Biomedical Engineering Reference
In-Depth Information
1
r
i
n
||
2
−
p
2
ʳ
(
k
+
1
)
i
s
(
k
)
i
)
[
ʳ
−
1
]
−
1
2
F
E
p
(ʳ
i
|
y
,
s
(
k
)
i
=
||
(6.25)
i
The associated M-step is then readily computed using
d
ʳ
ₒ
ʳ
i
L
T
1
ʳ
i
L
i
L
T
s
(
k
+
1
)
i
i
)
−
1
i
(Σ
+
y
.
(6.26)
i
=
s
(
k
+
1
)
=
s
(
k
+
1
)
is then the
th estimate of the source activity. Each iteration
of this procedure decreases the cost function of Eq. (
6.23
) and converges to some
fixed point is guaranteed. From a computational stand point, the
s
-
MAP
updates are
of the same per-iteration complexity as the
A
(
k
+
1
)
-
MAP
updates. Roughly speaking, the
EM iterations above can be viewed as coordinate descent over a particular auxiliary
cost function dependent on both
s
and
ʳ
ʳ
. There exists a formal duality between
s
-
MAP
and
-
MAP
procedures that is beyond the scope of this chapter, but details of
which are elaborate in [
10
,
14
].
Given
n
ʳ
e
i
, where each
e
i
is a standard indexing vector of zeros
with a “1” for the ith element, we get,
=
1, and
A
i
=
ʳ
(
k
+
1
)
i
k
i
L
i
(ʻ
k
)
]
L
T
)
−
1
y
]
(
2
−
p
)
ₒ[
ʳ
I
+
L
diag
[
ʳ
(6.27)
where
L
i
is the
i
th column of
L
. We then recover the exact FOCUSS updates when
p
1. Note however, that
while previous applications of MCE and FOCUSS to electromagnetic imaging using
n
ₒ
0 and a FOCUSS-like update for MCE when
p
=
1 require a separate iterative solution to be computed at each time point in
isolation, here the entire
s
can be computed at once with
n
=
>
1 for about the same
computational cost as a single FOCUSS run.
The nature of the EMupdates for
s
-
MAP
, where an estimate of
is obtained via the
E-step, suggest that this approach is indirectly performing some form of covariance
component estimation. But if this is actually the case, it remains unclear exactly
what cost function these covariance component estimates are minimizing. This is
unlike the case of
ʳ
ʳ
-
MAP
where it is more explicit. The fundamental difference
ʳ
between
s
-
MAP
and
-
MAP
lies in the regularization mechanism of the covariance
ʳ
components.Unlike
-
MAP
the penalty term in Eq. (
6.23
), is a separable summation
that depends on the value of
p
to affect hyperparameter pruning; and importantly there
is no volume-based penalty. For
p
and hence, every
local minimum of Eq. (
6.23
) is achieved at a solution with at most rank
<
1 the penalty is concave in
ʳ
d
y
non-zero hyper parameters. Rather than promoting sparsity at the level of individual
source elements at a given voxel and time (as occurswith standardMCE and FOCUSS
when
n
(
y
)
d
y
=
1
,
C
i
=
e
i
, here sparsity is encouraged at the level of the pseudo-sources,
operates on the Frobenius norm of each
S
i
and favors solutions
s
. The function
g
i
(.)
with many
0 for many indices of
i
. Notably though, by virtue of the
Frobenius norm over time, within a non-zero
||
s
||
F
=
s
i
temporally smooth (non-sparse)
