Linear inverse solutions with optimal resolution kernels applied to electromagnetic tomography

This paper discusses the construction of inverse solutions with optimal resolution kernels and applications of them in the reconstruction of the generators of the EEG/MEG. On the basis of the framework proposed by Backus and Gilbert [1967], we show how a family of well-known solutions ranging from the minimum norm method to the generalized Wiener estimator can be derived. It is shown that these solutions have optimal properties in some well-defined sense since they are obtained by optimizing either the resolution kernels and/or the variances of the estimates. New proposals for the optimization of resolution are made. In particular, a method termed "weighted resolution optimization" (WROP) is introduced that deals with the difficulties inherent to the method of Backus and Gilbert [1967], from both a conceptual and a numerical point of view. One-dimensional simulations are presented to illustrate the concept and the interpretation of resolution kernels. Three-dimensional simulations shed light on the resolution properties of some linear inverse solutions when applied to the biomagnetic inverse problem. The simulations suggest that a reliable three-dimensional electromagnetic tomography based on linear inverse solutions cannot be constructed, unless significant a priori information is included. The relationship between the resolution kernels and a definition of spatial resolution is emphasized. Special consideration is given to the use of resolution kernels to assess the properties of linear inverse solutions as well as for the design of inverse solutions with optimal resolution kernels.