## Estimating differential quantities using polynomial fitting of osculating jets

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Citations: | 87 - 2 self |

### BibTeX

@MISC{Cazals_estimatingdifferential,

author = {F. Cazals and M. Pouget},

title = {Estimating differential quantities using polynomial fitting of osculating jets},

year = {}

}

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### 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.

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Citation Context ...tion on the Gauss curvature of the underlying smooth surface [BCM02]. The estimation methods providing convergence guarantees are all concerned with first and second order differential quantities. In =-=[AB99]-=-, an error bound is proved on the normal estimate to a smooth surface sampled according to a criterion involving the skeleton. The surface area of a mesh and its normal vector field versus those of a ... |

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Citation Context ...| f (n+1) (x)|. Then for k = 0,...,n: |A k − B k | ≤ hn−k+1 c k!(n − k + 1)! . Proof. This result is a simple application of the analysis of the Lagrange interpolation remainder which can be found in =-=[EK66]-=-. Let Rn(x) = f (x) − J A (x), theorem 1 p.289 states that ∀k = 0,...,n and ∀x ∈ [−h,h]: R (k) n−k n (x) = ∏(x − ξ j ) j=0 f (n+1) (η) (n + 1 − k)! 11swith x,ξ j ,η ∈ [−h,h]. For x = 0, this leads to:... |

234 |
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Citation Context ...er of K and d min be the supremum of the diameter of disks inscribed in K. Also denote D k is the differential of order k. The result is a direct consequence of Theorem 2 of [CR72] or remark 3.4.2 of =-=[QV94]-=-, which states that sup{||D k f (x,y) − D k J A,n (x,y)||;(x,y) ∈ K} = O(h n−k+1 ). (13) 8sRephrasing it with our notations yields: |B k− j, j − A k− j, j | = 1 j!(k − J)! |Dk ( f − J A,n ) (0,0) (1,0... |

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Citation Context ...of the method. Figure 21 displays random patches on the Mechanic model, a 12,500 points model reconstructed from the output of a range scanner. In spite of the coarse sampling, patches and 5 See also =-=[HCV52, p197]-=- as well as [HGY + 99]. 14sprincipal directions provide faithful information. In a similar vein, approximation fitting with large neighborhoods Fig. 22 features a noisy triangulation of a graph. In sp... |

113 | Restricted delaunay triangulations and normal cycle
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Citation Context ...a sampled surface for several methods are given in [MW00]. Based upon the normal cycle and restricted Delaunay triangulations, an estimate for the second fundamental form of a surface is developed in =-=[CSM03]-=-. Another striking fact about the estimation of first and second order differential quantities is that for plane curves, these quantities are often estimated using the osculating circle, while for sur... |

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Citation Context ...lies on n lines, since they define an algebraic curve of degree n. One the other hand, triangular lattices yield poised problems for every degree. These results and further extensions can be found in =-=[GS00]-=- and references therein. p Figure 2: Two quadrics whose intersection curve I projects onto the parabola C : x = y2 . Interpolation points located on I do not uniquely define an interpolating height fu... |

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Citation Context ...lving a third point r which is not a local neighbor of p cannot minimize r circ and are also discarded. A more formal argument advocating the choice of the triangle with minimum r circ is provided in =-=[She02]-=-, where it is shown that the worst error on the approximation of the gradient of a bivariate function by a linear interpolant precisely involves r circ . 6.2 Solving the fitting problem The next stage... |

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Citation Context ...found in [Tau95, CW00]. Ridges of polyhedral surfaces as well as cuspidal edges of the focal sets are computed in [WB01]. Geodesics and discrete versions of the Gauss-Bonnet theorem are considered in =-=[PS98]-=-. Out of all these contributions, few of them address the question of the accuracy of the estimates proposed or that of their convergence when the mesh or the sample points get denser. This lack of so... |

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Citation Context ...everal definitions of normals, principal directions and curvatures over a mesh can be found in [Tau95, CW00]. Ridges of polyhedral surfaces as well as cuspidal edges of the focal sets are computed in =-=[WB01]-=-. Geodesics and discrete versions of the Gauss-Bonnet theorem are considered in [PS98]. Out of all these contributions, few of them address the question of the accuracy of the estimates proposed or th... |

48 |
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Citation Context ...tages methods. Fitting a 2-jet provides estimates of the tangent plane and the curvature related information. These steps can be carried out sequentially or simultaneously. Following the guideline of =-=[SZ90]-=-, most of the methods already mentioned proceed sequentially. The provably good algorithm we propose proceeds simultaneously. Along its analysis, we also provide theoretical results on the accuracy of... |

