Image Processing Reference
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by projections in two axes in the case of an MEG spherical model, in which the
silent radial to the skull component has been removed, or even just in a single
axis if normality to the cortical surface is assumed. The ECD model made it
possible to derive a tractable physical model linking neuronal activation and
observed E/MEG signals. In case of
K
simultaneously active sources at time
t
,
the observed E/MEG signal at the sensor
x
positioned at
p
can be modeled as
j
j
K
∑
ˆ (
rq
,,=
t
)
Gt
(
r
( )
, ⋅
p
)
q
( )
t
+
ε
(8.1)
x
j
i
i
i
j
i
i
where
G
is a lead-field function that relates the
i
th dipole and the potential (EEG)
or magnetic field (MEG) observed at the
is the sensor noise. In
the given formulation, function returns a vector, where each element
corresponds to the lead coefficient at the location
j
th sensor, and
ε
Gt
i
(()
rp
,
)
j
p
generated by a unit-strength
j
dipole at position
r
(
t
) with the same orientation as the corresponding projection
i
axis of
. The inner product between the returned vector and dipole strength
projections on the same coordinate axes yields a
θ
i
j
-th sensor the measurement
generated by the
th dipole.
The forward model (Equation 8.1) can be solved at substantial computational
expense using available numerical methods [22] in combination with realistic
i
structural information obtained from the MRI data (see
Section 8.1
). This high
computational cost is acceptable when the forward model has to be computed once
per subject and for a fixed number of dipole locations, but it can be prohibitive
for dipole fitting, which requires a recomputation of the forward model for each
step of nonlinear optimization. For this reason, rough approximations of the head
geometry and structure are often used, e.g., the best-fit single-sphere model, which
has a direct analytical solution [23], or the multiple-spheres model to accommodate
the difference in conductivity parameters across different tissues. Recently pro-
posed MEG forward modeling methods for realistic isotropic volume conductors
[24,25] seem to be more accurate and faster than BEM, and hence may be useful
substitutes for both crude analytical methods and computationally intensive finite-
element numeric approximations. Generally, the solution of the forward problem
is crucial for performing source localization using E/MEG, which is the main topic
of the following subsection.
8.2.3
T
I
P
HE
NVERSE
ROBLEM
8.2.3.1
Equivalent Current Dipole Models
The E/MEG inverse problem is very challenging (see Hämäläinen
et al. [
6
] and
Baillet et al. [
26
] for an overview of methods.) First, it relies on the solution of
the forward problem, which can be computationally expensive, especially in the
case of realistic head modeling. Second, the lead-field function
G
from
Equation 8.1 is nonlinear in
, so that the forward model depends nonlinearly on
the locations of activations. It is because of this nonlinearity that the inverse
r
i
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