Dual multiwavelet frames with high balancing order and compact fast frame transform

Abstract

An interesting method called oblique extension principle (OEP) has been proposed in the literature for constructing compactly supported MRA tight and dual wavelet frames with high vanishing moments and high frame approximation orders. Many compactly supported MRA wavelet frames have been recently constructed from scalar refinable functions via OEP. Despite the great flexibility and popularity of OEP for constructing compactly supported MRA wavelet frames in the literature, however, the associated fast frame transform is generally not compact and a deconvolution appears in the frame transform. Here we say that a frame transform is compact if it can be implemented by convolutions, coupled with upsampling and downsampling, using only finite-impulse-response (FIR) filters. In this paper we shall address several fundamental issues on MRA dual wavelet frames and fast frame transforms. Basically, we present two complementary results on dual wavelet frames which are obtained via OEP from scalar refinable functions (= refinable function vectors with multiplicity one) and from truly refinable function vectors with multiplicity greater than one. On the one hand, by a nontrivial argument, we show that from any pair of compactly supported refinable spline functions ϕ and ϕ ˜ (not necessarily having stable integer shifts) with finitely supported masks, if we require that the associated fast frame transform be compact, then any compactly supported dual wavelet frames derived via OEP from ϕ and ϕ ˜ can have vanishing moments at most one and the frame approximation order at most two. On the other hand, we prove in a constructive way that from any pair of compactly supported refinable function vectors ϕ and ϕ ˜ with multiplicity at least two and with finitely supported masks, then we can always build a pair of compactly supported dual multiwavelet frames in L 2 ( R ) with the following properties: (i) The associated fast frame transform is compact; therefore, no deconvolution appears in the fast frame transform. (ii) All the frame generators achieve the highest possible order of vanishing moments; the pair of dual multiwavelet frames has the highest possible frame approximation order. (iii) The pair of dual multiwavelet frames and its fast frame transform have the highest possible balancing order; therefore, the difficulty of approximation inefficiency facing most multiwavelet transforms does not appear here in the associated fast frame transform. In short, the two desirable properties (i) and (ii) of dual wavelet frames obtained via OEP from scalar refinable functions are generally mutually conflicting, while they coexist very well for truly refinable function vectors and multiwavelets with the additional property of high balancing orders in (iii). In this paper, we shall present a comprehensive study of the frame approximation order, balancing order and the fast frame transform associated with the dual multiwavelet frames derived from refinable function vectors via OEP. One of the key ingredients in our study of MRA multiwavelet frames is an interesting canonical form of a matrix mask which greatly facilitates the investigation of refinable function vectors and multiwavelets. An algorithm is given in this paper for constructing pairs of dual wavelet frames with compact fast frame transforms and high balancing orders. An example of balanced spline dual wavelet frames with compact fast frame transforms is provided to illustrate the algorithm and results of this paper.