Level-Dependent Interpolatory Hermite Subdivision Schemes and Wavelets

Cotronei, Mariantonia; Moosmüller, Caroline; Sauer, Tomas; Sissouno, Nada

doi:10.1007/s00365-018-9444-4

Cited by 24 publications

(16 citation statements)

References 24 publications

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“…Section 5 introduces our Hermite prediction-correction scheme for manifoldvalued data, which is a direct generalization of [7] and makes use of natural tools in nonlinear geometries such as the exponential map and the parallel transport operator. In this section we also prove that the wavelet coefficients at level n decay as 2 −2n for dense enough input data, showing that manifold-valued Hermite wavelets have similar properties as their linear counterparts [6].…”

Section: Introductionmentioning

confidence: 69%

“…This paper focuses on multiwavelets of Hermite-type, meaning that the multiscaling functions satisfy Hermite conditions [6,7,38]. Such wavelet systems can find applications in contexts where Hermite data need to be processed, for example for compression or denoising reasons.…”

Section: Introductionmentioning

confidence: 99%

“…The main result of this paper is a wavelet coefficient decay property of such manifold-valued wavelets, which mimics the linear case [6] and can be considered as an extension of [17] to Hermite-type interpolatory wavelets.…”

Section: Introductionmentioning

confidence: 99%

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Hermite multiwavelets for manifold-valued data

Cotronei¹,

Moosmüller²,

Sauer³

et al. 2021

Preprint

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In this paper we present a construction of interpolatory Hermite multiwavelets for functions that take values in nonlinear geometries such as Riemannian manifolds or Lie groups. We rely on the strong connection between wavelets and subdivision schemes to define a prediction-correction approach based on Hermite subdivision schemes that operate on manifoldvalued data. The main result concerns the decay of the wavelet coefficients: We show that our manifold-valued construction essentially admits the same coefficient decay as linear Hermite wavelets, which also generalizes results on manifold-valued scalar wavelets.

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Section: Introductionmentioning

confidence: 69%

Section: Introductionmentioning

confidence: 99%

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Hermite multiwavelets for manifold-valued data

Cotronei¹,

Moosmüller²,

Sauer³

et al. 2021

Preprint

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“…, φ r;r−1 ) T . The refinement property (2) makes Hermite B-splines particularly interesting in the context of vector multi-resolution analysis, multi-wavelets, and Hermite subdivision schemes [3][4][5][6][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

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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“…Hermite subdivision schemes find applications in geometric modeling (if derivatives are of interest) [16,31,32], in approximation theory (linear and manifold-valued) [19,22,25,26], and they can be used for the construction of multiwavelets [7,8,21] and the analysis of biomedical images [6,30].…”

Section: Introductionmentioning

confidence: 99%

A note on spectral properties of Hermite subdivision operators

Moosmüller

2019

Computer Aided Geometric Design

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In this paper we study the connection between the spectral condition of an Hermite subdivision operator and polynomial reproduction properties of the associated subdivision scheme. While it is known that in general the spectral condition does not imply the reproduction of polynomials, we here prove that a special spectral condition (defined by shifted monomials) is actually equivalent to the reproduction of polynomials. We further put into evidence that the sum rule of order ℓ > d associated with an Hermite subdivision operator of order d does not imply that the spectral condition of order ℓ is satisfied, while it is known that these two concepts are equivalent in the case ℓ = d.

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Level-Dependent Interpolatory Hermite Subdivision Schemes and Wavelets

Cited by 24 publications

References 24 publications

Hermite multiwavelets for manifold-valued data

Hermite multiwavelets for manifold-valued data

Hermite B-Splines: n-Refinability and Mask Factorization

A note on spectral properties of Hermite subdivision operators

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