Biomedical Engineering Reference
In-Depth Information
this mapping will be unique. If fold overs occur, the subdivision process should
be continued until all fold overs disappear. When the maximum distance between
a triangle and the associating patch is less than two voxels, fold overs cannot
occur. More details about this parametrization algorithm and its characteristics
can be found in [17].
The parameters obtained by this algorithm uniquely map shape points to
triangular faces. The mapping is continuous but not smooth. To obtain a smooth
parametrization, the parameters obtained here should be used as initial values to
the nonlinear optimization described by Brechb uhler
et al.
[3]. The surface-fitting
method used in this work, however, does not require a smooth parametrization
of the points. It only requires that the parameters vary continuously.
If the vertices of a triangular mesh approximating a digital shape are used as
the control points of a RaG surface and the parameters at mesh vertices are used
as the nodes of the surface, a smooth parametric surface can be obtained that
approximates the shape. The surface obtained in this manner only approximates
the mesh vertices. We can improve this shape recovery process by making the
surface interpolate the mesh vertices. In the following section, a least-squares
method that determines the control points of a RaG surface interpolating the
mesh vertices is described.
7.2.3
Least-Squares Computation of the Control Points
Suppose a digital shape is available and the shape voxels are parametrized
according to the procedure outlined in the preceding section. Also, suppose
the shape is composed of
N
voxels:
{
P
j
:
j
=
1
,...,
N
}
with parameter co-
ordinates
{
(
u
j
,v
j
):
j
=
1
,...,
N
}
. We would like to determine a RaG surface
with control points
{
V
i
:
i
=
1
,...,
n
}
that can approximate the shape points
optimally in the least-squares sense. Let's suppose
P
j
=
(
X
j
,
Y
j
,
Z
j
),
P
(
u
,v
)
=
[
x
(
u
,v
)
,
y
(
u
,v
)
,
z
(
u
,v
)], and
V
i
=
(
x
i
,
y
i
,
z
i
). Then the sum of squared distances
between the voxels and the approximating surface can be written as
j
=
1
{
[
x
(
u
j
,v
j
)
−
X
j
]
2
N
E
2
+
[
y
(
u
j
,v
j
)
−
Y
j
]
2
+
[
z
(
u
j
,v
j
)
−
Z
j
]
2
(7.5)
=
}
N
N
N
[
x
(
u
j
,v
j
)
−
X
j
]
2
[
y
(
u
j
,v
j
)
−
Y
j
]
2
[
z
(
u
j
,v
j
)
−
Z
j
]
2
(7.6)
=
+
+
j
=
1
j
=
1
j
=
1
=
E
x
+
E
y
+
E
z
.
(7.7)
