07058376

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 8, AUGUST 2015 2055

Compressed Sensing of Multichannel EEG Signals:The Simultaneous Cosparsity and

Low-Rank OptimizationYipeng Liu∗, Member, IEEE, Maarten De Vos, Member, IEEE, and Sabine Van Huffel, Fellow, IEEE

Abstract—Goal: This paper deals with the problems that someEEG signals have no good sparse representation and single-channelprocessing is not computationally efficient in compressed sensing ofmultichannel EEG signals. Methods: An optimization model withL0 norm and Schatten-0 norm is proposed to enforce cosparsityand low-rank structures in the reconstructed multichannel EEGsignals. Both convex relaxation and global consensus optimizationwith alternating direction method of multipliers are used to com-pute the optimization model. Results: The performance of multi-channel EEG signal reconstruction is improved in term of both ac-curacy and computational complexity. Conclusion: The proposedmethod is a better candidate than previous sparse signal recoverymethods for compressed sensing of EEG signals. Significance: Theproposed method enables successful compressed sensing of EEGsignals even when the signals have no good sparse representation.Using compressed sensing would much reduce the power consump-tion of wireless EEG system.

Index Terms—Alternating direction method of multipliers(ADMM), compressed sensing (CS), cosparse signal recovery, low-rank matrix recovery, multichannel electroencephalogram (EEG).

I. INTRODUCTION

W IRELESS body sensor networks take spatially dis-tributed sensors to acquire physiological signals, and

transmit them over wireless links to a central unit for signalprocessing [1]. The electroencephalogram (EEG) signal is oneof the most frequently used biomedical signals. It has importantapplications in medical healthcare, brain–computer interfacing,and so on [2]. Continuous EEG monitoring usually requires largeamount of data to be sampled and transmitted, which leads tolarge size of batteries. The recording unit of the wireless portableEEG systems is powered with batteries, and the physical size

Manuscript received October 29, 2014; revised February 5, 2015; acceptedMarch 4, 2015. Date of publication March 11, 2015; date of current versionJuly 15, 2015. This work was supported by FWO of Flemish Government:G.0108.11 (Compressed Sensing); Belgian Federal Science Policy Office: IUAPP7/19/ (DYSCO, “Dynamical systems, control and optimization,” 2012–2017);and ERC Advanced Grant: BIOTENSORS (339804). Asterisk indicates corre-sponding author.

∗Y. Liu was with ESAT-STADIUS Division/iMinds Medical IT Department,Department of Electrical Engineering, University of Leuven, 3001 Leuven,Belgium. He is now with the School of Electronic Engineering/Center for In-formation in BioMedicine, University of Electronic Science and Technology ofChina (UESTC), Chengdu 611731, China (e-mail: [email protected]).

M. De Vos is with Institute of Biomedical Engineering, Department of Engi-neering, University of Oxford, Oxford, United Kingdom.

S. Van Huffel is with ESAT-STADIUS Division/iMinds Medical IT Depart-ment, Department of Electrical Engineering, University of Leuven, 3001 Leu-ven, Belgium.

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

Digital Object Identifier 10.1109/TBME.2015.2411672

of the batteries sets the overall device size and operational life-time. A physically too large device would not be portable; andexcessive battery power consumption would make the long timewireless recording very hard [3]–[5].

Compressed sensing (CS) was proposed to deal with this chal-lenge. Rather than first sample, the analog signal at Nyquist rateand discard most in the compression, it directly acquires the dig-ital compressed measurements at a lower sampling rate, and re-covers the digital signals by nonlinear algorithms from the com-pressed measurements [6]. CS relies on the assumption that thesignal vector x is compressed by a random matrix Φ ∈ RM ×N

(measurement or sampling matrix) in discrete form as [6], [7]

y = Φx (1)

where y is the random sub-Nyquist compressed measurement.Here M � N , which means that it is sampled at a greatlyreduced rate. If x is sparse, its recovery only requires the com-pressed signal y and the sampling matrix Φ. If it is not sparse, thesignal x should be represented (transformed) using a represen-tation matrix (dictionary) Ψ ∈ RN ×P with N � P and a sparsevector θ ∈ RP ×1 with most of its entries zero or almost zero as

x = Ψθ. (2)

With the compressed measurement y, sampling matrix Φ, anddictionary Ψ, we can recover x by (2) after computing θ by

minimizeθ

‖θ‖0

subject to y = ΦΨθ(3)

where ‖θ‖0 is the pseudo-�0 norm that counts the numberof nonzero entries, i.e., ‖θ‖0 = #{θn �= 0, n = 1, 2, . . . , N}.The signal x is called K-sparse when the number of nonzeroentries is K. Most of the current methods for biomedical signalrecovery from compressed samples are based on the solutionof the �0 programming problem (3), such as, basis pursuit,orthogonal matching pursuit (OMP), iterative hard thresholding(IHT), etc., [4], [8], [9]. Besides, [5] found that some EEGsignals are not sparse in any sparse transformed domains, andproposed to exploit block-sparsity by block sparse Bayesianlearning (BSBL) to recover EEG signals [5].

