Image Processing Reference
In-Depth Information
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FIGURE 4.2: Chain codes and their enumeration in defining a (irreducible)
digital curve segment.
4.1 Digital Straightness
In Euclidean space, a straight line consists of points lying evenly on it-
self [27]. A digital straight line (DSL) is a sequence of digital points satisfying
certain straightness properties in an appropriate sense. With a similarity lying
in their constitution by points, there arise fundamental differences in their very
definitions and related properties compared to those of a Euclidean straight
line. Some of these are stated below.
Similar to a Euclidean straight line, the property of evenness of points lying
on a line, which was stated as the definition of a real straight line by Euclid,
was reiterated by both Freeman and Rosenfeld [87, 115, 176] in the 1960s-
70s. The difference is that Euclid stated the evenness as a definition, whereas
the concept of evenness for DSL was formalized and proved in [176]. Clearly,
this indicates evenness as a strong necessary condition in order that a digital
curve segment is digitally straight. A thorough discussion of the formalization
of evenness of points constituting a digital straight segment (DSS) or a DSL
is provided later in Sec. 4.1.2. 2 A DSL is infinitely long and a DSS is a finite
segment of a DSL. A DSS is an irreducible digital curve segment that is
digitally straight. An irreducible digital curve segment (referred to as a discrete
arc in Section 3.1 of Chapter 3) is a sequence of digital points having two
distinct endpoints, each with one neighbor (in 8-neighborhood), and each other
point having two neighbors from the curve segment (Fig. 4.2).
Another interesting aspect that distinguishes a DSS (DSL) from a
real/continuous straight line segment (straight line) is the cutting syndrome,
which is as follows. If a real line segment is cut into two (or more) segments,
then each segment remains straight. This is also true in digital geometry; if a
DSS is cut into two parts, then each of them would still be digitally straight.
However, from the two subsequences of digital points representing the cut-off
parts, the correspondence is not straightforward. The reason is as follows. Let
p and q be two digital points; pq denote their connecting real line segment, and
DSS(p,q) denotes the sequence of digital points obtained by the digitization
of pq. See Sec. 4.1.1, Eq. 4.1 in particular, for digitization of a straight line
2 A DSS is also referred to as DSLS, as in Chapter 3.
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