Information Technology Reference
In-Depth Information
8.7.3 Linearly Constrained Minimum Variance (LCMV)
Beamforming
Beamformers, as used in the field of NSI, are spatial filtering algorithms that scan
each source point independently to pass source signals at a location of interest while
suppressing interference from other regions using only the local gain vectors and the
measured covariance matrix. One of the most basic and often used linear beamform-
ers is the linearly constrained minimum variance (LCMV) beamformer, which at-
tempts to minimize the beamformer output power subject to a unity gain constraint:
tr
Ω
i
:
Σ
B
Ω
i
:
subject to
T
min
Ω
i
:
Ω
i
:
L
:
i
=
I
,
(8.41)
where
Σ
B
is the empirical data covariance matrix,
L
:
i
is the
d
b
by three gain matrix
of the
i
th source point, and
Ω
i
:
is the three by
d
b
spatial filtering matrix [95]. The
solution to this problem is given by
=
L
:
i
Σ
B
−
1
L
:
i
−
1
L
:
i
Σ
B
−
1
Ω
(
lcmv
)
i
:
.
(8.42)
The parametric source activity at the
i
th source point is given by
S
(
lcmv
)
i
:
=
Ω
i
:
B
.
This can be performed at each source point of interest to yield a score map of ac-
tivity. Note that these maps, like those obtained by sLORETA and dSPM, are not
real current density estimates. This beamforming approach can be expanded to a
more general Bayesian graphical model that uses event timing information to model
evoked responses, while suppressing interference and noise sources [104]. This ap-
proach uses a variational Bayesian EM algorithm to compute the likelihood of a
dipole at each grid location.
8.7.4 Synthetic Aperture Magnetometry (SAM)
Synthetic aperture magnetometry (SAM) is a nonlinear beamformer in which an
optimization algorithm is used to the find the dipole orientation at each source point
that maximizes the ratio of the total source power over noise power, the pseudo-Z
deviate
p
i
n
i
,
T
i
:
Ω
i
:
Σ
B
Ω
z
i
=
i
:
=
(8.43)
Ω
i
:
Σ
ϒ
Ω
where
Σ
ϒ
is the noise covariance, usually based on some control recording or as-
sumed to be a multiple of the identity matrix [97]. This maximization generates a
scalar beamformer with optimal dipole orientations in terms of SNR. This improves
the spatial resolution of SAM relative to that of LCMV beamforming. To generate
statistical parametric maps between an active task period (
a
) and a control period
(
c
), the so-called pseudo-T statistic can be computed as
