lem concerns to 
tive positioning, 
lation of relative 
n use absolute 
terrain or the 
)0]) for metric 
rivileged role as 
n. 
form a sweep of 
se a sequence of 
as. "There ‘exist 
quadrilaterals in 
jn, as the frames 
ways ([Har00]). 
ompare different 
yut parallel lines. 
altiple (triple or 
ng quadrilaterals 
n inside). So, we 
] by perspective 
compare (and 
sive images. To 
 quadrilateral we 
B-- DC) ^ AC 
ween homologue 
of homologue 
ry ([Har00]). The 
imposes strong 
elements along a 
ansformation is 
nslation vector 
re of gravity of 
from a standard 
econd of latitude 
on of parameters 
ctors. Hence, we 
with a common 
ue quadrilaterals 
is which give an 
etween sampled 
ferences between 
e quadrilaterals 
estimator of the 
sed quadrilateral 
map. Trapezoids 
two orthogonal 
nal view. So, we 
s parallel to the 
ved by discarding 
e of a segment 
tive optimisation 
mulation and the 
thod is based on 
energy functional 
ion of L'- and L’- 
1al potential and 
kinetic energies. In our case, we incorporate an additional 
elastic term to the energy functional which provides a more 
flexible tool for Digital Elevation Maps in the 3D Case. 
i 
; ; AL di ; 
n - p^y(a *)pe! -av1]-2p^ Y. ET +22(1+ 5%). 
i 1 j 
2 
where 4; and 4, are the intensity values of two pixels, p is the 
correlation between those two pixels and à, b, and b, are 
parameters of both images. 3) the estimation of the disparity 
gradient to evaluate Lagrangian optimal conditions, and 4) the 
extension of usual superimposed Delaunay triangulations to 
give solutions compatible with elastic models. The quadratic 
part is the responsible for the data adjustment, the linear term 
penalties meshes with a large number of vertices and the elastic 
term is in charge of an adaptive behaviour towards a Pareto 
optimal solution. In our case, solutions of this multiobjective 
optimisation adjust in an adaptive way to the boundary of a 
quadrangular region. So, we extend the ordinary classical 
approach based on ordinary L'- and L’-norms. Optimal 
solutions for the classical case are given by Delaunay 
triangulations which maximize the minimal angle ([Ede91]). In 
our case, we can not expect an optimal result as good as above, 
because multiobjective optimisation have no necessarily a 
unique solution, but only Pareto optimal solutions with 
smoothness and adaptive properties. 
6. Conclusions and future work 
We have developed grouping and tracking methods based on 
quadrilaterals. This approach provides tools which are more 
naturally linked to the contours structure contained in aerial 
images. So, we avoid some ambiguity problems linked with 
more traditional triangulations. We introduce a functional for 
multiobjective optimisation, and we reinterpret some properties 
of solutions in terms of superimposed structures. The next step 
to be given is to evaluate the invariance in terms of histograms 
of frequencies linked to the same region seen under different 
orientations and under different lightening or atmospheric 
conditions. Instead of looking at the invariance in terms of 
rotationally invariant Hamiltonian fields (as in the Euclidean 
case) we must look at invariance in terms of vector fields 
preserved by the affine group. The robustness and accuracy of 
this method is the challenge for the next future. 
Hence, some specific ingredients of our approach to the 
Multiobjective Optimisation are based on 1) the construction of 
a disparity map along pairs of consecutive images, 2) the 
correlation estimation, considering the maximum likelihood 
estimator of p ([Ybe01 ]): 
A, A; T 2p A, A; 
2 =0, 
i b,b, Llp b,b, 
7. References 
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O.Schwarzkopf: “Computational Geometry. Algorithms 
and Applications”, Springer-Verlag, 1997. 
e [DeP02] I.De Paz: “Mathematical Models for GIS”, M.Sc. 
Thesis, Univ of Valladolid, June 2002. 
eo [Ede91] H.Edelsbrunner and T.S.Tan: “4 quadratic time 
algorithm for the minmax length triangulation”, Proc. 32™ 
IEEE Symp. Found of Computer Science, IEEE Press, 
1991, 414-423. 
e [Fin02] J.Finat, M.Gonzalo-Tasis: “Dynamic trapezoidal 
maps for coarse perspective models in indoor scenes” 
Proceedings ISPRS 2002, Corfu, Greece, September 2002.. 
e [Har00] R.Hartley and A.Zisserman: “Multiple View 
Geometry”, Cambridge Univ. Press, 2000. 
e [Kre97] M. van Kreveld, J.Nievergelt, T.Roos and 
P.Widmayer, eds: “Algorithmic Foundations of GIS”, 
Springer-Verlag, 1997. 
e [Rus99] J.C.Russ: “The Image Processing Handbook (3™ 
ed)”, CRC Press, 1999. 
e  [Vil02] A.Viloria, J.Finat and M.Gonzalo-Tasis: “A fast 
self-organized iconic segmentation and grouping based in 
color", Proceedings ISPRS 2002, Corfü, Greece, 
September 2002. 
®  [Ybe01] Y.Belgued et.al.: “An accurate radargrammetric 
chain for DEM Generation", Alcatel Space Industries, 
ESA-ESTEC Publications, 2001. 
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