Biomedical Engineering Reference
In-Depth Information
solutions are favored. This is why, one obtains continuous source estimates over
time, as opposed to application of FOCUSS or MCE at each individual time-point.
Even though sparsity bounds for
s
-
MAP
and
-
MAP
are somewhat similar, the
actual performance in practical situations is quite distinct. This is because of global
minimum convergence properties of
s
-
MAP
that are impacted by the choice of
ʳ
g
i
(.)
or
p
. For instance, although setting
p
1 leads to a convex cost function devoid of
non-global minima, and the update rules for generalized-MCE will converge to the a
global minimum, the resultant estimate can have problems recovering the true source
estimate, especially for conditions related to MEG and EEG source reconstructions.
This is because lead-fields
L
required for MEG and EEG are highly correlated, and
violate the restricted isometric properties (RIP) that are required for accurate perfor-
mance of
=
1-norm procedures. More details on this point can be found in [
14
]. These
theoretical restrictions essentially render the conditions for MCE performance to
reconstruction of 1-2 dipoles at best, with no localization bias guarantees. The prob-
lem with MCE is not the existence of local minima, rather, it is that the global mini-
mum may be unrepresentative of the true source distribution even for simple dipolar
source configurations. In this situation, and especially when lead field columns are
highly correlated, the MCE solution may fail to find sufficiently sparse source rep-
resentations consistent with the assumption of a few equivalent current dipoles, and
this issue will also persist with more complex covariance components and source
configurations. Nevertheless, the unimodal, convex nature of the generalized MCE
like procedures are its attractive advantage.
Furthermore, if
p
z
2
<
g
i
(
)
is concave in z) then more
pseudo sources will be pruned at any global solution, which often implies that those
which remain may be more suitable than the MCE estimate. Certainly this is true
when estimating dipolar sources, but it likely holds in more general situations as
well. However, local minima can be an unfortunate menace with
p
1 (which implies that
<
1. For instance
when
p
0, we get the the canonical FOCUSS algorithm, and it has a combinatorial
number of local minima satisfying
ₒ
d
s
−
1
C
d
y
+
1
# of FOCUSS Local Minima
d
s
C
d
y
.
(6.28)
which is a huge number for practical lead-field matrices, and this property largely
explains the sensitivity of FOCUSS to initialization and noise. While the FOCUSS
cost function can be shown to have zero localization bias at the global solution,
because of the tendency to become stuck at local optima, in practice a bias can be
observed when recovering even a single dipolar source. Other selections of p between
zero and one can lead to a similar fate. In the general case, a natural trade-off exists
with
s
-
MAP
procedures: greater sparsity of solutions at the global minimum the
less possibility that this minimum is biased, but the higher the chance of suboptimal
convergence to a biased local minimum, and the optimal balance could be application
dependent. Nevertheless,
s
-
MAP
is capable of successfully handling large numbers
of diverse covariance components, and therefore simultaneous constraints on the
source space.
