A filtering approach to the two-dimensional volume conductor forward and inverse problems.

Abstract

A new formulation for the forward and inverse problems concerning the propagation of potentials through two-dimensional volume conductors is presented. These problems are important in the study of frequently encountered problems in cardiology. They are solved in a computationally efficient manner by representing the extracellular medium as an equivalent filter: the medium filter. This approach is effective for the forward problem: estimating the potential induced by a known source, since it allows simple filtering operations to replace more cumbersome methods such as the solution of an integral equation derived with Green's theorem. It is also useful for the inverse problem: estimating the source potential from measurements of the field, since it allows the inverse operator to be easily obtained by algebraically inverting the medium filter, a far simpler operation than the inversion of an integral equation. However, the inverse problem is ill-conditioned and must be regularized. Due to the filtering approach, regularization is easily implemented by multiplying the inverse filter by an additional function. It is shown that Tikhonov regularization, constrained least-squares regularization, and stochastic regularization using a Wiener filter all lead to the same spatial low-pass regularizing filter. An algorithm for the estimation of the number and locations of bioelectric sources in a horizontally layered volume conductor is also presented. Again the solution is facilitated by modeling the intervening medium as an equivalent filter. This reduces the estimation of location into the estimation of the parameters of this filter. To estimate both the number and location of multiple sources requires the extension of the algorithm to a composite multiple-hypothesis test. The location estimation algorithm was found to be quite robust, even in extremely high noise conditions. Even though depth estimation is a nonlinear estimation problem, the estimator was found to be efficient: its performance approached the Cramer-Rao bound for all but the most extreme signal-to-noise conditions. Even in these high noise situations, small errors were generally observed.