Biomedical Engineering Reference
In-Depth Information
similarity measures have the advantage that they do not require any feature
extraction process.
The second term is often referred to as the regularization or penalty func-
tion which can be used to constrain the transformation relating the source
and target images. In the case of rigid or affine registration this term is nor-
mally ignored and only plays a role in nonrigid registration. For example, in
elastic or fluid registration the regularization term (linear elasticity model)
forms an integral part of the registration. Other regularization models are the
Laplacian or membrane model:
7,41
2
2
2
T
T
T
-------
-------
-------
xdydd
x
y
z
and the biharmonic or thin-plate model:
16,42
2
T
2
2
T
2
2
T
2
---------
---------
---------
x
2
y
2
z
2
2
2
2
2
2
2
2
T
T
T
---------
---------
---------
xdydd
xy
xz
yz
Both models have an intuitive physical interpretation. While the Laplacian
model approximates the energy of a membrane (such as a rubber sheet)
which is subjected to elastic deformations, the biharmonic term approxi-
mates the energy of a thin plate of metal which is subjected to bending defor-
mations.
43
For example, Rueckert et al.
22
combine a similarity function based
on mutual information with thin-plate regularization.
13.3
Applications
There are a large number of applications for nonrigid registration. Areas of
considerable interest for nonrigid registration are the applications discussed
in more detail in Chapter 5, in which the geometry during image acquisition is
unknown or distorted, and include correction for scaling,
2
gantry tilt,
3
and
magnetic field inhomogeneity.
44
Other areas of nonrigid registration can be
classified into either the registration of deformable structures of the same indi-
vidual (intrasubject registration) or the registration across individuals (inter-
subject registration). Due to the different nature of these registration tasks, the
algorithms developed to solve them have quite different characteristics.
