Image Processing Reference
In-Depth Information
chain code string. We restrict ourselves to the standard situation (to be defined
later) with commonly used 26-connected chain code strings on the cubic grid
of discrete points. This is followed by a discussion on proper geometric or
algebraic characterization of a 3-D DSLS. This characterization enables us to
define an algorithm to detect whether a given set of 3-D grid points can be
the digitization of a 3-D CSLS.
3.3.1 Geometric Preliminaries, Digitization, and Character-
ization
A straight line in 3-D may be specified in several possible ways. For exam-
ple, coordinates of two points in 3-D uniquely determine a line. Similarly, the
slope of the straight line together with the Cartesian coordinates of a point
lying on the line also represents it uniquely. The slope, however, can be stated
(Fig. 3.9) either in terms of direction cosines commonly denoted by l, m, n,
or it can be given by θ and φ, where φ is the angle made by the projection of
the line on the XY plane with the X-axis and θ is the angle between the line
and its projection on the XY plane.
The quantities l, m, n, and θ, φ are related by the following equations:
l =
cosθ cosφ
m =
cosθ sinφ
n =
sinθ
and it is well known that
l
2
+ m
2
+ n
2
= 1.
Thus, a 3-D line segment can be specified by the initial point (x
1
,y
1
,z
1
),
the slope of the line given by φ, and θ and x
2
, the X-coordinate of the final
point.
Lemma 3.6. Any straight line segment, L, in 3-D (not parallel to Y or Z-
axis ) can be uniquely identified by its projections on XY and XZ planes and
vice versa [39].
Proof: Without any loss of generality, we may assume that line segment L
lies in the first octant and is contained within the planes x = 0 and x = x
2
.
Let the initial point of L be (0,y
1
,z
1
). If θ and φ together denote the slope
of L, then the equation of L
Y
, the projection of L on XZ plane, is given by
z = xsecφtanθ + z
1
. Similarly, the equation for L
Z
, the projection on XY
plane, is y = xtanθ + y
1
. Since the 4-tuple (θ,φ,y
1
,z
1
) is fixed for L, the
above equations are unique. Also, L
Y
, L
Z
are line segments on the y = 0 and
z = 0 planes, respectively, enclosed within the lines x = 0 and x = x
2
.
Conversely, given the equations of the projection segments y = px+y
1
and
z = qx + z
1
between x = 0 and x = x
2
, the slope of the original line in 3-D
