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Estimating Differential Quantities Using Polynomial Fitting of Osculating Jets

Frédéric Cazals 1 Marc Pouget 1
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : This paper addresses the pointwise estimation of differential properties of a smooth manifold S --a curve in the plane or a surface in 3D --assuming a point cloud sampled over S is provided. The method consists of fitting the local representation of the manifold using a jet, by either interpolating or approximating. A jet is a truncated Taylor expansion, and the incentive for using jets is that they encode all local geometric quantities --such as normal or curvatures. On the way to using jets, the question of estimating differential properties is recasted into the more general framework of multivariate interpolation/approximation, a well-studied problem in numerical analysis. On a theoretical perspective, we prove several convergence results when the samples get denser. For curves and surfaces, these results involve asymptotic estimates with convergence rates depending upon the degree of the jet used. For the particular case of curves, an error bound is also derived. To the best of our knowledge, these results are among the first ones providing accurate estimates for differential quantities of order three and more. On the algorithmic side, we solve the interpolation/approximation problem using Vandermonde systems. Experimental results for surfaces of R3 are reported. These experiments illustrate the asymptotic convergence results, but also the robustness of the methods on general Computer Graphics models.
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Frédéric Cazals, Marc Pouget. Estimating Differential Quantities Using Polynomial Fitting of Osculating Jets. Eurographics Symposium on Geometry Processing, Jun 2003, Aachen, Germany. pp.177-187. ⟨hal-00103047⟩

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