On the refinement matrix mask of interpolating Hermite splines

Romani, Lucia; Viscardi, Alberto

doi:10.1016/j.aml.2020.106524

Cited by 9 publications

(6 citation statements)

References 11 publications

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“…Lagrange and Hermite subdivision schemes have been extensively studied in [3,4,[6][7][8]10,11,14,15,[23][24][25][27][28][29] and references therein, while multivariate generalized Hermite subdivision schemes of type Λ have been systematically studied in [16].…”

Section: Strengthen Lagrange Hermite and Generalized Hermite Subdivis...mentioning

confidence: 99%

Vector Subdivision Schemes for Arbitrary Matrix Masks

Han¹

2022

Preprint

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Employing a matrix mask, a vector subdivision scheme is a fast iterative averaging algorithm to compute refinable vector functions for wavelet methods in numerical PDEs and to produce smooth curves in CAGD. In sharp contrast to the well-studied scalar subdivision schemes, vector subdivision schemes are much less well understood, e.g., Lagrange and (generalized) Hermite subdivision schemes are the only studied vector subdivision schemes in the literature. Because many wavelets used in numerical PDEs are derived from refinable vector functions whose matrix masks are not from Hermite subdivision schemes, it is necessary to introduce and study vector subdivision schemes for any general matrix masks in order to compute wavelets and refinable vector functions efficiently. For a general matrix mask, we show that there is only one meaningful way of defining a vector subdivision scheme. Motivated by vector cascade algorithms and recent study on Hermite subdivision schemes, we shall define a vector subdivision scheme for any arbitrary matrix mask and then we prove that the convergence of the newly defined vector subdivision scheme is equivalent to the convergence of its associated vector cascade algorithm. We also study convergence rates of vector subdivision schemes. The results of this paper not only bridge the gaps and establish intrinsic links between vector subdivision schemes and vector cascade algorithms but also strengthen and generalize current known results on Lagrange and (generalized) Hermite subdivision schemes. Several examples are provided to illustrate the results in this paper on various types of vector subdivision schemes with convergence rates.

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Section: Strengthen Lagrange Hermite and Generalized Hermite Subdivis...mentioning

confidence: 99%

Vector Subdivision Schemes for Arbitrary Matrix Masks

Han¹

2022

Preprint

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“…so, in the case of arity n = 2, 3, the positive indexed coefficients are given by: A Note that these are the same masks as obtained in [14] [Example 4.2, 4.3], but the computational effort for our construction is less.…”

Section: Spectral Condition and Computation Of The Maskmentioning

confidence: 99%

“…The construction proposed in [13], for example, relies on a recursive procedure for evaluating the explicit expression of the Hermite Bspline vectors of any order. The case of a general dilation factor has been recently studied in [14] and it exploits the refinability properties of the scalar cardinal B-splines with simple knots. Our computation strategy represents a simpler alternative to [13,14].…”

Section: Introductionmentioning

confidence: 99%

“…The case of a general dilation factor has been recently studied in [14] and it exploits the refinability properties of the scalar cardinal B-splines with simple knots. Our computation strategy represents a simpler alternative to [13,14]. It is a direct consequence of the polynomial reproduction properties of the Hermite B-splines, which, in turn, are linked to the spectral condition or sum rule property of the associated Hermite subdivision scheme [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

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Hermite B-Splines: n-Refinability and Mask Factorization

Cotronei

Moosmüller

2021

Mathematics

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This paper deals with polynomial Hermite splines. In the first part, we provide a simple and fast procedure to compute the refinement mask of the Hermite B-splines of any order and in the case of a general scaling factor. Our procedure is solely derived from the polynomial reproduction properties satisfied by Hermite splines and it does not require the explicit construction or evaluation of the basis functions. The second part of the paper discusses the factorization properties of the Hermite B-spline masks in terms of the augmented Taylor operator, which is shown to be the minimal annihilator for the space of discrete monomial Hermite sequences of a fixed degree. All our results can be of use, in particular, in the context of Hermite subdivision schemes and multi-wavelets.

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“…The excellent approximation capabilities and minimal-support property of the Hermite splines [30] give a strong incentive to investigate more general multi-spline spaces. The bicubic Hermite splines are the backbone of many computer-graphics applications and closely linked to Bézier curves [31,32,33,34]. Schoenberg and Lipow also found two fundamental functions to reconstruct any function in S 4 +S 5 from its samples and the samples of its first-order derivative.…”

Section: Multi-splinesmentioning

confidence: 99%

Shortest-support Multi-Spline Bases for Generalized Sampling

Goujon¹,

Aziznejad²,

Naderi³

et al. 2020

Preprint

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Generalized sampling consists in the recovery of a function f , from the samples of the responses of a collection of linear shift-invariant systems to the input f . The reconstructed function is typically a member of a finitely generated integer-shift-invariant space that can reproduce polynomials up to a given degree M . While this property allows for an approximation power of order (M + 1), it comes with a tradeoff on the length of the support of the basis functions. Specifically, we prove that the sum of the length of the support of the generators is at least (M +1). Following this result, we introduce the notion of shortest basis of degree M , which is motivated by our desire to minimize the computational costs. We then demonstrate that any basis of shortest support generates a Riesz basis. Finally, we introduce a recursive algorithm to construct the shortest-support basis for any multi-spline space. It provides a generalization of both polynomial and Hermite B-splines. This framework paves the way for novel applications such as fast derivative sampling with arbitrarily high approximation power.

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On the refinement matrix mask of interpolating Hermite splines

Cited by 9 publications

References 11 publications

Vector Subdivision Schemes for Arbitrary Matrix Masks

Vector Subdivision Schemes for Arbitrary Matrix Masks

Hermite B-Splines: n-Refinability and Mask Factorization

Shortest-support Multi-Spline Bases for Generalized Sampling

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