In: Paparoditis N.. Pierrot-Deseilligny M.. Mallet C.. Tournaire О. (Eds). 1APRS. Vol. XXXVIII. Part ЗА - Saint-Mandé, France. September 1-3. 2010 
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of the vanishing points are mutually statistically independent, we 
have the full statistical information from all relevant line seg 
ments. 
We now treat the coordinates (x y , E x . x; ) of the vanishing points 
achieved from the individual detection, boosting and estimation 
as observations, which need to be corrected. Based on approx 
imate values x", which in the first iteration are identical to x y . 
we again obtain x y = Xj + v 7 = x° + Ax ? -, j = 1,2,3 
to fulfill the orthogonality constraint. The model for enforcing 
the three orthogonality constraints is g(x,) = 0 with g\ — 
xlx 3 , g-2 — xjxi, 93 = x|x2. After reducing the obser 
vations x r j = Jx (x 7 ) x_,, j — 1,2,3 in order to be able to 
handle the singularity of the covariance matrices S x ., . w'e ob 
tain the reduced model g(x r -i) = 0 with g\ = x J r2 Xrz = 0. 
g-2 = xj 3 x r i = 0, gz = xJiX r 2 — 0. The linearized model 
therefore is c g (x a ) -+ Bj.Ax r = 0 or explicitely 
xf x; 
xf Scf 
xi' T X2 
+ 
0' 
The reduced covariance matrices E“ .... of the observations are 
derived following (6) and (8), first transforming the rotation into 
the approximate point and second transforming them to the north 
(or south) pole and omitting the third, the zero component. 
Minimizing Q = Y'' 
v, 
v r i under the three 
1 rj {^X r jX r j) v rj 
constraints yields the classical solution for the update for the 
fitted observations Ax r - Y% rjXrj B r (Bj'£ XrjXrj 8 r ) -1 (c s + 
Bj.x r ) + x r - They are used to obtain improved approximate val 
ues for the fitted values of the vanishing point coordinates, as in 
(24). In spite of the low redundancy of R = 3 it useful to deter 
mine and report the estimated variance factor So = Q/3. 
5 EXPERIMENTS 
5.1 Used data 
We perform two tests, one using uncalibrated images for inves 
tigating reliabity of the vanishing point detections and a second 
using partially calibrated images, where the principal distance is 
known. 
In both cases we automatically derive straight line segments. 
They are represented by their centroid xq [pel], their length l 
|pel], their direction 0. This allows to give the stochastical prop 
erties by the standard deviation cr q of the centroid across the line 
and the standard deviation <7$ of the direction. They are derived 
from a ML-estimation using the edge elements and are approxi 
mately 
1 / 12 
&e ? — A/ iQ 7 ® e (25) 
Vi 
i 3 -i 
where <j e is standard deviation of an edge element, which de 
pends on the manner of subpixel positioning and always is 
smaller than the rounding error l/\/T2 [pel]. In our context 
mainly the angular accuracy is relevant. Using the techniques 
described by Meidow et al. [2009] the spherically normalized co 
ordinates 1 := I s of all line segments together with their singular 
covariance matrix En are determined. For testing we always take 
a high significance level of S = 0.9999. We employ the adaptive 
determination of the number of trials in the RANSAC procedure 
as described by Hartley and Zisserman [2000]. 
The processing time for each image is in the order of a few' sec 
onds. including the edge detection program, written in C, and the 
vanishing point detection, using non optimized Matlab code. 
5.2 Detecting vanishing points 
The first group of experiments addresses the quality of the van 
ishing point detection. We evaluate the detection procedure on 
three levels of accuracy. 
Visual evaluation. First w'e check the reliability of the vanish 
ing point detection. We downloaded 140 Google images named 
’building’ or 'batiment’ (cf. Fig. 3) with a minimum side length 
of 768 pixels. No interior orientation is known, some images 
are images sections, some are graphics, some show significant 
lens distortion. For each image we visually identified the num- 
1 
Ax r \ 
' 0 ' 
Ax v 2 
= 
0 
- 
Axr3 
0 
КЗ 
V 
Figure 3: 12 of 140 building images, taken from Google (’build 
ing’, ’batiment'). Such images are used for evaluating the van 
ishing point detection. 
her of vanishing point, a human could find, and - by inspecting 
the color coded line segments and the directions to the vanishing 
points - the number of correctly found vanishing points. The re 
sult is showm in the table 1. From the 102 images, w'here 3 or 
more vanishing points could be detected, in only 7 images the 
system found only one vanishing points, whereas in 28 images 
two vanishing points were detected. From the 95 images, mostly 
with facades, where only tw'o vanishing points could be detected, 
in 90 % the system could find both vanishing points. In three 
images no vanishing points could be detected even by a human. 
This is coherent with the experiment on the eTRIMS-data base 
[ Korc and Forstner, 2009]. where in all 60 images of facades both 
vanishing points could be detected. 
0 
1 
2 
3 
4 
5 
6 
0 
3 
0 
0 
0 
0 
0 
0 
1 
0 
0 
3 
4 
0 
0 
0 
2 
0 
0 
92 
25 
2 
0 
I 
3 
0 
0 
0 
65 
4 
0 
1 
Table 1: Horizontal: number of vanishing points a human could 
detect. Vertical: the number of vanishing points correctly de 
tected by the algorithm. 140 Google images and 60 eTRIMS 
images. 
The software gives an internal estimate for the accuracy of the 
vanishing points. We compared this with the number of line 
segments supporting a vanishing point. In the Google data 
set of 40 images on an average 90 lines supported a vanishing 
point, the mean standard deviation of the direction is appr. 0.3°. 
On an average we obtain an internal estimate for the accuracy 
ad ~ 2.5 0 /y/ri; which is a lower bound for the real accuracy. 
Of course, no check on the orthogonality of the vanishing points 
could be performed as the intrinsic parameters are not available 
for these images.