The concept of ICA was introduced by Herault and Jutten 25 , especially in order to solve the BSS problem. In the mid 1990s, Comon 26 presented a mathematical formulation of ICA. During the past 25 years, a wealth of algorithms have been proposed and ICA-based methods are now extensively and successfully applied to solve many practical real-life problems such as biomedical signal analysis and processing, wireless communications, data mining, speech enhancement, image recognition, etc. (see 3 for details). More precisely, assuming the instantaneous linear observation model of Equation (2) and the hypotheses H1 to H3, the goal of ICA is to find a (N × P), full rank, separator matrix, W , such that the output signal

y ' [ m ] = W T x ' [ m ] , ∀ m ∈ N

is an estimate of the source vector s [ m] up to a multiplicative trivial matrix ΛΠ (where Λ is a diagonal invertible matrix and Π is a permutation matrix). Our previous work showed that, compared to 12 other ICA methods, the CoM2 algorithm 3 27 offers the best performance/complexity compromise for the denoising of Mu-like simulated EEG activity in the context of brain computer interface. This justifies the use of this method in this article. CoM2 starts with a prewhitening of the observations x [ m] and relies on a maximization of a contrast function derived from Fourth-Order (FO) cumulants:

ϒ C O M 2 z ' = ∑ p = 1 P C p , p , p , p , z ' 2

where z ' m represents the whitened random vector process associated with x ' m and C p , p , p , p , z ' is the FO cumulant of the pth random process z p ' m . In practice, CoM2 method maximizes the contrast function (3) iteratively by applying a planar Givens rotation to every whitened signal pair until convergence as in Jacobi sweeping algorithm for matrix diagonalization. More precisely, in our case where the signals are real, the optimal rotation angle (optimizing (3)) can be found by rooting a polynomial of degree 4.

Originally proposed by Hotelling 28 , CCA is a method that measures the linear relationship between two multi-dimensional random variables. Friman et al. 29 showed that CCA can be used to solve the BSS problem by taking the source vector as the first multi-dimensional random variable and the temporally delayed version of the source vector as the second multi-dimensional random variable. Thus, assuming the instantaneous linear observation model of Equation (2), the spatial statistical decorrelation of sources and the hypotheses H2 to H4, CCA aims at extracting a P-dimensional vector of sources s [ m] by both minimizing the spatial correlation between sources and maximizing their temporal correlation. To do so, let us consider the two vectors

y [ m ] = W x [ m ] a n d y [ m - τ ] = W x [ m - τ ] = W z [ m ] , ∀ m ∈ N

where y [ m] is an estimate of the source vector s [ m]. The problem is to find a (N × P), full rank, separator matrix W that maximizes the following function:

ρ = W R x ' z ' W T ( W R x ' x ' W T ) ( W R z ' z ' W T )

The maximum of ρ with respect to W is called the maximum canonical correlation, R x ' x ' and R z ' z ' are the autocovariance matrices of x ' and z ' , respectively, and R x ' z ' the cross-covariance matrix of x ' and z ' . After some manipulations, the demixing matrix W can easily be calculated by solving the eigen-matrix equation

R x ' x ' − 1 R x ' z ' R z ' z ' − 1 R z ' x ' w x ' = ρ 2 w x '

where the eigen-vectors w x ' are the columns of W .

EMD was originally introduced in the late 1990s to study water surface wave evolution 30 . Since then, it has been used in various fields such as biomedical signal analysis 31 , Hurst exponent estimation 32 , speech processing 33 , or texture analysis 34 . EMD aims at decomposing sequentially a given signal x n [ m ] into a sum of Amplitude and Frequency Modulated (AM/FM) zero mean oscillatory signal, s nk [ m ] , referred to as Intrinsic Mode Functions (IMFs), plus a non-zero mean-low-degree polynomial remainder. More precisely, EMD sequentially computes K IMFs s nk [ m ] and the corresponding trends r n [ m ] , such that

x n [ m ] = r n [ m ] + ∑ k = 1 K s nk [ m ] , ∀ m ∈ N

In practice, each IMF, s nk [ m ] , is calculated using an iterative procedure, called sifting process. At each instant m, the jth sifting iteration consists: (i) in computing the mean envelope, M ( s n k , j [ m ] ) , of the residual signal, and (ii) in extracting the detail s n k , j + 1 [ m ] = s n k , j [ m ] - M ( s n k , j m ) from the residue. This process is repeated until the stopping criteria is reached (Cauchy-like criterion used in 35 ). Several EMD methods have been proposed in the literature 30 32 34 35 , depending on the way the mean envelope M s n k , j [ m ] , ∀ m ∈ N is calculated. In the following, we will use the 2T-EMD method recently proposed by our team 35 . Briefly, the signal mean trend M . is redefined as the signal which interpolates the barycenters of particular oscillations, called elementary oscillations (see 35 for more details). We also show in 35 that, in practice, a robust computation of the mean trend is obtained by averaging two envelopes: a first envelope interpolates the even indexed barycenters, and a second envelope interpolates the odd indexed barycenters. Compared to other approaches the 2T-EMD algorithm is simpler, has a lighter computational cost, and enables both mono and multivariate decompositions without any change in the core of the algorithm.

