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B 2<
M
N
is called the
demixing
or
separating matrix
. This is what we want
to estimate in order to recover the sources from EEG. Hereafter, the hat indicates a
statistical estimation. Although this is the classical BSS model, we need a few
clari
where
cations for the EEG case: By
gð
t
Þ
, we model
instrumental
noise only. In the
following, we drop the
gð
t
Þ
term because the instrumental (and quantization) noise
of modern EEG equipment is typically low (<1
V). On the other hand,
biological
noise (extra-cerebral artifacts such as eye movements and facial muscle contrac-
tions) and
environmental
noise (external electromagnetic interference) may obey a
mixing process as well; thus, they are generally modeled as components of
μ
,
along with cerebral ones. Notice that while biological and environmental noise can
be identi
sð
t
Þ
, hence removed, source estimation
will be affected by the underlying cerebral
background noise
propagating with the
same coef
ed as separated components of
sð
t
Þ
cients as the signal (Belouchrani and Amin
1998
).
8.4
A Suitable Class of Solutions to the Brain BSS Problem
To tackle problem (
8.2
) assuming knowledge of sensor measurement only, we need
to reduce the number of admissible solutions. In this paper, we are interested in
weak restrictions converging toward condition
sð
t
Þ
¼
Gsð
t
Þ
;
ð
8
:
3
Þ
sð
t
Þ
where
s
(
t
) holds the time course of the true (unknown) source processes and
our
estimation, and the
system matrix
G
¼
BA KP
ð
8
:
4
Þ
approximates a signed scaling (a diagonal matrix
) and permutation (
P
) of the
Λ
rows of
(
t
). Equation (
8.3
) is obtained by substituting (
8.1
)in(
8.2
) ignoring the
noise term in the former. Whether condition (
8.3
) may be satis
s
ed is a problem of
identi
ability
, which establishes the theoretical ground of BSS theory (Tong, Ino-
uye and Liu
1993
; Cardoso
1998
; Pham and Cardoso
2001
; Pham,
2002
). We will
come back on how identi
ability is sought in practice with the proposed BSS
approach. Matching condition (
8.3
) implies that we can recover faithfully the
source
waveform
, but only out of a
scale
(including sign) and
permutation
(order)
indeterminacy. This limitation is not constraining for EEG, since it is indeed the
waveform that bears meaningful physiological and clinical information. Notice the
correspondence between the
m
th source, its
separating vector
(
m
th
row
of B), and
its
scalp spatial pattern
(mixing vector), given by the
m
th
column
of
þ
.
Hereafter, superscript + indicates the Moore
-
Penrose pseudo-inverse. The mono-
dimensionality of those vectors and their sign/energy indeterminacy implies the
explicit modeling of the orientation and localization parameters of the
mth
source,
but not its moment. This is also the case of inverse solutions with good source
A
¼
B
