

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SOUIDENE et al.: NEW LOOK TO MULTICHANNEL BLIND IMAGE DECONVOLUTION 1491

In this paper, we choose among the MRE equalizers torestore the original image as it provides the best restoration per-formance compared to extremal shift equalizers and asmentioned previously. Let us write this equalizer as

(11)

and let be the IFM associated with and defined as in (3)and (4). Since is a large block Toeplitz matrix with Toeplitzblocks, it can be approximated by a large block circulant matrix

with circulant blocks. This approximation leads to the fol-lowing estimation:

(12)

where denotes the original image in vector form shiftedby :

. denotes the th observed image invector form: . Let defineas a block circulant matrix with circulant blocks

...

Equation (12) becomes

...

As mentioned in Section IV, we propose to truncate thelargest eigenvalues of to avoid noise amplification whenrestoring the original image. A well known property of cir-culant matrices is that their eigenvalues can be expressed asa function of the elements of the first column. This propertycan be extended to block circulant matrices by considering thefirst column of each column block. Since is a block circulantmatrix with circulant blocks , it can be proved [23] that itseigenvalues are given by

(13)

where is a vector containing the eigenvalues of is aknown sparse matrix and is the number of pixels in the re-stored image. In order to truncate the largest eigenvalues of ,we apply a nonlinear filter represented by a projection matrix

. Let where and

ifotherwise

(14)

is a predetermined threshold. Here, we choose

(15)

where is a chosen scalar in the range whose value dependson the image type. Let be the error between theoriginal vector of eigenvalues and the truncated one,

. The regularized MRE criterion can be then expressed as

(16)

with

and (17)

(18)

In (18), represents a scalar factor that controls the amountof regularization we would like to incorporate into the MREcriterion. Note that is a nonlinear criterion since de-pends nonlinearly on . However, once is fixed, becomesquadratic and it can be easily solved by means of a classical min-imization algorithm. Here, we use a two step optimization pro-cedure. First, we compute the MRE equalizer that mini-mizes the MRE criterion without regularization. Then the equal-izer is used to fix the thresholding matrix . Once isfixed, the R-MRE criterion becomes quadratic and, hence,can be minimized using a classical optimization scheme. Notethat the solution set of the R-MRE criterion contains the de-sired filters but also, undesired “blocking” filters (denotedin vector form) that satisfy in the noiseless case

(19)

For example, the filter given by and, is a blocking filter as it satisfies. Consequently, we search for a so-

lution that minimizes the criterion (16) and maximizes the re-stored image energy. Indeed, a blocking filter output is given bynoise term only while the desired restoration filter would pro-vide us with an approximate version of the original image hatis assumed to have a much larger energy than noise. To achievethe previous objective, we simply constrain this energy to beunitary, i.e., we solve criterion (16) under the following energyconstraint:

(20)

where denotes the matrix Kronecker product. This is equiva-lent to minimizing the Rayleigh quotient

(21)

which solution is given by the principal generalized eigenvectorof .

V. PSF BLIND IDENTIFICATION TECHNIQUES

The second approach to perform MBD consists in first iden-tifying the degradation filters and then inverting them in orderto restore the original image. In this section, we are interested inidentifying the blur functions in each channel that would be usedlater for image restoration by means of TV-based regularizationtechnique. More precisely, we first carry out a comparative studyof several existing deterministic techniques for multichannelblind image identification identification techniques, namely, thesubspace method (SS), the minimum noise subspace method

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(MNS), and the cross relation (CR) method. These methods arequite similar in the principle, in fact they compute some “cor-relation” and “redundance” between the different acquisitionsin order to retrieve the degrading filters. Then, we propose anew identification algorithm based on a smoothing least squarestechnique. In this section, we present the principle and the algo-rithm of each method before comparing their performance usingcomputer simulation experiments.