The concept of WT was formalized in early 1980s and, since then, it has extensively been applied in a large variety of fields, such as biomedical signal processing, image compression, astronomy (see, e.g., 36 37 38 39 40 41 42 ). The WT is defined as the inner product or cross correlation of the signal x n [ m ] with the scaled and time shifted wavelet Ψ a , b [ m ] :

W T x n [ a , b ] = x n , Ψ a , b

where Ψ a , b [ m ] = a − 1 2 Ψ m - b / a (a and b are the scale and translation parameters, respectively). The WT provides a decomposition of the signal x n [ m ] in different scales, where the obtained wavelet coefficients represent a measure of similarity between the signal x n [ m ] and an appropriate wavelet function Ψ a , b [ m ] . Among the existing WT approaches, we used a most common form of the DWT which employs the dyadic grid ( a = 2 j and b = k 2 j where j and k are integers) because of its: (i) good temporal localization properties, (ii) fast calculation, and (iii) zero redundancy. In practice, the DWT can be computed by successively passing the signal x n [ m ] through: (i) a high-pass filter producing the detail coefficients, and (ii) a low-pass filter giving the approximation coefficients 43 . More precisely, the DWT decomposition is recursively (D levels) applied on the output of the low-pass filter and generate D details and one approximation.

For both CoM2 and CCA, the components that represent the sources of interest were selected by calculating the autocorrelation of each component (for a time lag τ = 1) and then by classifying them in a descending order according to their respective autocorrelation values. Consequently, as the autocorrelation of muscle artifacts is relatively low with respect to that of epileptic spikes, the components representing muscle artifacts are expected to be among the last components. In turn, CoM2 or CCA components of interest will be classified among the first components which facilitate their visual selection. Then, the signal is reconstructed by keeping only the components accounting for the sources of interest (epileptic spikes).

Regarding the 2T-EMD method, the visual selection of the relevant IMFs is quasi-impossible and an automatic selection of the IMFs of interest is needed. In the past, some interesting EMD procedures have been proposed to automatically select and classify the IMFs of interest. They can be divided into two categories: (i) methods based on low-pass filtering 44 45 46 and (ii) methods based on thresholding filtering 47 48 . EMD based on low-pass filtering considers that the IMFs are decomposed into two sets: the first set of IMFs represents only the signal of interest and the second set stands for the noise and artifacts. Nevertheless, in some practical cases noise or artifacts can be distributed over all IMFs. Thus, the low-pass filtering methods remove the high-frequency IMFs corresponding to the component of interest and keep the low-frequency IMFs related to noise and/or artifacts. The second family of procedures is inspired from wavelet thresholding methods and evaluates the noise level for each IMF using a suitable threshold as in wavelet analysis. Due to the properties of the muscle activity (wide spectral distribution, and variable topographical distribution), we chose to use a thresholding filter method, as proposed in 47 . Concisely, authors propose in 47 to modify the universal wavelet threshold proposed by Donoho 49 in order to fit the specificities of each IMF. The authors show also that the denoising performances are improved by averaging several denoised versions of the signal (see 50 for details). Note also that this method is advantageous in the sense it does not require any reference signal.

Signal denoising using DWT requires three successive steps. First, we decompose the original signal by choosing the number of levels D. Second, we threshold the obtained D details. Finally, we reconstruct the denoised signal using the D altered detail coefficients and approximation coefficients of level D. Thus, in DWT schemes, the mother wavelet, shrinkage rule, and noise level rescaling approach must be designated. Indeed, there are some general rules about the choice of these parameters: (i) the mother wavelet is selected based on its similarity to the desired signal, (ii) rescaling approach and shrinkage rule are chosen according to the nature of the noise (white or colored) and the variance of the noise. In our case, it is thus reasonable to use the Daubechies family that is interictal spikes-like. Regarding the thresholding method, among the different algorithms, the Stein’s Unbiased Risk Estimate (SURE) 51 shrinkage rule and a soft thresholding strategy gave the best results (on simulated signals). This choice can be justified because the signal-to-noise ratio (SNR) is weak and an algorithm which offers a low thresholding as SURE seems to be appropriate for our application. Indeed, contrary to the universal thresholding, in the SURE algorithm the threshold depends on the signal and not only the estimated noise.

It should be noted that, the number of extracted IMFs (K = 10) for 2T-EMD, the levels of decomposition (D = 6) for DWT and the mother wavelet (Daubechie order 4) are fixed using the simulated data in order to obtain the desired frequency resolution. These values are also used in the case of real data. We also point out that for DWT the reconstruction of the denoised signal has been achieved without the coefficients of the first level.

The performance of CoM2, CCA, 2T-EMD, and DWT was first evaluated by computing the following normalized mean-squared error (NMSE):

N M S E x n = ∑ ℓ = 1 L ∑ m = 1 M x n m - x ^ n ℓ m 2 L ∑ m = 1 M x n m 2

where x n m is the original EEG observation on the nth electrode (EEG without muscle activity), x ^ n ℓ m is the reconstructed surface EEG after denoising from the ℓ t h run, L is the number of Monte Carlo runs and M is the data length.

