Image Processing Reference
In-Depth Information
(41, 5)
0
7
1
0
7
1
0
7
1
4
0
7
1
(A)
2
0
0
5
10
15
20
25
30
35
40
(42, 5)
0
7
1
exception
0
7
1
0
8
1
4
0
7
1
(B)
2
0
0
5
10
15
20
25
30
35
40
(41, 4)
0
9
1
0
9
1
4
0
9
1
(C)
2
0
0
5
10
15
20
25
30
35
40
FIGURE 4.3: Periodicity in chain-code composition of DSS with integer
endpoints.
lying very close to this real line segment. In other words, we have to verify
whether there exists any real line segment—whose endpoints need not be
digital points—whose digitization produces S. In particular, we have to verify
whether S is a portion of some DSL, which is again the digitization of some
real line (of infinite length). Solutions for this problem and some related ones
may be seen in [50, 73, 74, 122, 142, 166, 195].
Several interesting works have revealed that DSL and DSS have a close
relationship with continued fractions [119, 143, 209]. Fig. 4.3 shows a few
examples of DSS whose two endpoints are digital points. For each of these,
we have a real line segment whose two endpoints coincide with the endpoints
of the concerned DSS. Thus, there exists a real line l with rational slope
corresponding to this DSS such that the digitization of l contains an infinite
concatenation of the DSS. In essence, given the real line l (with rational slope),
its DSL has a periodicity in its constitution, which can be obtained by an
analysis of continued fractions. For the DSS shown in Fig. 4.3(a), the slope of
the concerned real line / DSL is
5
41
, which can be expressed as
5
41
=
1
8 +
1
5
= [8,5].