A. Subspace Method (SS)

The SS method for image restoration was first introduced in[25]. It exploits the fact that, in the noiseless case, all vectors

are in the subspace scanned by the column vectors of(the image data subspace). Hence, we can estimate the columnrange space of from the observed data. In prac-tice, in order to take into account the additive noise, we estimate

as the subspace spanned by the principal eigen-vectors of , the covariance matrix of . On the other hand, itis shown in [25], that characterizes uniquely the pa-rameter vector (up to a constant factor).2 As a consequence,we can estimate the unknown channel parameters by fittingthe range space of to the image data subspace measured fromthe observation. The best fit is reached when is preserved (un-altered) by orthogonal projection onto . This can beperformed by minimizing the least squares criterion

(22)

where represents the matrix of minor(least) eigenvectors of referred to as the noise sub-space. Note that is unitary and, hence,

. As is a linear function of , thecriterion in (22) is a quadratic form of that can be written as

(23)

where is obtained from by using therelation where is a function of that can beobtained by straightforward algebraic manipulations. The lattercriterion is minimized under unit-norm constraint, i.e., ,to avoid the trivial solution , so that corresponds to theleast eigenvector of . The SS algorithm is summarized inTable I.

B. Minimum Noise Subspace Method (MNS)

The MNS method is a simplified version of the SS one. It isbased on the same principle but it does not use all the noise sub-space but a minimum number of noise space vectors. Indeed,it is shown in [26], that, instead of using the noise vectors

, only properly chosen noise space vectorsare sufficient to guarantee the uniqueness of the solution of (23)(up to a scalar factor). To estimate the noise space vectors, weconsider image pairs that form a tree structure. For eachpair , we estimate the least eigenvector of the covariancematrix of which is zero padded (as

2The constant factor is an inherent ambiguity in blind system identification asthe exchange of a scalar between � and � does not affect the observation, i.e.,� � �� � �� � ��������.

TABLE ISS ALGORITHM

shown in Table II) to form a noise space vector. By doing so,we avoid the computationally expensive eigen-decompositionof required in the SS method. Moreover, the noisespace vectors can be computed in a parallel scheme if a par-allel architecture is available. These noise space vec-tors, , are then used in (23) to form the LS cri-terionthat is solved similarly to the SS method, under unit-norm con-straint. The MNS algorithm is summarized in Table II. In theoriginal MNS method, some observed images are used morethan others, depending on the chosen set of image couples. Thismight lead to poor estimation performance if the system out-puts that were chosen correspond to the “worst observed im-ages.” This raises the problem of the best choice of the appro-priate set of outputs and motivates the development of a Sym-metric MNS method (SMNS). In SMNS technique, we guar-antee a certain symmetry in the choice of the set of observedimage couples. For instance, for channels, we use thefollowing channel couples: (1,2), (2,3), (3,4), (4,1). This choicemakes the SMNS method more robust than MNS one in termsof estimation accuracy. Hence, SMNS algorithm is the same asthe MNS except for the fact that we use pairs of observed im-ages: , instead of an ad-hocchoice of pairs forming a tree structure.

C. Cross Relation Method (CR)

The CR method was first introduced in [27]. It is based onthe commutativity of the convolution operator. In fact, in thenoiseless case, we observe that: .Therefore

We can write this relation for any pair of channels. We have atotal of pairwise equations corresponding to

(24)

where

...

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TABLE IIMNS ALGORITHM

TABLE IIICR ALGORITHM

.... . .

......

. . .

Inversely, it is shown in [28] that (24) characterizesuniquely—up to a constant factor—the unknown channelparameter vector . Hence, the CR method consists in esti-mating by solving (24) in the LS sense

Again, the solution of the above LS criterion under unit-normconstraint of , is given by the least eigenvector of . Thismethod presents a real computational gain as it requires only oneeigenvector computation whereas the other methods require twoor more eigen-decompositions. Moreover, we can further reducethe computational cost by choosing, as for the MNS and SMNSmethods, only or pairs of cross relations. The CRalgorithm is summarized on Table III.

D. Least Squares Smoothing Method (LSS)

The LSS method is an estimation technique that exploits theisomorphism between the input and the output spaces. LSS hasbeen introduced in [13] in the 1-D case. We propose, here, togeneralize it for the first time to the 2-D signals. Indeed, in thenoiseless case, one can write the data matrix as follows:

(25)

(26)

with

Since is full column rank, the row space of (outputspace) coincides with the row space of (input space). Thisallows us to construct appropriate projection subspaces usingonly observed data. More precisely, let be a row subspacethat contains all rows of except the th one (referred to as

