Biomedical Engineering Reference
In-Depth Information
y
and
z
coordinates. Minimizing
E
w
with respect to each homogeneous control
point coordinate yields the equivalent matrix formulation:
B
T
WXBΨ
=(
B
T
WB
)
P
ψ
x
,
(15)
B
T
WYBΨ
=(
B
T
WB
)
P
ψ
y
,
(16)
B
T
WZBΨ
=(
B
T
WB
)
P
z
,
(17)
where the diagonal matrices
X
,
Y
,
Z
, and
W
are formulated from the sets
D
and
W
.
Combining the previous equations and performing certain matrix manipula-
tions, the resulting system is derived:
B
·
P
=
0
,
(18)
where
B
is the 4
n
×
4
n
block matrix
B
T
W
2
B0
−
B
T
W
2
XB
0
T
W
2
B0
−
B
T
W
2
YB
0
,
(19)
B
T
W
2
B
−
B
T
W
2
ZB
0
0
0
0
0
M
M
is the
n
×
n
matrix given by
M
=
B
T
W
2
X
2
B
+
B
T
W
2
Y
2
B
+
B
T
W
2
Z
2
B
−
(
B
T
W
2
XB
)(
B
T
W
2
B
)
−
1
(
B
T
W
2
XB
)
−
(
B
T
W
2
YB
)(
B
T
W
2
B
)
−
1
(
B
T
W
2
YB
)
−
(
B
T
W
2
ZB
)(
B
T
W
2
B
)
−
1
(
B
T
W
2
ZB
)
,
(20)
and
P
Ψ
]
T
. A similar derivation follows from
employing a non-Cartesian NURBS description.
In this form, calculation of the weights can be performed separately from
calculation of the control points by first solving the homogeneous system
x
,
y
,
z
,
is the 4
n
×
1 vector [
P
P
P
M
·
Ψ
=
0
. In general, the nontrivial solution can be solved using singular-value
decomposition. However, this could lead to negative weights causing singularities
in the B-spline model. Therefore, we look for an optimal solution consisting of
strictly positive weights. This leads to the quadratic programming [23] formulation
mi
Ψ
M
·
Ψ
2
subject
to
{
ψ
i
≥
ψ
◦
:
∀
ψ
i
∈
Ψ
}
,
(21)
where
ψ
◦
is a positive constant. Since only the relative weighting of control points
is important, and to be consistent with the formulation in the original publication
[22], we choose
ψ
◦
=1.
