Image Processing Reference
In-Depth Information
Note that, in continued fraction, [q
1
,q
2
,...,q
n
] implies
1
.
1
q
1
+
1
.
.
.
q
2
+
1
1
q
n
q
n−1
+
As explained in the coming section, we get the corresponding chain-code
(Fig. 4.2; see also Chapter 1) representation from this as 0
8
(0
7
1)
4
(here, k
consecutive 0s are written as 0
k
for brevity), which defines the period of the
DSL. It indicates that there are four consecutive runs of identical composition,
i.e., 0
7
1, following (and preceding) a single run of 0
8
. A run is given by the
maximum sequence of contiguous digital points lying on the same horizontal
or vertical line. The first and the last runs of the DSS should be ignored, since
they are partial runs with respect to the concerned DSL. In fact, if we produce
the real line segment in both directions, then the partial runs would grow into
complete runs; and for the DSS of Fig. 4.3(a), both the left and the right runs
would become 0
8
.
In Fig. 4.3(b), the DSL has slope
5
42
=
1
8 +
2
5
1
=
= [8,2,2],
1
2 +
1
2
8 +
wherefore its period becomes 0
8
10
7
10
8
1(0
7
1)
2
, a portion of this being con-
tained in the DSS. Notice that, for this DSS, we get 0
8
1 as an exceptional run
among the runs of 0
7
1, which is the predominant run. The exceptional run
for the DSL in Fig. 4.3(a) was also 0
8
1, which was not present in the DSS as
0
8
1 was less frequent. The occurrence of such exceptional runs have been ad-
dressed with utmost importance to conceptualize digital straightness from the
perspective of word theory and number theory. The predominant run and the
exceptional run may change if there is a slight change of slope, as exemplified
in Fig. 4.3(c). Here the DSL has slope
4
41
=
1
10 +
1
4
= [10,4],
which yields the period 0
10
1(0
9
1)
3
, thus giving 0
10
1 and 0
9
1 as the respective
exceptional and predominant runs.
