Image Processing Reference
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.