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to integrate knowledge on the
structure in 3D reconstruction.
In the man-made world, there are quantities of simple geometric structures
(lines, planes, ...), which are strongly related to each other by Euclidean
relationships (orthogonality, parallelism, coplanarity...). Taking such constraints
into account for 3D reconstruction would be a great help: they enforce precise
Euclidean and metric properties to the reconstructed structure.
These constraints are easy to describe in a sentence, but more difficult to integrate into a system estimation procedure. The most attractive way of doing it would be to build a description of the structure observed that already include the relationships and uses a minimal number of free parameters. The main problem is to do it automatically, and to be able to deal with any system of points, lines, planes, etc. with any set of (realistic) relations.
An interesting approach is to translate the relations into polynomial equations on the variables of the system, then to make use of Gröbner bases or resultants to transform the set of equations into a subset of free monomials and dependency equations for the other variables. For the present problem, it has two major drawbacks. First, the computations involved get more and more difficult as you work in a space of higher dimensions (the set of possible monomials increase dramatically with a large number of initial variables). Then, the system obtained may be far from the original; doing the translation back into the geometric parameters can be difficult and the geometric meaning of the equations sometimes remain unclear.
I have therefore preferred an approach based on the geometric relations, that will take profit of their properties. The technique used is a set of rewriting steps, that will progressively remove constraints by creating a new equivalent system that integrate them. Each step is tracked to build the transformation function that maps the initial system of points and constraints into a parametric system with enforced relationships. Do deal with multiple elements sharing relations, we need to create relation graphs, determine cyclic structures in them, and then follow an appropriate path for the transformation.
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A practical example: the relation graphs obtained from a description of the MOVI house.
At the end, i obtained a method that can transform any set of points, lines and planes along with any (realistic) ensemble of relations of coplanarity, colinearity, orthogonality and parallelism. The system generated will end up most of the time with the minimal number of free parameter possible, and all quantities involved are geometric (angles, distances, points). The transformation function is continuous and differentiable, and is built automatically with the system. The computations involved take a few 10 milliseconds.
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original view in the image |
side view |
top view |
Three views of the MOVI house, after geometric reduction
Acknowledgements
This work has been done during my PhD in the MIRAGES
Team at Inria Rocquencourt.
I would like to thank the MOVI team
at Inria Rhône-Alpes for the House sequence, as usual.
Further details
A parametric scene
reduction algorithm from geometric relations, P. L. Bazin, Proceedings of
Vision Geometry IX, SPIE's 45th annual meeting, 2000. (postscript).
