Discrete, 3D distributed, linear imaging methods of electric neuronal activity. Part 1: exact, zero error localization

This paper deals with the EEG/MEG neuroimaging problem: given measurements of scalp electric potential differences (EEG: electroencephalogram) and extracranial magnetic fields (MEG: magnetoencephalogram), find the 3D distribution of the generating electric neuronal activity. This problem has no unique solution. Only particular solutions with “good” localization properties are of interest, since neuroimaging is concerned with the localization of brain function. In this paper, a general family of linear imaging methods with exact, zero error localization to point-test sources is presented. One particular member of this family is sLORETA (standardized low resolution brain electromagnetic tomography; Pascual-Marqui, Methods Find. Exp. Clin. Pharmacol. 2002, 24D:5-12; http://www.unizh.ch/keyinst/NewLORETA/sLORETA/sLORETA-Math01.pdf). It is shown here that sLORETA has no localization bias in the presence of measurement and biological noise. Another member of this family, denoted as eLORETA (exact low resolution brain electromagnetic tomography; Pascual-Marqui 2005: http://www.research-projects.unizh.ch/p6990.htm), is a genuine inverse solution (not merely a linear imaging method) with exact, zero error localization in the presence of measurement and structured biological noise. The general family of imaging methods is further extended to include data-dependent (adaptive) quasi-linear imaging methods, also with the exact, zero error localization property.

💡 Research Summary

The paper addresses the classic EEG/MEG inverse problem, where scalp electric potentials and extracranial magnetic fields are used to infer the three‑dimensional distribution of neuronal current sources inside the brain. Because the lead‑field matrix (L) has far fewer rows (sensor channels) than columns (brain voxels), the problem is severely ill‑posed and admits infinitely many solutions. Traditional approaches impose regularization (e.g., Tikhonov) or Bayesian priors to obtain a unique estimate, but these often introduce a systematic localization bias.

The authors introduce a general family of linear imaging operators that achieve exact zero‑error localization for point‑test sources. The key idea is to incorporate the noise covariance matrix (Σε) directly into the inverse operator. By defining a standardized weighting matrix S = diag(√diag(Lᵀ Σε⁻¹ L)), the operator

 W = S⁻¹ Lᵀ Σε⁻¹

produces an estimate ŴV whose expected value equals the true source J₀, regardless of measurement noise or uncorrelated biological noise. This operator underlies sLORETA (standardized low‑resolution brain electromagnetic tomography). The paper proves mathematically that sLORETA is unbiased: E