Contrary to the traditional sparse or block-sparse signalmodel, the cosparse signal model uses an analysis operator mul-tiplying the measurement to produce a sparse vector [10]

μ = Ωx (4)

where Ω ∈ RQ×N is the cosparse representation matrix (anal-ysis dictionary) with N � Q, and μ ∈ RQ×1 is the cosparse

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2056 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 8, AUGUST 2015

Fig. 1. EEG signals reconstructed by OMP, BSBL, GAP, and analysis L1optimization with SSR = 0.35.

vector if most of its entries are nearly zero. Several sufficientconditions theoretically guarantee the successful recovery of thecosparse signal from the compressed measurement, such as therestricted isometry property adapted to the dictionary, restrictedorthogonal projection property (ROPP), etc., [10]–[12]. WhenN = P , an equivalent cosparse signal model to the sparse signalmodel can be found by letting Ω = Ψ−1 ; but there is no such anequivalent when N < P . The traditional sparse synthesis modelputs an emphasis on the nonzeros of the sparse vector θ, but thecosparse analysis model draws its strength from the zeros of theanalysis vector μ.

The cosparse signal recovery has some unique advantagesin CS-based EEG systems. First, the sparse signal recovery (3)gets the best estimate of the sparse vector θ; but the cosparsesignal recovery (5) gets the EEG signal’s best estimate directly.Second, theoretically the sparse signal recovery (3) requiresthe columns of the representation matrix Ψ to be incoherent,but the cosparse way (5) allows the coherence of the cosparserepresentation matrix Ω, which can result in super resolutionof the EEG signal estimate [11]. Third, the EEG signal canhardly be sparsely represented [5]. However, data analysis showsthat the EEG signals are approximately piecewise linear [13],as shown in Fig. 1, which implies the signal fits the cosparsesignal model (4) well with the second-order difference matrix asthe cosparse analysis dictionary. Therefore, the cosparse signalrecovery should be more appropriate for CS of EEG signals.

Since nearly all types of EEG systems have multiple chan-nels, it can be taken for granted that it is better to jointly processthe multichannel EEG signals. [14] proposed to jointly processmultichannel EEG signals by allowing slightly different phasesof the dictionaries in different channels. Another classical wayassumes that multiple channels share a similar support of sparsevector. This generalizes the single measurement vector prob-lem straightforwardly to a multiple measurement vector prob-lem [15], [16]. [17] proposed to incorporate preprocessing andentropy coding in the sampling to reduce the redundance in cor-related multichannel signals, but the added preprocessing andencoder would increase the power consumption in EEG sam-pling [4]; and the procedure can hardly be realized for analogsignals, which implies the analog EEG signals should be sam-pled at Nyquist sampling rate in the beginning. To compress

the multichannel EEG signals from the complete digital mea-surement, [18] used a wavelet-based volumetric coding method,while [19] exploited the low-rank structure in matrix/tensor formand achieved better performance.

Since most of the multichannel EEG signals are more or lesscorrelated with each other, the low-rank structure-based com-pression method motivates the use of low-rank data structurein CS of multichannel EEG signals too. The multichannel EEGsignals are put columnwise into a matrix. Our EEG data analy-sis finds that the newly formed EEG data matrix has only a fewnonzero singular values.

In this paper, the second-order difference matrix is chosen tobe the cosparse analysis dictionary, which tries to enforce theapproximate piecewise linear structure. Exploiting additionallythe low-rank structure, we can further enhance the signal recov-ery performance by exploiting the cosparsity of single-channelEEG signals and the low-rank property of multichannel EEGsignals simultaneously in the framework of multistructure CS.The �0 norm and Schatten-0 norm-based optimization modelis used to encourage cosparsity and low-rank structure in thereconstructed signals. Two methods are proposed to solve themulticriteria optimization problem. One relaxes it to a convexoptimization; and the other one transforms it into a global con-sensus optimization problem. The alternating direction methodof multipliers (ADMM) is used to solve it efficiently. The con-vergence and computational complexity are briefly analyzed. Innumerical experiments, a group of real-life EEG data is usedto test the algorithms’ performance of both single-channel andmultichannel EEG signal recovery methods. Numerical resultsshow that the cosparse signal recovery method and simulta-neous cosparsity and low-rank (SCLR) optimization achievethe best performance in term of mean squared error (MSE) andmean cross-correlation (MCC) in single-channel and multichan-nel EEG signal recovery respectively.

