5 EEG Source Imaging and Multimodal Neuroimaging
95
The matrix multiplication of the forward and inverse operator, G M, is referred to as
the resolution matrix and ideally equals to the identity matrix I, representing a perfect
inverse solution. The inverse problem then amounts to the following generalized cost
function:
min
M
G ˆ
J + n
− V
2
p + λL
ˆ
J
(5.26)
where · · p denotes the L p -norm and L
ˆ
J
is some regularization scheme or
a priori constraint on the estimated sources with a scalar regularization parameter
λ. The regularization parameter λ represents the balance between maximizing the
goodness of fit and minimizing the constraint term L
ˆ
J
. λ is typically selected
using, among others [35], two common approaches: the generalized cross-validation
(GCV) method [32] and the L-curve method [40]. The optimal value for λ is obtained
in the GCV method by minimizing a function of λ:
GC V (λ)
G ˆ
J(λ) + n
− V
2
(tr(I − G M))
2
(5.27)
The numerator represents the residual error resulting from a solution ˆ
J regularized
by a particular λ value, while the denominator depicts the inaccuracy in the resolution
matrix. On the other hand, the L-curve is a log-plot of the norm of the residual error
term
G ˆ
J + n
− V against the norm of the regularized solution term L
ˆ
J
at
multiple values for the regularization parameter λ. Figure 5.4 demonstrates the Lcurve plot and the effect of different λ values on the inverse solution. Typically, a
regularization parameter λ is chosen to be near the “characteristic corner” of the
L-curve and, as such, generally yields a good balance between a small residual norm
and a small solution norm.
5.2.2.1 The Minimum-Norm Estimates
The minimum-norm estimates (MNE) [39] produces the inverse solution that minimize the overall power of the estimated source activity. Here, the L 2 -norm is applied
on the error term, and L
ˆ
J
ˆ
J
2
2 , yielding the solutions as:
M M N E G
T
GG
T + λI
−1
(5.28)
In such expression, the source and noise covariance matrices, R, and C, are
assumed to be an identity matrix I. A more generalized expression of MNE that
explicitly accounts for the covariance matrix is given as [17]:
M M N E RG
T
G RG
T + λC
−1
(5.29)
95
The matrix multiplication of the forward and inverse operator, G M, is referred to as
the resolution matrix and ideally equals to the identity matrix I, representing a perfect
inverse solution. The inverse problem then amounts to the following generalized cost
function:
min
M
G ˆ
J + n
− V
2
p + λL
ˆ
J
(5.26)
where · · p denotes the L p -norm and L
ˆ
J
is some regularization scheme or
a priori constraint on the estimated sources with a scalar regularization parameter
λ. The regularization parameter λ represents the balance between maximizing the
goodness of fit and minimizing the constraint term L
ˆ
J
. λ is typically selected
using, among others [35], two common approaches: the generalized cross-validation
(GCV) method [32] and the L-curve method [40]. The optimal value for λ is obtained
in the GCV method by minimizing a function of λ:
GC V (λ)
G ˆ
J(λ) + n
− V
2
(tr(I − G M))
2
(5.27)
The numerator represents the residual error resulting from a solution ˆ
J regularized
by a particular λ value, while the denominator depicts the inaccuracy in the resolution
matrix. On the other hand, the L-curve is a log-plot of the norm of the residual error
term
G ˆ
J + n
− V against the norm of the regularized solution term L
ˆ
J
at
multiple values for the regularization parameter λ. Figure 5.4 demonstrates the Lcurve plot and the effect of different λ values on the inverse solution. Typically, a
regularization parameter λ is chosen to be near the “characteristic corner” of the
L-curve and, as such, generally yields a good balance between a small residual norm
and a small solution norm.
5.2.2.1 The Minimum-Norm Estimates
The minimum-norm estimates (MNE) [39] produces the inverse solution that minimize the overall power of the estimated source activity. Here, the L 2 -norm is applied
on the error term, and L
ˆ
J
ˆ
J
2
2 , yielding the solutions as:
M M N E G
T
GG
T + λI
−1
(5.28)
In such expression, the source and noise covariance matrices, R, and C, are
assumed to be an identity matrix I. A more generalized expression of MNE that
explicitly accounts for the covariance matrix is given as [17]:
M M N E RG
T
G RG
T + λC
−1
(5.29)