). In order to construct , first, note that each column ofmatrix consists in blocks containing

elements each one. So that, each row indexin matrix can be written as

and . Let be the index of the row to be se-lected: . In order to extract the th row of ma-trix , we first eliminate the rows in all row blocks of exceptthose in blocks , i.e., rows corresponding to blocks in therange and . In our paper, we con-sider , for which the previousrow vectors belong to the subspace generated by the rows of

andrespectively. Then,

we eliminate the rows in the same block as row , i.e., rows in theth block corresponding to indices in the range and

(for and). These rows live in the subspace spanned by the row vec-

tors ofand ,respectively. Finally, is constructed from the row space gen-erated by . This subspace will beused to project the data matrix . In practice, to take into ac-count additive noise, the desired row space is estimated fromthe principal right singular vectors of . The projection error of

onto this row space allows us to extract the th columnvector of , i.e.,

where is a rank one matrix with principal singular vectorequal to up to a scalar factor. By selecting a column vectorof that contains all channel coefficients, one can estimate thelatter through the estimation of as the principal left singulareigenvector of . Note that many columns of contain allchannel coefficients. Using (3), one can observe that the columnblocks in the range contain all block matrices

. Similarly, (4) showsthat the column vectors of in the rangecontain all the coefficients . Con-sequently, the column index should be in the set

(which justifiesthe choice of ). The LSS algorithm issummarized in Table IV.

Remark: All the considered identification methods requirethe a priori knowledge of the PSF size, i.e., and . A mainadvantage of the LSS method is the possibility to relax this con-straint in a joint channel parameter and channel size estimation

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TABLE IVLSS ALGORITHM

framework. This implementation has already been consideredin [13] in the 1-D case and can be used in the 2-D case, as well.

VI. IMAGE RESTORATION USING IDENTIFIED FILTERS

In this section, we are interested in a restoration techniquewhich consists in retrieving the original image using both theset of observed images and the identified filters in Section V.We propose to generalize the total variation (TV) method, firstintroduced in [29] in the monochannel case, to the multichannelcase and to use it to control the noise amplification during theinverse filtering. To this end, we consider a regularized leastsquares criterion given by

(27)

is a scalar parameter that controls the amount of the de-sired regularization and, hence, measures the tradeoff betweena good fit and the regularity of the solution. denotes the vec-torized version of original image . is the filter matrix as-sociated to the th PSF estimate. is the th observed image.

is the well known TV of the original image which is ex-pressed in the continuous 2-D domain as [29]

(28)

where represents a differentiable 2-D function and its spa-tial support. The minimization of the regularized least squarescriterion (27) necessitates the derivation of which raisesthe problem of differentiability of . In fact, is non-differentiable at zero point. Consequently, the TV term is ap-proximated as follows [29]:

(29)

where is a small scalar parameter which allows to disturbthe gradient value around zero point in order to guarantee itsdifferentiability. In order to numerically estimate the deriva-tive of , it is necessary to compute a discrete version ofthe TV term. Then, classical numerical algorithms, such as gra-dient descent optimization, could be used to solve criterion (27)and estimate the desired solution. The discrete gradient ofimage is expressed using derivative operators and inthe column direction and the row one , respectively. Theexpressions of the corresponding matrices and dependon the approximation technique adopted to compute the gradient[30]. These operators have the following matrix representation:

and where representsthe Kronecker product. Several approximations can be used tocompute this gradient. Here, we use the central derivative in thecentral part of the image and we consider the forward derivativeon the image boundaries. Using these operators, the TV termcan be expressed in the discrete form as

(30)

where and represent the thelement of vector and , respectively.

Let be an element of vector , computing the derivative ofwith respect to leads to

(31)

where

(32)

where and represent the th entries of ma-trices and , respectively. The estimate of the originalimage satisfies

(33)

When using numerical algorithms in order to solve (33),two difficulties are encountered: i) the nonlinearity of terms

and , ii) the nonregularity of thedesired solution. In fact, concerning point ii), it is well knowthat the grey levels of a digital image present some jumps andsteep gradients especially at edge locations. So, the solutionof (33) must satisfy such properties. Numerical optimizationalgorithms such as Raphson–Newton suffer from local minimaconvergence when used to optimize functions with discontinu-ities. Consequently, we propose to cope with these problemsby using the primal dual Newton method that is shown to

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Fig. 3. Blurred noisy observed images with PSFs (a) motion filter, (b) average filter, (c) gaussian filter with variance 1, (d) gaussian filter with variance 1.5 and��� � �� dB.

