An efficient Jacobi-like deflationary ICA algorithm: alication to EEG denoising Seideh Hajiour, Laurent Albera, Mohammad Bagher, Isabelle Merlet To cite this version: Seideh Hajiour, Laurent Albera, Mohammad Bagher, Isabelle Merlet. An efficient Jacobi-like deflationary ICA algorithm: alication to EEG denoising. IEEE Signal rocessing Letters, Institute of Electrical and Electronics Engineers, 2015. <hal-01245250> HAL Id: hal-01245250 htts://hal.archives-ouvertes.fr/hal-01245250 Submitted on 17 Dec 2015 HAL is a multi-discilinary oen access archive for the deosit and dissemination of scientific research documents, whether they are ublished or not. The documents may come from teaching and research institutions in France or abroad, or from ublic or rivate research centers. L archive ouverte luridiscilinaire HAL, est destinée au déôt et à la diffusion de documents scientifiques de niveau recherche, ubliés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires ublics ou rivés.
An efficient Jacobi-like deflationary ICA algorithm: alication to EEG denoising Seideh Hajiour Sardouie, Student Member, IEEE, Laurent Albera, Senior Member, IEEE, Mohammad Bagher Shamsollahi, Senior Member, IEEE, and Isabelle Merlet 1 Abstract In this aer, we roose a Jacobi-like Deflationary ICA algorithm, named JDICA. More articularly, while a rojection-based deflation scheme insired by Delfosse and Loubaton s ICA technique DelL Ê is used, a Jacobi-like otimization strategy is roosed in order to maximize a fourth order cumulant-based contrast built from whitened observations. Exerimental results obtained from simulated eiletic EEG data mixed with a real muscular activity and from the comarison in terms of erformance and numerical comlexity with the FastICA, RobustICA and DelL Ê algorithms, show that the roosed algorithm offers the best trade-off between erformance and numerical comlexity when a low number 12 of electrodes is available. Index Terms Indeendent Comonent Analysis, deflation, higher order statistics, Jacobi-like otimization, ElectroEncehaloGrahy, denoising, interictal eiletic data. I. INTRODUCTION INDEENDENT Comonent Analysis ICA [8], [9] is a very useful tool in signal rocessing esecially to rocess biomedical signals such as ElectroEncehaloGrahic EEG data [1] [5]. The ICA roblem consists of retrieving unobserved realizations of a -dimensional random vector s = [s 1,...,s ] T from observed realizations of an N-dimensional random vectorx = [x 1,...,x N ] T that can linearly be modeled as follows: x = a s +ν = As+ν 1 =1 where ν reresents an N-dimensional noise indeendent of s. The fundamental assumtion of ICA is that the unknown random variables s called sources are statistically indeendent, i.e. their joint robability Density Function DF can be factorized as the roduct of their marginal DFs. ICA algorithms can be divided into two grous: i "joint" or "symmetric" aroaches jointly extract the indeendent comonents ii "deflationary" techniques estimate sources one by one. Joint algorithms seem to converge to the exected solution in ractice, but no theoretical result is available. On the other hand, the convergence of most of deflationary algorithms have been roved analytically [4], [6], [10]. In addition, in deflationary algorithms, a enalty term can be S. Hajiour Sardouie is with Inserm, UMR 1099, Rennes, F-35000, France, LTSI, University of Rennes 1, Rennes, F-35000, France and BiSIL, Sharif University of Technology, Tehran, Iran. L. Albera is with Inserm, UMR 1099, Rennes, F-35000, France, LTSI, University of Rennes 1, Rennes, F-35000, France and INRIA, Centre Inria Rennes-Bretagne Atlantique, 35042 Rennes Cedex, France. M. B. Shamsollahi is with BiSIL, Sharif University of