Adaptive Beamformers - Electromagnetic Brain Imaging - page 45

the nonadaptive spatial filters. 5 A representative and basic nonadaptive spatial filter
is the minimum-norm filter, which is the spatial filter version of the minimum-norm
method described in Chap. 2 .
The minimum-norm source reconstruction method is formulated as a nonadaptive
spatial filter in the following manner. In Eq. ( 2.26 ) , FF T is expressed such that [ 14 ]

N

FF T L T L T = L ( r n ) ( r n ) ≈ L ( r ) ( r ) d r = G , (3.73)

n = 1

Ω

indicates the source space. That is, FF T is equal to the gram matrix G ,if
we ignore the voxel discretization error. Thus, let us rewrite Eq. ( 2.26 ) as
⊡
⊣

where Ω

⊤
⊦ =

⊡
⊣

⊤
⊦

L T

(

r 1 ,

) ( r 1 )

s t

L T

(

r 2 ,

)

s t ( r 2 )

F T G − 1 y G − 1 y ( t ) = ( t ). (3.74)

. .

L T

s ( r N , t )

( r N )

We can see that, at each voxel location r n , the relationship,

L T G − 1 y s ( r n , t ) = ( r n ) ( t ), (3.75)

holds.
The equation above has the same form as the vector-type beamformer in Eq. ( 3.46 ),
i.e., Eq. ( 3.75 ) is rewritten as

W T

s ( r , t ) = ( r ) y ( t ), (3.76)

where the weight matrix W ( r ) is given by:

G − 1 L W ( r ) = ( r ). (3.77)

The two equations above indicate that the minimum-norm method is formulated as
a nonadaptive spatial filter, in which the weight matrix is given by Eq. ( 3.77 ). Also,
the L 2 -regularized version of the weight matrix is given by:

) − 1 L W ( r ) = ( G + ʾ I ( r ), (3.78)

where a scalar is the regularization constant. The spatial filter whose weight is

expressed in either Eq. ( 3.77 )or( 3.78 ) is called the minimum-norm filter.

ʾ

5

They might be also called as the nonadaptive beamformers, but this usage is uncommon.