Image Processing Reference
In-Depth Information
problem is generally treated by nonlinear optimization methods, which can lead
to solutions being trapped in local minima. In the case of Gaussian sensor noise,
the best estimator for the reconstruction quality of the signal is the squared error
between the obtained and modeled E/MEG data:
K
t
M
2
∑∑∑
=
1
E
(
rq
,
)
=
(
x
( )
t
−
ˆ (
rq
,
,
t
))
2
+
λ
f
(
rq
,
)
(8.2)
x
j
j
i
i
i
tt
j
where
0 is often introduced to regularize the solution, i.e., to obtain the
desired features of the estimated signal (e.g., smoothness in time or in space, and
lowest energy or dispersion), and
f
(
r
,
q
)
>
0 is used to vary the trade-off between the
goodness of fit and the regularization term.
This least-squares model can be applied to the individual time points (
λ
>
t
=
t
)
1
2
(“moving dipole” model) or to a block (
t
<
t
) of data points. If the sources are
1
2
assumed not to change during the block (
t
,
t
), then the solution with time
1
2
constant
q
i
(
t
)
q
i
is the target.
Other features derived from the data besides pure E/MEG signals as the
argument of Equation 8.1 and Equation 8.2 are often used, e.g., ERP/ERF wave-
forms that represent averaged E/MEG signals across multiple trials, mean map in
the case of stable potential or field topography during some period of time, or
signal frequency components to localize the sources of the oscillations of interest.
Depending on the treatment of Equation 8.2, the inverse problem can be pre-
sented in a couple of different ways. The brute-force minimization of Equation 8.2
in respect to both parameters and and the consideration of different
K
neuronal
sources is generally called ECD fitting. Because of nonlinear optimization, this
approach works only for cases in which there is a relatively small number of sources
K
, and therefore the inverse problem formulation is overdetermined, i.e.,
Equation 8.1 cannot be solved exactly ( ). If fixed time locations of the
target dipoles can be assumed, the search space of nonlinear optimization is reduced,
and the optimization can be split into two steps: (a) nonlinear optimization to find
locations of the dipoles and then (b) analysis to determine the strength of the dipoles.
This assumption constitutes the so-called spatiotemporal ECD model.
Two other frameworks have been suggested as means of avoiding the pitfalls
associated with nonlinear optimization: distributed ECD (DECD) and beamform-
ing. We discuss these two approaches in detail in the following subsections.
=
x
r
q
E
(
rq
,
)
>
0
8.2.3.2
Linear Inverse Methods: Distributed ECD
In the case of multiple simultaneously active sources, an alternative to solving
the inverse problem by ECD fitting is a distributed source model. We will use
the label DECD to refer to this type of model. The DECD is based on a spatial
sampling of the brain volume and distribution of the dipoles across all plausible
and spatially small areas that could be a source of neuronal activation. In such
cases, fixed locations (
r
i
) are available for each source or dipole, removing the
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