Image Processing Reference
In-Depth Information
necessity for nonlinear optimization as in the case of the ECD fitting. The forward
model (Equation 8.1) can be presented for a noiseless case in the matrix form
XGQ
=
(8.3)
where , lead-field matrix, is assumed to be static in time. The
j
,
i
(th)
entry of describes how much a sensor is influenced by a dipole , where
varies over all sensors while varies over every possible source, or to be more
specific, eve
ry
axis-aligned component of every possible so
ur
ce:
G
MLN
×
G
j
i
j
i
gG
j
ı
=,
(
rp
)
.
i
j
The vector
ı
co
nt
ains indices of
L
such projections, i.e.,
ı =,
[
33 13
ii
+,
i
+
2
]
when
L
=
3
, and
ı =
i
when the dipole has a fixed known orientation. Using this
notation,
G
ı
corresponds to the lead matrix for a single dipole
q
i
. The
MT
×
matrix
X
holds the E/MEG data, while the
LN
×
T
matrix
Q
(note that
Qq
ı
=
i
t
()
)
t
corresponds to the projections of the ECD's moment onto orthogonal axes.
The solution of Equation 8.3 relies on finding an inverse
L
G
+
of the matrix
G
Q
to express the estimate
in terms of
X
ˆ
QGX
=
+
(8.4)
ˆ
and will produce a linear map . Other than being computationally conve-
nient, there is not much reason to take this approach. The task is to minimize the
error function (Equation 8.2), which can be generalized by weighting the data to
account for the sensor noise and its covariance structure:
XQ
L
() (
Q
=
XGQWXGQ
X
−
)
−
1
(
−
))
,
(8.5)
where is a weighting matrix in sensor space.
A zero-mean Gaussian signal can be characterized by the single covariance
matrix . In the case of a nonsingular we can use the simplest weighting
scheme to account for nonuniform and possibly correlated sensor noise.
Such a brute-force approach solves some problems of ECD modeling, specif-
ically the requirement for a nonlinear optimization, but, unfortunately, it introduces
another problem: the linear system (Equation 8.3) is ill-posed and underdeter-
mined, because the number of sampled possible source locations is much higher
than the dimensionality of the input data space (which cannot exceed the number
of sensors), i.e., Thus, there is an infinite number of solutions for the
linear system because any combination of terms from the null space of will
satisfy Equation 8.4 and fit the sensor noise perfectly. In other words, many
different arrangements of the sources of neural activation within the brain can
produce any given MEG or EEG map. To overcome this ambiguity, a regulariza-
tion term is introduced into the error measure
W
−1
C
ε
C
ε
WC
X
=
ε
NM
>>
.
G
L
r
()
QQ Q
=
L
()
+ λ
f
()
(8.6)
where
λ ≥
0
controls the trade-off between the goodness of fit and the regular-
ization term
f
()
Q
.
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