Technology, Tehran, Iran. I. Merlet is with Inserm, UMR 1099, Rennes, F-35000, France and LTSI, University of Rennes 1, Rennes, F-35000, France. added to the contrast function [7] to force the algorithm to extract the sources of interest during the early stes. Besides when the number of all sources largely encomasses the number of sources of interest, the comutational comlexity of the deflationary algorithms is greatly reduced. In this aer, we roose an efficient Jacobi-like Deflationary ICA algorithm, called JDICA, based on second and Fourth Order FO statistics. The deflation rocedure of our algorithm is insired by [4]. The gradient-based ICA algorithm called DelL Ê throughout this aer roosed in [4], estimates the sources one by one using a smart rojection-based deflation scheme. According to its gradient-based structure, the ste size must be recisely chosen to guarantee accetable results, esecially with noisy data. A multi-initialization rocedure can even be necessary in some ractical contexts. In order to overcome these drawbacks, we roose a Jacobi-like algorithm to maximize the contrast function comuted from the FO cumulants of the whitened observations. We have examined the effectiveness of JDICA in denoising of simulated interictal eiletic data when a low number of electrodes is available as for children. The comarison in terms of erformance and numerical comlexity with classical deflationary ICA algorithms, namely FastICA [6], RobustICA [10] and DelL Ê shows that JDICA offers a better accuracy than DelL Ê and a lower numerical comlexity than FastICA and RobustICA. II. METHODOLOGY We assume that we have some realizations of the real-valued random vector x 1. Since JDICA, like a large grou of ICA algorithms, needs a rewhitening ste [4] without loss of generality, we assume that vector x denotes the rewhitened observation random vector and matrix A = [a 1,...,a ] is a real-valued orthogonal mixing matrix. The aim of our method is then to estimate the columns a of A and the corresonding sources such that s = a T x. More articularly, vector a can be identified by maximizing the following contrast function: Fg = 1 4 [C 4y ] 2 = 1 4 [C 4g T x] 2 2 with resect tog wherec 4 y is the FO marginal cumulant of y = g T x. The advantage of defining such a contrast function is that the arguments of the local maxima of F on the unit shere are the vectors {±a } =1,..., [4]. This roerty ensures our maximization 2 to converge to one column of the matrix A. Consequently one of the sources is extracted. Thus a rojection deflation rocedure is alied to subtract the
2 contribution of the extracted source from the mixture. These two stes require a articular arametrization of the elements of the unit shere which is given by: Lemma 1: Each unit norm column vector g R whose last comonent g is strictly ositive can be reresented as the last column of an orthogonal matrix given by: Gt = G 1 t 1...G 2 t 2 G 1 t 1 3 where the 1 real-valued elements of t = [t 1,...,t 1 ] T corresond to tangents of uniquely defined angles belonging to ] π/2,π/2[ and G t is a Givens rotation of size derived from an identity matrix for which the,-th,,-th,,-th,,-th comonents are relaced with 1/2, 1/2, t 1/2 and t 1/2, resectively. roof derives from [4, lemma 2.2] by exressingcosθ and sinθ as a function of t = tanθ. This arametrization differs from that of [4] and allows us both to reformulate the contrast 2 as a rational function and to consider other otimization strategies such as a Jacobi-like rocedure. To extract the first source, we then roose to comute a matrix G 1 t such that its last column, g 1 t, maximizes the contrast function 2 with resect to t. Our Jacobi-like otimization rocedure consists of decomosing G 1 t as a roduct of 1 