From Hermite to stationary subdivision schemes in one and several variables
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From Hermite to stationary subdivision schemes in one and several variables Jean–Louis Merrien · Tomas Sauer
Received: 30 March 2010 / Accepted: 28 October 2010 / Published online: 29 September 2011 © Springer Science+Business Media, LLC 2011
Abstract Vector and Hermite subdivision schemes both act on vector data, but since the latter one interprets the vectors as function values and consecutive derivatives they differ by the “renormalization” of the Hermite scheme in any step. In this paper we give an algebraic factorization method in one and several variables to relate any Hermite subdivision scheme that satisfies the so– called spectral condition to a vector subdivision scheme. These factorizations are natural extensions of the “zero at π ” condition known for the masks of refinable functions. Moreover, we show how this factorization can be used to investigate different forms of convergence of the Hermite scheme and why the multivariate situation is conceptionally more intricate than the univariate one. Finally, we give some examples of such factorizations. Keywords Subdivision · Hermite · Taylor expansion · Factorization Mathematics Subject Classifications (2010) 41A60 · 65D15 · 13P05
Communicated by T. N. T. Goodman. J.-L. Merrien INSA de Rennes, 20 av. des Buttes de Coesmes, CS 14315, 35043 Rennes Cedex, France e-mail: [email protected] T. Sauer (B) Lehrstuhl für Numerische Mathematik, Justus–Liebig–Universität Gießen, Heinrich-Buff-Ring 44, 35392 Gießen, Germany e-mail: [email protected]
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J.-L. Merrien, T. Sauer
1 Introduction Subdivision algorithms are iterative methods for producing curves and surfaces with a built-in multiresolution structure. They are now used in curve and surface modelling in computer-aided geometric design, video games, animation and many other applications. Stationary and homogeneous subdivision schemes iterate the same subdivision operator and use rules that, independently of the location and the iteration step, compute new values of a refined discrete sequence only from a certain amount of local and neighboring data. This data can be scalar or vector even matrix valued, the matrix case been done by doing vector schemes columnwise, and the stationary, i.e. local and neighboring rule is always acting on a vector sequence c as SAc = A (· − 2α) c(α), α∈Zs
where the finitely supported sequence of coefficients is referred to as the mask of the subdivision scheme. This operator is iterated, leading to a sequence SnA c, n ∈ N, of vector valued sequences, which, when related to the finer and finer grid 2−n Zs , converges to a limit vector field with certain properties. Such schemes have been investigated for example in [4, 12, 15, 19, 20] and there exists quite a substantial amount of literature on vector subdivision schemes in one and several variables meanwhile, investing convergence as well as smoothness of the limit functions or the multiresolution structures generated by these functions. If the vector data to be processed represents the value and
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