Wavelets for multichannel signals

In this paper, we introduce and investigate multichannel wavelets, which are wavelets for vector fields, based on the concept of full rank subdivision operators. We prove that, like in the scalar and multiwavelet case, the existence of a scaling function with orthogonal integer translates guarantees the existence of a wavelet function, also with orthonormal integer translates. In this context, however, scaling functions as well as wavelets turn out to be matrix-valued functions.     

Full rank interpolatory subdivision schemes: Kronecker, filters and multiresolution

In this extension of earlier work, we point out several ways how a multiresolution analysis can be derived from a finitely supported interpolatory matrix mask which has a positive definite symbol on the unit circle except at −1. A major tool in this investigation will be subdivision schemes that are obtained by using convolution or correlation operations based on replacing the usual matrix multiplications by Kronecker products.     

Full rank interpolatory subdivision: A first encounter with the multivariate realm

We extend our previous work on interpolatory vector subdivision schemes to the multivariate case. As in the univariate case we show that the diagonal and off-diagonal elements of such a scheme have a significantly different structure and that under certain circumstances symmetry of the mask can increase the polynomial reproduction power of the subdivision scheme. Moreover, we briefly point out how tensor product constructions for vector subdivision schemes can be obtained.     

Full rank positive matrix symbols: interpolation and orthogonality

We investigate full rank interpolatory vector subdivision schemes whose masks are positive definite on the unit circle except the point z=1. Such masks are known to give rise to convergent schemes with a cardinal limit function in the scalar case. In the full rank vector case, we show that there also exists a cardinal refinable function based on this mask, however, with respect to a different notion of refinability which nevertheless also leads to an iterative scheme for the computation of vector fields. Moreover, we show the existence of orthogonal scaling functions for multichannel wavelets and give a constructive method to obtain these scaling functions. AMS subject classification (2000) 42C40, 65T60, 65D05     

A Method for the Construction of Families of Biorthogonal Filters with Prescribed Properties

Some necessary and sufficient conditions are given in order to obtain biorthogonal wavelet filters with properties which are usually required in applications, such as symmetry and regularity order, by appropriately choosing the parameters in the lifting scheme. A special class of filters extending the biorthogonal B-spline filters is then proposed and analyzed.     

Multifilters with and without Prefilters

To explore the full approximation order and thus compression power of a multifilter, it is usually necessary to incorporate prefilters. Using matrix factorization techniques, we describe an explicit construction of such prefilters. Although in the case of approximation order 1 these prefilters are simply bi-infinite block diagonal matrices, they can become very intricate as soon as one aims for higher approximation order. For this reason, we introduce a particular class of multifilters which we call full rank multifilters. These filters have a peculiar structure which allows us to obtain approximation order without the use of prefilters. The construction of such filters via the lifting scheme is pointed out and examples of the performance of these filters for image compression are given.     

An algebraic construction of k-balanced multiwavelets via the lifting scheme

Multiwavelets have been revealed to be a successful generalization within the context of wavelet theory. Recently Lebrun and Vetterli have introduced the concept of “balanced” multiwavelets, which present properties that are usually absent in the case of classical multiwavelets and do not need the prefiltering step. In this work we present an algebraic construction of biorthogonal multiwavelets by means of the well-known “lifting scheme”. The flexibility of this tool allows us to exploit the degrees of freedom left after satisfying the perfect reconstruction condition in order to obtain finite k-balanced multifilters with custom-designed properties which give rise to new balanced multiwavelet bases. All the problems we deal with are stated in the framework of banded block recursive matrices, since simplified algebraic conditions can be derived from this recursive approach. This revised version was published online in June 2006 with corrections to the Cover Date.     

Factorization of Hermite subdivision operators preserving exponentials and polynomials

In this paper we focus on Hermite subdivision operators that act on vector valued data interpreting their components as function values and associated consecutive derivatives. We are mainly interested in studying the exponential and polynomial preservation capability of such kind of operators, which can be expressed in terms of a generalization of the spectral condition property in the spaces generated by polynomials and exponential functions. The main tool for our investigation are convolution operators that annihilate the aforementioned spaces, which apparently is a general concept in the study of various types of subdivision operators. Based on these annihilators, we characterize the spectral condition in terms of factorization of the subdivision operator.     

Multifilters and prefilters: Uniqueness and algorithmic aspects

In this paper we study the structure of prefilters which ensure a certain approximation order of a given multifilter system. We show how all prefilters can be obtained from the generic one and that in general these prefilters are necessarily rational filters, but point out how a judicious choice of parameters can be used to obtain FIR filters for the approximation orders 1,2,3$1,2,3$. Moreover, we illustrate the results by giving prefilters for the standard examples of multiwavelets.