A filtering approach to the two-dimensional volume conductor forward and inverse problems (biomedical application)

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Abstract
The two-dimensional layered volume conductor forward and inverse problems are solved by modelling the medium as an equivalent two-dimensional filter relating the source and measured potentials. This extends previous results to the multi-dimensional multi-layered case. In principle, the inverse problem can be solved as easily as the forward problem, using 2D FFT algorithms. In fact, it is ill-conditioned and must be regularized. The authors show that Tikhonov regularization, constrained least-squares regularization, and stochastic regularization using a Wiener filter all lead to the same spatial low-pass regularizing filter. New contributions include: (1) extension of the medium filter concept from one to two dimensions; (2) a simple layer-recursive formula for computing the medium filter for a layered medium; and (3) application of various regularization techniques, in the form of regularizing filters, to this problem.