. Istanbul 2004 
Vi, Va. A pair 
1e management 
s, we introduce 
ees linked to 
third vanishing 
uadrilateral 2D 
which provide 
management of 
lanar templates 
ame way, the 
med by octrees 
oids. À cuboid 
by an affine 
map linked to 
icremental way 
to elementary 
ing process of 
¢ described by 
he number of 
ed by means of 
ns inform to us 
ncave features. 
uble, triple and 
in our case). 
pond to a) two 
bout ceiling or 
rs of inserted 
T-type double 
e perspective 
le junctions, d) 
anishing points 
junctions. The 
ctions along a 
, by including 
e to the relative 
veen regions in 
hism between 
candidate to be 
ie 3D scene is 
let us suppose 
y quadrilaterals 
ntrant corner is 
ypical Y-triple 
arising from 
isible segments 
litectural scene 
rtex for three 
aterals give an 
ected to three 
"he hexagon H 
by compatible 
nt at the triple 
hen the central 
t is an inverted 
egion w.r.t. the 
ecewise linear 
nant behaviour 
continuous or 
ovides tools to 
asily verifiable 
scene w.r.t. the 
spective lines 
erals, which are 
x each pair of 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
vanishing points V; and V; we generate a map of quadrilaterals; 
this map is specialized to a rectangular grid for orthogonal 
perspective and a trapezoidal grid for frontal view with a 
vanishing point in the view. The segments lying onto each 
quadrilateral of such a map make part of the pencil i and j of 
perspective lines through the vanishing points V; and V; . A 
coarse management of almost flat facades in outdoor scenes or 
flat walls in indoor scenes is performed with map of 
quadrilaterals. However, the sudden jumps in relative depth or 
the alternance between concave and convex regions (typical in 
Baroque style) reduce the performance of quadrilaterals maps. 
The solution of this problem involves the labelling of vertices 
onto the map of quadrilaterals as multiple junctions. 
2.4.1 Automatic generation of superimposed perspective 
models 
The intersection of two pencils i and j of perspective lines 
through two vanishing points V; and V; determine a map of 
quadrilaterals. In the same way as for trapezoids in frontal 
perspective in indoor scenes [Fin02], we represent each 
quadrilateral by means of a bivector given as one half of the 
external product of the diagonal of the quadrilateral multiplied 
by the difference of vectors arising from a selected vertex of the 
diagonal. Thus, quadrilaterals inheritate a natural orientation 
depending on the sweep out process and labelling processes. 
Each map of quadrilaterals generated by two planar pencils 
from perspective lines provides a support for the automatic 
grouping and for their associated bilinear maps. In fact, 
optimization processes can be performed directly onto the space 
of bilinear maps, with the additional advantage linked to 
similarity relations between each map of quadrilaterals. In fact, 
propagation of similarities simplifies the matching process 
between candidates to homologue quadrilaterals. Above process 
can be performed for each pair of vanishing points. In this way, 
we obtain three maps of quadrilaterals that are matched together 
between them thanks to the existence of cuboids linked to three 
non-aligned vanishing points. An oriented triple product 
algebraically represents every cuboid. The set of lines through a 
vanishing point can be interpreted as the.support of a radial 
vector field centered at the vanishing point. In the same way, 
each map of quadrilaterals (resp. cuboids) can be interpreted as 
the support of 2D (respectively, 3D) distribution D of vector 
fields whose singular points are located at vanishing points. The 
transformation from cuboids to quadrilaterals it is automatically 
performed by applying the contraction of distribution D along 
the vector field supported onto the set of lines through a 
vanishing point. The relative position of vanishing points in 
each view modify the vector field according to the semidirect 
product of translations by the special unitary group SU(2) (the 
universal double covering of the most common spccial 
orthogonal group SO(3) linked to rigid transformations in 
ordinary space. Maps of quadrilaterals provide the support for 
reading the information relative to the  infinitesimal 
transformations of the Lie algebra su(2) of the group SU(2). 
3. EXPERIMENTAL RESULTS 
The program developed by ourselves is helpful to experiment 
and visualize varied estimators behaviours within different 
geometrical and statistical conditions under a flexible and 
friendly context. We have applied some robust estimators to 
several images obtained of Santa Ana's Cloister. 
SC 
e ener dt 
Liaara TI 
      
Faro (teg i cs d 
TIAE enr eh nn en 
PF PFY PFZ 
[355 [3634 | 85255 
[1315 [120987 ]| 4x4 
E 
mm 
P 
E ! = = E 
» ve, 
ed 
  
  
Fig. 3: /nterface of the program 
The results of perspective lines and vanishing points are 
expressed as a residuals (V) and weights (W) in the following 
tables, in which we have 3 blunders errors: 10, 11, 12. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
VANISHING LSQ HUBER 
LINES V NV V AV 
1 1.084 1 0.3836 0.0008 
= 0.2714 1 0.0007463 0.1837 
3 0.07606 1 0 1 
4 0.2352 1 0.0164 0.02501 
3 0.6622 1 0.02153 0.00629 
6 0.3377 1 0.0009933 0.2603 
7 0.1513 1 0.0002925 0.3673 
8 2.22 1 2.008 0.000192 
9 1.525 1 1.176 0.000329 
10 1.501 1 1.782 0.0002288 
11 2.025 1 2.027 0.0001978 
12 6.61 1 6.426 6.223e-005 
  
  
  
  
Fig. 4: LSQ vs. Huber estimator in vanishing points detection 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
VANISHING LSQ MINIMUM SUM 
EINES V W V Ww 
1 1.084 1 0.385 0.02 
2 0.2714 1 0.00074 4.593 
3 0.07606 1 0 77.41 
4 0.2352 1 0.0164 0.6254 
5 0.6622 1 0.02153 0.157 
6 01177 1 0.00099 6.508 
7 0.1513 1 0.00029 9.182 
8 222 1 2.008 0.0047 
9 1525 1 1.176 0.0082 
10 1.501 1 1.782 0.0057 
11 2.028 1 2.027 0.0049 
12 6.61 1 6.426 0.0016 
  
  
  
  
  
  
  
Fig. 5: LSQ vs. Minimum Sum estimator in vanishing points 
detection