Biomedical Engineering Reference
In-Depth Information
Chapter 2
Minimum-Norm-Based Source Imaging
Algorithms
2.1 Introduction
In this chapter, we describe the minimum-norm and related methods, which are
classic algorithms for electromagnetic brain imaging [
1
,
2
]. In this chapter, the
minimum-normmethod is first formulated based on the maximum-likelihood princi-
ple, and the properties of the minimum-norm solution are discussed. This discussion
leads to the necessity of regularization when implementing the minimum-norm
method. We discuss two different representative regularization methods: the
L
2
-norm regularization and the
L
1
-norm regularization. Theminimum-normmethod
is, then, formulated based on Bayesian inference—Bayesian formulation providing
a form of the minimum-norm method where the regularization is already embedded.
2.2 Definitions
In electromagnetic brain imaging, we use an array of sensors to obtain
bioelectromagnetic measurements. We define the output of the
m
th sensor at time
t
as
y
m
(
t
)
, and the column vector containing outputs from all sensors, such that
⊡
⊣
⊤
⊦
,
y
1
(
t
)
y
2
(
t
)
(
)
=
y
t
(2.1)
.
y
M
(
t
)
where
M
is the total number of sensors. This column vector
y
expresses the outputs
of the sensor array, and it may be called the data vector or array measurement.
(
t
)
