Biomedical Engineering Reference
In-Depth Information
A wide number of alternative choices for radial basis functions including
multiquadrics and Gaussians exists.
17, 12
Modeling deformations using thin-
plate splines has a number of advantages. For example, they can be used to
incorporate additional constraints such as rigid bodies
18
or directional
constraints
19
into the transformation model, and they can be extended to
approximating splines where the degree of approximation at the landmark
depends on the confidence of the landmark localization.
20
13.2.2.2 B-Splines
In general radial basis functions have infinite support. Therefore each basis
function contributes to the transformation and each control point has a global
influence on the transformation. In a number of cases the global influence of
control points is undesirable since it becomes difficult to model local defor-
mations. Furthermore, for a large number of control points the computational
complexity of radial basis function splines becomes prohibitive. An alterna-
tive is to use freeform deformations (FFDs) which have been widely used for
animations in computer graphics. FFDs based on locally controlled functions
such as B-splines are a powerful tool for modeling 3D deformable objects
21
and have been used successfully for image registration.
22-24
The basic idea of
FFDs is to deform an object by manipulating an underlying mesh of control
points. The resulting deformation controls the shape of the 3D object and pro-
duces a smooth and continuous transformation. In contrast to radial basis
function splines which allow arbitrary configurations of control points, spline-
based FFDs require a regular mesh of control points with uniform spacing.
A spline-based FFD is defined on the image domain
C
2
{(
x
,
y
,
z
)
0
x
X
, 0
y
Y
, 0
z
Z
} where
denotes an
n
x
n
y
n
z
mesh of control
. In this case the displacement field
u
defined by the FFD can be expressed as the 3D tensor product of the familiar
1D cubic B-splines:
25
points
i,j,k
with uniform spacing
3
3
3
u
x
,
y
,
z
(
)
l
()
m
v
u
()
n
w
()
i
+
l
,
j
+
m
,
k
+
n
(13.8)
l
0
m
0
n
0
y
y
x
y
z
x
z
x
z
where
i
1,
j
1,
k
1,
u
---
,
v
---
,
w
---
,
---
---
---
---
---
---
l
represents the
l
-th basis function of the B-splines:
25
and
3
0
s
()
(
1
s
)
6
3
s
3
6
s
2
1
s
()
(
4
)
6
s
3
3
s
2
2
s
()
(
3
3
s
1
)
6
s
3
3
s
()
6
As mentioned previously, FFDs are controlled locally, which makes them com-
putationally efficient even for a large number of control points. In particular,
