Image Processing Reference
In-Depth Information
TABLE 6.5: % Error computation between the slices obtained from MSR
using a digital sphere and the MSR using an approximated Euclidean sphere
for cross-section plane (-1,1,-1).
Distance Functions
Dumbel
Dumb-Cone
Sph-Cone
{1}
11.24
21.15
32.01
{112}
4.14
6.48
6.99
{113}
3.88
4.42
4.46
{12}
5.18
5.34
3.80
{122}
8.02
6.40
5.89
{123}
9.74
7.56
7.98
{13}
5.98
5.06
8.40
{133}
9.79
8.42
13.70
{2}
16.14
10.46
17.99
{223}
16.07
11.10
16.66
{23}
16.02
11.47
15.42
{233}
16.31
11.95
16.87
{3}
17.36
12.19
19.82
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.
a sphere using conventional techniques. Once again in [151], it was reported
that MATs obtained from distances such {113}, {123}, and {13} are better
in approximating such rendering results. Typical results on volume rendering
of objects with known geometry, such as CYLINDER, CONE, and SPHERE,
using diffrent representations are shown in Fig. 6.25. The representations un-
der consideration are voxel, octree, and MAT of the distance function {123}.
For rendering voxels, surfaces of each unit cube of the 3-D array are rendered
independently. Similarly, the leaf nodes of an octree represent a cubic box.
Hence, they are shaded following the same principle. The shading was carried
out assuming an illumination from parallel incident rays along the direction
(−1,−1,0). In all the cases, all other shading parameters, such as reflection
coe
cients of surfaces, shading interpolation techniques, etc., are kept the
same. For a better perception of the quality of shading, benchmark shaded
images are also rendered from an analytical boundary surface representation
of these objects. In this representations, the algebraic expressions for repre-
senting a cylinder, a cone, and a sphere are used. It could be easily observed
that the quality of shaded images using MAT is much better than the other
two representation schemes.
6.8 Summary
Medial Axis Transform (MAT) is a useful representation for 2-D and 3-D
objects. It exploits the advantage of shape representation in terms of overlap-
ping circles and spheres in 2-D and 3-D, respectively. In the discrete space,
