A Probabilistic Approach to Geometric Hashing using Line Features

CHAPTER 4. INVARIANT MATCHING USING LINE FEATURES 41
This representation is not unique, since ça, for any ç 6= 0, is also a representation of the
same line.
A transformation T in image space changes the coordinate of every point w on a line
to w 0 by w 0 = Tw. Substituting w = T ,1 w 0 in a t w =0,we get
a t T ,1 w 0 =0
and hence
èèT ,1 è t aè t w 0 =0;
or
a 0 tw 0 =0; where a 0 =èT ,1 è t a:
This shows that the change of the coordinate of the point in3-D parameter space is
given by
a 0 = Pa:
where P =èT ,1 è t .
4.1.3 Change of Coordinates in èç; rè Space
Line features of an image are usually extracted by the Hough transform. A common
implementation of the Hough transform applies the normal parameterization suggested by
Duda and Hartë15ë, in the form
x cos ç + y sin ç = r;
where r is the perpendicular distance of the line to the origin and ç is the angle between
a normal to the line and the positive x-axis.
This unique parameterization of lines relates to the preceding parameterization of lines.
Let F : R 2 7! R 3 be a mapping such that
Fèèç; rè t è = ècos ç; sin ç; ,rè t
If we restrict the domain of ç to be in ë0;çè, then F ,1 exists. Deæne another mapping
G : R 3 7! R 2 by F ,1 as
Gèèa 1 ;a 2 ;a 3 è t è=
8
é
é:
F ,1 èèp 1
; p a 2
; p a 3
è t è
a 2 1 +a2 a
2
1 +a2 a
2
1 +a2 2
if a 2 ç 0
F ,1 èèp 1
; p ,a 2
; p ,a 3
è t è
a 2 1 +a2 a
2
1 +a2 a
2
1 +a2 2
otherwise