Biomedical Engineering Reference
In-Depth Information
13.2.4
Fluid Registration
Registration based on elastic transformations is limited by the fact that highly
localized deformations can not be modeled, since the deformation energy
caused by stress increases proportionally with the strength of the deformation.
In fluid registration these constraints are relaxed over time, which enables the
modeling of highly localized deformations including corners. This makes fluid
registration especially attractive for intersubject registration tasks (including
atlas matching) which have to accommodate large defor-mations and large
degrees of variability. At the same time the scope for misregistration increases,
as fluid transformations have a vast number of degrees of freedom.
Elastic deformations are often described in a
Lagrangian
reference frame,
i.e., with respect to their initial position. In contrast to that, fluid deforma-
tions are more conveniently described in a
Eulerian
reference frame, i.e., with
respect to their final position. In this Eulerian reference frame, the deforma-
tions of the fluid registration are characterized by the Navier-Stokes partial
differential equation,
2
v
x
,
y
,
z
(
)
(
)
(
v
x
,
y
,
z
(
)
)
f
x
,
y
,
z
(
)
0
(13.11)
similar to Equation (13.9) except that differentiation is carried out on the
velocity field
v
rather than on the displacement field
u
and is solved for each
time step. The relationship between the Eulerian velocity and displacement
field is given by:
u
x
,
y
,
z
,
t
(
)
----------------------------------
v
x
,
y
,
z
,
t
(
)
v
x
,
y
,
z
,
t
(
)
u
x
,
y
,
z
,
t
(
)
(13.12)
t
Christensen et al.
34
suggested to solve Equation (13.11) using successive over
relaxation (SOR).
31
However, the resulting algorithm is rather slow and requires
significant computing time. A faster implementation has been proposed by
Bro-Nielsen et al.
35
Here, Equation (13.11) is solved by deriving a convolution
filter from the eigenfunctions of the linear elasticity operator. Bro-Nielsen et al.
35
also pointed out that this is similar to a regularization by convolution with a
Gaussian as proposed in a nonrigid matching technique by Thirion
36
in which
the deformation process is modeled as a diffusion process. However, the solu-
tion of Equation (13.11) by convolution is only possible if the viscosity is
assumed constant, which is not always the case. For example, Lester
37
has pro-
posed a model in which the viscosity of the fluid is allowed to vary spatially,
and therefore allows for different degrees of deformability for different parts of
the image. In this case Equation (13.11) must be solved using conventional
numerical schemes such as SOR.
31
13.2.5
Registration Using FEM and Mechanical Models
As mentioned previously, the PDE for elastic deformations can be solved by
finite element methods (FEM) which also form the topic of Chapter 15. A sim-
plified version of an FEM model has been proposed by Edwards et al.
38
to
