LI et al.: INCL UDING SIGNAL INTENSITY INCREASES THE PERFORMANCE OF BLIND SOURCE SEPARATION ON BRAIN
IMAGING DATA 553
1) Version A of the Intensity-Term (for SIM-A): As a con-
vention in our work, the intensity of a source in each individual
observed-signal
is termed its indi vidual intensity,andthe
intensity of a source in the whole dataset
is termed the total
intensity. The total intensity is defined as the sum of all indi-
vidual intensities in every observ ed-signal. Letting
denote th e individual intensity of the th source
in the th observed-signal and denotes the total inten-
sity of
,weobtain
(8)
Now the question is how to determine the value of
.
Consider (1), which represents the equation for the mixture
model of BSS. In the model, e v e ry source is weighted by the
coefficient
before it is combined. A larger indicates
exists more significantly in . T herefore, can be rep-
resented by the squared weight coefficient
(9)
Then, (8) can be written as
(10)
To write
as an explicit formula of the demixing v ector ,
we first represent
as a formula of per the following.
Given (1), the co-variance of
and can be represented
by
(11)
Because the sources are uncorrelated with each other, i.e.,
if , we obtain
from (11). Then, can be written as an explicit formula of
(12)
Inserting (12) into (10), we arrive at
(13)
Because (13) holds for every
, the subscript can be removed
(14)
The next mission is to write the total intensity provided in
(14) as an explicit formula of the demixing vector
. This could
be achieved by inserting the demixing m od el (3) in to (1 4) if
the original data are directly fed to the BSS process. However,
because the data of OI and fMRI have large dimensionality, the
original data are not directly analyzed . In our work, t he original
data are first dimensionally red uced using PCA. T he underlying
sources are demixed from the dimensionally reduced data, not
directly from the original data. Therefore, the demixing model
(3) should be rewritten by replacing the observed data
by the
dimensionally reduced data.
Let
be the dimensionally reduced
data, where
is one of the retained
principle components after PCA and
has been normalized,
i.e.,
. Replacing in (3) by , the new demixing
model can be represented as
(15)
It should be noted that the symbol
in (14) cannot be replaced
by
. This is because the total intensity is defined as signal in-
tensity in the observed data, bu t not the dimension ally reduced
data.
Inserting (15) into (14), the total i nten sity can be represented
as an explicit form ula of
(16)
According to the theor y of PCA, the whitene d pr in cipl e com-
ponent can be represented as
,where and
are the th eigenvector and eigenvalue of , respectively
[10]. We then have
diag
(17)
where
and .
Without losing generality, the diagonal elements of
are ar-
ranged in descending order, i.e.,
is the maximal eigenvalue.
Inserting (17) into (16), the representation of the total inten-
sity
can be reduced to
(18)
It is easy to f urther obtain that the value of
is in the interval
of
. If it i s then rescaled to the interval of [0, 1], the inten-
sity-term
from (6) can then be defined as the normalized total
intensity
(19)
Sequentially inserting (19), (7) and (15) into (6), we o btai n
the expression of the objective fu nction
(20)
where
.Underthe
constraint
, it can be determined that the that make
Obj approach its extrema are the eigenvectors of
.
2) Version B of the In tensity-Term (for SIM-B): In the brain
imaging data, a practical source varies smoothly over both time
and space [26]. This variation indicates that the source is smooth
and that its mixing coefficient varies smoothly across different