Biomedical Engineering Reference
In-Depth Information
2
U
=
0 at different scales. Solut
ions are
expressed as
z
(
x
,
y
)
=−
U
(
r
)
=−
r
2
log(
r
2
) i
n dimension
2 (with
r
=
solutions of the biharmonic equation
x
2
+
y
2
)
and
z
(
x
,
y
)
=|
r
|
in dimension 3 (with
r
=
x
2
+
z
2
). This spline transfor-
mation ensures the matching of landmarks as well as a smooth interpolation
of the deformation. Chui and Rangarajan have proposed the TPS-RPM algo-
rithm [30] where they address both the correspondence and the transformation
problem. They propose the softassign algorithm to solve the correspondence
problem and the TPS for the transformation.
Another approach is the use of free-form deformations [123]. Initially intro-
duced to model and deform objects [6, 148], they have also been used to model
deformations [43, 64, 70, 96, 107, 117, 129, 134]. Splines models are quite powerful
to extrapolate deformations indeed.
+
y
2
8.2.3
Photometric Methods
The number of features that can be extracted reproductively among a population
of subjects is rather low. Therefore, photometric (also called “intensity-based” or
iconic) methods have been developed to take into account the entire information
of the volume. Photometric methods rely on a similarity (or dissimilarity) that
measures the dependency between two volumes. We have chosen to present the
registration methods according to the following classification: methods that de-
rive from the laws of continuum mechanics; methods that use cross-correlation;
the demon's method; methods based on optical flow, and finally methods that
estimate jointly an intensity correction and a geometrical transformation.
8.2.3.1
Models Based on Continuum Mechanics
Considering two MR images of two different subjects, the estimation of a “plau-
sible” transformation must be sought. The notion of a “plausible” transforma-
tion in this context being particularly difficult to state, some authors have pro-
posed to comply with the laws of continuum mechanics, either elastic or fluid
(Section 8.2.3).
8.2.3.2
Elastic Models
Elastic models have been introduced by Broit [18] and extended by Bajcsy and
Kovacic [4, 5]. These models are nowadays used by various authors [39, 38, 54,