Fig. 4. (a) MRE restored image; (b) R-MRE restored one for observed imagesin Fig. 3.

have better convergence performance as compared to standardRaphson–Newton technique (see details in [31]). We introducetwo vectors of extra variables, one for each dimension, and eachof them substituting the “most nonlinear” part of the system inthe corresponding direction, namely

(34)

(35)

Let us define the vectors of extra variable as

and

For (34), we define a residual term as:or equivalently:

where .A similar expression is obtained for the -direction. Takinginto account equations (32), (34), and (35), one has to solve thelinear system

(36)

This system is globally “more linear” than the one in (31). Wecompute the Jacobian of this system by means of Taylor series

expansion and linearization of the residual terms. So, we canwrite the linearized system as

(37)

where

and

Now, to solve this system only with respect to , we apply blockelimination to (37) leading to the computation of the Schur com-plement of matrix . This transformation is possible

as the block is invertible. We divide the coeffi-

cient matrix of system (37) into four sub-blocks

Let denote the Schur complement of in

Let denote the difference between matrices and

(38)

We note that if is symmetric, then the whole matrix issymmetric, too. can be written as

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(39)

This expression shows that is a symmetric matrix sinceand are diagonal matrices. Consequently,

this transformation allows us to solve the system by means ofwell known iterative methods such as the minimum residual one(Minres) [32], [33]. The Minres function attempts to find a min-imum norm residual solution x to the system of linear equations

when matrix is symmetric. This condition fits thesystem that we are studying. In order to solve the consideredsystem, one can multiply the two bottom row blocks of by

and then subtract from the top block to obtain

After eliminating the variables and as they representsmall residuals, the transformed set of equations to solve to findthe descent direction becomes

(40)

Therefore, the primal-dual Newton method used here consistsin finding an appropriate descent direction for the criterionto be minimized. This solution guarantees the decrease ofthe criterion with respect to all spatial direction when matrix

is positive (semi)-definite. Thesum of matrices is a positive semi-definite matrix.We have to prove that this is also the case for . Let bea nonzero vector, we introduce and

. According to (39), we can write

(41)

Let denote and . Equation(41) can be re-written as

(42)

Using Cauchy–Schwartz inequality, and the fact that all ele-ments of diagonal matrices and are in the interval

, we can write

(43)

Given the inequality in (43) and the (42), we conclude that fora given nonzero vector . Thus, matrix ispositive semi-definite and so is . As mentioned earlier, thisproperty guarantees the convergence of the iterative algorithmto the global minimum.

VII. SIMULATION RESULTS

In the following, we test the image restoration performanceusing regularized MRE and TV-based algorithms. Experimentswere carried out for two different images: a portion of parrotimage which has a lot of features and details to be preserved byrestoration and the cameraman one which has a homogeneousbackground with a man in the middle. These images are ade-quate to measure the ability of the developed algorithms to re-store the image details and edges as well as the homogeneousarea. In all experiments, the degraded images are altered by a setof PSFs which simulate a camera motion, an averaging actionand a gaussian filtering with and , respectively.The number of observed images corresponding to the numberof independent PSFs is and the PSFs’ size is 5 5.The degraded images are corrupted by white gaussian additivenoise of power . The signal-to-noise ratio (SNR) is defined as

. For the plots in Figs. 9, 11,and 12, the statistics are evaluated over 100 Monte Carlo runs.

We first propose a comparison between the regularized ver-sion of the MRE algorithm and the nonregularized one for a setof degraded images with dB. The degraded imagesare shown in Fig. 3. The image restoration results are depictedin Fig. 4. This experiment confirms the inefficiency of MRE al-gorithm in the noisy case. It demonstrates the importance of theregularization to avoid the noise amplification phenomenon as-sociated with the image deconvolution problem.

