# Project : coprin

## Section: New Results

### Resolution of geometrical constraints systems by rigidity, consistency and interval analysis

#### Uncertainty management

Keywords : distance equations, uncertainty.

Participants : Carlos Grandon, Bertrand Neveu.

We have studied how to manage small uncertainties in the parameters of a system of distance equations. The main aim is to approximate the sub-spaces of solutions as precise as possible. Classical interval solvers have difficulties for describing it, because solutions are not isolated points but continuous subspaces. Our work is based on the following idea: we have the solutions of distance equation system without uncertainties (or we compute them), then we use these solutions to determine a dynamic splitting policy of the domains (using these solutions as N-dimensional points to compute a subspace like Voronoi diagrams). Filtering algorithms are then applied to each subspace to obtain boxes containing the solutions of the system with uncertainties.

We have studied the limitations of this method. Despite that the method is complete (no region with solutions is lost), we have found the following difficulties:

• A given distance system without uncertainties may have no solution while introducing uncertainties allows one to get solutions. The difficulty is then to find the different continuous subspaces.

• The solutions found when uncertainties are taken into account do not always correspond to an extension around the solution points without uncertainties. If new solution regions appear when the uncertainties are taken into account, the method is not able to isolate them.

• When regions overlap, our splitting technique will separate two regions that will be each smaller than the extensions of solution points. A method to identify this case consists in searching for a solution in the hyperplane that separates the solutions. If no solution is found, then there is no overlapping between these regions.

We are now studying how to solve these difficulties and how to extend the method to other systems of equations.

#### Scene Modeling Based on Constraint System Decomposition Techniques

Keywords : computer vision, scene reconstruction, geometric constraints, decomposition.

Participants : Gilles Trombettoni, Marta Wilczkowiak.

This work has been performed in collaboration with Marta Wilczkowiak working in the MOVI project at INRIA Rhônes-Alpes.

In 2002 and 2003, we had designed and implemented a new approach to 3D scene modeling based on geometric constraints (published in 2003 in the main conferences in constraint programming and computer vision). Contrary to the existing methods, we can quickly obtain 3D scene models that respect exactly the given constraints. Our system can describe a large variety of linear and non-linear constraints in a flexible way.

In 2004, this work has been continued in two ways:

• In computer vision, the tool was improved and a full description appears in the PhD thesis of Marta Wilczkowiak (defended in April 2004; this work represents about the half of the thesis contribution) and has been submitted to a journal in computer vision.

• feedback on real-world constraint problems (modeling scenes) has encouraged us to better understand the properties of the main algorithm called GPDOF. GPDOF can be viewed as a general algorithm for decomposing geometric constraint systems and has been presented to the French community working in the CAD and geometric constraint fields .

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