Biomedical Engineering Reference
In-Depth Information
i
denotes the location of the control point in the target image and
denotes the location of the corresponding control point in the source image.
There are a number of different ways to determine the control points. For
example, anatomical or geometrical landmarks which can be identified in
both images can be used to define a spline-based mapping function, which
maps the spatial position of landmarks in the source image into their corre-
sponding position in the target image.
10
In addition, Meyer et al.
11
suggested
updating the location of control points by optimization of a voxel similarity
measure such as mutual information. Alternatively, control points can be
arranged with equidistant spacing across the image, forming a regular
mesh.
12
In this case the control points are only used as a parameterization of
the transformation and do not correspond to anatomical or geometrical land-
marks. Therefore they are often referred to as
pseudo
- or
quasi
-landmarks.
where
i
13.2.2.1 Thin-Plate Splines
Thin-plate splines are part of a family of splines that are based on radial basis
functions. They have been formulated by Duchon
13
and Meinguet
14
for the
surface interpolation of scattered data. In recent years they have been widely
used for image registration.
10,15,16
Radial basis function splines can be defined
as a linear combination of
n
radial basis functions
(
s
).
n
)
b
j
(
(
x
,
y
,
z
)
tx
,
y
,
z
(
)
a
1
a
2
x
3
y
4
z
(13.5)
j
j
1
Defining the transformation as three separate thin plate spline functions
T
T
yields a mapping between images in which the coefficients
a
characterize the affine part of the spline-based transformation, while the
coefficients
b
characterize the nonaffine part of the transformation. The inter-
polation conditions in Equation (13.4) form a set of 3
n
linear equations. To
determine the 3(
n
(
t
1
,
t
2
,
t
3
)
4) coefficients uniquely, 12 additional equations are
required. These 12 equations guarantee that the nonaffine coefficients
b
sum
to zero and that their crossproducts with the
x
,
y
and
z
coordinates of the con-
trol points are likewise zero. In matrix form this can be expressed as
0
b
a
(13.6)
T
0
Here
a
is a 4
3 vector of the affine coefficients
a
,
b
is a
n
3 vector of the non-
affine coefficients
b
, and
). Solving
for
a
and
b
using standard algebra yields a thin-plate spline transformation
which will interpolate the displacements at the control points.
The radial basis function of thin-plate splines is defined as
is the kernel matrix with
ij
(
i
j
s
2
in 2D
in 3D
log
s
()
s
()
s
(13.7)
