The Studies Of Numerical Methods For The Inverse Problems Based On Electroencephalogram

Brain science is one of the most prospective filed among the active branches of life science. Since their importance, functional imaging methods based on electroencephalogram (EEG) have been extensively studied during last several decades. This thesis concentrates on two aspects of inverse problems based on EEG signal: EEG inverse problems of source localization for determining the electrical activities inside the brain, and EIT inverse problems for estimating the impendence distribution of brain.Firstly, the basic theories of inverse problem based on EEG signal are concisely reviewed, the perspective of applications is illuminated, and the current numerical computation methods of both EEG inverse problems and EIT inverse problems are comprehensively analyzed in depth.Hence therefore, the EIT forward problems, which are the basis of EIT inverse problems, are studied. Concretely, the finite element method (FEM) is adopted, the EIT field is mathematically described, the computations are carried out for two dimensional circle models and simulated head models, and the numerical computation accuracy are satisfied.The modified Newton-Raphson (MNR) method for EIT static imaging is thoroughly studied, and the improved regularization method is applied for conquering the ill-poseness of the inverse problems undertaking. The selections of regularization matrixes and the regularization parameters are deliberated with careful comparisons, then the best regularization parameter selection method is determined, in which the self-decision of appropriate regularization parameter during the iterations is realized. The improved MNR method has the advantages of fast convergence and high reconstruction precisions, however the results are highly depending on the initial estimation in some occasions.Therefore, the genetic algorithm (GA) and the differential evolution (DE) method are used for head EIT imaging studies, while it's the first time that DE is applied in EIT inverse problems. The philosophy of GA is evaluated, the genetic operations are analyzed and compared, and improvements are proposed and tested. Particular attention is paid to DE methods and their implementation, and the performances are superior. Comparing with the GA, DE has the advantages of simpler operation, faster convergence and better effect.Furthermore, a new combinational method is presented for head EIT imaging based on the analysis of properties of various methods mentioned above. While the perfect results can be obtained in any case using DE methods, their computation takes longer time. On the contrary, the computation time taken by MNR is much shorter but the reconstruction results are highly depending on the initial distribution. Thus we presented a novel method by combining DE and MNR. The simulations demonstrated that the convergence is guaranteed, the computation time is shortened, and the efficiency is improved.In the studies of EEG forward problems, primarily the EEG computation models are established, the analysis solution for concentric circle models and the boundary element method (BEM) for piecewise humongous models are developed. These provide the basis for EEG inverse problems.In the solving of EEG inverse problems, the nonlinear inverse solutions of EEG based on equal current dipole (ECD) model are investigated, where the DE method is cooperated originally. The reconstructions for single dipole source are validated, and the outcomes revealed that DE is of the benefit of simpler operations with more accurate localization ability.Lastly the voxel imaging methods of EEG based on distribution current model are presented, the low resolution brain tomography algorithm (LORETA) is closely explored for localizing sources inside the brain and on the cortex, with proper regularizations being applied. The perfect results are obtained using the improved cortical LORETA.