Image Processing Reference
In-Depth Information
effectively make the time complexity of O(n). As polygon filling [84] is done
e ciently with the help of special hardware and architecture by graphics pro-
cessors, we may consider it as a unit task. Moreover, since the polygons of
the disks are convex, the computation could be further optimized. In our dis-
cussion, we consider the cost of polygon filling is a unit cost, ignoring its
coverage.
Since the vertices of the rotated convex polygons are discretized, and the
vertices themselves are computed using approximations as discussed in Section
2.5.5.2 of Chapter 2, it is of interest to observe how the reconstructed objects
differ from the true rotations of each and every pixel of our object. As the
latter acts as the reference, the rotated objects are smoothed by filling the
erroneously created pockets or gaps using simple checking of its 4-neighbors.
Experiments [129] were carried out to observe the quality of reconstruction by
rotating an object about the center of an image by a degree θ. Suppose the set
of object points after rotation is Σ pix (θ). The object is similarly rotated by
rotating its medial disks and reconstructed through polygon filling. Suppose
this set is denoted by Σ MAT (θ). Then the error of rotation E rot (θ) is expressed
as the percentage of pixels that differ with respect to the pixels of Σ pix (θ).
This is expressed in the following:
| Σ pix (θ) −Σ MAT (θ) | + | Σ MAT (θ)−Σ pix (θ) |
| Σ MAT (θ) |
E rot (θ) =
×100%. (6.5)
It has been reported in [129] that the average E rot (θ) of objects (in an image
of size 256×256) lies within 3%.
Similar experimentation was also carried out for transforming 3-D objects.
In this case, rotation around the z-axis was performed using both voxel and
MAT (in 3-D) representations. For reconstructing objects with the transforma-
tion over MAT representation, digital disks in the form of convex polyhedra
are filled by computing intersections with planes parallel to the XY -plane.
The image sizes were taken as 128 × 128 × 128. The reported error margins
(E rot (θ)) lie within 8%.
6.4.1 Approximate Transformation by Euclidean Disks
If the digital disks are replaced by Euclidean counterparts with the same
set of centers and radii, the computation becomes much simpler and faster.
In this case, the transformation is applied to the centers of disks only. In 2-D,
the polygon filling is replaced by computing the set of pixels inside a circle
and similarly, 3-D computation involves to get the point set in the interior
of a sphere. Naturally, the closer a digital disk is to a Euclidean one (of the
same radius), the better the approximation of transformation. In [129], the
same experiments were carried out by replacing digital disks with a Euclidean
one using MAT of different octagonal distance functions, in both 2-D and 3-D.
Average errors of the transformations for different metrics are shown in Tables
6.1 and 6.2 for 2-D and 3-D, respectively.
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