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Cite as: “R.D. Pascual-Marqui: Discrete, 3D distributed, linear imaging methods of electric neuronal activity. Part 1: exact, zero
error localization. arXiv:0710.3341 [math-ph], 2007-October-17, http://arxiv.org/pdf/0710.3341 ”
Page 7 of 16
The particular case of interest here will only consider a
structured block-diagonal
weight matrix
W, where all matrix elements are zero except for the diagonal sub-blocks
denoted as
, the i-th voxel, with
.
Note that for
, this is a genuine solution, in the sense that
is a direct
estimator for the current density, and it reproduces exactly the measurements. In other
words, for
:
Eq. 29:
The current density estimator at the i-th voxel then is:
Eq. 30:
Based on the results of the previous section (entitled “A family of discrete, 3D
distributed linear imaging methods with exact, zero error localization”), by comparing Eq. 30
with Eq. 15, exact, zero error localization is attained with weights satisfying:
Eq. 31:
This result is easily derived by noting that Eq. 30 matches Eq. 15 when:
Eq. 32:
and:
Eq. 33:
The weights satisfying the system of equations given by Eq. 31
define the eLORETA
method, which is a genuine solution to the inverse problem (not merely a linear imaging
method), and attains exact, zero error localization.
Additionally, eLORETA is standardized
by definition, meaning that its theoretical expected variance is unity.
Furthermore, following the derivations as in the previous section entitled “Unbiased
localization for sLORETA”, it can easily be shown that eLORETA is unbiased in the presence
of measurement and structured biological noise of the form:
Eq. 34:
Unfortunately, such a structure on background brain activity (the so-called biological
noise) is determined by the physics properties of the head model
and the laws of
electrodynamics, and might have little relation to electrophysiological reality. This might be
seen as a disadvantage of eLORETA as compared to sLORETA.