370 research outputs found
From Hermite to stationary subdivision schemes in one and several variables
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
Ellipse-preserving Hermite interpolation and subdivision
We introduce a family of piecewise-exponential functions that have the
Hermite interpolation property. Our design is motivated by the search for an
effective scheme for the joint interpolation of points and associated tangents
on a curve with the ability to perfectly reproduce ellipses. We prove that the
proposed Hermite functions form a Riesz basis and that they reproduce
prescribed exponential polynomials. We present a method based on Green's
functions to unravel their multi-resolution and approximation-theoretic
properties. Finally, we derive the corresponding vector and scalar subdivision
schemes, which lend themselves to a fast implementation. The proposed vector
scheme is interpolatory and level-dependent, but its asymptotic behaviour is
the same as the classical cubic Hermite spline algorithm. The same convergence
properties---i.e., fourth order of approximation---are hence ensured
Hermite subdivision schemes, exponential polynomial generation, and annihilators
We consider the question when the so--called spectral condition} for Hermite
subdivision schemes extends to spaces generated by polynomials and exponential
functions. The main tool are convolution operators that annihilate the space in
question which apparently is a general concept in the study of various types of
subdivision operators. Based on these annihilators, we characterize the
spectral condition in terms of factorization of the subdivision operator
Extended Hermite Subdivision Schemes
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
Level-dependent interpolatory Hermite subdivision schemes and wavelets
We study many properties of level-dependent Hermite subdivision, focusing on
schemes preserving polynomial and exponential data. We specifically consider
interpolatory schemes, which give rise to level-dependent multiresolution
analyses through a prediction-correction approach. A result on the decay of the
associated multiwavelet coefficients, corresponding to a uniformly continuous
and differentiable function, is derived. It makes use of the approximation of
any such function with a generalized Taylor formula expressed in terms of
polynomials and exponentials
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