41 |
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Citation Context ...nd its normal vector field versus those of a smooth surface are considered in [MT02]. Asymptotic estimates for the normal and the Gauss curvature of a sampled surface for several methods are given in =-=[MW00]-=-. Based upon the normal cycle and restricted Delaunay triangulations, an estimate for the second fundamental form of a surface is developed in [CSM03]. Another striking fact about the estimation of fi... |

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Citation Context ...lies on n lines, since they define an algebraic curve of degree n. One the other hand, triangular lattices yield poised problems for every degree. These results and further extensions can be found in =-=[GS00]-=- and references therein. Figure 2: Two quadrics whose intersection curve I projects onto the parabola C : x = y 2 . Interpolation points located on I do not uniquely define an interpolating height fun... |

37 | On multivariate Lagrange interpolation
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Citation Context ...sed iff the set of nodes X is not a subset of any algebraic curve of degree at most n, or equivalently the Vandermonde determinant formed by the interpolation equations does not vanish. As noticed in =-=[SX95]-=-, the set of nodes for which the problem is not poised has measure zero, hence it is almost always poised. However let us illustrate non-poised cases or degenerate configurations of nodes, together wi... |

30 |
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Citation Context ......,N , dmax be the diameter of K and d min be the supremum of the diameter of disks inscribed in K. Also denote D k is the differential of order k. The result is a direct consequence of Theorem 2 of =-=[CR72]-=- or remark 3.4.2 of [QV94], which states that sup{||D k f (x,y) − D k J A,n (x,y)||;(x,y) ∈ K} = O(h n−k+1 ). (13) 8sRephrasing it with our notations yields: |B k− j, j − A k− j, j | = 1 j!(k − J)! |D... |

16 |
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Citation Context ...uget@inria.fr 1sdiscretized surface [MT02], or the angular defect at a vertex of a triangulation which usually does not provide any information on the Gauss curvature of the underlying smooth surface =-=[BCM02]-=-. The estimation methods providing convergence guarantees are all concerned with first and second order differential quantities. In [AB99], an error bound is proved on the normal estimate to a smooth ... |

15 | Computational aspects of multivariate polynomial interpolation
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Citation Context .... 7sAlternatives for the interpolation case. An alternative to the Vandermonde system consists of using the basis of Newton polynomials. Resolution of the system can be done using divided differences =-=[Sau95]-=-, a numerically accurate yet instable method [Hig96]. 4 Surfaces 4.1 Problem addressed Let S be a surface and p be a point of S. Without loss of generality, we assume p is located at the origin and we... |

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Citation Context ......,N , dmax be the diameter of K and d min be the supremum of the diameter of disks inscribed in K. Also denote D k is the differential of order k. The result is a direct consequence of Theorem 2 of =-=[CR72]-=- or remark 3.4.2 of [QV94], which states that sup{||D k f (x,y) − D k J A,n (x,y)||;(x,y) ∈ K} = O(h n−k+1 ). (13) 8Rephrasing it with our notations yields: |B k− j, j − A k− j, j | = 1 j!(k − J)! |D... |

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Citation Context ...is, 2004 route des Lucioles, F-06902 Sophia-Antipolis; Frederic.Cazals@inria.fr § INRIA Sophia-Antipolis, 2004 route des Lucioles, F-06902 Sophia-Antipolis; Marc.Pouget@inria.fr 1sdiscretized surface =-=[MT02]-=-, or the angular defect at a vertex of a triangulation which usually does not provide any information on the Gauss curvature of the underlying smooth surface [BCM02]. The estimation methods providing ... |

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Citation Context ...s is chosen arbitrarily out of the kth ring. The point-cloud case. The normal at p is first estimated, and the neighbors of p are further retrieved from a power diagram in the estimated tangent plane =-=[BF02]-=- —a provably good procedure if the samples are dense enough. If the number of neighbors collected is less than N, we recursively collect the neighbors of the neighbors. Collecting the points therefore... |

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Citation Context ... (x,y) = k ∑ j=0 B k− j, j x k− j y j . (4) Notice that in Eq. (3), the term O(||(x,y)|| n+1 ) stands for the remainder in Taylor’s multivariate formula. Borrowing to the jargon of singularity theory =-=[BG92]-=- , the truncated Taylor expansion J B,n (x) or J B,n (x,y) is called a degree n jet, or n-jet. Since the differential properties of a n-jet match those of its defining curve/surface up to order n, the... |

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