through the vanishing points in two orthogonal directions on a 
plane providing the vanishing line or the image of the line at in- 
finity 1... The projective transformation relating image to plane 
coordinates is a homography, expressed as follows in homoge- 
neous representation (Hartley & Zisserman, 2000): 
c 
a c 
Hx-X or d f| x5|2|X, (5) 
g 1 
=> € 
whereby H is a 3x3 matrix with 8 coefficients, x ^ (xi, x», 1)" is 
an image point, X = (X;, X,, 1)" is a point on the ground plane. 
The vector (g, h, 1)" describes the vanishing line of the ground 
plane (L,) and is found from the cross product of two vanishing 
points (Liebowitz et al., 1999). The knowledge of L, allows to 
remove pure projectivity, i.e. allows affine rectification (correct 
length ratio on parallel lines). The orthogonality of the two di- 
rections restores angles. Here one faces a 1D problem, i.e. one is 
interested in measuring distances in a single direction to estima- 
te vehicle speed. Hence, there is no need to correct aspect ratio 
for securing uniform scale. Only the scale along the road axis 
has to be restored, for which one known ground distance in this 
direction suffices. 
3.2.2 Affine rectification from one vanishing point However, 
the main question posed here concerns the kind of rectifications 
possible in case only one vanishing point is available. Indeed, in 
several cases lines do not suffice for determining two vanishing 
points. In Fig. 7, which presents an example of the images used 
here, parallel lines have been identified only along the road axis 
(ground Y-axis), as lane demarcations cannot always be trusted 
as regards the orthogonal direction (ground X-axis). In this con- 
text, two main assumptions can be made. 
  
Figure 7. The vanishing point of the road. 
e [tis first assumed that the image plane is practically parallel 
to the ground X-axis (as in the present case; cf. Fig. 7). Hence, 
the second vanishing point is at infinity, i.e. lines orthogonal to 
the road axis are imaged parallel to the x-image axis. Vanishing 
points F, in the direction of the road axis and F,, assumed at in- 
finity, are expressed as follows on the image plane: 
f, 1 
Fi = f, F, =|0 
0 0 
Points F, and F, define I, (cross product), a line parallel to the 
image x-axis. The point at infinity in the direction orthogonal to 
the road axis is thus forced to be transformed through H to the 
point at infinity of the image x-axis: 
(6) 
= 
ooc- 
Il 
© © = 
As mentioned, the orthogonality constraint provides the remain- 
ing coefficients of H. Having set coefficients c and f (since they 
concern pure translation) to zero, it can be proved that finally: 
abc 1 — fi/f5 0 
H-|ld c f[-lo 1 9 (7) 
2 h 1 à VE 4 
As already pointed out, no need arises here for correcting aspect 
ratio. However, visual reasons might dictate its modification. A 
scale factor m, suitably chosen in each case, will ‘improve’ the 
aspect ratio of the rectification (without actually restoring it). 
This modification is described by matrix M, contributing to the 
final transformation matrix Hy: 
— H,,-MH (8) 
s 
cos 
OS — © 
— o © 
A measured distance along the road axis relates the pixel size of 
the rectified image to the ground units in this direction. 
However, a direct approach is also possible. Similar to Eq. (6), 
the transformation for the finite vanishing point F, is: 
f [o 
Hi flzl1 (9) 
1 0 
From Eqs. (6) and (9), H may be directly determined in one step 
without computing the vanishing line etc. 
e In case the previous assumption is invalid, i.e. both vanish- 
ing points are finite but only one can be identified on the image, 
with the above approach ground distances can be measured only 
on the specific line of the known reference length. Parallelism 
of lines in the direction of the road axis (ground Y-axis) will be 
restored, however their scale will not be uniform. The answer to 
this can be provided by the ‘construction’ of a second vanishing 
point from two known lengths. 
3.2.3 Construction of a second vanishing point As mentioned 
already, a second vanishing point is needed to allow affine recti- 
fication when the image planes are not parallel to the ground X- 
axis. Following Fig. 8, assume two known distances d and p on 
ground lines D and P running parallel to the road axis (Y-axis); 
on the image they converge to point F;. By exploiting the cross 
ratio, from length p one may compute a ground length q = d on 
line P. Evidently, these two equal and parallel line segments d 
and q define two parallel ground lines. On the image, these lines 
define their vanishing point F, which lies on the horizon. 
The above are illustrated in Fig. 9. On the left, an image patch is 
rectified from only one vanishing point F,. Parallelism has been 
restored only along the y-axis. Extraction of metric information 
—24—