Image Processing Reference
In-Depth Information
This method, defined as
cortex-based ICA
, was proposed to remove from the
analysis the signal dynamics due to the voxels belonging to white matter,
cerebrospinal fluid, ventricles, and other uninteresting structures. The method
may enhance the localization power of ICA decomposition because it can work
on reduced voxel numbers with the same number of independent components.
Moreover, it speeds up the computation operation.
16.5.4
M
ODEL
V
ALIDATION
In Reference 35 the validity of the ICA model in the spatial domain was investi-
gated. The validity of the hypotheses of linear mixing of the components, and the
effect of dimension reduction by means of PCA were studied. In each voxel
ν
i
,
the minus log-likelihood of the data given in the model was calculated, resulting
in a value
u
(
ν
i
). A spatial-smoothing operation was performed to achieve a map
of the goodness of fit of the model in the different brain regions. Once the unmixing
matrix
W
is found as well as the estimate of the original sources, and if we consider
the unmixing matrix invertible, it is possible to reconstruct the data from the
components by means of
XW
1
=
−
(16.50)
where
S
are the estimated independent components. The likelihood of observing
the data under the model specified by
W
-1
and
S
can be written as
P
(
X
i
=
det(
W
)
P
(
S
i
) where
X
i
is the
i
th column or the
i
th voxel's time series and
S
i
is the
i
th column of the
S
matrix. Using the statistical independence among the com-
ponents contained in the rows of
S
, it is possible to write
P
(
X
i
|
W
)
|
W
)
=
det(
W
)
P
(
S
i
)
n
det(
W
) .The quantity is the probability of the
i
th point in the
k
th component maps and is derived from an estimation of the probability density
functions of the
k
th component. In Reference 35 a smoothed version of the
histogram of each component was used. The minus log-likelihood function can
be written as The loga-
rithm is usually used because of the exponential form of several probability
density functions. In the work of McKeown it is verified that the ICA model fits
better for white matter than for gray matter in real data sets. This is supposed to
be related to the difference in the number of spatially independent components
for the different regions or to a nonlinear mixing of the sources in the gray matter.
Moreover, the validity of the model in simulated and real data sets against different
degrees of dimensionality reduction by means of PCA was investigated. The
simulated data sets were created using 50 eigenimages and mixed randomly.
Gaussian random noise was then added. The method showed different behavior
in real and simulated data sets: in the simulated data set the
u
(
=
k
1
P
()
P
k
()
k i
=
n
−
log(
P
(
XW
|
))
≈ −
log(det(
W
))
−
=
Σ
1
log(
P
(
S
)
)
=
u
(
ν
).
i
k i
i
k
ν
i
) function
decreases steeply if more components than the actual number are chosen, and in
the real data set this function decreases slowly showing that the smallest variance
components are unlikely to have a Gaussian structure.
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