Stationary subdivision

Cavaretta, A. S.; Dahmen, Wolfgang; Micchelli, Charles A.

doi:10.1090/memo/0453

Cited by 410 publications

(600 citation statements)

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“…In [6], Daubechies and Lagarias showed that up to a scalar multiple the refinement equation (1) Moreover, they also showed the nonexistence of a C ∞ -refinable function with compact support in one dimension when the number λ and all the d j are integers. In [2], Cavaretta et al extended this result to higher dimensions by a matrix method. When λ is non-integer, 'the regularity question becomes more complicated and perhaps more interesting from the viewpoint of pure analysis' [4].…”

Section: Introductionmentioning

confidence: 95%

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Refinement equations and spline functions

Dubickas

2008

Adv Comput Math

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In this paper, we exploit the relation between the regularity of refinable functions with non-integer dilations and the distribution of powers of a fixed number modulo 1, and show the nonexistence of a non-trivial C ∞ solution of the refinement equation with non-integer dilations. Using this, we extend the results on the refinable splines with non-integer dilations and construct a counterexample to some conjecture concerning the refinable splines with non-integer dilations. Finally, we study the box splines satisfying the refinement equation with non-integer dilation and translations. Our study involves techniques from number theory and harmonic analysis.

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Section: Introductionmentioning

confidence: 95%

“…For simplicity, we shall call the function f (x) satisfying (1) with f (0) = 1 a λ-refinable function with translations {d j | 0 ≤ j ≤ N }. It plays a fundamental role in the construction of compactly supported wavelets and in the study of subdivision schemes in CAGD ( [2,7]). …”

Section: Introductionmentioning

confidence: 99%

Refinement equations and spline functions

Dubickas

2008

Adv Comput Math

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“…Hence, the order of smoothness of the target surface is determined by that of the refinable function φ. If this refinable function is not a compactly supported piecewise polynomial with prescribed smoothness joining property (called a bivariate spline), the order of smoothness of φ can be analyzed by applying the theory of shift-invariant spaces [3,8,15,19,17,27].…”

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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“…The convergence properties of linear interpolatory subdivision schemes is a well understood subject nowadays (see e.g. [12,17] and references therein), as well as their tendency to reconstruct 'discontinuous' discrete data while creating spurious oscillations in the process. In recent years, several nonlinear subdivision schemes have been proposed (see e.g.…”

Section: Introductionmentioning

confidence: 99%