Linear inverse solutions with optimal resolution kernels applied to electromagnetic tomography

Abstract

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.

Localization bias and spatial resolution of adaptive and non-adaptive spatial filters for MEG source reconstruction

“…The resolution kernel (Backus and Gilbert, 1968;de Peralta Menendez et al, 1997) can play this role. We introduce the resolution kernel, derived by combining Eqs.…”

Functional brain imaging of human sleep

“…For this reason, we chose to consider distributed solutions. With this approach, each and every brain voxel is a possible current source and results are presented as clouds of electrical activity, as it is implemented in the Low Resolution Electromagnetic Tomography (LOR-ETA) method (Pascual-Marqui and Michel 1994) or other techniques (Grave de Peralta Menendez et al 1997). Figure 2 shows the results obtained during sleep in a child presenting with continuous spike-and-wave discharges during sleep and acquired cognitive deterioration.…”

Recursive penalized least squares solution for dynamical inverse problems of EEG generation

“…In this approach, a discretization of brain volume into a set of voxels is employed, each of which is considered to be the location of a current vector. In order to obtain a unique solution, various constraints have been suggested in previous studies: as prominent examples we mention optimal resolution (Backus and Gilbert 1968;Grave de Peralta Menendez et al 1997;Grave de Peralta Menendez and Gonzalez Andio 1999), L 2 minimum norm (Hämäläinen and Ilmoniemi 1984), L 1 minimum norm (called 'selective minimum norm') (Matsuura and Okabe 1995) and maximum spatial smoothness (called 'low resolution brain electromagnetic tomography', LORETA) ).…”