49 research outputs found

    C1C^1 Interpolatory Subdivision with Shape Constraints for Curves

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    International audienceWe derive two reformulations of the C1C^1 Hermite subdivision scheme introduced in [12]. One where we separate computation of values and derivatives and one based of refinement of a control polygon. We show that the latter leads to a subdivision matrix which is totally positive. Based on this we give algorithms for constructing subdivision curves that preserve positivity, monotonicity, and convexity

    Hermite Subdivision Schemes and Taylor Polynomials

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    International audienceWe propose a general study of the convergence of a Hermite subdivision scheme H\mathcal H of degree d>0d>0 in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called Taylor subdivision (vector) scheme S\cal S. The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of S\mathcal S is contractive, then S\mathcal S is C0C^0 and H\mathcal H is CdC^d. We apply this result to two families of Hermite subdivision schemes, the first one is interpolatory, the second one is a kind of corner cutting, both of them use Obreshkov interpolation polynomial

    Extended Hermite Subdivision Schemes

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    International audienceSubdivision schemes are efficient tools for building curves and surfaces. For vector subdivision schemes, it is not so straightforward to prove more than the Hölder regularity of the limit function. On the other hand, Hermite subdivision schemes produce function vectors that consist of derivatives of a certain function, so that the notion of convergence automatically includes regularity of the limit. In this paper, we establish an equivalence betweena spectral condition and operator factorizations, then we study how such schemes with smooth limit functions can be extended into ones with higher regularity. We conclude by pointing out this new approach applied to cardinal splines

    From Hermite to stationary subdivision schemes in one and several variables

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    International audienceVector and Hermite subdivision schemes both act on vector data, but since the latter one interprets the vectors as function values and consecutive derivatives they differ by the "renormalization" of the Hermite scheme in any step. In this paper we give an algebraic factorization method in one and several variables to relate any Hermite subdivision scheme that satisfies the so-called spectral condition to a vector subdivision scheme. These factorizations are natural extensions of the "zero at π" condition known for the masks of refinable functions. Moreover, we show how this factorization can be used to investigate different forms of convergence of the Hermite scheme and why the multivariate situation is conceptionally more intricate than the univariate one. Finally, we give some examples of such factorizations

    Hermite Subdivision with Shape Constraints on a Rectangular Mesh

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    International audienceWe study a two parameter version of the Hermite subdivision scheme introduced in [7], wish gives C1C^1 interpolants on rectangular meshes. We prove C1C^1-convergence for a range of the two parameters. By introducing a control grid we can choose the parameters in the scheme so that the interpolant inherits positivity and/or directional monotonicity from the initial data. Several examples are given showing that a desired shape can be achieved even if we use only very crude estimates for the initial slopes

    Dual Hermite subdivision schemes of de Rham-type

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    International audienceThough a Hermite subdivision scheme is non-stationary by nature, its non-stationarity can be of two types, making useful the distinction between Inherently Stationary and Inherently Non-Stationary Hermite subdivision schemes. This paper focuses on the class of inherently stationary, dual non-interpolatory Hermite subdivision schemes that can be obtained from known Hermite interpolatory ones, by applying a generalization of the de Rham corner cutting strategy. Exploiting specific tools for the analysis of inherently stationary Hermite subdivision schemes we show that, giving up the interpolation condition, the smoothness of the associated basic limit function can be increased by one, while its support width is only enlarged by one. To accomplish the analysis of de Rham-type Hermite subdivision schemes two new theoretical results are derived and the new notion of HC-convergence is introduced. It allows the construction of Hermite-type subdivision schemes of order d + 1 with the first element of the vector valued limit function having regularity ≥ d

    Fourier analysis of 2-point Hermite interpolatory subdivision schemes

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    Two subdivision schemes with Hermite data on Z are studied. These schemes use 2 or 7 parameters respectively depending on whether Hermite data involve only first derivatives or include second derivatives. For a large region in the parameters space, the schemes are C1 or C2 convergent or at least are convergent on the space of Schwartz distributions. The Fourier transform of any interpolating function can be computed through products of matrices of order 2 or 3. The Fourier transform is related to a specific system of functional equations whose analytic solution is unique except for a multiplicative constant. The main arguments for these results come from Paley-Wiener-Schwartz theorem on the characterization of the Fourier transforms of distributions with compact support and a theorem of Artzrouni about convergent products of matrices

    Interpolants d'Hermite

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    We propose a two point subdivision scheme with parameters to draw curves that satisfy Hermite conditions at both ends of [a,b]. We build three functions f,p and s on dyadic numbers and, using infinite products of matrices, we prove that, under assumptions on the parameters, these functions can be extended by continuity on [a,b], with f'=p and f''=s

    Approximation de problemes faiblement non lineaires par des schemas de differences finies superconvergents

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    SIGLECNRS T 58283 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc
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