Information Technology Reference
In-Depth Information
Maximizing the log posterior rather than the posterior, illustrates how these MAP
approaches are equivalent to algorithms that minimize
p
-
norm
-like measures
ϒ
)
−
1
B
LJ
i
=
1
J
i
:
d
n
d
b
d
v
2
2
F
+
p
q
+
2
2
min
J
(
2
σ
−
sgn
(
p
)
log
(
2
πσ
ϒ
)
.
(8.11)
2
ϒ
,
σ
2
parameter used in Tikhonov regu-
larization, which can be fixed to a value, stabilized (e.g., using the empirical L-curve
or generalized cross validation methods [56]), learned from the data, or adjusted to
achieve a desired representation error
2
ϒ
The noise variance
σ
is equivalent to the
λ
ε
using the discrepancy principle,
B
LJ
2
F
=
ϒ
2
F
−
=
ε
.
(8.12)
2) are equivalent to noise-regularized
minimum-
l
2
-norm solutions, often called minimum-norm estimates (MNE),
MAP estimates using a Gaussian prior (
p
=
L
T
LL
T
I
−
1
B
J
2
ϒ
=
+
σ
,
(8.13)
which are widely used in the field [2,98,21]. This basic model assumes homoscedas-
tic uncorrelated noise. Heteroscedastic uncorrelated noise can be modeled by replac-
ing
2
ϒ
I
with a diagonal matrix containing the estimated variance of each channel
on the diagonal. To suppress correlated noise, the matrix
σ
2
ϒ
σ
I
can be replaced with
a non-diagonal noise covariance matrix
Σ
ϒ
obtained from the measurements, which
is equivalent to performing whitening.
The point estimates obtained with the Gaussian prior are spatially distributed
and suffer from depth bias (i.e., deep source distributions tend to mislocalize to
more superficial source points). Many different types of weighted minimum-
l
2
-norm
algorithms can be used to partially compensate for this depth bias by assuming an a
priori source covariance other than the identity matrix, which is the assumed source
covariance in the standard minimum-
l
2
-norm approach. In its more general form,
the inverse operator using a Gaussian prior is given by
Ω
(
map
−
L
2
)
=
Σ
J
L
T
L
+
Σ
ϒ
−
1
Σ
J
L
T
,
(8.14)
where
Σ
J
is the source covariance matrix. Depth bias compensation is often im-
plemented by setting
WW
T
, where
W
is a diagonal matrix [e.g.,
W
Σ
J
=
=
diag
L
:
i
−
1
/
2
, 3D Gaussian function, or fMRI priors]
[38, 20]. More generally, and to include the case of unconstrained dipole orienta-
tions, the source covariance matrix can be defined as
L
:
i
−
2
,
W
diag
=
L
:
i
−
2
F
, where
L
:
i
is the gain matrix for the
i
th source point containing one column per dipole com-
ponent (indexed by the vector
i
), and this value is assigned to all variance diagonal
elements corresponding to the
i
th source point. A
diag
Σ
J
=
value between 0.5 and 0.8 is
usually adopted to avoid overcompensating with a full normalization (
κ
κ
=
1). Non-
diagonal
Σ
J
matrices can be used to incorporate source covariance and smoothness
