Estimating differential quantities using polynomial fitting of osculating jets

Abstract

This paper addresses the point-wise 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, and either interpolation or approximation. A jet is a truncated Taylor expansion, and the incentive for using jets is that they encode all local geometric quantities -such as normal, curvatures, extrema of curvature.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 R 3 are reported. These experiments illustrate the asymptotic convergence results, but also the robustness of the methods on general Computer Graphics models.

Principal curvatures from the integral invariant viewpoint

“…For this reason, Section 3.3 is rather short, as numerical results have already been presented in that paper. They compare the principal curvatures obtained via the ball and sphere neighbourhoods with other methods: -Principal components of the patch neighbourhood according to Clarenz et al (2004b) and Pauly et al (2003); -The method of normal cycles of Cohen-Steiner and Morvan (2003); -A fitting method (osculating jets by Cazals and Pouget (2003)). Yang et al (2006) report that PCA of the patch neighbourhood is quite sensitive to noise, normal cycles less so.…”

“…the work by Bajaj and Xu (2003), Clarenz et al (2004b), and Osher and Fedkiw (2002)). Local methods, using smooth approximations of the data in an appropriate neighbourhood, are presented by Cazals and Pouget (2003), Goldfeather and Interrante (2004), Ohtake et al (2004), Taubin (1995), and Tong and Tang (2005). In either case, the preservation of features which may not be considered as noise is not an easy task and requires especially adapted algorithms.…”

Generating parametric models of tubes from laser scans

“…This technique is commonly used for estimating normals and curvatures; Pouget [22] gives a detailed analysis of this popular technique, and Lukácz [23] compares it to other methods for estimating curvature in point clouds. Local fitting of polynomials is also used for defining a distance function to the various definitions of moving least squares surfaces, see [7,2].…”

Multigrid convergent principal curvature estimators in digital geometry

“…a cloud of points), the most natural way to approach curvature(s) is to fit a polynomial surface of degree two at least. Perhaps the best representative of these techniques is the osculating jets of Cazals and Pouget [16]. The authors provide O(δ 2 ) convergence results in the case of where data is a surface sampling, assuming δ is the density of points.…”