A. R-MRE Versus TV-Based Algorithms

In this experiment, we compare the performance of the twoproposed algorithms for the restoration of the parrot and theCameraman images, respectively. The R-MRE algorithm withparameters for the regularization parameter (16) and

% for the relative threshold (15) has been applied to thedegraded images shown in Fig. 3. The result is compared withthe one obtained by the TV-based restoration approach usingthe channel estimate given by the CR method. The regulariza-tion parameters are and . To highlight theperformance gain due to the MC processing, we also present inFig. 5 the image restored by the monochannel technique in [18]applied to image (a) in Fig. 3. In this example, we note that theR-MRE algorithm mitigates the noise effect but alters the detailsof the original image, whereas the TV based method decreasesthe noise effect while preserving the image texture details. Themono-channel method is almost inefficient since the restoredimage is similar to the degraded one. The same experiment wasconducted on Cameraman image. The set of Cameraman de-graded images is shown in Fig. 6 and the restoration results inFig. 7. The parameter values used for the R-MRE algorithm are

for the regularization parameter and % forthe relative threshold. Concerning the TV regularization term,the regularization parameter is and the disturbingscalar is . We also display in Fig. 8 a log-compressedversion of the degraded and restored images in order to highlightthe restored image details. Note that details like the cameramanhand is missing in the degraded image whereas it is visible inthe TV-based restored image.

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Fig. 5. Restored images using different algorithms: (a) a regularized monochannel method; (b) R-MRE algorithm; (c) TV-based algorithm.

Fig. 6. Blurred noisy observed images with PSFs (a) motion filter; (b) average filter; (c) gaussian filter with variance 1; (d) gaussian filter with variance 1.5 and��� � �� dB.

Fig. 7. Restored images using different algorithms (a) restored image using a regularized monochannel method applied to image (a) of Fig. 6; (b) R-MRE restoredimage; (c) TV-based restored image.

Fig. 8. Log-compressed images for visual purposes corresponding to (a) onedegraded image; (b) R-MRE restored image; (c) TV-based restored image.

The last part of this experiment is a comparative study of theperformance of R-MRE and TV-based algorithms versus theSNR values in the range dB. To avoid manual choice ofthe regularization parameters, we have chosen them accordingto the following rule: %and . Indeed, this “ad hoc” choice is to reducethe weight of the regularization term when the SNR level in-creases. Fig. 9 shows the evolution of the restored image qualitywith respect to the SNR. The objective quality of the restoredimage is evaluated by means of a structural objective imagequality index namely the Structural Similarity Index Measure(SSIM) introduced in [34]. For this experiment, some remarksare done and several conclusions could be drawn.

Remark 1: This experiment has been carried on two differentkinds of images. Parrot image has many details (parrot face de-tails). On the opposite, cameraman image has a homogeneousbackground with a cameraman in the middle. It is expected thatthe performance of both algorithms is different for each kind ofimage. In order to see this difference, we fixed the whole param-eters the same way for both images. Namely, parameters and

depend only on the SNR.Remark 2: Concerning R-MRE algorithm, when the images

are too noisy (low SNR) if the same “big amount” of regulariza-tion is applied to restore parrot and cameraman images, it seemsthen predictable that the parrot image have worse quality thancameraman. In fact, in this case the regularization (large valueof ) also “destroys” the details in parrot image. This remark ex-plains the fact that for low SNRs, parrot image similarity indexis in the interval % % whereas cameraman image simi-larity index is in the interval % % .

Remark 3: The performance of R-MRE and TV-basedrestoration algorithms highly depends on the image nature,the amount of noise and the parameters selection. In futurework, we will thoroughly evaluate the sensitivity of the latterwith respect to the choice of different parameters. However, in

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Fig. 9. Objective quality of the restored image by means of R-MRE and TVregularization, respectively, versus SNR for (a) parrot image; (b) cameramanimage.

Fig. 10. Choice of restoration method with respect to SNR value and imagetype.

this stage of the work, we could propose to use the algorithmsdepending on the image nature and the SNR as depicted inFig. 10.

