Image Processing Reference
In-Depth Information
3.4.4.3
Non-linear Estimators ..................... 124
3.5
Summary ......................................................... 124
Exercises ......................................................... 125
Line drawings represent a large class of pictorial data used in such diverse areas
as cartography, computer graphics, alphanumeric text, engineering drawing,
and so on. Straight lines are particularly important because of their simplicity
and their ability to approximate any curve to any desired accuracy. More than
two thousand years ago, Euclid introduced straight lines through an axiomatic
approach and made them the basic elements in his geometry. It is no wonder
that studies in discrete geometry concentrated on defining and characterizing
discrete counterparts of this vital concept.
With the rapid growth of three-dimensional scene analysis, the study of
the geometry of 3-D digital images has raised considerable interest. In 3-D,
straight lines and planes are the most elementary geometric objects. Hence,
the properties of their discrete forms are fundamental to this geometry and
are discussed in this chapter. These include characterization of straight lines
and planar segments in 2-D and 3-D digital grids. We also discuss how we can
obtain their algebraic description in the Euclidean space, and estimate their
lengths and areas in respective dimensions.
3.1 2-D Discrete Straight Line Segments
A discrete straight line segment (DSLS) is defined as a set of dis-
crete points obtained from the digitization of some continuous straight line
segment (CSLS) from (x
1
, y
1
) to (x
2
,y
2
) in R
2
.
There are various digitization schemes available in the literature. We dis-
cuss three such schemes for 2-D CSLS.
(a) Nearest integral coordinates (NIC): In this digitization scheme, we con-
sider the digital image of a point p belonging to a straight line in a
Euclidean space, which has integral coordinates with the nearest inte-
gers to the real coordinates of p.
(b) Inner digitization: Let l be a straight line. Whenever l crosses a coordi-
nate line, the nearest digital point to the right of the point of crossing a
digital grid line with respect to the direction of traversing l becomes a
point of D(l), the digital representation of l. We assume that the slope
of the line l lies between 0 and 1.
(c) Object Boundary Quantization (OBQ): Intuitively, whenever a closed
figure is placed on a square array of points, the grid points that are
