Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets

Abstract

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∈Zs∖{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.

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

“…It is well-known (e.g., see [15,19,28,42]) that the interpolatory refinable function can be also used to construct biorthogonal wavelets in L 2 (R). The main idea is to factorize the mask A m into a pair of dual masks.…”

“…In order to make the construction work, it is important to make sure that both corresponding refinable functions are in L 2 (R). This idea of [15] is also used to construct biorthogonal wavelets from pseudo-splines by [23] and multivariate biorthogonal wavelets by [27,28,42]. However, since we allow one of φ orφ to be a distribution when we consider pairs of dual Riesz wavelet systems in (H …”

A note on the construction of nonseparable wavelet bases and multiwavelet matrix filters of 𝐿²(ℝⁿ), where 𝕟≥2

“…It is known (see [3,7,9]) that if T H0 denote the transition operator associated with H 0 and defined by…”

Quincunx fundamental refinable functions and quincunx biorthogonal wavelets

“…Quincunx biorthogonal wavelets are useful in image processing [21] because of their special properties. For biorthogonal wavelets, the reader is referred to [2,3,4,5,6,12,13,16,21,22,26,27] and references therein.…”