Multivariate refinable Hermite interpolant

Han, Bin; Yu, Thomas P.-Y.; Piper, Bruce

doi:10.1090/s0025-5718-03-01623-5

Cited by 42 publications

(44 citation statements)

References 33 publications

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“…Our third example is a refinable vector of functions with Hermite interpolation properties (see [16]). Such refinable functions are useful in computer aided geometric design.…”

Section: Subdivision and Transition Operatorsmentioning

confidence: 99%

Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets

Jia¹,

Jiang²

2003

SIAM J. Matrix Anal. & Appl.

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The purpose of this paper is to investigate spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions.Let, where M is an expansive integer matrix. We assume that M is isotropic, i.e., M is similar to a diagonal matrix diag(σ 1 , . . . , σ s ) withThe smoothness of Φ is measured by the critical exponentwhereWe assume that the mask a is finitely supported, i.e., the set suppa := {α ∈ Z Z s : a(α) = 0} is finite. Note that each a(α) is an r × r complex matrix. Let A := α∈Z Z s a(α)/d, where d := | det M |. We assume that spec (A) (the spectrum of A) is {η 1 , η 2 . . . . , η r }, where η 1 = 1 and η j = 1 for j = 2, . . . , r., where ⊗ denotes the (right) Kronecker product. Suppose the highest degree of polynomials reproduced by Φ is k − 1. LetThe main result of this paper asserts that if Φ is stable, then λ(Φ) = − log d ρ k s/2, where. This result is obtained through an extensive use of linear algebra and matrix theory. Three examples are provided to illustrate the general theory. All these examples have background of practical applications.

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“…Our third example is a refinable vector of functions with Hermite interpolation properties (see [16]). Such refinable functions are useful in computer aided geometric design.…”

Section: Subdivision and Transition Operatorsmentioning

confidence: 99%

Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets

Jia¹,

Jiang²

2003

SIAM J. Matrix Anal. & Appl.

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“…(See also [13] for the construction of Hermite interpolation schemes by the parametric approach.) In the present paper, we consider several issues related to the matrix-valued subdivision.…”

Section: Figure 2: Subdivision Templates Of the Catmull-clark Scheme mentioning

confidence: 99%

Matrix-valued subdivision schemes for generating surfaces with extraordinary vertices

Chui

Jiang

2006

Computer Aided Geometric Design

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Subdivision templates of numerical values are replaced by templates of matrices in this paper to allow the introduction of shape control parameters for the feasibility of achieving desirable geometric shapes at those points on the subdivision surfaces that correspond to extraordinary control vertices. Formulation of the matrix-valued subdivision surface is derived. Based on refinable bivariate spline function vectors for matrix-valued subdivisions, the notion of characteristic map introduced by Reif is extended from (scalar) surface subdivisions to matrix-valued subdivisions. The C 1 -and C k -continuity of Reif and Prautzsch for matrix-valued subdivisions are discussed. To illustrate the general theory, the smoothness of matrix-valued triangular subdivision schemes for extraordinary vertices with valences 3 and 4 is analyzed. The issue of effective choices of the shape control parameters will also be discussed in this paper.

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“…in [11,18]. In [2,19,20], the lifting scheme has been applied for biorthogonal multiwavelets generated from Hermite splines also yielding wavelets with arbitrary order of vanishing moments.…”

Section: Lifting Schemementioning

confidence: 99%

On interpolatory divergence-free wavelets

Bittner

Urban

2006

Math. Comp.

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Abstract. We construct interpolating divergence-free multiwavelets based on cubic Hermite splines. We give characterizations of the relevant function spaces and indicate their use for analyzing experimental data of incompressible flow fields. We also show that the standard interpolatory wavelets, based on the Deslauriers-Dubuc interpolatory scheme or on interpolatory splines, cannot be used to construct compactly supported divergence-free interpolatory wavelets.

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Multivariate refinable Hermite interpolant

Cited by 42 publications

References 33 publications

Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets

Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets

Matrix-valued subdivision schemes for generating surfaces with extraordinary vertices

On interpolatory divergence-free wavelets

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