Biomedical Engineering Reference
In-Depth Information
FIGURE 13.1
Example of different types of transformations of a square: (a) identity transformation,
(b) rigid transformation, (c) affine transformation, and (d) nonrigid transformation.
For example, the quadratic transformation model is defined by second
order polynomials
a
00
…
a
08
a
09
x
2
y
2
1
x
a
10
…
a
18
a
19
y
T
x
,
y
,
z
(
)
(13.2)
z
1
a
20
…
a
28
a
29
0
…
01
whose coefficients determine the 30 degrees of freedom of the transforma-
tion. In a similar fashion this model can be extended to higher order polyno-
mials such as third (60 DOF), fourth (105 DOF), and fifth-order polynomials
(168 DOF).
5
However, their ability to recover anatomical shape variability is
often quite limited since they can model only global shape changes and cannot
accommodate local shape changes. In addition, higher order polynomials
tend to introduce artifacts such as oscillations;
6
therefore, they are rarely used
for nonrigid registration.
13.2.1
Registration Using Basis Functions
Instead of using a polynomial as a linear combination of higher order terms,
one can use a linear combination of basis functions
to describe the defor-
i
mation field:
a
00
…
a
0
n
1
x
,
y
,
z
(
)
x
y
a
10
…
a
1
n
T
x
,
y
,
z
(
)
(13.3)
z
1
n
x
,
y
,
z
(
)
a
20
…
a
2
n
1
0
…
1
A common choice is to represent the deformation field using a set of (orthonor-
mal) basis functions such as Fourier (trigonometric) basis functions
7, 8
or wavelet
