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In-Depth Information
4.1 Affine Registration
Affine registration may correct global differences between moving and static im-
age. Several common spatial transformations applied to images can be expressed
in terms of an affine equation. Author used scaling, rotation and translations.
Those three transformations in
R
have together five degrees of freedom (two for
scaling, one for rotation and two for translation). The error function (
1 - CC(A,B)
)
was minimize with simplex method.
4.2 FFD Registration
The affine transformation captures only the global motion of the brain. The nature
of that local deformation of the brain can vary significantly across patients and
with their age. In order to model local deformation, the free form deformation
(FFD) model, based on B - splines can be used. The basic idea of FFD's is to
deform an object by manipulating an underlying mesh of control points.
3
3
==
FFD
(
x
,
y
)
=
β
(
u
)
β
(
v
)
φ
(4.2)
l
m
i
+
l
,
j
+
m
l
0
m
0
φ
The parameter
is the set of the deformation coefficients that is defined on a
sparse, regular grid of control points placed over the moving image. The functions
i
,
j
β
β
are the third-order spline polynomials [28]. The minimum of
error function (
1 - CC(A,B)
) was found with gradient descent method.
through
0
3
4.3 Thirion's Demons Algorithm
Thirion's algorithm can be treated as a numerical solution of Optical Flow model
[31]. In Optical Flow there is an assumption, that each pixel
x
o
f moving image
M
can be generated from static image
S
by displacement field
ν
(
x
)
. The displace-
ments are computed by constraining the brightness of images pixels to be con-
stants in time. Wang et al [34] made some modification to the original Demons
algorithm to make the registration more efficient (this modification is called Ac-
tive Demons registration algorithm). Displacement that maps a voxel position in S
to its position in M is found using an iterative method:
∇
S
∇
M
(4.3)
ν
(
x
)
=
G
⊗
ν
−
(
M
−
S
)
+
n
+
1
σ
n
2
2
∇
S
+
(
M
−
S
)
2
∇
M
+
(
S
−
M
)
2
⊗
Where
is the convolution operator.
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