Research On Multivariate Wavelet Construction With Lifting Scheme

In recent years, multivariate wavelet has generated considerable interest because of their widely applications. And the construction of multivariate non-separable wavelet systems with good synthetic performance becomes a very important aspect on the theory of wavelet analysis. On the other hand, the lifting scheme has recently emerged as a new way for wavelet construction. The main feature of lifting is that it provides an entirely spatial-domain interpretation of the transform, as opposed to the more traditional frequency-domain based constructions. It is worthwhile to simplify the higher dimensional non-separable wavelet construction to some extent.Motivated by the pioneering work on the construction of multivariate Lagrange interpolating wavelet with lifting schemes, in this paper, we focus on the construction of multivariate nonseparable wavelet and nonseparable multiwavelet. The main results include the following:1) We study the wavelet construction based on multi-step lifting scheme, and systematically investigate the relationship between the lifting operators and some important properties of wavelet system. And then, we propose a general framework based on lifting for constructing filter banks and wavelets in any dimension, for any lattice with desired properties such as linear phase, short support length and high order vanishing moment. And it also presents a guideline for the selection of optimal lifting scheme to satisfy the given wavelet properties.2) We study the construction of multivariate Lagrange multiwavelet based on lifting scheme. A new multifilter named as Multi-Neville filter was defined for prediction, and the relations between the Multi-Neville filter and the properties of multiwavelet were systematically investigated. And then, we propose a general recipe for constructing biorthogonal balanced multiwavelets in any dimension, for any lattice with any multiplicity and any order of vanishing moment. A series of biorthogonal multiwavelet with some desired properties were obtained. To our knowledge, there is no formal publication on such systematic construction. 3) We firstly define a new predictor named as Hermite-Neville filter, and from which a general design framework is developed for building Hermite interpolation filter banks of any derivative order for any lattice in any dimension with any number of primal and dual vanishing moments. And almost all of the existed multivariate interpolation wavelets become a special case of our algorithm. Moreover, based on lifting scheme, a novel pre-filter design for arbitrary dimensional Hermite interpolation multiwavelet with arbitrary multiplicity is proposed.