B. Identification Performance

Our objective here is the performance study of the blind iden-tification of the channel parameter vector given the blurredimages . Indeed, channel estimation errors may re-sult in significant degradation of the restored image quality.Hence, we disturb the channel vector by adding where

is a varying positive scalar and a fixed random vectorof unit norm. Hence, represents the blur func-tion used for image restoration by means of TV-based methodin VI. The restored image quality is measured using the SSIM.Fig. 11 illustrates the degradation of the restored image qualitydue to channel identification disturbance. As we can observe,this degradation becomes significant when the channel estima-tion error is larger than ( dB). In our paper,we have used several methods to identify the PSF coefficients.A comparison between these methods is given in Fig. 12. Wecompare the channel estimation performance of the consideredmethods for different SNR values. More precisely, we plot themean value of normalized mean square error (NMSE) betweenthe estimated and the actual PSF vector parameter:

as a function of the SNR for the SS, MNS,SMNS, CR and LSS methods. To do so, we fix the originalimage, but we generate randomly the filters at each MonteCarlo run. Surprisingly, the CR method (which is the less ex-pensive one) outperforms the other methods. This is due prob-ably to the fact that all other methods use the eigen-subspaces

Fig. 11. Impact of the channel identification precision on the restored imagequality for (a) parrot image; (b) cameraman image.

Fig. 12. Channel estimation MSE versus SNR: ��- SMNS,� � � SS, � �

MNS,��- CR, -LSS.

of large ill-conditioned matrices and, hence, are very sensitiveto noise effect.

C. Comparison to the State of the Art

In this section we carry out a comparative study between themethods proposed in this article and the iterative MBD methodin [10]. The first experiment is performed with the cameramandegraded images in Fig. 6. The result is depicted in Fig. 13.

This figure shows that while the R-MRE methods removesmore noise it smooths image edges. TV based method and theiterative one proposed in [10] have similar results. When wecompare the cameraman jacket on Fig. 13(b) and (c), we cansee that the TV-based method removes more noise from the de-graded images.

VIII. CONCLUSIONS AND FUTURE WORK

In this paper, we focus on efficient techniques aiming atrestoring an original image using several degraded renditionsof it. This paper introduces two multichannel restoration tech-niques with regularization. The first one is a direct restorationtechnique based on regularized MRE algorithm. The MREalgorithm ensures a perfect restoration of the original imagein the noiseless case, but is inefficient in presence of noise.Hence, in the noisy case, the MRE algorithm was used jointlywith an appropriate regularization technique that improves sig-nificantly its performance even at low and moderate SNRs. Thesecond restoration technique consists in the identification of thedegradation filters before their inversion. Several deterministicidentification algorithms were tested including a version ofthe LSS method that is generalized here from the 1-D to the2-D case. The estimated PSFs are then exploited to restore theoriginal image using a least squares criterion jointly with a

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Fig. 13. Objective quality of the restored image by means of (a) R-MRE; (b) TV regularization; (c) iterative MBD method in [10] with ten iterations.

total variation term that mitigates the noise amplification effect.Efficient optimization with convergence study of the TV-basedcriterion was considered. Simulation comparisons of the twoproposed restoration techniques are provided for different SNRranges and for different image types using an objective imagequality metric. In a future work, several issues will be deepened.We will study the sensitivity of the proposed methods towardsthe selection of parameters. We will also use real-life images tofurther evaluate the performance of the proposed methods.

REFERENCES

[1] G. A. Showman and J. H. McClellan, “Blind polarimetric equalizationof ultrawideband synthetic aperture radar imagery,” presented at theInt. Conf. Image Processing, 2000.

[2] A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “Hyperparameter esti-mation for satellite image restoration using a mcmc maximum likeli-hood method,” Pattern Recognit., vol. 35, pp. 341–352, 2002.

[3] Y. V. Zhulina, “Multiframe blind deconvolution of heavily blurred as-tronomical images,” Appl. Opt., vol. 45, no. 28, pp. 7342–7352, Oct.2006.

[4] S. Filip, S. Stanislava, F. Jan, and S. Tomás, “Multichannel blind de-convolution of the short-exposure astronomical images,” in Proc. Int.Conf. Pattern Recognition, Sep. 2000, vol. 3, pp. 53–56.

[5] M. Vrhel and B. L. Trus, “Multi-channel restoration of electron micro-graphs,” in Proc. Int. Conf. Image Processing, Oct. 1995, vol. 2, pp.516–519.

[6] H. J. Trussel, M. I. Sezan, and D. Tran, “Sensitivity of color LMMSErestoration of images to the spectral estimation,” IEEE Trans. SignalProcess., vol. 39, no. 1, pp. 248–252, Jan. 1991.