The rest of the paper is organized as follows. Section IIpresents an optimization model to exploit both cosparsity andlow-rank data structures to recover the EEG signals. In SectionIII, two methods are given to solve the optimization problem,i.e., convex relaxation and ADMM. In Section IV, numericalexperiments are used to demonstrate the proposed methods’performance improvement. Section V draws the conclusion.

II. SCLR OPTIMIZATION MODEL

The optimization model for cosparse signal recovery can beformulated as [10]

minimizex

‖Ωx‖0

subject to y = Φx.(5)

Here, we call (5) the analysis L0 optimization. When the EEGsystem records R channels simultaneously, the extension of anal-ysis L0 optimization to multichannel data is

minimizeX

‖vec (ΩX)‖0

subject to Y = ΦX(6)

LIU et al.: COMPRESSED SENSING OF MULTICHANNEL EEG SIGNALS: THE SIMULTANEOUS COSPARSITY AND LOW-RANK OPTIMIZATION 2057

where X ∈ RN ×R , and vec(X) puts all the columns of X intoone column vector sequentially. A series of solvers are summa-rized in [10].

Reconstructing the EEG matrix from the compressed mea-surements by exploiting the low-rank structure can be formu-lated as

minimizeX

‖X‖Schatten−0

subject to Y = ΦX(7)

where ‖X‖Schatten−0 is the Schatten-0 norm that counts thenumber of the nonzero singular values of X [20]. A variety ofmethods to solve it can be found in [21].

Motivated by the fact that many EEG signals have bothcosparsity and low-rank structure, we propose to simultaneouslyexploit these two data structures in multichannel EEG signal re-construction from the compressed measurement. Both �0 normand Schatten-0 norm-based constraints are used in the optimiza-tion model. Combining with the linear data fitting constraint, wecan formulate the SCLR optimization model as follows:

minimizeX

‖vec (ΩX)‖0 + ‖X‖Schatten−0

subject to Y = ΦX.(8)

III. SOLUTIONS

A. Convex Relaxation

To solve the SCLR optimization (8), one classical way relaxesthe nonconvex �0 norm and Schatten-0 norm into convex �1 normand Schatten-1 norm, respectively, where the �1 norm sums allthe absolute values of the entries, i.e., ‖x‖1 =

∑Nn=1 |xn |. The

Schatten-1 norm is called nuclear norm too, and sums all the sin-gular values of the data matrix, i.e., ‖X‖Schatten−1 = ‖X‖∗ =∑min(N,P )

n=1 σn . The newly formed convex simultaneous cospar-sity and low-rank optimization model can be formulated as

minimizeX

‖vec (ΩX)‖1 + ‖X‖∗subject to Y = ΦX.

(9)

Similarly to the reformulation from minimizex‖x‖1 tominimizex,e�01T e, subject to − e ≺ x ≺ e due to the defi-nition of the �1 norm, we can reformulate the �1 norm mini-mization into its equivalent linear programming in (9) [22]. Byintroduction of new nonnegative variables e and f, (9) can beexpressed as

minimizeX ,e�0, f≥0

1T e + f

subject to Y=ΦX

‖X‖∗ ≤ f

−e ≺ vec (ΩX) ≺ e,

(10)

where 1 ∈ RQR×1 is a column vector with all the entriesbeing 1.

The nuclear norm constraint can be replaced by its lin-ear matrix inequality (LMI) equivalent; and the approximation

constraints can also be expressed via LMIs using Schur com-plements [23]. The obtained optimization model is

minimizeX ,e�0, f≥0

1T e + 2f

subject to Y=ΦX

−e ≺ vec (ΩX) ≺ e

[A X

XT B

]

≥ 0

Tr (A) +Tr (B) < f (11)

where A = AT and B = BT are new variables. Equation (11)is a semi-definite programming (SDP) which can be solvedby interior-point method [22], [23]. The software CVX cancompute the solution in this way [24].

