Image Processing Reference
In-Depth Information
(a)
(b)
(c)
FIGURE 6.24: Cross-sections obtained by intersecting a plane passing
through the center and with a normal along the direction (−1,1,−1) with
the objects using Euclidean approximation of medial spheres of MAT repre-
sentation from {113}: (a) DUMBCONE, (b) DUMBLE, and (c) SPHCONE.
Reprinted from
Pattern Recognition Letters
, 21(2000), J. Mukherjee et al., Fast Computation of Cross-Sections of
3-D Objects from Their Medial Axis Transforms, 605-613, Copyright (2000), with permission from Elsevier.”
ular, for the distance {113} these values are reported to be within 5% in most
cases. In Fig. 6.24, typical results obtained from Euclidean approximation are
shown using the MAT obtained from the distance function {113}. The results
are quite similar to those shown in Fig. 6.23.
6.7 Shading of 3-D Objects
The MAT is also useful for rendering 3-D objects, and in [151] it is reported
to be the better representation for volume rendering compared to the schemes
using an octree [186] or voxels. From the MAT of an object, each medial sphere
is rendered independently. The process may be aided by the z-buffering fea-
tures [84] of the graphics hardware environment, so that the hidden surface
elimination is managed. For rendering a medial sphere, the shaded color values
of each face of the polyhedron are computed following different shading in-
terpolation techniques, such as flat rendering, and Goraud's [93] and Phong's
[164] interpolation techniques. For applying the shading interpolation, it is
required to compute the normals at the faces and vertices of the polyhedron.
Computation of normals at faces is trivial, as it is the vector formed by the
centers of the face and the sphere, respectively. Similarly, the direction of the
normal at a vertex is simply given by the vector from the center of the sphere
and the vertex itself. The technique may be further refined by approximating
each digital sphere as Euclidean and the shading is performed as it is done for
