Biomedical Engineering Reference
In-Depth Information
formulation of a torus, which is closed along both
u
and
v
. Staib and Duncan
[25] make a torus that closes at two points and separate the segment between
the points by selecting proper weights in the formulation of the torus. An al-
ternative method [13] is to transform a torus to a sphere by giving the exterior
and interior circles that define the torus the same center and the same radius,
allowing parametrization of an object with spherical topology using parameter
coordinates of the torus.
The standard deviation of Gaussians in formulas (7.3) and (7.4) determines
the smoothness of a generated surface. A surface with a smaller standard
deviation represents local details better than a surface with a larger standard
deviation. The larger the standard deviation, the smoother the obtained surface.
When the control points are the voxels representing a closed 3D region, the
region can be represented by a parametric surface by mapping the voxels to a
sphere. RaG surfaces described by Eqs. (7.1), (7.2) and (7.4) represent surfaces
with a spherical topology. Assuming parameters
φ
∈
[
−
π/
2
,π/
2] and
θ
∈
[0
,
2
π
]
represent spherical coordinates of voxels defining an object, we will need to set
u
=
(
φ
+
2
)
/π
and
v
=
θ/
2
π
in the equations of a half-closed RaG surface. In
the following section, we will show how to spherically parametrize voxels in a
closed digital shape, and in the subsequent section, we will show how to find
the control points of a RaG surface in order to approximate a digital shape while
minimizing the sum of squared errors.
7.2.2
Parametrizing the Shape Voxels
Brechb uhler
et al.
[2, 3] describe a method for mapping simply connected shapes
to a sphere through an optimization process. Although this method can find pa-
rameters of voxels in various shapes, the process is very time consuming. We
use the coarse-to-fine method described in [17] to parametrize a digital shape. In
this method, first, a digital shape is approximated by an octahedron and at the
same time a sphere is approximated by an octahedron. Then, correspondence
is establishes between triangles in the shape approximation and triangles in
the sphere approximation. By knowing parameters of octahedral vertices in the
sphere approximation, parameters of octahedral vertices in the shape approxi-
mation are determined. This coarse approximation step is depicted in Fig. 7.2.
The process involves placing a regular octahedron inside the shape and extend-
ing its axes until they intersect the shape and replacing the octahedral vertices