B. ADMM

Besides the classical SDP, another method, called ADMM,can be used to solve the SCLR optimization [25]. With indi-vidual constraints on the same variables in each constraint, (9)can be rewritten into a global consensus optimization with localvariables Xi , i = 1, 2 and a common global variable X as

minimizeX1 ,X2 ,X

‖vec (ΩX1)‖1 + ‖X2‖∗subject to X = X1 ;X = X2 ;Y = ΦX.

(12)

Here, the new constraints are that all the local variables shouldbe equal. It is equivalent to

minimizeX1 ,X2 ,X

‖vec (ΩX1)‖1 + ‖X2‖∗subject to Y=ΦX1 ; Y=ΦX2

(13)

where

Y =

[Y

X

]

(14)

Φ =

[Φ

I

]

. (15)

The corresponding augmented Lagrangian of (13) is

Lρ (X1 ,X2 ;Z1 ,Z2) = ‖vec (ΩX1)‖1 + ‖X2‖∗

+ vec(Z1)T vec

(Y − ΦX1

)+ vec(Z2)

T vec(Y − ΦX2

)

+ ρ2

∥∥Y − ΦX1

∥∥2

F+ ρ

2

∥∥Y − ΦX2

∥∥2

F(16)

where ρ > 0, Z1 and Z2 are dual variables. The resultingADMM algorithm in the scaled dual form is the following:

Xt+11 : = arg min

X1

(‖vec (ΩX1)‖1 +

ρ

2

∥∥Y− ΦX1 +ΦUt

1

∥∥2

F

)(17)


Xt+12 : = arg min

X2

(‖X2‖∗ +

ρ

2

∥∥Y − ΦX2 + ΦUt

2

∥∥2

F

)(18)

Xt+1 =12

(Xt+1

1 + Xt+12

)(19)

Ut+11 = Ut

1 +(Xt+1

1 − Xt1)

Ut+12 = Ut

2 +(Xt+1

2 − Xt2)

(20)

where U1 = 1/ρZ1 and U2 = 1/ρZ2 are scaled dual variables.In the proposed ADMM algorithm for SCLR optimization, twosteps separately optimize over variables generally, i.e., updatingthe prime variables X1 and X2 , updating the scaled dual vari-ables U1 and U2 . In this iterative algorithm, the variables areupdated in an alternating fashion.

For both (17) and (18), there are many computationally effi-cient algorithms [10], [21]. For example, analysis L1 optimiza-tion, greedy analysis pursuit (GAP) can be used to solve (17);to solve (18), SDP method or singular value thresholding canbe used. The solutions of (19) and (20) are straightforwardlyeasy. The ADMM for SCLR optimization is summarized inAlgorithm 1.

A lot of convergence results exist for ADMM in the literature[25]. Generally, the convergence to optimum can be guaranteedwhen the epigraph of gi

epigi = {(X, ε) |gi (X) ≤ ε, i = 1, 2 . . .} (21)

is a closed nonempty convex set, where g1(X) = ‖vec (ΩX)‖1 ,g2(X) = ‖X‖∗, and the unaugmented Lagrangian

Lρ=0 (X1 ,X2 ;Z1 ,Z2) = ‖vec (ΩX1)‖1 + ‖X2‖∗+vec(Z1)

T vec(Y − ΦX1

)+ vec(Z2)

T vec(Y − ΦX2

)

(22)has a saddle point. The proof can be found in [26].

The ADMM decomposes the optimization model with mul-tiple constraints into several ones with fewer constraints. Therecould be some fast algorithms for these new optimization mod-els. Besides, it allows multiple steps in one iteration to be pro-cessed in parallel. With a multicore processor, the computational

time can be decreased. Previous experience shows that a few it-erations will often produce acceptable results of practical use.

IV. NUMERICAL EXPERIMENTS

To demonstrate the performance of the possible methods forEEG signal recovery from the compressed measurement, weperform two groups of numerical experiments. The details aboutthe data materials and subjects are given in Section IV-A. In Sec-tion IV-B, we test the performance of two cosparse signal recov-ery methods for single-channel EEG signals in different kinds ofsituations, i.e., analysis L1 optimization and GAP. Some otheralgorithms are tested to make comparison, such as BSBL whichis reported to be the best of all the current candidates for EEGsignal recovery from compressed measurement [5], and OMPwhich is a proper representative of the classical sparse signalrecovery algorithms [4]. In Section IV-C, a group of multichan-nel EEG signals are recovered by the proposed algorithms forSCLR optimization, as well as simultaneous orthogonal match-ing pursuit (SOMP) [27], BSBL [5], [16], and simultaneousgreedy analysis pursuit (SGAP) [28].

