Biomedical Engineering Reference
In-Depth Information
is the
necessity of satisfying the
Schoenberg-Whitney
conditions [24,
?
] to ensure that
the matrix
Implicit in the assignment of parameter values to the points in set
D
B
T
B
is positive definite and well conditioned. Suppose that the knot
vector of the fitting spline is given by
U
=
{
U
1
,U
2
,...,U
m
}
and the para-
meterization assigned to the
s
sample points of
D
for least-squares fitting is
U
D
=
{
. Compliance with the Schoenberg-Whitney conditions
implies that there exists a set
u
1
,u
2
,...,u
s
}
U
D
=
{
u
1
,...,u
m
}⊂
U
D
such that
U
i
< u
i
<U
i
+
d
+1
i
=1
,...,m
−
(
d
+1)
.
(22)
This requirement in the presence of knots with multiplicity (
d
+1)is relaxed to
U
i
≤
u
i
<U
i
+
d
+1
U
i
=
...
=
U
i
+
d
<U
i
+
d
+1
,
(23)
U
i
< u
i
≤
U
i
+
d
+1
U
i
<U
i
+1
=
...
=
U
i
+
d
+1
.
(24)
This requirement is easily extended to the case for higher-order NURBS objects
[
?
].
3.2. Least-Squares Fitting with Polar NURBS
Closed periodic NURBS using the Cartesian formulation simply require repe-
tition of the first
d
control points, where
d
is the degree of spline. However, straight-
forward application to the
φ
coordinate of the cylindrical or prolate spheroidal-
based NURBS model would yield erroneous results because of the discontinuity
at
φ
=
−
π
. This requires subtraction or addition of 2
π
to the appropriate con-
trol point
φ
coordinates when the ordered set of control points crosses over this
discontinuity. Essentially, this is a type of
phase unwrapping
to ensure a smooth
transition across the
φ
-coordinate.
From a set of sample points, one of which is denoted by
≤
φ
i
<π
), with the corresponding parametric value
u
i
, one can construct a
d
-
degree closed NURBS curve using polar coordinates consisting of
n
distinct control
points. (The periodicity issue using both cylindrical and prolate spheroidal-based
closed NURBS is illustrated using polar coordinates. Polar coordinates are simply
the 2D analog of cylindrical coordinates and contain the
φ
coordinate inherent to
both 3D cooordinate systems.)
p
i
(
r
i
,φ
i
)(
−
π
Assume, without loss of generality, that
−
π
≤
φ
1
n
<π
, where
φ
denotes the
φ
coordinate of the control point
P
.
Due to the local support property of B-splines, each sample point influences, at
most, the location of (
d
+1)control points. If this ordered set of control points,
beginning with
< ... <
P
P
P
π
, a suiTable solution
is to add or subtract 2
π
from the
φ
coordinate of the appropriate control points.
This necessitates modifying the general equation for calculating
φ
i
(
u
i
) to
P
k
, crosses over the discontinuity at
φ
=
−
k
+
d
j
=
k
N
h,d
(
u
i
)
ψ
h
Q
φ
j
φ
i
(
u
i
)=
,
(25)
k
+
d
j
=
k
N
h,d
(
u
i
)
ψ
h
