Image Processing Reference
In-Depth Information
W
Q
can also account for different spatial features:
•
normalizes the columns of the matrix to
account for deep sources by penalizing voxels too close to the sensors
[34,35].
WW
==
(diag(
GG
))
−
1
G
Q
n
•
, where the
i
th diagonal element incorporates the gray matter
content in the area of the
i
th dipole [36], i.e., the probability of having
a large population of neurons capable of creating the detected E/MEG
signal.
WW
Q
=
gm
•
, where rows of represent averaging coefficients
for each source [37]. So far only geometrical [38] or biophysical
averaging matrices [29] have been suggested.
W
=
(
W
W
)
−
1
W
a
Q
a
a
•
incorporates the first-order spatial derivative of the image [39] or
Laplacian form [40].
W
Q
Features defined by the diagonal matrices (e.g., and ) can be com-
bined through the simple matrix product. An alternative approach is to present
in terms of a linear basis set of the individual
W
n
W
gm
W
Q
W
Q
factors, i.e.,
W
Q
= µ
1
W
n
, with later optimization of via the EM algorithm [36].
To better condition the underdetermined linear inverse problem (Equation 8.4),
Philips et al.
[36] suggested to perform the inverse operation (Equation 8.4) can
be performed in the space of the largest eigenvectors of the . Such preprocessing
can also be done in the temporal domain, when a similar subspace selection is
performed using prior temporal covariance matrix, thus effectively selecting the
frequency power spectrum of the estimated sources.
Careful selection of the described features of data and source spaces helps
to improve the fidelity of the DECD solution. Nevertheless, the inherent ambiguity
of the inverse solution precludes achieving a high degree of localization precision.
It is for this reason that additional spatial information about the source space,
readily available from other functional modalities such as f MRI and PET, can
help to condition the DECD solution (Section 8.4).
+ µ
2
W
gm
+
µ
i
W
Q
8.2.3.3
Beamforming
Beamforming (sometimes called a spatial filter or a virtual sensor) is another way
to solve the inverse problem, which actually does not directly minimize Equation 8.2.
A beamformer attempts to find a linear combination of the input data ,
which represents the neuronal activity of each dipole in the best possible way
one
at a given time. As in DECD methods, the search space is sampled, but, in
contrast to the DECD approach, the beamformer does not try to fit all the observed
data at once.
The linearly constrained minimum variance (LCMV) beamformer [41] looks
for a spatial filter defined as
q
=
Fx
i
i
q
i
F
i
of size
ML
×
minimizing the output energy
CF
i
under the constraint that only
q
i
is active at that time, i.e., that there
F
i
X
is no attenuation of the signal of interest:
FG
k
ı
= δ
I
, where the Kronecker
ki
L
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