ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision", Graz, 2002 
  
  
d 
— 
— 
— 
Ain 
Api 
Api 
dn aml 
Amn 
A 
bu . Din m 0 
. = ; [16] 
bz Dan 0 
Apn 
di Wi 0 
dm 0 
Simplifying [16] gives: 
EídizA4Ap;. B'ApzO0; Wid) [17] 
Where A is a (m by n)- matrix containing the partial 
derivatives for all observations, and B* is a (2 by n)- 
matrix to ensure that the two Plücker conditions from 
equations [6] and [7] are fulfilled after updating the 
parameters, and J/ is the weighting matrix. 
By least squares minimisation of the distance d, 
weighted by the gradients, the solution leads to the 
corrections to the parameters of the object line. The 
parameters of the object line are updated in an iterative 
process: 
(k+1) (k) ^ (K) 
=p; +Ap, [18] 
Representation [17] is in the form of observation 
equations with conditions on the parameter vector. 
Now the least squares solution can be obtained in a 
two-step calculation: 
1 Ap, = (AWA) Awd [19] 
2Ap -[ - Qu, B(B'Q., B) B']AD,, 
with Q,,, - (A'WA) . [20] 
In the first step one considers equation [17] without 
the conditions on the parameters. In the second step 
the temporary solution is projected from the R® space 
into the R^ space, restricting the six parameter values 
such that the updated 3D line p^ *  fulfils the two 
Plücker conditions. As one can see in figure 4 the 
fitting algorithm fits line segments to the strongest 
edge, shown for nadir data (raw image pixels projected 
on local plane). 
Visually the succes of the algorithm can best be seen 
in figure 5: fragmented 3D lines representing one 
object line converge to one optimal solution. The 
accuracy of the reconstruction of the object lines after 
matching is about 50-60 centimeter in horizontal and 
100 centimeter in vertical direction and after fitting 
about 15 centimeter in horizontal and 25 centimeter in 
vertical direction. So, the fitting algorithm increases 
A- 112 
the accuracy of the reconstructed object lines with a 
factor between 3 and 4. 
  
Figure 4: Fitting to strongest edge: matched 
line segments (left), back projected fitted line 
segments (right). 
A - C n7 = 
= 2 ms 
7 VA ef i, 
-— = "f ; rd e = 
~~ — iat” Et” / 
2m A Val 
Se. t 7 T 
e ex > rd == \ 
x NS. me / n, \ 
Figure 5: 3D view on the fitted line segments. 
5. CONCLUSIONS 
The use of digital data instead of analogue images is a 
major step towards fully automatic object 
reconstruction. Three-line scanners proof to produce 
suitable data for automatic 3D line reconstruction. 
This suitability is expressed in high geometric and 
radiometric resolution, together with the reliability of 
a triple viewing angle. The line perspective geometry 
causes distortions in the rectified images, which 
depend on the object height above the projection 
plane. Height information can be obtained by 
matching corresponding features in all three images. 
Most of the mismatches in the two-view stereo 
configuration are removed when using three viewing 
angles. When using rectified images, feature-based 
matching algorithms produce 3D lines, which still 
contain errors. We proposed to use the geometry of the 
recording situation in a least squares fitting step in 
order to remove these errors, finally resulting in 
accurate 3D line equations. Fitting results showed the 
convergence of fragmented reconstructed lines to one 
object line. The automatic reconstruction of 3D object 
lines shows the potential of airborne digital sensors to 
reconstruct complete object models in a fully 
automatic approach. 
6. REFERENCES 
Bignone, F., O. Henricsson, P. Fua, M. Stricker, 
“Automatic extraction of generic house roofs from