Biomedical Engineering Reference
In-Depth Information
(
,
)
and the source vector
s
r
t
is estimated as
G
−
1
L
)
]
−
1
/
2
L
T
G
−
1
y
W
T
L
T
s
(
r
,
t
)
=
(
r
)
y
(
t
)
=[
(
r
)
(
r
(
r
)
(
t
).
(3.85)
The weight matrix for the
L
2
-regularized version is given by,
)
−
1
L
L
T
)
−
1
L
)
]
−
1
/
2
W
(
r
)
=
(
G
+
ʾ
I
(
r
)
[
(
r
)(
G
+
ʾ
I
(
r
.
(3.86)
Although the standardization makes the sLORETA filter to have no localization bias
[
6
], the theoretical basis of this standardization is not entirely clear.
3.9 Recursive Null-Steering (RENS) Beamformer
3.9.1 Beamformer Obtained Based on Beam-Response
Optimization
T
r
)
In the beamformer formulation, the product,
, is called the beam response,
which expresses the sensitivity of the beamformer pointing at
r
to a source located
at
r
. We wish to derive a beamformer that only passes the signal from a source at the
pointing location and suppresses the leakage from sources at other locations. Such a
beamformer may be derived by imposing a delta-function-like property on the beam
response. That is, the weight vector is obtained using
w
(
r
)
l
(
w
2
T
r
)
−
ʴ(
r
)
d
r
.
w
(
r
)
=
argmin
w
(
r
)
(
r
)
l
(
r
−
(3.87)
Ω
The weight obtained from the optimization above is expressed as
G
−
1
l
w
(
r
)
=
(
r
).
(3.88)
The beamformer in Eq. (
3.88
) is exactly equal to the minimum-norm filter.
Variants of the minimum-norm filter can be obtained by adding various constraints
to the optimization in Eq. (
3.87
)[
17
]. For example, the optimization
2
T
r
)
−
ʴ(
r
)
d
r
w
(
)
=
w
(
)
(
−
r
argmin
w
(
r
l
r
r
)
Ω
T
subject to
w
(
r
)
l
(
r
)
=
1
,
(3.89)
leads to the weight expression
l
T
G
−
1
l
)
]
−
1
G
−
1
l
w
(
r
)
=[
(
r
)
(
r
(
r
).
(3.90)
