A constructive algebraic strategy for interpolatory subdivision schemes induced by bivariate box splines
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A constructive algebraic strategy for interpolatory subdivision schemes induced by bivariate box splines Costanza Conti · Luca Gemignani · Lucia Romani
Received: 23 March 2012 / Accepted: 3 October 2012 / Published online: 7 November 2012 © Springer Science+Business Media New York 2012
Abstract This paper describes an algebraic construction of bivariate interpolatory subdivision masks induced by three-directional box spline subdivision schemes. Specifically, given a three-directional box spline, we address the problem of defining a corresponding interpolatory subdivision scheme by constructing an appropriate correction mask to convolve with the threedirectional box spline mask. The proposed approach is based on the analysis of certain polynomial identities in two variables and leads to interesting new interpolatory bivariate subdivision schemes. Keywords Interpolatory subdivision schemes · Subdivision symbols · Bezout equation · Bivariate polynomial · Matrix polynomial · Box splines Mathematics Subject Classifications (2010) 65F05 · 68W30 · 65D05 · 41A15
Communicated by: T. Lyche. C. Conti (B) Dipartimento di Energetica “Sergio Stecco”, Università di Firenze, Viale Morgagni 40/44, 50134 Firenze, Italy e-mail: [email protected] L. Gemignani Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy e-mail: [email protected] L. Romani Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via R. Cozzi 53, 20125 Milano, Italy e-mail: [email protected]
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1 Introduction Interpolatory subdivision schemes are useful practical tools to generate graphs of functions, curves and surfaces interpolating given sets of points. As any subdivision scheme, they consist in the repeated application of linear rules determining successive refinements of coarse initial meshes and converging to continuous (interpolatory) limits. In the “standard” bivariate situation, the refinement rules are simple average rules with the average coefficients defining the so called subdivision mask, a finite sequence of real numbers hereafter denoted by a = {aα , α ∈ Z2 }. In the interpolatory case, the mask a satisfies a2α = δα,0 , for all α ∈ Z2 . The subdivision mask a is associated with the bivariate Laurent polynomial, or subdivision symbol, a(z1 , z2 ) =
aα z α ,
zα = zα1 1 zα2 2 ,
z1 , z2 ∈ C ,
α∈Z2
whose algebraic properties translate into analytical properties of the corresponding subdivision scheme and of its limit (see [2, 3, 10, 21] and references therein). For example, a convergent subdivision scheme is interpolatory if and only if its symbol satisfies a(z1 , z2 ) + a(−z1 , z2 ) + a(z1 , −z2 ) + a(−z1 , −z2 ) = 4 , or a “shifted” version of it. This is actually the feature that we are going to exploit in this paper to provide an algebraic procedure for constructing an interpolatory bivariate symbol from a given non–interpolatory one. In particular, in this paper we confine our attention to the case of a three-directional box spline symbol i.e., a(z1
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