[7] U. A. Al Suwailem and J. Keller, “Multichannel image identificationand restoration using continuous spatial domain modeling,” presentedat the Int. Conf. Image Processing, Oct. 1997.

[8] F. Sroubek and J. Flusser, “Multichannel blind deconvolution of spa-tially misaligned images,” IEEE Trans. Image Process., vol. 14, no. 7,pp. 874–883, Jul. 2005.

[9] N. P. Galatsanos, A. K. Katsaggelos, R. T. Chin, and A. D. Hillery,“Least squares restoration of multichannel images,” IEEE Trans.Acoust., Speech, Signal Process., vol. 39, no. 10, pp. 2222–2236, Oct.1991.

[10] F. Sroubek and J. Flusser, “Multichannel blind iterative image restora-tion,” IEEE Trans. Image Process., vol. 12, no. 9, pp. 1094–1106, Sep.2003.

[11] C. A. Ong and J. A. Chambers, “An enhanced NAS-RIF algorithm forblind image deconvolution,” IEEE Trans. Image Process., vol. 8, no. 7,pp. 988–992, Jul. 1999.

[12] W. Souidene, K. Abed-Meraim, and A. Beghdadi, “Deterministic tech-niques for multichannel blind image deconvolution,” presented at theProc. ISSPA, Aug. 2005.

[13] L. Tong and Q. Zhao, “Joint order detection and blind channel estima-tion by least squares smoothing,” IEEE Trans. Signal Process., vol. 47,no. 9, pp. 2345–2355, Sep. 1999.

[14] E. Fishler, A. Haimovich, R. Blum, D. Chizhik, L. Cimini, and R.Valenzuela, “MIMO radar: An idea whose time has come,” presentedat the IEEE Radar Conf., Apr. 2004.

[15] B. Hassibi, B. M. Hochwald, and T. L. Marzetta, “Multi-antenna wire-less communications—From theory to algorithms,” presented at theIEEE ICASSP, May 2002.

[16] D. G. Karakos and P. E. Trahanias, “Generalized multichannel image-filtering structures,” IEEE Trans. Image Process., vol. 6, no. 7, pp.1038–1045, Jul. 1997.

[17] G. B. Giannakis and R. W. Heath, “Blind identification of multichannelfir blurs and perfect image restoration,” IEEE Trans. Image Process.,vol. 9, no. 11, Nov. 2000.

[18] D. S. C. Biggs and M. Andrews, “Acceleration of iterative imagerestoration algorithms,” Appl. Opt., vol. 36, no. 8, 1997.

[19] A. Beghdadi and B. Pesquet-Popescu, “A new image distortion mea-sure based on wavelet decomposition,” presented at the ISSPA, Jul.2003.

[20] W. Souidene, K. Abed-Meraim, and A. Beghdadi, “Blind multichannelimage deconvolution using regularized MRE algorithm: Performanceevaluation,” presented at the ISSPIT, Dec. 2004.

[21] Wirawan, “Multichannel image blind restoration: Second ordermethods,” Ph.D. dissertation, 2002.

[22] M. K. Ng, R. J. Plemmons, and S. Qiao, “Regularization of RIF blindimage deconvolution,” IEEE Trans. Image Process., vol. 9, no. 6, pp.1130–1134, Jun. 2000.

[23] A. Mayer, A. Castiaux, and J. P. Vigneron, “Electronic green scatteringwith n-fold symmetry axis from block circulant matrices,” Comput.Phys. Commun., vol. 109, pp. 81–89, Mar. 1998.

[24] G. J. Tee, “Eigenvectors of block circulant and alternating circulantmatrices,” New Zealand J. Math., vol. 8, pp. 123–142, 2005.

[25] Wirawan, K. Abed-Meraim, H. Maitre, and P. Duhamel, “Blind mul-tichannel image restoration using subspace based method,” presentedat the ICASSP, 2003.

[26] Y. Hua, K. Abed-Meraim, and M. Wax, “Blind system identificationusing minimum noise subspace,” IEEE Trans. Signal Process., vol. 45,no. 3, Mar. 1997.

[27] G. Harikumar and Y. Bresler, “Exact image deconvolution from mul-tiple fir blurs,” IEEE Trans. Image Process., vol. 8, no. 6, pp. 846–862,Jun. 1999.

