Image Processing Reference
In-Depth Information
is that
S
i
(
k
) is a monotonically decreasing function of
k
and, thus, there exists
some frequency
k
c
such that
S
i
(
k
)
<<
E
i
(
k
)
for
k
>
k
c
(15.6)
Thus, a function
H
(
k
) such that
1 for
k
<
k
c
H
(
k
)
≈
0
for
k
>
k
c
(15.7)
would be an ideal filter. Indeed, multiplying
H
(
k
) by
I
i
((
k
) would result in noise
suppression with minimal effect on the signal
S
i
(
k
). However, the presence of
regionally specific activation implies that high-spatial-frequency components of
the signal are present in
S
i
(
k
) as well and, thus, besides reducing the noise
contribution, spatial smoothing will also decrease the effective spatial resolution
of the functional analysis.
These two contrasting effects influence the detection of activation regions
and have to be balanced. Intuitively, when the activated brain regions extend over
clusters of several voxels, spatial smoothing will strengthen the signal relative to
the noise (see
Figure 15.3
). Conversely, when focal regions of the brain are
activated, they might no longer be discernible after spatial smoothing. Further-
more, according to the matched filter theorem, the signal is best detected by
smoothing with a filter whose width matches that of the signal. In practical cases,
however, because both focal and broad activation regions may be present in the
same data set and their real extent cannot be known, the width and type of spatial
filter are chosen on the basis of a trade-off between the desired spatial resolution
and the expected enhancement of the functional contrast-to-noise ratio [14]. High
effective spatial resolution is especially important in individual studies and, thus,
little or no spatial smoothing is suggested. In multisubject studies, on the other
hand, a high degree of spatial smoothing is necessary even after normalization
to a standard stereotaxic space [17] in order to reduce the anatomical differences
between subjects and to allow the correct use of statistical tools [10].
Thermal noise and high-temporal-frequency fluctuations arising from spon-
taneous activity can be attenuated by temporal smoothing of each voxel's time
series [7]. In the case of temporal smoothing, the choice of the bandwidth of the
filter is driven by a trade-off between the expected enhancement of functional
contrast-to-noise ratio and the loss of temporal resolution (see
Figure 15.4
)
unavoidably caused by the filtering. Indeed, despite the fMRI response being
governed by slow hemodynamics, neural information on the order of a few
hundreds of milliseconds may be present in BOLD signals collected using event-
related protocols, and may be washed out by the temporal smoothing. Further, it
should be noted that temporal smoothing introduces a high degree of dependency
between subsequent samples of the time courses (temporal autocorrelation). This
temporal autocorrelation has to be appropriately taken into account when per-
forming the statistical analysis (see the following section), in order not to artifi-
cially inflate the significance levels of the tests.
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