Thank You Bob Thatcher !
mathematics of LORETA. J = currents in units of uA/cm^3 and phi =
electrical potentials at the scalp surface in uV. This is the standard
math for the lead field and the minimum norm. Roberto added the
physiological constraint of the 3D Laplacian operator which regularized
the matrix (reduced the dimensionality) so that a single solution for the
elements of the source matrix can be obtained. The Laplacian operator
maximally smoothed the values in the local vacinity of each source and
approximates the tendency of the brain to synchronize large numbers of
neurons, thus the term "low resolution", i.e., a smearing of the currents.
This does not mean that LORETA has low spatial accuracy because the
voxel with maximum current is not distorted by the operator and can still
be detected and this single voxel is spatially accurate and near to the
center of the neural activation.
I know that only a few on the forum are interested in the math, but this
flow chart has a beauty all of its own and represents the combined
thoughts of hundreds of people over the last few centuries. A good source
for the history is: Malmivuo J, Plonsey R. Bioelectromagnetism, New York,
NY: Oxford. University Press, 1995.