elementary Givens rotations G 1 t and of sequentially identifying the 1 corresonding arameters t. The 1-dimensional otimization roblem is thus relaced with 1 sequential mono-dimensional otimization roblems. In ractice, several swees of the 1 arameters are necessary to achieve convergence. More recisely, let us consider the -th mono-dimensional maximization roblem of a swee of our Jacobi-like rocedure. It consists in comuting matrix G 1+ t defined by G 1+ t = G 1 t G 1 such that its last column, g 1+ t, maximizes the contrast function 2, where G 1 is the roduct of all the elementary Givens rotations estimated reviously. Denoting the last column of G 1 by = [ 1,..., ]T, the last column of G 1+ can be written as: where: g 1+ t = [g 1+ 1 t,...,g 1+ t ] T = 4 [ 1,..., g 1+ t = g 1+ 1,g1+ 1 t = t t, +1,...,g1+ t ] T + + t 1 5 6 It aears that only the -th and -th comonents of g 1+ t deend on t. Then, we set the derivative of the contrast function with resect to t equal to zero to find the aroriate t value: C 4 y 1 2 /4 = 1 t 2 C 4y 1 C 4y 1 = 0 7 t which results in simly vanishing C 4 y 1 / t. Now using the multi-linearity roerty of cumulants, it is shown that C 4 y 1 can be written as follows: C 4 y 1 = d 40 g 1+ t 4 +d 31 g 1+ t 3 g 1+ t + d 22 g 1+ t 2 g 1+ t 2 +d 13 g 1+ t g 1+ t 3 + d 04 g 1+ t 4 +d 30 g 1+ t 3 +d 21 g 1+ t 2 g 1+ t +d 12 g 1+ t g 1+ t 2 +d 03 g 1+ t 3 +d 20 g 1+ t 2 +d 11 g 1+ t g 1+ t +d 02 g 1+ 2 +d 10 g 1+ +d 01 g 1+ +d 00 8 where the coefficients d ij are defined in aendix. Consequently, by comuting the derivative of 8, we obtain: 4 e m t m 3 + f n t n =0 9 m=0 n=0 where the coefficients e m and f n are given in aendix. Equation 9 can be simlified to an 8-th degree olynomial equation as follows: 8 m 6 m e n e m n t m f n f m n tm =0 10 m=0n=0 m =0n =0 By rooting 10,8 solutionsˆt are obtained. Then we calculate the contrast function 2 for all real-valued roots and we choose the root ˆt ot which maximizes it. Eventually, we calculate the matrices G 1 ot ˆt and G 1+ ˆt ot. This rocedure is erformed iteratively for all {1,..., 1} and for several swees until convergence. At this stage, the first column â 1 of the estimated mixing matrix is given by the last udate of g 1+ ˆt ot 1 and the first source is estimated by ŝ 1 = â T 1 x. After estimating the first source, we remove its contribution from the observations by rojecting the observations onto the subsace orthogonal to that sanned by â 1 by comuting x 1 = Π 1 x where Π 1 is a 1 rojection matrix built by stacking vertically the 1 first rows of the last udate of G 1+ ˆt ot T. Now to estimate the other sources, the same rocedure should be done by using equations 4 to 10. The only difference is that the vector of observations x should be relaced by the observation x 1 of reduced dimension +1 in order to extract the -th source. Note that the estimation of FO cumulants is not required at each iteration of our Jacobi-like rocedure. The N 4 FO cumulants C n1,n 2,n 3,n 4,x of vector x can be estimated at the beginning of the rocedure and sorted in a N 2 N 2 matrix, Q x, called quadricovariance [1]. The FO cumulants C of vector n1,n 2,n 3,n 4,x x can then be derived using the following formula Q x = HQ x H T where: 1 H = Π i Π i 11 i=1 with the Kronecker roduct oerator. III. NUMERICAL COMLEXITY In this section, we analyze the numerical comlexity of the roosed algorithm in terms of real-valued floating oint oerations flos. A flo corresonds to a multilication followed by an addition, but in ractice only the number of multilications is comuted. In the following comutations,
