Image Processing Reference
In-Depth Information
with 1-D (sinc) interpolation in the temporal domain or by Fourier transforming
the time series into a 1-D frequency representation, applying a phase shift to this
data, and then recovering the corrected data by applying a reverse 1-D Fourier
transform [7].
15.2.2
M
C
OTION
ORRECTION
The subject's motion poses a severe problem for the analysis of functional data.
Despite the use of physical constraints, head movements cannot be completely
eliminated during functional scanning. Head movements can be identified by
viewing successive volumes of the functional time series as a “movie.” Functional
time series with gross motion of the head (greater than the voxel size) can be
severely corrupted and, because they cannot be easily corrected with postprocess-
ing algorithms, they should be discarded from further analysis. Small head move-
ments (less than the voxel size) also produce effects that can mask the relatively
small BOLD signal changes and should be corrected using realignment algorithms.
In the following text, we describe the basic steps of these algorithms [8-15].
Let us consider
I
(
x
) and
I
(
x
) as two images (2-D or 3-D) collected at times
i
k
i
and
k
within a series of
T
repeated functional measurements. Let us suppose
that
I
(
x
) and
I
(
x
) are related by a geometric transformation
T
[
x
], so that
i
k
I
(
T
[
x
])
≈
I
(
x
).
(15.1)
k
i
Realignment algorithms deal with the problem of finding the transformation
T
that minimizes the differences between the two images due to the subject's
motion.
The most commonly adopted algorithms are based on iterative computation
of the rotation-translation parameters that reduce the mismatch between a reference
image (e.g., the T/2 scan of the time series) and the other images of the time series
[8-11]. These realignment procedures are based on the following steps:
•
Measurement of the spatial discrepancy between the transformed image
I
(
T
[
x
]) and the reference image
I
(
x
)
k
T/2
•
Evaluation of the parameters that define
T
•
Evaluation of the new values of
I
after
T
has been determined (inter-
k
polation method)
]
to be a rotation-translation transformation based on the rigid-motion hypothesis
[8]. With this hypothesis, the transformation
A robust method, commonly adopted in fMRI data analysis, considers
T
[
x
] is defined by three parameters
in the case of realignment of 2-D images (two translation offsets and one rotation
angle) and, by six parameters in the case of 3-D images (three translation offsets
and three rotation angles).
Defining
T
[
x
r
(
x
)
=
I
(
T
[
x
])/
I
(
x
)
(15.2)
K
T/2
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