36 BINARY DIGITAL IMAGE PROCESSING

example [45]). This concept is first related to its continuous counterpart via the

definition of a simple digitisation scheme. Different characterisations of a dig-

ital straight segment in the 8-neighbourhood space are then given. Extensions

of the characterisations of straightness in the particular cases of the 4- and 16-

neighbourhood spaces are also presented. Furthermore, complementary results

concerning efficient representations of the digitisation of a continuous segment

are given.

Discrete convexity is presented in Section 2.3 as a natural extension of

discrete straightness. On the square lattice, different definitions of discrete con-

vexity have been given. Their inter-relationships are detailed in Section 2.3.2.

Discrete convex hulls based on these definitions are also presented.

Remark 2.1:

Based on the argument developed in Section 1.~.~, triangular lattices will not be

detailed here. For such a study, the interested reader is referred to [53].

Curvature is another important geometrical concept in Euclidean geometry. Its

discrete counterpart results in the geometrical characterisation of discrete circu-

lar objects. Few advances have been made on this topic in discrete geometry.

These are summarised in Section 2.4. Finally, for the sake of completeness, Sec-

tion 2.5 briefly considers inter-relationships such as parallelism and orthogonality

that can be characterised between discrete objects.

2.2 Discrete straightness

In the following, a real point is a point of the continuous space IR 2, a

discrete point is an integer point on the square lattice (i.e. a point in the discrete

space ~ 2). The continuous segment [a,/3] is the part of the straight line (c~,/3)

in IR 2 which passes by these two points.

In the continuous space IR 2, straightness can be characterised in different

equivalent ways. Analytically, in the plane (x, y), a straight line is the repre-

sentation of a linear function L which takes the form L : y =

ax

+ p, with

(a, #) E IR 2, where a is called the slope of the straight line. Geometrically, a

continuous straight line is a continuous set of points such that any pair of real

points within it defines the same slope.

In the discrete space, straightness is referred to as discrete straightness.

The following introductory definition for this concept is given.

Definition

2.2" Digital straight segment

A discrete set of points is a digital straight segment if and only if it is the dig#

tisation of at least one continuous straight segment.

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