Image Processing Reference
In-Depth Information
m l = 1/b + (1 −1/a−1/b)(1 + b/a) s ,s = ⌊(k−1)/2⌋,k ≥ 0.
Therefore, c l < c u and m l < m u if and only if min(a,b) = 1. Now it is easy
to see that the chord property for the particular kind of digital set reduces to
1/a+ 1/b > 1 or min(a,b) = 1. Hence, in this case, consistency and the chord
property are equivalent. It may be noted that in the absence of the chord
property, c u
→ +∞ violate consistency.
Next we illustrate the domain of the DSLS of Example 3.1.
→−∞ and m l
Example 3.3. n = 4,D o = {0,1,1,1,2}.m o = 0.4 and c o = 0.6. From
Example 3.1, m l = 1/4;m u = 1/2,c l = 1/2 and c u = 1. Diagrammatically,
we show the domain in Fig. 3.7.
FIGURE 3.7: The domain Dom(D o ) and the bounding rectangle R ul of D o
for m o = 0.4 and c o = 0.6 (Example 3.3). The domain has been computed
using Theorem 3.10. The original parameter value has been shown by an '*.'
Reprinted from Pattern Recognition Letters , 12(1991), S. Chattopadhyay et al., A new method of analysis for discrete
straight lines, 747-755, Copyright (1991), with permission from Elsevier.
It can be shown that for any valid DSLS D o , Domain(D o ) is a quadri-
lateral or a triangle formed in the (c,m)-plane. First, we prove the following
lemma.
Lemma 3.5. The points (c l ,m u ), (c,m
l (c)), and (c PR ,m PR ) are collinear
for all c,c l ≤c ≤ c PR .
Similarly, we can show that ∀c,c PR ≤c < c u , and the points (c PR ,m PR ),
(c,m
l (c)), and (c u ,m l ) are collinear.
Thus, the lower boundary of the region Domain(D o ) is a pair of straight
lines in the (c,m)-plane defined by the three points (c l ,m u ), (c PR ,m PR ), and
(c u ,m l ).
Search WWH ::




Custom Search