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 2 of 16
scalar accounting for the physics nature of electric potentials
determined up to an arbitrary constant; and 1 denotes a vector of ones, in this case
In Eq. 1, the structure of K is:
where the superscript T denotes transposition; and
, corresponds to the scalp potentials at the i-th electrode due to three orthogonal
unit strength dipoles at voxel j, each one oriented along the coordinate axes x, y, and z. For
instance, in infinite homogeneous medium with conductivity
are position vectors for the i-th scalp electrode and for the
respectively. As another example, for the case of a homogeneous conducting sphere in air,
the lead field is:
In the previous equations, the following notation was used:
denotes the trace, and X is any matrix or vector. If X
is a vector, then this is the
norm; if X is a matrix, then this is the squared Frobenius norm.
Note that K can also be conveniently written as:
, is defined as:
Ideally, the lead field should correspond to the real head (with realistic geometry and
conductivities). For the EEG problem, the voxels should correspond to cortical grey matter.
For other situations (e.g. EKG), appropriate volume conductor models and solution spaces
should be used.