Biomedical Engineering Reference
In-Depth Information
the basis functions of cubic B-splines have a limited support, i.e., changing
control point
i
,
j
,
k
affects the transformation only in the local neighborhood
of that control point.
13.2.3 Elastic Registration
Elastic registration techniques were proposed by Bajcsy et al.
26
for matching
a brain atlas with a CT image of a human subject. The idea is to model the
deformation of the source image into the target image as a physical process
which resembles the stretching of an elastic material such as rubber. This
physical process is governed by two forces. The first term is the
internal
force
which is caused by the deformation of elastic material (i.e., stress) and coun-
teracts any force which deforms the elastic body from its equilibrium shape.
The second term corresponds to the
external
force which acts on the elastic
body. As a consequence, the deformation of the elastic body stops if both forces
acting on it form an equilibrium solution. The behavior of the elastic body is
described by the Navier linear elastic partial differential equation (PDE):
2
u
x
,
y
,
z
(
)
(
)
(
u
x
,
y
,
z
(
)
)
f
x
,
y
,
z
(
)
0
(13.9)
Here
u
describes the displacement field,
f
is the external force acting on the
elastic body,
2
denotes the gradient operator, and
denotes the Laplace
are Lamé's elasticity constants which
describe the behavior of the elastic body. These constants are often inter-
preted in terms of Young's modulus
E
1
, which relates the strain and stress of
an object, and Poisson's ratio
E
2
, which is the ratio between lateral shrinking
and longitudinal stretching:
operator. The parameters
and
(
3
)
2
-------------------------------
E
1
E
2
-----------------------
(13.10)
(
)
2
(
)
The external force
f
is the force which acts on the elastic body and drives the
registration process. A common choice for the external force is the gradient of
a similarity measure such as a local correlation measure based on intensities,
26
intensity differences,
27
or intensity features such as edge and curvature.
28
An
alternative choice is the distance between the curves
29
and surfaces
30
of corre-
sponding anatomical structures.
The PDE in Equation (13.9) may be solved by finite differences and succes-
sive over relaxation (SOR).
31
This yields a discrete displacement field for each
voxel. Alternatively, the PDE can be solved for only a subset of voxels which
correspond to the nodes of a finite element model.
28,32
These nodes form a set
of points for which the external forces are known. The displacements at other
voxels are obtained by finite element interpolation. An extension of the elas-
tic registration framework has been proposed by Davatzikos
33
to allow for
spatially varying elasticity parameters. This enables certain anatomical struc-
tures to deform more freely than others.
