Biomedical Engineering Reference
In-Depth Information
Fourier contour, described as:
X
a
0
c
0
a
k
b
k
c
k
d
k
cos 2
∞
(
s
)
π
ks
X
(
s
)
=
=
+
(4.12)
Y
(
s
)
sin 2
π
ks
k
=
1
with the Fourier coefficients
a
0
,
c
0
,
a
k
,
b
k
,
c
k
,
d
k
. They are defined as
1
1
1
2
1
π
a
0
=
X
(
s
)
ds
,
a
k
=
X
(
s
)
cos 2
π
ks ds
,
π
0
0
1
1
π
b
k
=
X
(
s
)
sin 2
π
ks ds
(4.13)
0
A smooth representation can be obtained by truncating the series. The shape
translation is defined by the coefficients
a
0
and
c
0
. The subsequent terms follow
the parametric form of an ellipse and can be mapped to the standard properties of
an ellipse [
84
]. The parameters follow scale ordering, with low indices for global
properties and high indices describing local deformations.
To incorporate prior knowledge, Staib and Duncan [
83
] use a Bayesian approach.
A prior probability approach is defined by manual delineation and the consequent
parameterization of Fourier coefficients using a converted ellipse parameter set. In
this way, a mean and variance can be calculated for each parameter.
4.6.2 Deformable Models Using Modal Analysis
Modal analysis has been introduced by Pentland and Horowitz [
85
]. Deformable
models using it are similar to deformable Fourier models but their basis functions
and nominal values are derived from templates. The objects consist of finite elements,
stacked in the vector
X
. Displacements are stored in
U
, so that the new state after
a deformation step is given by
X
+
U
. They are constrained using the following
differential equation
M
d
2
U
dt
2
C
dU
+
dt
+
KU
=
f
(4.14)
with
M
as mass,
C
as damping and
K
as stiffness matrix. The vector
f
contains the
external forces and
f
and
U
are defined as functions of time.
