Image Processing Reference
In-Depth Information
as the voxel-by-voxel ratio between the two images, the algorithm then estimates
the degree of misregistration between them by considering the mean m
and the
r
standard deviation
σ
of
r
(x) over all the voxels for which {
I
(
x
)
>
=
0.215
r
T/2
max(
)} (i.e., over all the intrabrain voxels)
The ratio
I
T/2
E
=
σ
/m
(3)
r
r
is then used to measure the degree of misregistration between the template and
the target image. When
I
(
T
[
x
]) is realigned to
I
(
x
), then
r
(x) is constant and,
K
T/2
consequently, the ratio
E
is small; conversely, when
I
(
T
[
x
]) is not realigned to
k
is large and new iterations are computed.
In the original version of the algorithm, Newton's method [15] was used
separately for each parameter of
I
(
x
), then the ratio
E
T/2
T
to minimize the ratio
E
, and trilinear interpo-
lation was used to calculate the new values of
on the grid defined at each
iteration. Other implementations of Wood's algorithm use more complex multi-
dimensional minimization schemes and different interpolation methods (e.g., sinc
interpolation).
The algorithm in [9], for example, differs from the Wood's algorithm in that
it utilizes the Euclidean norm in L
I
k
2
as the mismatch function:
∑
E
=
(([ )
I
T x
/
−
I
()
x
)
2
(15.4)
k
T
2
xbrain
∈
and the Levenberg-Marquardt algorithm [15] for the optimization of the rotation-
translation parameters.
Other realignment algorithms emphasize the importance of removing the
additional effects of the subject's movements on the magnetic spin excitation
history (e.g., by correcting with an autoregression moving average (ARMA)
model [11]), and other residual effects remaining after image realignment [14].
15.2.3
S
T
F
PATIAL
AND
EMPORAL
ILTERING
Spatial and temporal filtering of fMRI time series aims to reduce the effects of
the confounding factors that arise from instrumentation and spontaneous physi-
ological activity on the detection of brain activation.
The high-spatial-frequency noise, mainly from the scanner devices, can be
attenuated by spatially “smoothing” the fMRI time series with low-pass filters
(Gaussian, Hamming, and Fermi filters) [16].
Let us express the acquired data as
I
(k)
=
S
(
k
)
+
E
(
k
)
(15.5)
i
i
i
where
k
is the 3-D spatial-frequency span,
S
(
k
) the spatial-frequency domain
i
representation of a functional volume at scan
) the noise contribution
(physiological and electronic). The underlying assumption of spatial smoothing
i
, and
E
(
k
i
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