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
).