Cortical Functional Mapping by High-Resolution EEG - Quantitative EEG Analysis Methods and Clinical Applications - page 358

Ax += n b (13.1)

n matrix with the number of rows equal to the number of sensors
and the number of columns equal to the number of modeled sources. We denote
with A

where A is an m ×

j the potential distribution over the m sensors due to each unitary j th cortical
dipole. The collection of all of the m -dimensional vectors A

⋅

j ,( j =1,…, n ) describes
how each dipole generates the potential distribution over the head model, and this
collection is called the lead field matrix A . This is a strongly underdetermined linear
system, in which the number of unknowns, the dimension of the vector x , is greater
than the number of measurements b by about one order of magnitude. In this case,
from the linear algebra we know that infinite solutions for the x dipole strength vec-
tor are available, explaining in the same way the data vector b . Furthermore, the lin-
ear system is ill-conditioned as a result of the substantial equivalence of several
columns of the electromagnetic lead field matrix A . In fact, we know that each col-
umn of the lead field matrix arose from the potential distribution generated by the
dipolar sources that are located in similar positions and have orientations along the
cortical model used. Regularization of the inverse problem consists in attenuating
the oscillatory modes generated by vectors that are associated with the smallest sin-
gular values of the lead field matrix A , introducing supplementary and a priori infor-
mation on the sources to be estimated.
In the following, we use the term source space to characterize the vector space in
which the “best” current strength solution x will be found. Data space is the vector
space in which the vector b of the measured data is considered. The electrical lead
field matrix A and the data vector b must be referenced consistently.
Before we proceed to the derivation of a possible solution for the problem, we
recall a few definitions from algebra that will be useful. A more complete introduc-
tion to the theory of vector spaces is outside the scope of this chapter, and the inter-
ested readers could refer to related textbooks [9, 10]. In a vector space provided with
a definition of an inner product (·, ·), it is possible to associate a value or modulus to
a vector b by using the notation

⋅

()

bb , = b (13.2)

Any symmetric positive definite matrix M is said to be a metric for the vector
space furnished with the inner product (·, ·), and the squared modulus of a vector b
in a space equipped with the norm M is described by

2

b = b

T

b (13.3)

M

With this in mind, we now face the problem of deriving a general solution of the
problem previously described under the assumption of the existence of two distinct
metrics N and M for the source and the data space, respectively. Because the system
is undetermined, infinite solutions exist. However, we are looking for a particular
vector solution x that has the following properties: (1) It has the minimum residual
in fitting the data vector b under the norm M in the data space, and (2) it has the
minimum strength in the source space under the norm N . To take into account these