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Dynamic trapezoidal maps for coarse perspective models in indoor scenes
Javier Finat, Margarita Gonzalo-Tasis, Maria J. Antolinez Susana Aguilar
Javier Finat, Dept of Algebra and Geometry, ETS Computer Science Eng, Univ of Valladolid,
47011 Valladolid, Spain, e-mail: jfinat@agt.uva.es
Keywords: Vision Sciences, Geometry, Navigation, Representation, Visualization, Motion
Abstract
A major problem in Automatic Navigation is how to update and maintain coherent information arising from the Computer Vision
techniques. In this paper we address the problem of generating and updating perspective models for indoor scene with a mobile
uncalibrated camera in absence of additional information. We develop a bottom-up approach based on a coarse paraperspective
model. Trapezoidal maps are constructed from a perspective representation. Resulting structures are overimposed to the original
views to simplify grouping criteria. Trapezoids are symbolically represented as pairs of bivectors associated to two evolving oriented
areas. Each pair of bivectors represents a shear deformation along a privileged direction, and inversely. Privileged directions linked
to trapezoids play the role of dynamical organizers for propagation models. Its linearization is inverted following the tangent space of
a shear transformation to predict, generate and validate a coarse mobile 3D Reconstruction.
Keywords: Trapezoidal maps, Egomotion, Shear transformations
1. Introduction
The relative motion of a mobile platform can be captured from a
sequence of images captured from an embarked camera.
Classical approach are based on Optic Flow ([Hor81]) and
Detection-tracking of Geometric Features ([Fau93]). The Optic
Flow is the 2D projection onto the image of the true 3D
distributions of motion vector fields linked to mobile objects.
Hence, to avoid the indeterminacy about the lifting of a 2D
vector field to a 3D vector field we must add information about
the scene, the camera or some coarse motion characteristics of
objects with real or apparent motion (see [Zha92], [Fau93],
[Har00] for details and additional references).
A good initialization is convenient to select a spatial
arrangement of geometric data which can be tracked in a simple
way along the sequence of views. Usual geometric techniques to
update information about a changing scene are based on easily
identifiable 0-dimensional elements. Triangular decompositions
provide general techniques for the grouping of such elements in
static scenes. The tracking problem is far more difficult in the
case of dynamic scenarios where several objects can move
simultaneously. In this case, the relative motion is composed of
an evaluation of egomotion, and an evaluation of trajectories of
mobile objects. Some recent advances ([Avi00]) provide
information about trajectories of mobile objects along lines by
overimposing a triangulation for dynamic scenarios, also.
Unfortunately, a simple computation of parameters give us five
views to determine such trajectory, and it would be interesting
to find a segmentation requiring less views to capture the
motion by using easily trackable regions inside pairs of triples
of views. Our approach intends to introduce a regional
segmentation based onto mid-level overimposed structures
(instead of using the color as usually).
Nevertheless the advances for egomotion evaluation produced
along the nineties (see [Tia96] for an excellent comparison
between them), the mise in correspondence of changing shapes
and some "dislocations" linked to depth changes make difficult
a mobile segmentation and clustering of regions. We have
privileged a robust instead of a precise strategy for the
algorithms design. Robustness allow us to simplify the
estimation, grouping, updating and propagation models. It is
well known that lines verify all these conditions, instead of
points. Constraints relative to lines provide local compatibility
criteria which can be lifted to a global coherence 3D structure.
Triangular decompositions support also an information relative
to lines to be grouped, but the updating and tracking requires to
overimpose epipolar constraints. Epipolar constraints are need
for a precise reconstruction, but they have a high computational
effort. To avoid it and maintain the global coherence of the
geometric structure, we have introduced a trapezoidal map
(following [Ber97]). In this work we have selected
1) asimple indoor scene with an apparent motion arising
from the egomotion of the mobile platform
2) an on-board processing for an autonomous embarked
navigation;
3) a generation of trapezoidal maps for grouping and
updating information.
The first constraint minimizes the discontinuities of the
optic flow field arising from the depth discontinuities. The
second constraint avoids the design of protocols linked to
the control of communications. Main novelties of our
approach concern to the third aspect ([Agu01 |).
Trapezoids provide mid-level primitives for grouping with a
simple and robust characterization. Simple character arises from
the bilinear maps associated to segmentation and from their
geometric representation as pairs of bivectors. We construct
unbiased estimators (by extending some arguments of [Coe92]).
Furthermore, the propagation model based on trapezoids
simplifies the tracking, provides natural bounds for the field of
view and allow us to maintain clustering devices with a
minimal number of new events.
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