UPING 
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modelled in terms 
1 the above levels 
unctual and local 
e minimization of 
ignal response. To 
ite the correlation 
ations acting onto 
ields arising from 
ns under uncertain 
to obtain a coarse 
's of the Earth. 
perspective views. 
addressed to find 
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level and the local 
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nogeneous regions 
ose already known. 
ic conditions or the 
Il take always a 
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| orthographic view 
ther a sequence of 
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w.rt. the relative 
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entation. Regional 
homogeneity depends on lightening conditions. To avoid an 
excessive dependence w.r.t. illumination conditions and, in 
absence of a natural system of chromaticity coordinates, we 
have developed a hybrid method having in account geometric 
elements linked to OD (corners, multiple junctures) and 
segments connecting them through polygonals. A 
discrimination based on the colour allows us to delete segments 
which are contained in the interior homogenised regions, 
provided they are not connected to the boundaries (details in 
[DeP02]). 
A mid-term goal for this hybrid approach is to find pattern 
correlations between punctual data relative to homogenous 
regions and local primitives used for grouping. In a more 
advanced stage, this approach would must be correlated with 
modifications of histograms arising from real motion of camera. 
Nevertheless the partial character of current results, our 
contribution relative to convex quadrilaterals for tracking and 
motion estimation is meaningful for the quasi-static case, where 
the position-orientation of the camera is changing along an 
unknown trajectory with a uniform movement. Inversely, the 
geometric invariance linked to the contours extraction suggests 
that there exists a correlation between relative orientations and 
transformations between histograms corresponding to different 
views of the same region under skew (non-orthographic) 
projections. A particular case of quadrilaterals given by 
trapezoidal maps has been used for egomotion estimation in 
indoor scenes in another paper ([Fin02]). So, the 
superimposition of quadrilateral maps linked to the contours 
allows to obtain information about changes of the relative 
orientation for a visual navigation in absence of good network 
of geographic or SAR information. The incorporation of metric 
structure will allow the generation of maps based on a visual 
navigation. 
2. Extracting and grouping data 
Extracting and grouping is usually performed in terms of local 
and global filters based on reinforcing 1D discontinuities and 
2D continuities in a digitalized image ([Rus99]). Global filters 
involve all pixels in the same way, and are based in usual 
morphological mathematical operators. Elimination of small 
regions is performed by composing a local erosion procedure 
with a dilatation to fill regions under a threshold size .Global 
filters degrade in an irreversible way the high quality of current 
images for GIS. Hence, we have concentrated our attention on 
local filters with a special regard to the detection of meaningful 
(significant) points, the 1D contours extraction and the 
characterization of regions. We have used standard tools for 
grouping based on Canny's detector as prototype of the 
Laplacian of a Gaussian operator (LOG). Laplacian operator is 
more robust in several aspects, including an invariant approach 
w.rt. the rigid transformations (rotations and translations) 
involving to changes in the relative localization of the observer. 
The correspondence between homologue 2D regions is 
simplified by applying a colour discrimination ([Rus99]). We 
select a palette of 256 colours based on a processing of 8 bits, 
and we apply a non-linear median filter to homogenise regions 
with size higher than a threshold. An adaptation of the vector 
representation for the Laplace's operator allows comparing 
homologue data in pairs of successive images taken along a 
sequence of views . 
A vector version of Laplacian operator allows to establish a 
method of correspondence. We try to find a translation 
u—(uj,u5)^, which transforms an image 7 (the template), in an 
image /, the reference, such that (x) =T(x— u(x)). In 
continuous variables the reference I and the template T can be 
represented via functions from QcR* — R®, which associate 
to the pixel (x,x2) € R° , the (Tr(X1X2), Tr(X1,X2), Tr(X1,X2)) OT 
(Ip(x1,x2), Ir(X1X2), Ir(X1,X2)) levels. I is a fixed image; so, we 
have a mapping M=T-¢, where @(x,u)=x-u(x), which goes from 
the set of translations into the set of digital images. To solve this 
colour correspondence problem we try then to find the 
displacement value, u, such that M(u)=I. To find an 
approximate solution to the problem we measure the L’ - 
difference in O of the two images, A(u)-—T(x-u)-I(x): 
Dw) =| hw) I}, = =[(T(x-u(x)-1(x) dQ. 
where D(u) is the functional of squared minima w.r.t. 1, and this 
is non-linear. Through the vector Laplace operator, we try to 
give a solution to the following functional: 
= NX, 
I, = (V2 D@®)W) 
which approximates / after a given number of iterations on k as 
a solution of a diffusion operator: 
- V?y(x) - V(div(v(x))) » f (u^), 
for xeQ and where v(x)=0 for xe0N. We are currently working 
in an extension of this approach to the spatio-temporal case, by 
replacing the Laplacian by the D’Alembert operator. Some 
troubles of their implementation are linked to the development 
of integro-differential operators, with their corresponding 
prediction and validation models in the time-space domain 
given by an appropriate Kalman filter. 
Grouping criteria are usually based on the characterization of 
regions which present some degree of homogeneity. The 
contours extraction and the labelling of closed regions bounded 
by these contours simplify compression and transmission of 
data. Extracted contours can be described as a piecewise linear 
array of pixels or, following an ideal mathematical model, as the 
integral curve of a planar vector field. This integral curve is the 
directrix curve of a cone with vertex the current position of the 
satellite in the 3D space. The tangent planes along a transversal 
slice of the cone give a uniparametric family of planes which is 
projected onto tangent lines which rectify the silhouette given 
by the planar projection of the apparent contour. Hence, the 
cones associated to a uniform trajectory of the satellite are 
modelled as a uniparametric family of cones with their 
corresponding tangent planes. Last version simplifies a 
theoretical approach for the contours tracking, because every 
contour is the projection of the inverse image associated to a 
rigid uniform movement of the satellite. In addition, such 
trajectory is known a priori, and it can be described in terms of 
vector fields. The deformation of homologue contours along a 
sequence of frames is performed along transversal directions to 
the boundaries. These transversal directions are theoretically 
given as the integral curves of the gradient vector fields 
obtained as an interpolation between the perceived contours. So, 
gradient vector fields provide tools for the deformations 
tracking of apparent contours along a sequence. Instability of 
resulting gradient vector fields must be corrected with a more 
robust approach linked to the superimposition of additional 
discrete structures. 
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