Image Processing Reference
In-Depth Information
FIGURE 7.13: 3-D plot of error versus k and radius r of a digital circle.
Note that with higher values of k, the resultant 2k-gon covering the digital
circle has fewer errors. (See color insert.)
the gain for the run length finding approach using the number-theoretic tech-
nique continues to increase as the radius increases. For a very large radius,
exceeding 1000 or so, the number-theoretic technique contributes substantial
improvements to a circle generation procedure. The technique is based on the
distribution of perfect squares (square numbers) in integer intervals. Owing
to such a unique characterization, the problem of constructing a digital circle
or a circular arc maps to the domain of number theory. Given an integer ra-
dius and an integer center, the corresponding digital circle can be constructed
e
ciently by using number-theoretic properties only. Some of these relevant
number-theoretic properties discussed here.
While generating the digital circle C
1
(o,r) with center at o = (0,0) and
radius r as a positive integer, a decision is made to select between east pixel
p
E
(i + 1,j) or southeast pixel p
SE
(i + 1,j − 1), standing at the current pixel
p(i,j), depending on which one between p
E
and p
SE
is closer to the point
of intersection of the next ordinate line (i.e., x = i + 1) with the real circle
