Theory of the EEG Inverse Problem - Quantitative EEG Analysis Methods and Clinical Applications - page 129

At this point, the basic EEG inverse problem for the discrete, three-dimensional
distributed case consists of solving (5.9) for the unknown current density J and con-
stant c , given the lead field K and measurements .

5.4.1 The Reference Electrode Problem
As a first step, the reference electrode problem is solved by estimating c in (5.9).
Given Φ and KJ, the reference electrode problem is

1 2 min

c

Φ− KJ − c (5.14)

The solution is

1
11 Φ

T

( )

c = − KJ (5.15)

T

Plugging (5.15) into (5.9) gives

H Φ= J (5.16)

where

T

11
11

HI =− (5.17)

T

is the average reference operator, also known as the centering matrix, and
I R N EE is the identity matrix. This result establishes the fact that any inverse

solution will not depend on the reference electrode. This applies to any form of the
EEG inverse problem, including the inverse dipole fitting problems in (5.7) and
(5.8).
Henceforth, it will be assumed that the EEG measurements and the lead field
are average reference transformed, that is,

×

∈

ΦΦ H

KHK

←
←

⎩ ⎭

(5.18)

and (5.9) is then rewritten as follows:

Φ= KJ (5.19)

Note that H plays the role of the identity matrix for EEG data. It actually is the
identity matrix, except for a null eigenvalue corresponding to an eigenvector of
ones, accounting for the reference electrode constant.

5.4.2 The Minimum Norm Inverse Solution
In 1984 Hämäläinen and Ilmoniemi [23] published a technical report with a partic-
ular solution to the inverse problem corresponding to the forward equation of the