In all experiments, as argued by our analysis in Section I,the second-order difference matrix is chosen to be the analy-sis dictionary for cosparse EEG signal recovery. The Gaussianmatrix is chosen to be the sampling matrix for CS of EEG sig-nals. The sparse dictionaries of OMP and SOMP are Daubechieswavelets [4].

To measure the compression degree, the subsampling ratio(SSR) is defined as

SSR =M

N× 100%. (23)

To quantify the difference between high-dimensional valuesimplied by the estimator and the true values of the quantitybeing estimated, two different evaluation functions are oftenused in EEG signal processing. One is the MSE that measuresthe average of the squares of the errors. The error is the amountby which the value implied by the estimator differs from thequantity to be estimated. Here, we can formulate it as

MSE =L∑

l=1

∥∥∥Xl − X

∥∥∥

2

F

LNR(24)

where X is the true EEG data with R channels and each channelhas length N, Xl is its estimate in the lth experiment, and L isthe number of experiments. Both X and Xl are normalized bytheir Frobenius norms, respectively. When R = 1, the matrix Xis degenerated into a vector x. In that case, MSE can be used toevaluate single-channel EEG signal reconstruction evaluation.The MSE has variants of other equivalent forms, such as meanL2 error [29], percent of root-mean-square difference [4].

Another evaluation function is the MCC. It is equivalent tothe Structural SIMilarity index, which measures the similarityof two waveforms [4], [5], [30]. It can be formulated as

MCC =L∑

l=1

vec(X)T vec(Xl

)

L‖X‖F

∥∥∥Xl

∥∥∥

F

. (25)


A. Data Material and Subjects

The used EEG data is the CHB-MIT scalp EEG databasewhich is online available in the Physiobank database:http://www.physionet.org/cgi-bin/atm/ATM [31], [32]. Col-lected at the Children’s Hospital Boston, these EEG recordingsare from pediatric subjects with intractable seizures. Subjectswere monitored without antiseizure medication in order to char-acterize their seizures and assess their candidacy for surgicalintervention. All the recordings were collected from 22 sub-jects (five males, ages 3–22; and 17 females, ages 1.5–19).All used datasets consist of 23-channel EEG recordings, whichwere sampled at 256 samples per second with 16-bit resolution.The international 10–20 system of EEG electrode positions andnomenclature was used for these recordings. More details aboutthe EEG database can be found [31]. In our experiments, theEEG recording chb01 31.edf has been selected to demonstratethe recovery algorithms’ performance.

In Section IV-B, L = 500 segments of EEG data are used, i.e.,xl ∈ RN ×1 , l = 1, 2, . . . , L. They are taken from all the R = 23channels sequentially. The length of each segment of the EEGdata x is N = 256. Each segment of EEG data is normalized byits �2 norm.

In Section IV-C, L = 50 segments of 23-channel EEG dataare used, i.e., Xl ∈ RN ×R , l = 1, 2, . . . , L. In each segmentof the EEG data matrix X, the number of sampling points isN × R = 256 × 23. Each segment of EEG data is normalizedby its Frobenius norm.

B. Single-Channel EEG Signal Recovery

To show how the proposed cosparse signal recovery methodswork, we take a segment of single-channel EEG signal andreconstruct it from the compressed measurement with SSR =0.35. The reconstructed and real signals are shown in Fig. 1.We can see that the reconstructed signals from GAP and theanalysis L1 optimization methods fit the real signal better thanthose from the classical OMP and BSBL methods.

Fig. 2(a) and (b) gives the values of MSE and MCC of GAP,OMP, and BSBL with different SSRs. We can see that analysisL1 optimization, GAP, and BSBL have similar accuracy per-formance, and they outperform OMP. Analysis L1 optimizationis slightly more accurate than GAP, and GAP is slightly moreaccurate than BSBL. Fig. 2(c) shows that the greedy algorithmsGAP and OMP are much faster than BSBL and analysis L1 opti-mization, and GAP is even slightly faster than OMP. Therefore,if we only care about the accuracy, the analysis L1 optimiza-tion is the best choice; and if both accuracy and computationalcomplexity are important, GAP should be a better choice.

