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Efficient Computation of Lead Field Bases and Influence Matrix for the FEM-based EEG and MEG Inverse Problem. Part I: Complexity Considerations
Carsten H. Wolters, Lars Grasedyck and Wolfgang Hackbusch
The inverse problem in Electro- and Magneto-EncephaloGraphy (EEG/ MEG) aims at reconstructing the underlying current distribution in the human brain using potential differences and/or magnetic fluxes that are measured non-invasively directly, or at a close distance, from the head surface. The simulation of EEG and MEG fields for a given dipolar source in the brain using a volume-conduction model of the head is called the forward problem. The Finite Element (FE) method, used for the forward problem, is able to realistically model tissue conductivity inhomogeneities and anisotropies, which is crucial for an accurate reconstruction of the current distribution. So far, the computational complexity is quite large when using the necessary high resolution FE models.
In this paper we will derive algorithms for the efficient computation of EEG and MEG lead fields bases which exploit the fact that the number of sensors is generally much smaller than the number of reasonable dipolar sources.
The state-of-the-art forward approach will be speeded up by a factor of more than 100 for a realistic choice of the number of sensors and sources.
Our approaches can be applied to inverse reconstruction algorithms in both continuous and discrete source parameter space for EEG and MEG. In combination with algebraic multigrid solvers, the presented approach leads to a highly efficient solution of FE-based source reconstruction problems.