Successive Estimation Method of Locating Dipoles based on QR Decomposition using EEG Arrays

Abstract

<p> EEG is a noninvasive technique useful for the human brain mapping and for the estimation of neural electrical activities in human brain. A goal of processing EEG signals of a subject is the localization of neural current sources in human brain known as dipoles. Although this location estimation problem can be modeled as a particular kind of parameter estimation problem as in array signal processing, the nonlinear structure of an EEG electrode array, which is much more complicated than a traditional sensor array, makes the problem more difficult. </p> <p> In this thesis, we formulate the inverse problem of the forward model on computing the scalp EEG at a finite set of sensors from multiple dipole sources. It is observed that the geometric structure of the EEG array plays a crucial role in ensuring a unique solution for this problem. We first present a necessary and sufficient condition in the model of a single rotating dipole, that guarantees its location to be uniquely determined, when the second-order statistic of the EEG observation is available. In addition, for a single rotating dipole, a closed-form solution to uniquely determine its position is obtained by exploiting the geometrical structure of the EEG array. </p> <p> In the case of multiple dipoles, we suggest the use of the Maximum Likelihood (ML) estimator, which is often considered optimum in parameter estimation. We propose an efficient localization algorithm based on QR decomposition. Depending on whether or not the probability density functions of the dipole amplitude and the noise are available, we utilize the non-coherent ML or the LS as the criterion to develop a unified successive localization algorithm, so that solving the original multi-dipole optimization problem can be approximated by successively solving a series of single-dipole optimization problems. Numerical simulations show that our methods have much smaller estimation errors than the existing RAP-MUSIC method under non-ideal situations such as low SNR with small number of EEG sensors. </p>