Image Processing Reference
In-Depth Information
TABLE 6.3: Average percentage normal computational errors using MAT
with digital metric (for Nthresh=1).
Octagonal Distances
Average E n Octagonal Distances
Average E n
{1}
1.83
{12}
3.19
{1112}
1.32
{122}
5.15
{112}
1.67
{1222}
6.08
point P ∈ C, its contribution at that point toward the error measure (E n ) is
not considered, and subsequently the point p is excluded from the set C.
In Table 6.3, the average error measures for the geometric objects of Fig.
6.17 as reported in [153] are shown for normal computation using MAT with
different digital distances. As expected, computation with {112} and {1112}
distances yield low errors.
To demonstrate the effectiveness of the technique, in Fig. 6.19, examples
of normal computation for arbitrary 2-D objects are shown. In this case, the
distance function used is {112}.
6.6 Computation of Cross-Sections of 3-D Objects
One of the important tasks of rendering a 3-D object is to show its cross-
section on an intersecting plane. This reveals internal structures of 3-D ob-
jects along that plane. In a voxel-based representation, the computation is
performed by selecting the set of voxels lying on the plane. In this case, each
voxel's distance from the plane is computed, and if it lies within a threshold,
the point is considered to be part of the plane. However, due to the discretiza-
tion of the point set, there would be gaps in the computed set, which needs
to be smoothed out in a post-processing stage. Typical examples are shown
in Figs. 6.20 and 6.21 to demonstrate this effect. In Figs. 6.20 and 6.21, a set
of 3-D objects and their corresponding cross-sections along a plane with the
normal in a direction (1,−1,1) passing through the center point of the voxel
array are displayed, respectively. The size of the synthetically generated 3-D
objects of Fig. 6.20 is 128 × 128 × 128. The cross-section images computed
from the voxel representation are marred by the presence of empty pockets,
which should have been otherwise occupied by an object point.
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