LINEAR INVERSE SOLUTIONS WITH OPTIMAL RESOLUTION KERNELS APPLIED TO ELECTROMAGNETIC TOMOGRAPHY

This paper discusses the construction of inverse solutions with optima l 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], sue show how a family of well-known sol utions ranging from the minimum norm method to the generalized Wiener estimator can be derived. It is shown that these solutions have optima l properties in some well-defined sense since they are obtained by opt imizing either the resolution kernels and/or the variances of the esti mates. New proposals for the optimization of resolution are made. In p articular, 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-dimen sional simulations shed light on the resolution properties of some lin ear inverse solutions when applied to the biomagnetic inverse problem. The simulations suggest that a reliable three-dimensional electromagn etic tomography based on linear inverse solutions cannot be constructe d, unless significant a priori information is included. The relationsh ip between the resolution kernels and a definition of spatial resoluti on is emphasized. Special consideration is given to the use of resolut ion kernels to assess the properties of linear inverse solutions as we ll as for the design of inverse solutions with optimal resolution kern els. (C) 1997 Wiley-Liss, Inc.