Biomedical Engineering Reference
In-Depth Information
N(
,Σ
)
are drawn independently from
. However, temporal correlations can easily
be incorporated if desired using a simple transformation outlined in [
1
]orusing
the spatio-temporal framework introduced in [
2
]. In this chapter, we will mostly
assume that
0
is known; however, robust procedures for its estimation can be found
should be noted that joint estimation of brain source activity and
Σ
Σ
potentially leads
to identifiability issues, and remains an unsolved problem to date.
To obtain reasonable spatial resolution, the number of candidate source locations
will necessarily be much larger than the number of sensors (
d
s
d
y
). The salient
inverse problem then becomes the ill-posed estimation of regions with significant
brain activity, which are reflected by voxels
i
such that
0; we refer to these
as
active
dipoles or sources. The severe underdetermine nature of this MEG (or
related EEG) source localization problem (since the mapping from source activity
configuration
s
|
s
i
|
>
s
1
,...,
s
d
s
]
T
to sensor measurement
y
is many to one), requires
the incorporation of prior assumptions when choosing an appropriate solution out of
an infinite set of candidates.
Bayesian approaches are useful in this capacity because they allow these assump-
tions to be explicitly quantified using postulated prior distributions. However, the
means by which these priors are chosen, as well as the estimation and inference pro-
cedures that are subsequently adopted to affect localization, have led to a daunting
array of algorithms with seemingly very different properties and assumptions.
While seemingly quite different in many respects, we present a generalized frame-
work that encompasses all of these methods and points to intimate connections
between algorithms. The underlying motivation here is to leverage analytical tools
and ideas from machine learning, Bayesian inference, and convex analysis that have
not as of yet been fully exploited in the context of MEG/EEG source localization.
Specifically, here we address how a simple Gaussian scale mixture prior with flexible
covariance components underlie and generalize all of the above. This process demon-
strates a number of surprising similarities or out-right equivalences between what
might otherwise appear to be very different methodologies. Theoretical properties
related to convergence, global and local minima, and localization bias are analyzed
and fast algorithms are derived that improve upon existing methods. This perspec-
tive leads to explicit connections between many established algorithms and suggests
natural extensions for handling unknown dipole orientations, extended source config-
urations, correlated sources, temporal smoothness, and computational expediency.
Specific imaging methods elucidated under this paradigm include weighted min-
imum
[
2-norm, FOCUSS [
3
], MCE [
4
], VESTAL [
5
], sLORETA [
6
], ReML [
7
]
and covariance component estimation, beamforming, variational Bayes, the Laplace
approximation, and automatic relevance determination (ARD) [
8
,
9
]. Perhaps sur-
prisingly, all of these methods can be formulated as particular cases of covariance
component estimation using different concave regularization terms and optimization
rules, making general theoretical analyses and algorithmic extensions/improvements
particularly relevant. By providing a unifying theoretical perspective and compre-
hensive analyses, neuroelectromagnetic imaging practitioners will be better able to
