Image Processing Reference

In-Depth Information

FIGURE 8
Localized skewed NL.

3.4 Finding Intersections in Cartesian Space

Each Hough line returned by the HT is described as (
ρ
,
θ
), where
ρ
is the length of the normal

from the line to the origin of the polar coordinate system and
θ
is the angle specifying the

rotation of the line about the origin. Given
l
1
= (
ρ
1
,
θ
1
) and
l
2
= (
ρ
2
,
θ
2
) in the polar coordinate

system, we need to find the intersection of these lines in the Cartesian system in order to find

the four intersection coordinates shown in
Figure 7
. If
θ
1
=
θ
2
and
ρ
1
=
ρ
2
, then
l
1
and
l
2
are the

same line and there are infinitely many intersections. If
θ
1
=
θ
2
and
ρ
1
≠
ρ
2
, then
l
1
and
l
2
are

parallel lines and have no intersections. If
l
1
and
l
2
do not coincide and are not parallel, there

AB
and AC, denoted as
n
1
and
n
2
, respectively (see
Figure 8
) to
l
1
and
l
2
. The normal
n
1
goes

from the origin
A
to point
B
on
l
1
whereas the normal
n
2
goes from the origin
A
to point
C
on

l
2
. The segment
BC
, denoted as
l
3
, completes the triangle
T
1
consisting of points
A
,
B
and
C
.

Another triangle
T
2
consists of points
B
,
C
, and
D
, where point
D
is the sought intersection of

l
1
and
l
2
. Consider the line segments
BD
and
CD
denoted as
l
4
and
l
5
, respectively (see
Figures

9
,
10
,
or
11
)
. Once
l
4
is found, point
D
is found by adding vectors
n
1
and
l
4
. Technical details of

FIGURE 9
Case 1: Intersection D is between two normals.

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