On Multivariate Compactly Supported Bi-Frames

Abstract

In this article, we construct compactly supported multivariate pairs of dual wavelet frames, shortly called bi-frames, for an arbitrary dilation matrix. Our construction is based on the mixed oblique extension principle, and it provides bi-frames with few wavelets. In the examples, we obtain optimal bi-frames, i.e., primal and dual wavelets are constructed from a single fundamental refinable function, whose mask size is minimal w.r.t. sum rule order and smoothness. Moreover, the wavelets reach the maximal approximation order w.r.t. the underlying refinable function. For special dilation matrices, we derive very simple but optimal arbitrarily smooth bi-frames in arbitrary dimensions with only two primal and dual wavelets.

Elementary factorisation of Box spline subdivision

“…But the condition given by our Theorem 1 for a Box spline subdivision to be uniformly elementary factorable is less restrictive. It would then be interesting to investigate what kind of bi-framelets could be built from these more exotic Box splines [6,4,23,11]. In particular, as the polyphase matrix is square, the wavelet bi-frames will suffer from the restrictions of the biorthogonal setting such as dual functions with poor properties [10], but the framework could still be useful in practice.…”

Symmetric tight frame wavelets with dilation factor M=4

“…Bandpass filter H 2 (z) is known from equation (36). Similar to Section 3.3, the filters {h 1 , h 5 , h 6 } are then obtained using a(z) and b(z) as polyphase components as follows:…”

“…The resulting limit functions are smooth and benefit from short supports. Multivariate dual frames based on the mixed oblique extension principle with six wavelets and M = 4 are discussed in [36]. In [37] the authors present complex symmetric orthonormal wavelets based on FIR filters with M = 4.…”

Wavelet bi-frames with few generators from multivariate refinable functions

“…In the univariate setting, one can derive compactly supported wavelet bi-frames with few generators from any pair of compactly supported refinable functions, see [6]. In arbitrary dimensions, the frame concept has already been applied to construct arbitrarily smooth compactly supported bi-frames with few generators satisfying a variety of optimality conditions, see [8,9]. There, the underlying refinable function ϕ =φ must be fundamental, i.e., ϕ(k) = δ 0,k , for all k ∈ Z d , and their Fourier transform must have a certain required factorization.…”