Image Processing Reference
In-Depth Information
step: this symmetric approach has the advantage of reducing the propagation of
estimation error from the first components to the last and requires a symmetric
orthogonalization of the matrix
W
=
(
w
1
,
w
2
,
…
,
w
n
)
T
.
16.5.2.5
Ambiguities in the ICA Model
The model (Equation 16.33) implies the existence of some ambiguities. In fact,
because both the mixing matrix and the sources are unknown, it is not possible
to determine the energies and the sign of the independent components. It is then
possible to overcome this ambiguity by constraining the independent components
to have unit variance, while the sign remains ambiguous. It can be shown that
introducing the constraint of unit variance of the estimated independent compo-
nents, along with the whitening operation, results in constraining
w
i
to lie in the
unit sphere, so that the algorithms are modified accordingly. After a new estima-
tion step of the directions of
w
i
, a normalization step has to be performed. Another
ambiguity concerns the order of the independent components, which cannot be
determined
a priori
: for this reason, we are not able to inter the significance of
a component just looking at its extraction order; that is ambiguous.
16.5.3
P
REPROCESSING
Because spatial-filtering as well as temporal-filtering operations do not affect the
validity of the ICA model [85], some filtering stages such as spatial smoothing
or temporal filtering can be applied. High-pass temporal filtering can be used to
remove low-frequency signal changes or drift. Spatial smoothing is usually per-
formed in order to enhance signal-to-noise ratio and reduce movement-related
effects. The filtering procedure may have a strong influence on the results because
some information in the data set may be lost: low-pass filtering in the time domain
may lead to loss of independence of the components, whereas high-pass filtering
may enhance independence because it allows the removal of low-frequency fluc-
tuations or drifts in the signal that may bias the independence of the components.
Low-pass filtering in the spatial domain is usually performed in order to enhance
signal-to-noise ratio and reduce motion-related effects. Centering of the variables,
i.e., the time points if we are applying spatial ICA or voxels in a temporal IC
model, is also carried out. The centering operation does not alter the mixing
matrix
A
, so that the mean can be added back to the independent components
s
by means of this operation: where
x
here refers to the observed
data before mean removal. A data reduction stage is usually included; in fact, the
basic ICA model assumes that the number of the sources that generate the data
equals the number of the observed variables, which means that applying spatial
ICA decomposition to a data set consisting of 100 time acquisitions of a volume
will result in the extraction of 100 components. This may not be true in general,
and the number of underlying sources is often supposed to be less than the number
of observed mixtures. In this case, the mixing matrix would be rectangular, and
so the basic ICA model would not hold. Moreover, trying to estimate more sources
ssAEx
←+
−1
{},
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