Image Processing Reference
In-Depth Information
17.2.3
N
ORMALIZATION
After realigning the data, a mean image of the series or other coregistered (e.g.,
a T
-weighted) image, is used to estimate some warping parameters that map it
onto a template that already conforms to some standard anatomical space (8). This
estimation can use a variety of models for the mapping, including: (1) a 12-
parameter affine transformation, where the parameters constitute a spatial trans-
formation matrix, (2) low-frequency spatial basis functions (usually a discrete
cosine set or polynomials), where the parameters are the coefficients of the basis
functions employed and, (3) a vector field specifying the mapping for each control
point (e.g., voxel). In the latter case, the parameters are vast in number and
constitute a vector field that is bigger than the image itself. Estimation of the
parameters of all these models can be accommodated in a simple Bayesian frame-
work, in which one is trying to find the deformation parameters
1
θ
that have the
maximum posterior probability
).
Put simply, one wants to find the deformation that is most likely, given the data.
This deformation can be found by maximizing the probability of getting the data
(assuming the current estimate of the deformation is true) times the probability of
that estimate being true. In practice, the deformation is updated iteratively using
a Gauss-Newton scheme to maximize
p
(
θ
|
y
) given the data
y
, where
p
(
θ
|
y
)
p
(
y
)
=
p
(
y
|
θ
)
p
(
θ
p
(
θ
|
y
). This involves jointly minimizing
the likelihood and prior potentials
). The
likelihood potential is generally taken to be the sum of squared differences between
the template and deformed image and reflects the probability of actually getting
that image if the transformation was correct. The prior potential can be used to
incorporate prior information about the likelihood of a given warp. Priors can be
determined empirically or motivated by constraints on the mappings. Priors play
a more essential role as the number of parameters specifying the mapping increases
and are central to high-dimensional warping schemes (9).
In practice, most people use an affine or spatial basis function warps and
iterative least squares to minimize the posterior potential. A nice extension of this
approach is that the likelihood potential can be refined and taken as the difference
between the index image and the best (linear) combination of templates (e.g.,
depicting gray, white, CSF, and skull tissue partitions). This models intensity
H
(
y
|
θ
)
=
−
ln
p
(
y
|
θ
) and
H
(
θ
)
=
−
ln
p
(
θ
differences that are unrelated to registration differences and allows different
modalities to be coregistered (see
Figure 17.2
).
17.2.4
C
F
OREGISTRATION
OF
UNCTIONAL
A
D
AND
NATOMICAL
ATA
It is sometimes useful to coregister functional and anatomical images. However,
with echo-planar imaging, geometric distortions of T
* images, relative to ana-
2
tomical T
-weighted data, are a particularly serious problem because of the very
low frequency per point in the phase-encoding direction. Typically, for echo-
planar fMRI magnetic field inhomogeneity, sufficient to cause dephasing of 2
through the slice, corresponds to an in-plane distortion of a voxel. “Unwarping”
1
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