Adaptive Beamformers - Electromagnetic Brain Imaging - page 46

3.8.2 Weight-Normalized Minimum-Norm Filter

The weight-normalized minimum-norm filter has been proposed by Dale et al. and
the method is often called as dynamic statistical parametric mapping (dSPM) [ 15 ].
The idea is to normalize the minimum-norm weight with its weight's norm to ensure
that the spatial distribution of the noise is uniform. The scalar-type weight is thus
given by

G − 1 l G − 1 l

( r ) ( r )

w ( r ) =

) =

l T . (3.79)

G − 1 l

( r

G − 2 l ( r ) ( r )

The idea of weight normalization can be extended to derive the weight matrix for
the vector-type spatial filter, such that

G − 1 L ( r )

W ( r ) = tr L T

) .

(3.80)

G − 2 L ( r ) ( r

Using this weight matrix, the source vector is estimated as

G − 1 y L T ( r ) ( t )

W T s ( r , t ) = ( r ) y ( t ) = tr L T

) .

(3.81)

G − 2 L

( r ) ( r

The weight matrix for the L 2 -regularized version is given by

) − 1 L ( G + ʾ I ( r )

W ( r ) = tr L T

) .

(3.82)

( )( + ʾ ) − 2 L (

r G I r

3.8.3 sLORETA Filter

Standardized low resolution electromagnetic tomography (sLORETA) was originally
proposed by Pascual-Marqui [ 16 ]. The method can be reformulated as a nonadaptive
spatial filte r. In this metho d, the minimum-norm filter outputs are normalized by the

quantity l T G − 1 l . This normalization is called as the standardization. The

scalar-type weight is given by

( r ) ( r )

lG − 1

w ( r ) = l T . (3.83)

G − 1 l ( r ) ( r )

The extension to the vector-type sLORETA filter results in the weight matrix
expressed as

G − 1 L L T G − 1 L ) ] − 1 / 2 W ( r ) = ( r ) [ ( r ) ( r , (3.84)