Image Processing Reference
In-Depth Information
been defined [88, 215, 216, 179, 109, 102, 174]. We mention one of the first
results spelled out by Rosenfeld [176], which is known as the celebrated chord
property. This property is a bijective relation between discrete arcs and dis-
crete straight line segments. A discrete arc in 2-D (also called a simple arc in
Section 1.2.1 of Chapter 1) is a sequence of discrete points p
i
, 0 ≤ i ≤n such
that each p
i
, except p
0
and p
n
, has exactly two 8-neighbours p
i−1
and p
i+1
.
p
0
and p
n
have single 8-neighbour p
1
and p
n−1
, respectively.
Theorem 3.1. Let G be a discrete arc and S ⊆ G. S is a DSLS if and only if
S is a discrete arc having the chord property. S has the chord property if for
all discrete points p,q ∈ S and for every point (x,y) belonging to the CSLS
joining p and q, there exists some (u,v) ∈ S such that
max{|x−u|,|y−v|}< 1
€
The chord property effectively means that all CSLSs connecting two arbi-
trary points of a DSLS lie closer to the discrete points of the DSLS than any
other point in the digital plane. An example illustrating the chord property is
given in Fig. 3.2.
In this figure, the points in the DSLS are represented by the round dots.
The CSLSs joining any two points of the DSLS are also shown in the figure.
If we consider any such CSLS, then we note that all points of that CSLS are
within a unit distance from some point of the DSLS in the X or Y direction.
FIGURE 3.2: Chord property of a DSLS.
.
Considering the possible run lengths and runs of runs in the principal
direction, Rosenfeld showed that there can be only two run lengths in the
chain code of a DSLS. These run lengths can differ by at most one, and one
