Biomedical Engineering Reference
In-Depth Information
(
)
×
Thus, the voxel source vector
x
t
, in this case, is an
N
1 column vector,
⊡
⊣
⊤
⊦
,
s
1
(
t
)
s
2
(
)
t
x
(
t
)
=
(2.13)
.
s
N
(
t
)
in which the
j
th component of
x
(
t
)
is
s
j
(
t
)
, which is the scalar intensity at the
j
th
voxel. In this topic, the same notation
x
(
t
)
is used to indicate either the 3
N
×
1 vector
in Eq. (
2.9
)orthe
N
1 vector in Eq. (
2.13
), unless any confusion arises.
In summary, denoting the additive noise in the sensor data
×
ʵ
, the relationship
between the sensor data
y
(
t
)
and the voxel source vector
x
(
t
)
is expressed as
y
(
t
)
=
Fx
(
t
)
+
ʵ
,
(2.14)
where
x
1 column vector in Eq. (
2.9
). When voxels have predetermined
orientations, using the augmented lead field matrix
H
in Eq. (
2.11
), the relationship
between
y
(
t
)
is a 3
N
×
(
t
)
and
x
(
t
)
is expressed as
y
(
t
)
=
Hx
(
t
)
+
ʵ
,
(2.15)
where
x
(
t
)
is an
N
×
1 column vector in Eq. (
2.13
).
2.5 Maximum Likelihood Principle and the Least-Squares
Method
When estimating the unknown quantity
x
from the sensor data
y
, the basic principle
is to interpret the data
y
as a realization of most probable events. That is, the sensor
data
y
is considered the result of the most likely events. We call this the maximum
likelihood principle. In this chapter, we first derive the maximum likelihood solution
of the unknown source vector
x
.
We assume that the noise distribution is Gaussian, i.e.,
2
I
ʵ
∼
N(
ʵ
|
0
, ˃
).
Namely, the noise in the sensor data is the identically and independently distributed
Gaussian noise with a mean of zero, and the same variance
2
. According to (C.1)
in the Appendix, the explicit form of the noise probability distribution is given by
˃
2
exp
2
1
1
(
ʵ
)
=
−
2
ʵ
.
p
(2.16)
2
M
/
(
2
ˀ˃
)
2
˃
