Adaptive Beamformers - Electromagnetic Brain Imaging - page 36

The beamformer that uses the above weight is called the diagonal-loading minimum-
variance beamformer [ 10 ]. The diagonal loading is a form of regularization used when
inverting the sample covariance matrix, and is nearly equivalent to adding noise to
the sensor data. Contrary to the L 2 -regularization in the minimum-norm method, the
diagonal loading is not needed when the SNR of the sensor data is low, because the
condition number of the sample covariance is generally low in such cases.
The diagonal loading is needed when the sensor data has high SNR, because,
in such cases, a significant SNR degradation is caused due to the problem of array
mismatch [ 6 ], which indicates a situation where the lead field used for computing the
beamformer's weight vector is different from the true lead field. When computing a
weight vector, the exact lead field is generally unknown and the lead field is usually
estimated from some kind of forward models such as the homogeneous conductor
model. Thus, we cannot completely avoid this array mismatch problem. The diagonal
loading can reduce the SNR degradation due to the array mismatch. On the other
hand, however, it degrades the spatial resolution of reconstructed source distribution,
so it provides a trade-off between the SNR degradation and spatial resolution [ 6 ].
The sample data covariance R , which is the maximum-likelihood estimate of the
model data covariance, is not necessarily the best estimate of

ʣ y . We can obtain
a better estimate by applying the Bayesian factor analysis described in Chap. 5 .
Applying the VBFA algorithm, we can get an estimate of the data covariance R ,as
shown in Eq. ( 5.104 ) . The beamformer with a weight expression computed using a
Bayesian-inferred data covariance matrix, (such as R in Eq. ( 5.104 ) ), is called the
Bayesian beamformer [ 11 ]. Note that a Bayesian-inferred data covariance matrix
has an intrinsic regularization term, and thus the regularization is embedded in the
Bayesian beamformer.

3.5 Scalar Adaptive Beamformer with Unknown
Source Orientation

3.5.1 Expressions for the Unit-Gain Constraint Beamformer

So far, we have derived the weight of the adaptive beamformer by assuming that the
source orientation is predetermined. The source orientation may be predetermined
using an accurate three-dimensional anatomical image of the subject, if available.
However, in general, the source orientation

is an unknown quantity, and should
be estimated from the data. There are two types of beamformers that can handle the
estimation of the source vector. One is the scalar-type and the other is the vector-
type adaptive beamformer. In the following, we first describe the scalar adaptive
beamformer [ 5 ].
In the scalar-type adaptive beamformer, the weight is first formulated using the
unknown source orientation

ʷ ( r )

ʷ , such that [ 5 ]