Bayesian Factor Analysis: A Versatile Framework for Denoising, Interference Suppression, and Source Localization - Electromagnetic Brain Imaging - page 101

K K

K
2

log | ʛ |

1
2

1
2

M
2

log | ʱ | y k ʛ z k ʓ ¯

F =

| ʓ | −

y k +

1 ¯

z k +

| ʨ | .

(5.131)

k = 1 k =

This free energy is used for monitoring the progress of the PFA-VBEM iteration.
Using the PFA algorithm, the estimate of the signal of interest, y k , is given by.

y k u k ]= ¯ A = E ( A , u ) [ Au k ]= E A [ A ] E u [ u k . ¯ (5.132)

The sample covariance matrix computed only using the signal of interest is given by

K ʛ − 1 tr R uu ʨ − 1

1 1

K ¯ AR uu ¯ A T

R

= + , (5.133)

where R uu is defined in Eq. ( 5.128 ). This R can be used in source imaging algorithms
such as the adaptive beamformers, and resultant images can be free from the influence
of interferences.

5.5 Saketini: Source Localization Algorithm
Based on the VBFA Model

5.5.1 Data Model

This section describes a virtual-sensor type source localization algorithm, called
Saketini [ 2 ]. The Saketini algorithm is based on Bayesian factor analysis, and enables
the estimation of source activity. In the Saketini algorithm, the sensor data y

( t ) is

modeled using

( ) = ( ) ( , ) + ( ) + ʵ ,

y t L r s r t Au t (5.134)

where s ( r , t ) is the source vector defined in Eq. ( 2.3 ) . An M × 3 matrix, L ( r ) ,isthe

lead field matrix at r , which is defined in Eq. ( 2.4 ) , and is the additive sensor noise.

In Eq. ( 5.134 ), the signal from the source activity at r is represented by L

ʵ

( r ) s ( r , t ) .

On the basis of the factor analysis model, the interference is modeled using Au ( t )

where A is an M × L factor mixing matrix, and u ( t ) is an L × 1 factor vector. Here,

Au represents all interference and source activities except the source activity at r .

As in the preceding sections, y

( t )

( t k ) , s ( r , t k ) , and u ( t k ) are denoted y k , s k , and u k ,

respectively. L ( r ) is denoted L for simplicity. Then, Eq. ( 5.134 ) is rewritten as

y k = Ls k + Au k + ʵ . (5.135)

The goal of this algorithm is to estimate s k from the data y k . Once s k is obtained, the
algorithm pointing location r is scanned over the whole brain region to obtain the spa-
tiotemporal reconstruction of the source activity over the whole brain. To formulate