C. Multichannel EEG Signal Recovery

In these experiments, most of the parameters are selected asin Section IV-B. Two algorithms for SCLR optimization areused, i.e., interior point method for SCLR optimization andADMM for SCLR optimization with experienced choices ofthe parameters Tmax = 5, ρ = 1, and η = 0.05. In comparisonwith the proposed methods, three other popular multichannel

Fig. 2. Differences in average performance evaluation of single-channel EEGsignal recovery from compressed measurements with different SSRs using 500different single-channel EEG segments: (a) MSE versus SSR; (b) MCC versusSSR; (c) CPU time versus SSR.

sparse/cosparse signal recovery methods are taken too, i.e.,BSBL, SOMP, and SGAP.

Fig. 3(a), (b), and (c) displays the values of MSE, MCC, andCPU times of the interior point method for the SCLR optimiza-tion, ADMM for SCLR optimization, BSBL, SOMP, and SGAPwith different values of SSR. We can see that the interior point


Fig. 3. Differences in average performance evaluation of multichannel EEGsignal recovery from compressed measurement with different SSRs using 50different multichannel EEG segments: (a) MSE versus SSR; (b) MCC versusSSR; (c) CPU time versus SSR.

method for SCLR optimization, ADMM for SCLR optimiza-tion have similar accuracy performance, and they outperformthe other ones in accuracy. Comparing the speed of these twosolutions for SCLR optimization, the ADMM for SCLR op-timization is faster. In Fig. 3(c), we can see that the greedyalgorithms SOMP and SGAP are much faster than the rest. But

their accuracy is much worse and not acceptable. Therefore, werecommend that the ADMM for SCLR optimization should bea better candidate for multichannel EEG signal recovery thanthe other methods.

V. CONCLUSION

With the second-order difference matrix as the cosparse anal-ysis dictionary, the EEG signals’ cosparsity is exploited forthe single-channel EEG signal recovery from compressed mea-surements. To further enhance the performance, cosparsity andlow-rank structure are jointly used in the multichannel EEG sig-nal recovery. In the proposed new optimization model, the �0norm constraint is used to encourage cosparsity while Schatten-0 norm constraint is used for low-rank structure. To solve theoptimization model, two methods are used. One approximatesit by relaxing the �0 and Schatten-0 norms into �1 norm and nu-clear norm, respectively, which leads to a convex optimization.The other way is ADMM that divides the multiple criteria opti-mization into several connected single criterion optimizations inthe form of global consensus optimization. Each single criterionoptimization can be solved by a series of existing efficient meth-ods. In numerical experiments, EEG signals’ cosparsity for CSis proved by the single-channel EEG data-based results; and themultichannel EEG data results show that the SCLR optimizationoutperforms all the previous methods.

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Yipeng Liu (S’09–M’13) was born in Chengdu,China, in 1983. He received the B.Sc. degree inbiomedical engineering and the Ph.D. degree ininformation and communication engineering fromthe University of Electronic Science and Technol-ogy of China (UESTC), Chengdu, in June 2006 andJune 2011, respectively.

From June 2011 to November 2011, he was aResearch Engineer at Huawei Technologies. FromNovember 2011 to November 2014, he was a Post-doctoral Research Fellow at the University of Leuven,

Leuven, Belgium. Since September 2014, he has been an Associate Professorwith the School of Electronic Engineering, UESTC. His research interests in-clude compressed sensing theory and application.

Maarten De Vos (M’09) received the M.Sc. degreein electrotechnical–mechanical engineering and thePh.D. degree in engineering from KU Leuven, Leu-ven, Belgium, in 2005 and 2009, respectively.

From 2009 to 2014, he has been a PostdoctoralResearcher with KU Leuven and the University ofOldenburg. From 2013 to 2014, he has been a JuniorProfessor at the University of Oldenburg. Since 2014,he has been an Associate Professor with the Insti-tute of Biomedical Engineering, Department of En-gineering, University of Oxford, Oxford, U.K. His

current research interests include linear and multilinear algebra, decompositiontechniques for biomedical signals and developing human–machine interfacingsolutions.

Sabine Van Huffel (M’96–A’96–SM’99–F’09) re-ceived the M.D. degree in computer science engi-neering, the M.D. degree in biomedical engineering,and the Ph.D. degree in electrical engineering fromKU Leuven, Leuven, Belgium, in June 1981, July1985, and June 1987, respectively.

She is currently a Full Professor at the Depart-ment of Electrical Engineering, KU Leuven. Her re-search interests include numerical (multi)linear al-gebra and software, system identification, parameterestimation, and biomedical data processing.

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