For data simulated from a single epileptic patch (Figure 2), the calculation, for a set of 50 trials, of the mean performance criterion (NMSE) calculated for all electrodes shows that 2T-EMD: (i) performs better than CoM2, CCA for very low SNR (–30 dB), (ii) gives comparable results with CoM2 and CCA in the case of SNR –25, and (iii) is less efficient than CCA and CoM2 for SNR ranging from –20 to –5 dB (see Figure 2A). The DWT-based method is less efficient than CCA, CoM2, and 2T-EMD across all SNRs. In addition, the performance of CCA, CoM2, and the DWT algorithms increases as SNR values increase. On the contrary the performance of the 2T-EMD algorithm remains stable across all SNRs, with a low variability within trials compared to the three other methods. As illustrated in Figure 2B, when the performance criterion is specifically calculated for electrode T3 (electrode facing the cortical epileptic patch), CCA and CoM2 clearly outperform 2T-EMD and the DWT for all SNRs. 2T-EMD outperforms the DWT method for SNR of –30 and –25 dB, while the DWT gives comparable results with 2T-EMD in the case of SNR –20 dB. Finally, the DWT-based method outperforms 2T-EMD for SNR ranging from –15 to –5 dB. It is noteworthy that for the CCA and CoM2 methods, the performance remains stable for all considered SNRs while the 2T-EMD and DWT methods become more efficient as the SNR in the simulated signals increases.

The results obtained on signals simulated from two epileptic patches are illustrated in Figure 3. In that case, the mean performance of the four algorithms calculated for 50 trials over all electrodes (Figure 3a) follows the same trend as for data obtained with a single patch (Figure 2a). Nevertheless, for SNR ranging from –20 to –5 dB, the performance of the CCA method is highly variable within trials compared to the other methods. Figure 3b shows that at electrode T3 (facing the location of the first epileptic patch), CCA and CoM2 have similar performance and slightly outperform 2T-EMD for SNRs of –30 and –25 dB, while the DWT shows the worse performance. The performance of CCA, CoM2, and 2T-EMD is quasi-identical for SNR –20 dB, and better than that of the DWT-based method. All methods show almost equivalent performance for SNR of –15 dB. Finally, CoM2 and the DWT algorithms give better results than the two other algorithms for SNRs of –10 and –5 dB. Note also that the performance of the DWT increases as SNR values increase. At CP5 (facing the second epileptic patch), the performance of CCA, CoM2, and 2T-EMD are comparable to that obtained at electrode T3. We also remark that, for SNRs ranging from –20 to –5 dB the DWT algorithm slightly outperform the other methods (Figure 3c).

Figures 2 aand 3a indicate that the mean performance criterion calculated for the four methods over all electrodes is of the same order of magnitude for data obtained with a single patch or with two patches. On the contrary, Figures 2b, 3b, and 3c show that when data were simulated from two epileptic patches the performance of CCA and CoM2 at electrodes facing the patches (T3 or CP5) is worse than that obtained for data simulated from a single patch.

As another performance criterion, source localization was applied on original and noisy simulated data before and after denoising by the CCA, CoM2, 2T-EMD, or the DWT algorithms. An example is given in Figure 4 for two levels of SNR. Figure 5 reports the ROC curves obtained on the set of 50 trials, both for single-patch and for double-patch simulated data. As compared to original data, sources of spikes localized from noisy data are misleading. This is observed both for spikes arising from a single or from two distinct epileptic patches, and for both SNRs (–25 and –15 dB). In the single patch configuration, the source of spikes is still mislocated after denoising of low SNR (–25 dB) data with CCA, CoM2, and DWT method. However, for 2T-EMD denoised data, the source of spikes is localized closer to the actual patch location (inferior temporal region).

ROC curves show that this trend is reversed for higher SNR values (Figure 5a). In particular, as also illustrated in Figure 4, localization results obtained after denoising the –15 dB data with CCA, CoM2, and DWT methods are very close to those obtained for original clean data, while the epileptic patch is still mislocalized from the 2T-EMD-denoised data.

Similarly, in the double-patch configuration, and for low SNR data (–25 dB), spikes arising from two distinct patches are also better localized after 2T-EMD than after CCA, CoM2, and DWT (see Figure 4). In particular, one of the two epileptic sources (inferior parietal) is retrieved from the 2T-EMD-denoised data while for the same SNR, source localization is misleading for CCA, CoM2, and DWT (source localized in-between both patches or in a remote region). As shown in Figure 5b, for SNR ranging from –30 and –25 dB, source localization from CoM2 and CCA and 2T-EMD-denoised data remain comparable, while localization results of DWT-based method are the worse. For an SNR of –20 dB, source localization results are similar, whatever the approach used for denoising. For SNRs of –15 to –5 dB the best results are obtained from data denoised with the DWT or with CoM2. It is noteworthy than for the two-patches source configuration, source localization results get better as SNR in noisy data increases. Nevertheless, for all SNR values, source localization of signals denoised by any of the considered approaches did not retrieve properly the second patch (superior temporal).