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Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets
Author(s) -
Ji Hui,
Riemenschneider Sherman D.,
Shen Zuowei
Publication year - 1999
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00108
Subject(s) - biorthogonal system , mathematics , wavelet , dual polyhedron , pure mathematics , interpolation (computer graphics) , mathematical analysis , wavelet transform , computer science , animation , computer graphics (images) , artificial intelligence
In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function φ which has sufficient regularity and which is fundamental for interpolation [that means, φ(0)=1 and φ(α)=0 for all α∈ Z s ∖{0}]. Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function φ. Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces. A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed.