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29
Estimating differential quantities using polynomial fitting of osculating jets
"... 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, and either interpolation ..."
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Cited by 87 (2 self)
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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, 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 wellstudied 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.
A chronology of interpolation: From ancient astronomy to modern signal and image processing
 Proceedings of the IEEE
, 2002
"... This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into histo ..."
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Cited by 61 (0 self)
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This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation. Keywords—Approximation, convolutionbased interpolation, history, image processing, polynomial interpolation, signal processing, splines. “It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it. ” 1 I.
Bivariate interpolation at Xu points: results, extensions and applications, Electron
 Trans. Numer. Anal
, 2006
"... Abstract. In a recent paper, Y. Xu proposed a set of Chebyshevlike points for polynomial interpolation on the square [−1, 1] 2. We have recently proved that the Lebesgue constant of these points grows like log 2 of the degree (as with the best known points for the square), and we have implemented a ..."
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Cited by 7 (3 self)
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Abstract. In a recent paper, Y. Xu proposed a set of Chebyshevlike points for polynomial interpolation on the square [−1, 1] 2. We have recently proved that the Lebesgue constant of these points grows like log 2 of the degree (as with the best known points for the square), and we have implemented an accurate version of their Lagrange interpolation formula at linear cost. Here we construct nonpolynomial Xulike interpolation formulas on bivariate compact domains with various geometries, by means of composition with suitable smooth transformations. Moreover, we show applications of Xulike interpolation to the compression of surfaces given as large scattered data sets. Key words. Bivariate polynomial interpolation, Xu points, Lebesgue constant, domains transformations, generalized rectangles, generalized sectors, large scattered data sets, surface compression. AMS subject classifications. 65D05 1. Introduction. The
Gröbner basis structure of finite sets of points
, 2003
"... We study the relationship between certain Gröbner bases for zerodimensional radical ideals, and the varieties defined by the ideals. Such a variety is a finite set of points in an affine ndimensional space. We are interested in monomial orders that “eliminate ” one variable, say z. Eliminating z ..."
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Cited by 7 (2 self)
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We study the relationship between certain Gröbner bases for zerodimensional radical ideals, and the varieties defined by the ideals. Such a variety is a finite set of points in an affine ndimensional space. We are interested in monomial orders that “eliminate ” one variable, say z. Eliminating z corresponds to projecting points in nspace to (n − 1)space by discarding the zcoordinate. We show that knowing a minimal Gröbner basis under an elimination order immediately reveals some of the geometric structure of the corresponding variety, and knowing the variety makes available information concerning the basis. These relationships can be used to decompose polynomial systems into smaller systems.
Polynomial Interpolation on the Unit Sphere and on the Unit Ball
, 2002
"... The problem of interpolation on the unit sphere S by spherical polynomials of degree at most n is shown to be related to the interpolation on the unit ball B by polynomials of degree n. As a consequence several explicit sets of points on S are given for which the interpolation by spheric ..."
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Cited by 6 (2 self)
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The problem of interpolation on the unit sphere S by spherical polynomials of degree at most n is shown to be related to the interpolation on the unit ball B by polynomials of degree n. As a consequence several explicit sets of points on S are given for which the interpolation by spherical polynomials has a unique solution. We also discuss interpolation on the unit disc of R for which points are located on the circles and each circle has an even number of points. The problem is shown to be related to interpolation on the triangle in a natural way.
Polynomial Interpolation On The Unit Sphere
 SIAM J. Numer. Anal
"... The problem of interpolation at (n + 1) points on the unit sphere S by spherical polynomials of degree at most n is studied. Many sets of points that admit unique interpolation are given explicitly. The proof is based on a method of factorization of polynomials. A related problem of interpolat ..."
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Cited by 6 (4 self)
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The problem of interpolation at (n + 1) points on the unit sphere S by spherical polynomials of degree at most n is studied. Many sets of points that admit unique interpolation are given explicitly. The proof is based on a method of factorization of polynomials. A related problem of interpolation by trigonometric polynomials is also solved.
Solving Toeplitz and Vandermondelike Linear Systems with Large Displacement Rank
, 2007
"... Linear systems with structures such as Toeplitz, Vandermonde or Cauchylikeness can be solved in O˜(α 2 n) operations, where n is the matrix size, α is its displacement rank, and O ˜ denotes the omission of logarithmic factors. We show that for Toeplitzlike and Vandermondelike matrices, this cos ..."
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Cited by 4 (2 self)
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Linear systems with structures such as Toeplitz, Vandermonde or Cauchylikeness can be solved in O˜(α 2 n) operations, where n is the matrix size, α is its displacement rank, and O ˜ denotes the omission of logarithmic factors. We show that for Toeplitzlike and Vandermondelike matrices, this cost can be reduced to O˜(α ω−1 n), where ω is a feasible exponent for matrix multiplication over the base field. The best known estimate for ω is ω < 2.38, resulting in costs of order O˜(α 1.38 n). We also present consequences for HermitePadé approximation and bivariate interpolation.
DIRECTIONS FOR COMPUTING TRUNCATED MULTIVARIATE TAYLOR SERIES
"... Abstract. Efficient recurrence relations for computing arbitraryorder Taylor coefficients for any univariate function can be directly applied to a function of n variables by fixing a direction in R n. After a sequence of directions, the multivariate Taylor coefficients or partial derivatives can be ..."
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Cited by 2 (1 self)
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Abstract. Efficient recurrence relations for computing arbitraryorder Taylor coefficients for any univariate function can be directly applied to a function of n variables by fixing a direction in R n. After a sequence of directions, the multivariate Taylor coefficients or partial derivatives can be reconstructed or “interpolated”. The sequence of univariate calculations is more efficient than multivariate methods, although previous work indicates a space cost for this savings and significant cost for the reconstruction. We completely eliminate this space cost and develop a much more efficient algorithm to perform the reconstruction. By appropriate choice of directions, the reconstruction reduces to a sequence of Lagrange polynomial interpolation problems in R n−1 for which a divided difference algorithm computes the coefficients of a Newton form. Another algorithm collects like terms from the Newton form and returns the desired multivariate coefficients. 1.