3, N and T are the number of sources, the number of observation channels and the number of time samles, resectively. f 4 = +1 +2 +3/24 is equal to the number of free entries in a fourth order cumulant tensor of dimension enjoying all symmetries.b = mint N2 2 +4N3 NT,2TN 2 is the number of flos required to erform satial whitening. R is the comlexity required to comute the roots of a real 8-th degree olynomial by using the comanion matrix technique we may take R = 972 flos. As a result the roosed ICA algorithm requires B+2T +2 +N 2 + 3 + 3Tf 4 + 1 =2 22 1 2 2 +1+ =2 2 T + + 1 2 + =2 It +1R+4 3 /3 7 2 +62/3+ 195+min 2T +84T +8,4+82 4 + 2 +7 flos to extract all sources. IV. ERFORMANCE ANALYSIS ON SIMULATED DATA A. Data generation The simulated eiletic EEG was generated using a realistic model develoed in our team [3]. We built a mesh of the cortical surface from a 3D MRI T1 image of a subject Brain- Visa, SHFJ, Orsay, France. This mesh is comosed of 40500 triangles of mean surface 5 mm 2. A current diole is laced at the barycenter of each triangle and oriented orthogonally to the triangle surface, leading to a field of current dioles. From this mesh, e distributed sources, called "atches", generating interictal sikes, are defined. Each atch is comosed of 100 diole sources to which we assigned hyer-synchronous sikelike activities generated from a model of neuronal oulations [3]. From this setu and considering 12 electrodes, namely F1, F2, C3, C4, T3, T4, O1, O2, F7, F8, T5 and T6, the forward roblem was then calculated using a realistic head model made of three nested homogeneous volumes shaing the brain, the skull and the scal ASA, ANT, Enschede, Netherlands. The aforementioned electrodes are commonly used to record EEG in ediatric atients. The eiletic activity at the level of these electrodes, namely the signal of interest, was then obtained by solving the forward roblem using a realistic head model and the Boundary Element Method BEM. In this aer, we considered a single atch localized in the left suerior temoral gyrus and 50 Monte Carlo simulations were generated. In addition a 12-dimensional signal of non-interest extracted from real 12-channel EEG and comosed of muscle activity, background EEG and instrument noise was added to each trial with a secified Signal-to-Noise Ratio SNR. B. Results We comared the erformance of the roosed JDICA algorithm with three deflationary ICA algorithms, namely FastICA, RobustICA, DelL Ê. Note that, unlike the three other algorithms, RobustICA does not require any rewhitening. The erformance was comuted as a function of comutational comlexity using the Normalized Mean-Squared Error NMSE as defined in [1]. In our exeriment, the data length is fixed to 5120 samles and the SNR value is 5 db. By varying the number of estimated sources ˆ in the range of 2 to 12, we make vary the comutational comlexity of each algorithm. Figure 1 shows then the average Error as a function of flos at the outut of the four algorithms. This figure illustrates that the JDICA algorithm offers the best comromise between erformance and numerical comlexity when a low number of electrodes is used even if RobustICA converges faster. It imlies that one iteration of RobustICA requires more flos than one swee of JDICA. Error 0.65 0.6 0.55 0.5 0.45 0.4 0.35 JDICA FastICA RobustICA 10 6 10 7 Flos Fig. 1. Average Error as a function of flos obtained by varying the number of estimated sources ˆ with SNR= 5dB. F1 F2 C3 C4 O1 O2 T3 T4 F7 F8 T5 T6 clean data noisy data JDICA denoised data a b c Fig. 2. Denoising of real interictal sikes data a a noise-free interictal sikes, b an eoch including sikes hidden in muscle activity and c EEG denoised by JDICA. The source localization results at the outut of 4-ExSo-MUSIC are deicted at the bottom of each column. V. ALICATION TO REAL DATA In this section we evaluate JDICA in the case of real data. The JDICA algorithm was alied to denoise interictal sikes obtained from a atient suffering from drug-resistant artial eilesy. Scal-EEG data were acquired from 12 electrodes at a samling frequency of 256 Hz. These data were reviewed in order to isolate an eoch of clean data containing interictal sikes figure 2a and an eoch of noisy EEG containing sikes hidden by muscle activity of high amlitude figure 2b. The same rocedure as for simulated data was alied to reconstruct the denoised EEG signals by using JDICA figure
