Biomedical Engineering Reference
In-Depth Information
3
3
3
2.5
2.5
2.5
2
2
2
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
0
0
-
2
-
1.5
-
1
-
0.5
0
0.5
1
1.5
2
-
2
-
1
0
1
2
-
2
-
1
0
1
2
(a)
(b)
(c)
Figure 4.
Cartesian versus non-Cartesian nonrational B-spline curves: order = 4; control
point coordinates
(
x, y
)
—
P
1
=(
−
1
.
5
,
0)
,
P
2
=(
−
0
.
5
,
0)
,
P
3
=(
−
0
.
5
,
3)
,
P
4
=
(0
.
5
,
3)
,
P
5
=(0
.
5
,
0)
,
P
6
=(1
.
5
,
0)
. Illustrated are (a) the Cartesian B-spline curve,
(b) the polar-based B-spline curve, and (c) their superposition constructed from the control
points.
2.1.2. Higher-dimensional NURBS objects
Equation (1) can be extended to create higher-order NURBS objects. A vol-
umetric NURBS object,
S
(
u, v, w
), is defined by the equation
i
=1
j
=1
k
=1
N
i,p
(
u
)
N
j,q
(
v
)
N
k,r
(
w
)
ψ
i,j,k
P
i,j,k
i
=1
j
=1
k
=1
S
(
u, v, w
)=
,
(13)
N
i,p
(
u
)
N
j,q
(
v
)
N
k,r
(
w
)
ψ
i,j,k
which consists of a triple summation involving three sets of basis functions span-
ning the volume of the NURBS model for the
u
,
v
, and
w
location parameters,
l
n
control points, and the corresponding weights,
ψ
i,j,k
. The respective
degree in each of the three parametric directions is
p
,
q
, and
r
. Additionally, we
can add a fourth B-spline basis function and parameter to compose a temporally
varying NURBS volumetric object. The 4D model created in our methodology is
constructed by
skinning
or
lofting
a temporal series of volumetric NURBS models
representing each frame of data [20].
×
m
×
3.
MATHEMATICAL PRELIMINARIES
One of the distinct attributes of the work presented in this chapter is the use
of NURBS to model left- and right-ventricular deformation. The motivation for
employing NURBS over uniform nonrational B-splines is that NURBS allow for
variation of the knot vector and weighting of the control points in addition to the
degree of the spline and the location and number of control points.
The degrees of freedom available in constructing the approximating model
should be optimized to produce the most accurate results. To construct the initial