[28] G. Harikumar and Y. Bresler, “Perfect blind restoration of imagesblurred by multiple filters: Theory and efficient algorithms,” IEEETrans. Image Process., vol. 8, no. 2, pp. 202–219, Feb. 1999.

[29] L. I. Rudin, S. J. Osher, and Fatemi, “Nonlinear total variation basednoise removal algorithms,” Phys. D, vol. 60, pp. 259–268, 1992.

[30] A. K. Jain, Fundamentals of Digital Image Processing, 1989.[31] T. F. Chan, G. H. Golub, and P. Mulet, “A nonlinear primal-dual

method for total variation-based image restoration,” in Proc. ICAOS,1996, vol. 219, pp. 241–252.

[32] C. Paige and M. Saunders, “Solution of sparse indefinite systems oflinear equations,” SIAM J. Numer. Anal., vol. 12, pp. 259–268, 1975.

[33] B. Fisher, Polynomial Based Iteration Methods for Symmetric LinearSystems, 1996.

[34] Z. Wang, E. P. Simoncelli, and A. C. Bovik, “Multi-scale structuralsimilarity for image quality assessment,” presented at the IEEEAsilomar Conf., Nov. 2003.

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Wided Souidene (M’06) was born in 1979. Shereceived the State Engineering degree from the ÉcolePolytechnique, Tunis, Tunisia, in 2002, the M.S.degree from Paris XI University, Orsay, France, in2003, and the Ph.D. degree from Paris XII Univer-sity, France, in 2007.

From 2006 to 2007, she was a Lecturer-Researcherwithin Paris XIII University. She worked on severalresearch projects related to image quality enhance-ment. Since 2007, she has been an Associate Pro-fessor within the Signal Processing Department, ES-

IGETEL. Her research interests are in signal and image processing, includingimage quality assessment and enhancement, multichannel blind image decon-volution and separation, speech and audio signal recognition, and classification.

Karim Abed-Meraim (SM’04) was born in 1967.He received the State Engineering degree from theÉcole Polytechnique, Paris, France, in 1990, theState Engineering degree from the École NationaleSupérieure des Télécommunications (ENST) Paris,in 1992, the M.S. degree from Paris XI University,Orsay, France, in 1992, and the Ph.D. degree fromENST in 1995.

From 1995 to 1998, he was a Research StaffMember with the Electrical Engineering De-partment, University of Melbourne, Melbourne,

Australia, where he worked on several research projects related to blind systemidentification for wireless communications, blind source separation, and arrayprocessing for communications. Since 1998, he has been an Associate Pro-fessor with the Signal and Image Processing Department, ENST. His researchinterests are in signal processing for communications and include systemidentification, multiuser detection, space-time coding, adaptive filtering andtracking, array processing, and performance analysis.

Azeddine Beghdadi (SM’05) received the“Maitrise” in physics and Diplome d’EtudesApprofondies in optics and signal processing fromthe University Orsay-Paris XI, Paris, France, in June1982 and June 1983, respectively (M.S. degree),the Ph.D. degree in physics (optics and signal pro-cessing) from the University Paris 6 in June 1986,and the “Habilitation a diriger des recherches” in1994 from the Université Paris 13.

He has been a Professor since 2000 at the Univer-sity of Paris 13 (Institut Galilée), France. He worked

at different places including the “Groupe d’Analyse d’Images Biomédicales”(CNAM Paris) and “Laboratoire d’Optique des Solides” (CNRS-UniversityParis 6). From 1987 to 1989, he has been a “Assistant Associé” (AssistantProfessor) at the University Paris 13. During 1987–1998, he was with theLPMTM-CNRS Laboratory working on scanning electron microscope (SEM)material image analysis. He ahs published over 100 international refereedscientific papers. He is a founding member of the L2TI Laboratory and leaderof “Visual Information Processing Group.” His research interests include imagequality enhancement and assessment, compression, and bio-inspired modelsfor image analysis, watermarking, and digital cinema.

Dr. Beghdadi has served as conference chair of ISSPA 2003 and technicalchair of ISSPA 2005. He also served as session organizer and a member of theorganizing and technical committees for many IEEE conferences. He has beeninvolved in many national and international research projects.