Quincunx fundamental refinable functions  and quincunx biorthogonal wavelets

Han, Bin; Jia, Rong-Qing

doi:10.1090/s0025-5718-00-01300-4

Cited by 41 publications

(30 citation statements)

References 27 publications

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“…It is straightforward to show that (35) holds when n = 1 (k 0 must be 0) for both c n,k 0 G n,k 0 and d n,k 0 H n,k 0 , since it reduces to the case m = (2n + 1, 0) for 1-dimensional interpolatory mask a 2n+1 and the case m = (2n − 1, 1) in [18].…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Han and Jia in ( [18], Theorem 3.3) constructed a family of quincunx interpolatory subdivision schemes associated with a family, {a (m,n) : m, n ∈ N 0 , m + n odd}, of 2-dimensional unique quincunx interpolatory masks, which can be viewed as the generalization of the family of Deslauriers and Dubuc's interpolatory masks in dimension one to dimension two in the following sense:…”

Section: Introductionmentioning

confidence: 99%

“…It should be pointed out that the results in [18] are already highly nontrivial. Many of the results are tailored for dimension two.…”

Section: Introductionmentioning

confidence: 99%

“…It was not clear whether techniques in that paper can be carried over directly to higher dimensions. In this paper, we develop new techniques that extend many results in [18] to arbitrary dimensions and give a complete picture of the unique family of quincunx interpolatory masks in arbitrary dimension d ∈ N. That is, we prove the following main theorem. (1) a m is a quincunx interpolatory mask; i.e., (6) holds.…”

Section: Introductionmentioning

confidence: 99%

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Quincunx Fundamental Refinable Functions in Arbitrary Dimensions

Zhuang

2017

Axioms

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Abstract:In this paper, we generalize the family of Deslauriers-Dubuc's interpolatory masks from dimension one to arbitrary dimensions with respect to the quincunx dilation matrices, thereby providing a family of quincunx fundamental refinable functions in arbitrary dimensions. We show that a family of unique quincunx interpolatory masks exists and such a family of masks is of real value and has the full-axis symmetry property. In dimension d = 2, we give the explicit form of such unique quincunx interpolatory masks, which implies the nonnegativity property of such a family of masks.

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Section: Proof Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…It should be pointed out that the results in [18] are already highly nontrivial. Many of the results are tailored for dimension two.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quincunx Fundamental Refinable Functions in Arbitrary Dimensions

Zhuang

2017

Axioms

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“…For the 1-to-4 split rule, the dilation matrix to be selected is simply 2I 2 , both for triangular and quadrilateral meshes. Other topological rules of interest include the √ 3 [22,23,20,28,21,6] and the √ 2 split [40,41,12,14,24] rules, with dilation matrices given, for example, by…”

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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Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets

Han

2001

Journal of Approximation Theory

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Multiwavelets are generated from refinable function vectors by using multiresolution analysis. In this paper we investigate the approximation properties of a multivariate refinable function vector associated with a general dilation matrix in terms of both the subdivision operator and the order of sum rules satisfied by the matrix refinement mask. Based on a fact about the sum rules of biorthogonal multiwavelets, a coset by coset (CBC) algorithm is presented to construct biorthogonal multiwavelets with arbitrary order of vanishing moments. More precisely, to obtain biorthogonal multiwavelets, we have to construct primal and dual masks. Given any primal matrix mask a and a general dilation matrix M, the proposed CBC algorithm reduces the construction of all dual masks of a, which satisfy the sum rules of arbitrary order, to a problem of solving a well organized system of linear equations. We prove in a constructive way that for any given primal mask a with a dilation matrix M and for any positive integer k, we can always construct a dual mask aõ f a such that a~satisfies the sum rules of order k. In addition, we provide a general way for the construction of Hermite interpolatory matrix masks in the univariate setting with any dilation factors. From such Hermite interpolatory masks, smooth Hermite interpolants, including the well known cubic Hermite splines as a special case, are obtained and are used to construct biorthogonal multiwavelets. As an example, a C 3 Hermite interpolant with support [&3, 3] is presented. Then we shall apply the CBC algorithm to such Hermite interpolatory masks to construct biorthogonal multiwavelets. Several examples of biorthogonal multiwavelets are provided to illustrate the general theory. In particular, a C 1 dual function vector with support [&4, 4] of the cubic Hermite splines is given. Academic Press

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Quincunx fundamental refinable functions and quincunx biorthogonal wavelets

Cited by 41 publications

References 27 publications

Quincunx Fundamental Refinable Functions in Arbitrary Dimensions

Quincunx Fundamental Refinable Functions in Arbitrary Dimensions

Matrix-valued subdivision schemes for generating surfaces with extraordinary vertices

Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets

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