4 2c. Since we do not know the ground truth to evaluate the erformance of the roosed method, a source localization rocess was erformed on the original clean signal considered as a reference, on the noisy data, as well as on data denoised by JDICA. The recent 4-ExSo-MUSIC algorithm [2] was used to achieve source localization. As shown in figure 2, the eiletic sikes maximal at temoral and frontotemoral electrodes T4, F8 on clean data are retrieved at the same electrodes on denoised data. In addition, the muscle activity visible on noisy data is strongly reduced by the JDICA rocedure at F8 and T4 and almost entirely removed at other channels. Source localization bottom of figure 2 of clean 2a and of denoised sikes 2c is similar right anterior temoral and consistent with the atient athology. For noisy data, the sike source is incorrectly localized. VI. CONCLUSION In this aer, we roosed a new deflationary ICA algorithm based on a Jacobi-like otimization rocedure to searate indeendent sources. We examined the effectiveness of the roosed algorithm in denoising of simulated ediatric eiletic data. The comarison in terms of erformance and numerical comlexity with the FastICA, RobustICA and DelL Ê algorithms shows that the roosed algorithm offers the best trade-off between erformance and numerical comlexity when a low number of electrodes is available, such as in ediatric atients. We also examined the feasibility of JDICA in the case of real interictal data and showed that the JDICA algorithm is able to roerly denoise real data as well as simulated ones. As a art of our future work, we will examine the roosed algorithm with higher number of electrodes which may lead to different results. AENDIX A d 40 =C,,,,x, d 31 =4C,,,,x, d 22 =6C,,,,x d 13 =4C,,,,x, d 04 =C,,,,x 1 d 30 =4 1 d 12 =12 1 d 20 =6 1 d 11 =12 1 d 02 =6 C,,,i1,x C,,,i1,x 1, d 21 =12 1, d 03 = 4 i C,,i1,,x 2 +2 C,,,i1,x C,,i1,,x = 1 i 2 +2 i 2 +2 1 1 d 10 =4 C,i1,,,x 3 +3, C,,,i1,x C,,,i1,x C,,i1,,x C,,i1,,x C,,i1,,x C k,i1,,,x 2 i +6 1 d 01 =4 d 00 = i 2 1 i 3 =1 i 3 C,i1,,i 3,x C,i1,,,x 3 +3 i +6 1 1 k 1 i 3 =1 i 3 i 2 1 i 3 =1 i 3 1 i 4 = 1 i 4 i 3 1, C,i1,,i 3,x C i1,,i 3,i 4,x C,i1,,,x 2 i 3 i 3 i 4 e 0 =d 11 2 d 11 g 1 2 +d 13 4 d 31 g 1 4 + 2d 22 4d 04 g 1 3 +4d 40 2d 22 g 1 3 3d 31 3d 13 2 2 +2d 20 2d 02 g 1 e 1 = 2d 02 2d 20 g 1 2 +2d 20 2d 02 2 + 2d 22 4d 04 4 +2d 22 4d 40 g 1 4 + 6d 31 10d 13 g 1 3 +6d 13 10d 31 g 1 12d 04 12d 22 +12d 40 g 1 2 2 4d 11 g 1 + 3 + e 2 = 3d 31 3d 13 g 1 4 +3d 31 3d 13 4 + 12d 04 12d 22 +12d 40 g 1 3 +12d 22 12d 04 12d 40 g 1 3 +18d 13 18d 31 2 2 e 3 = 2d 02 2d 20 g 1 2 +4d 04 2d 22 g 1 4 + 2d 20 2d 02 2 +4d 40 2d 22 4 + 6d 13 10d 31 g 1 3 +6d 31 10d 13 g 1 3 12d 22 12d 04 12d 40 e 4 = d 11 4d 04 2d 22 3d 31 3d 13 f 0 = d 10 2 2 4d 11 g 1 2 +d 13 g 1 4 d 11 2 d 31 d 01 2d 21 3d 03 3 2 +2d 22 4d 40 2 +2d 02 2d 20 g 1 +d 12 3 d 21 g 1 3 + + 4 + 3 + 2 +3d 30 2d 12 g 1 2 f 1 = 2d 12 3d 30 g 1 3 +2d 21 3d 03 6d 03 7d 21 g 1 2 d 10 f 2 = d 10 d 01 d 01 6d 03 7d 21 g 1 +3d 30 2d 12 3 3 + +6d 30 7d 12 +2d 21 3d 03 g 1 3 + 2 +7d 12 6d 30 g 1 f 3 = 2d 12 3d 30 g 1 2 d 10 g 1 d 21 3 d 01 +2d 21 3d 03 2 2 d 12 g 1 3 2
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