Image Processing Reference
In-Depth Information
mean image. Instead, we propose to represent longitudinal change by exploiting
the numerous transformations between subjects of similar ages, computed during
the construction of the atlas. The proposed method allows the derivation of a mean
growth model from these transformations, without additional intensity-driven reg-
istration of the spatio-temporal atlas time points.
We provide qualitative evidence that the constructed atlas is of higher anatom-
ical detail than other state-of-the-art neonatal brain atlases and that our growth
model allows the accurate modeling of brain growth during early development.
2
Methods
2.1
Parametric Diffeomorphic Registration
The proposed spatio-temporal atlas construction method is based on pairwise
image registration as schematically illustrated in Fig.
1a
. The brain MR images
are rigidly preregistered to a common image space. Registrations between all
pairs of images (
I
i
,I
j
) are then carried out in two stages. First, an ane regis-
tration is performed, followed by an inverse consistent registration which finds
the spline coecients of a stationary velocity field
v
ij
that minimizes the objec-
tive function. To avoid extrapolation of the initial global transformation outside
the finite control point lattice, we model the transformation
T
ij
between an
image pair as the sum of global and local velocity fields
T
ij
=exp(
v
ij
) = exp
v
global
+
v
local
ij
(1)
ij
where
v
global
ij
= log (
A
ij
). The logarithm of the 4x4 homogeneous coordinate
transformation matrix,
A
ij
, obtained by the initial ane registration, is com-
puted using a Pade approximation [
5
]. The velocity field
v
local
ij
represents the
local deformation to be optimized in the second stage and is given by
ʲ
x
−
x
c
ʴx
ʲ
y
−
y
c
ʴy
ʲ
z
−
z
c
ʴz
(
x
)=
m
(
c
)
ij
v
local
ij
ν
(2)
c
=1
The
m
control points are defined on a regular lattice with spacing (
ʴx, ʴy, ʴz
)
T
,
where
x
=(
x, y, z
)
T
and (
x
c
,y
c
,z
c
)
T
is the position of the
c
-th control point with
spline coecient vector
(
c
)
ij
) denotes the cubic B-spline function [
17
].
The use of a FFD model reduces the number of parameters of the stationary
velocity field to be optimized and allows the analytic derivation.
In order to remove bias due to the direction of registration, which can be sub-
stantial as shown for hippocampal volume measurements in [
22
], and to obtain
consistent pairwise transformations, we use a symmetric energy formulation.
Using normalized mutual information (NMI) as similarity measure, the energy
minimized by our registration with respect to the spline coecients of the local
velocity field component of
v
ij
ν
,and
ʲ
(
·
is given by
−ʻ
1
NMI
I
i
◦
+
ʻ
2
BE (
v
ij
)+
ʻ
3
JAC (
v
ij
)
T
−
0
.
5
ij
T
+0
.
5
ij
E
=
,I
j
◦
(3)
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