ITERATIVE ENHANCEMENT OF EVENT RELATED POTENTIALS THROUGH SPARSITY CONSTRAINTS N. Mourad ∗ , James P. Reilly ∗ , Laurel J. Trainor † , Bernhard Ross ‡ ∗ Department of Electrical & Computer Eng. † Department of Psychology, Neuroscience & Behaviour McMaster University, 1280 Main St. W., Hamilton, Ontario, Canada L8S 4K1 ‡ Rotman Research Institute for Neuroscience Baycrest Centre for Geriatric Care, Toronto, Canada. email: mouradna@mcmaster.ca, reillyj@mcmaster.ca, ljt@trainor.ca, bross@rotman-baycrest.on.ca ABSTRACT In this paper we propose an iterative technique that en- hances the average event related potential (ERP) by correct- ing the delay associated with the ERP in each trial. This correction is done in three steps: in the first step a sparse template function is estimated. In the second step, this tem- plate is utilized in estimating the inter–trial ERP delays. The ERPs from each trial are time-aligned using the estimated de- lays. In the third step, a new estimate of the ERP waveform is obtained by averaging these time–aligned signals over the trials. The algorithm iterates through these three steps until convergence. The sparse template is estimated in each it- eration through the minimization of a convex objective func- tion which compromises between the fit of the estimated ERP waveform to the template, and the sparsity of the estimated ERP waveform. Indexing Terms: Electroencephalogram (EEG), event re- lated potential (ERP), convex optimization. 1. INTRODUCTION The event related potential (ERP) is a brain response to an external stimulus. The ERP has a well defined pattern whose amplitude and latency can be used in the diagnosis of pos- sible brain injury or disorders in the central nervous system [1]. The ERP is characterized by a relatively low signal to noise ratio (SNR) with respect to the background brain EEG activity. Accordingly, the ERP is usually estimated by aver- aging over a large number of trials. The disadvantage of this approach is that it is based on the assumption that the ERP occurs exactly at the same time after the stimulus onset in all trials. In reality, this assumption is not realistic. A more realistic model for the measured EEG signal in the i-th trial can be represented mathematically by [2] x i (t )= a i s(t − d i )+ n i (t ), i = 1,..., N (1) where x i (t ) is the measured EEG signal, s(t ) is the ERP waveform, n i (t ) is the background brain activity, N is the to- tal number of trials, a i is a random variable representing the amplitude of the ERP, and d i is an unknown delay. Due to This work was supported in part by the Egyptian Ministry of High Ed- ucation. the variation in the d i ’s, the estimated ERP ˆ s(t ) obtained due to averaging over trials will be distorted. Thus, many pre- vious approaches have attempted to correct this problem by estimating the d i , and time–aligning the individual responses before averaging. There are many algorithms for solving the problem of de- lay estimation. These algorithms are based either on cross correlation [3], adaptive techniques [1], [2] or maximum likelihood (ML) [4]. Each of these algorithms assume dif- ferent characteristics of the signal and noise; e.g. in [4], it was assumed that the background EEG is a realization of a zero-mean stationary Gaussian process, while in [2] it was assumed to be a realization of a non–Gaussian process. Clearly, invoking some assumptions on the model may result in inaccurate estimate of the delays. A comparison between some of these algorithms is presented in [5]. In this paper, our objective is to enhance the ERP, through estimation of the delays. The challenge associated with this problem is the similarity between the ERP and the back- ground brain activity in such a way that, until now, it is not known precisely whether the ERP is a separate signal added to the background brain activity [6], or it is just a result of partial phase resetting in the background brain activity [7]. This implies that the waveforms of the ERP and the back- ground EEG signals have similar characteristics. We invoke a realistic assumption that ERP exists only in a small window after the stimulus onset, specified by the interval [d min , d max ], referred to as the the window of interest (WOI). This assump- tion implies that the ERP is a sparse signal. A signal is sparse if it has few non zero elements. In addition, we assume also that the ERP dominates the background EEG signal inside the WOI. In this paper, we make use of a sparse “template signal” p(t ), which is a good fit to the ERP inside the WOI. An opti- mization technique is utilized for estimating this template. Once p(t ) is estimated, the cross correlation between the template and each x i (t ) is utilized for estimating the delays d i . This paper is organized as follows: in Section 2 The de- velopment of the proposed method is presented. The pro- posed technique is presented in Section 3. The performance of the proposed algorithm is examined in Section 4. Finally,