De-Noising Electroencephalogram ( EEG ) Signal Using Iterative Clipping Algorithm

Signal de-noising has been a topic of great interest for a long period. EEG is used to detect the neurological diseases. In the process of EEG recording, signal is contaminated due to several factors. Hence, for analysis of EEG signal in order to detect the diseases, it is necessary that signal must be de-noised first. Here, de-noising of signal is expressed as an inverse problem with total variation. This is an optimization problem. The solution of this optimization problem is obtained by using the iterative clipping algorithm. In this article, iterative clipping algorithm is used for de-noising EEG signal. To measure the performance of method, signal to noise ratio(SNR) and root mean square error(RMSE) have been calculated. It has been observed that the approach used here, works well in denoising the EEG signal.

Brain is one of the most important organs of human body which controls the coordination of muscles and nerves.Electroencephalogram (EEG) signal is a non-stationary biological signal and provides a lot of important information about different activities of the brain.It is the recording of electrical activity along the scalp with several electrodes placed on it over a short period of time.EEG is used to detect several neurological disorders 1 , 2 .There are many factors which affect the recording and contaminate the EEG.Electrooculographic (EOG) is the most common physiological noise source that generates the EEG artefacts 3 .Therefore, denoising plays a vital role in analysis of the EEG signal.
The problem of de-noising a signal is simply the noise removal from that signal.During last few decades, many techniques have been used to denoise (remove artefacts) the EEG signals.Kalman filters 4 , adaptive filters 5,6 ), blind source separation (BSS) method 7 , independent component analysis (ICA) have been applied 8,9,10,11 in artifact removing.Lagerlund et al 12 explored the applications of principal component analysis (PCA) in removing artefacts from EEG signals.Wavelet and its several variants have been widely used in artefact removing from EEG 13,14 ).Empirical Mode Decomposition 15,16 ) and ensemble empirical mode decomposition (EEMD) have been used in 17 and 18 to remove different artefacts present in EEG.
Another method proposed by Rudin et al. [19], total variation denoising is used in signal and image processing.Rodriguez & Wohlberg 20 , Hu & Jacob 21 and Bredies et al 22 have applied it in image processing.In total variation regularization, minimization of a cost function produces the denoising filter.All the algorithms that solve the optimization problem of de-noising can be used to implement the total variation regularization.So many methods are used to solve this problem.Selesnick and Bayram 23 have developed an algorithm based on 24 which has been used in this article.
Inverse problems deal with determining an input that produces an observed output, or determining an input that produces a desired output, often in presence of noise.Mathematically, let X and Y be spaces having appropriate structures such as Banach space, Hilbert space.Let A : X '!Y be an operator which describes the relationship between the data y and the model parameter x.Direct problem: given the input x, find the output y; inverse problem: given an observed output y, find an input x that produces it.The solution minimizing is known as total variation regularized solution.Here L is a smoothing and ë is regularization parameter.Regularization parameter ë plays important role in solving an inverse problem 25,26 .
In this article, iterative clipping algorithms proposed in 23 have been used for de-noising EEG signal.Two performance measuring parameters, Signal to Noise Ratios (SNR) and Root Mean Square Errors (RMSE) have been calculated for different values of the regularization parameter ë.It has been observed that iterative clipping algorithm works well for de-noising the EEG signal.

Material
Data used here is taken from research project of Kocaeli University, Turkey, taken by Dr. Huya K. Sevindir on applications of wavelets methods to EEG data collected at the Hospital of Kocaeli University 27 .

Total Variation (TV) de-noising
The total variation of an N -point signal ( ) where D is an N-1 x N matrix.
Let ( ) arg min ( ) The regularization parameter ë controls the degree of smoothing 18 .

Algorithm for TV de-noising
Problem ( 2) is solved by replacing D by A and formulating the dual by using an auxiliary vector z 5 .The iterative clipping algorithm for TV de-noising is given by: ( where max ( ).

Numerical implementation and performance analysis
Total variation de-noising based on (3) is implemented by the MATLAB program described in 23 .Here á = 4 is set for de-noising.In the MATLAB program, D is implemented with the diff command.Number of iterations is kept fixed at 100 in algorithm.
To evaluate the performance, Signal to Noise Ratio (SNR) and Root Mean Square Error (RMSE) have been calculated for different values of regularization parameter ë in table1.
Figure1 shows the original data while Figure2, Figure3, Figure4 and Figure5 show the de-noised data.

RESULTS AND DISCUSSION
In this article, de-noising of EEG signal is modeled as an inverse problem with total variation and is expressed as the minimization of a nondifferentiable cost function.Minimization of a nondifferentiable cost function is a complex problem.It is minimized using the iterative clipping algorithm.It has been used for de-noising the EEG signal.More SNR implies more de-noising and on the contrary, more RMSE implies lesser de-noising.
Table1 shows the values of SNR and RMSE for different values of ë.From the result obtained, we observe that iterative clipping algorithm works well for de-noising the EEG signal.We also observe that as ë increases, SNR decreases and RMSE increases.Figure1 represents the original data.

CONCLUSIONS
Increased values of signal to noise ratio for different values of regularization parameter show that iterative clipping algorithm is suitable for de-noising the EEG signal.As the values of regularization parameter increase, the signal to noise ratio decreases and root mean square error increases.This shows that de-noising depends on the values of the regularization parameter.This n be the measured (noisy) data of the form ( ) ( ) ( )y n = x n + ε n , n= 0, 1, 2,..., N-1 ...(1)where x(n) is (approximately) piecewise constant signal and å(n) is white Gaussian noise.Then the inverse problem is to estimate x(n) given the noisy data y(n).TV de-noising estimates the signal x(n) by solving the optimization problem