Biomedical Engineering Reference
In-Depth Information
7. Previous focal source imaging techniques such as FOCUSS and MCE display
undesirable discontinuities across time as well significant biases in estimat-
ing dipole orientations. Consequently, various heuristics have been proposed
to address these deficiencies. However, the general spatiotemporal framework
of
s
-
MAP
,
-
MAP
, and
VB
handles both of these concerns in a robust, principled
fashion by the nature of their underlying cost function. The standard weighted
minimum norm can be seen as a limiting case of
ʳ
ʳ
MAP
.
8. As described in other chapters here, adaptive beamformers are spatial filters that
pass source signals in particular focused locations while suppressing interfer-
ence from elsewhere. The widely-used minimum variance adaptive beamformer
(MVAB) creates such filters using a sample covariance estimate; however, the
quality of this estimate deteriorates when the sources are correlated or the num-
ber of samples n is small. The simpler
-
MAP
strategy can also be used to
enhance beamforming in a way that is particularly robust to source correla-
tions and limited data [
15
]. Specifically, the estimated
ʳ
ʳ
-
MAP
data covariance
=
Σ
+
i
Σ
y
ʳ
i
L
C
i
L
T
matrix
can be used to replace the problematic sam-
yy
T
. This substitution has the natural ability to remove
the undesirable effects of correlations or limited data. When
n
becomes large
and assuming uncorrelated sources, this method reduces to the exact MVAB.
Additionally, the method can potentially enhance a variety of traditional signal
processing methods that rely on robust sample covariance estimates.
9. It can be shown that sLORETA is equivalent to performing a single iteration
of a particular
ple covariance
C
y
=
ʳ
-MAP optimization procedure. Consequently, the latter can be
viewed as an iterative refinement of sLORETA. This is exactly analogous to the
view of FOCUSS as an iterative refinement of a weighted minimum
2-norm
estimate.
10.
-MAP and VB have theoretically zero localization bias estimating perfectly
uncorrelated dipoles given the appropriate hyperprior and initial set of covariance
component.
11. The role of the hyperprior
p
ʳ
is heavily dependent on the estimation algorithm
being performed. In the
s
-
MAP
framework, the hyperprior functions through
its role in creating the concave regularization function
(ʳ)
. In practice, it is
much more transparent to formulate a model directly based on a desired
g
i
(.)
g
i
(.)
as
opposed to working with some supposedly plausible hyperprior
p
(ʳ)
and then
inferring the what the associated
-
MAP
and
VB
the opposite is true. Choosing a model based on the desirability of some
g
i
(.)
would be. In contrast, with
ʳ
g
i
(.)
can lead to a model with an underlying hyperprior
p
(ʳ)
that performs poorly.
Both
VB
and
ʳ
-
MAP
give rigorous bounds on the model evidence log
p
(
y
)
.
In summary, we hope that these ideas help to bring an insightful perspective to
Bayesian source imaging methods, reduce confusion about how different techniques
relate, expand the range of feasible applications of these methods. We have observed
a number of surprising similarities or out-right equivalences between what might
otherwise appear to be very different methodologies. Additionally, there are numer-
ous promising directions for future research, including time-frequency extensions,
