In this work we present some results and applications concerning the recent theory of multi-scaling functions and multiwavelets. In particular, we present the theory in compact notation with the use of some types of recursive block matrices. This allows a exible schematization of the construction of semi-orthogonal multiwavelets. As in the scalar case, an e cient algorithm for the computation of the coe cients of a multiwavelet transform can be obtained, in which r input sequences are involved. This is a crucial point: the choice of a good pre lter which can provide a good approximation of the true initial coe cient sequences, when applied to the input data, is critical in the context of multiwavelet analysis. We explore this problem with concrete examples, showing the strong dependence of the pre lter on the chosen multiwavelet basis. Finally, an application of the multiwavelet-based algorithm to signal compression is shown. The goal is both to compare the results obtained with di erent multiwavelet bases, and to test the e ectiveness of multiwavelets in this kind of problem with respect to scalar wavelets.

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.

The aim of the paper is to present Hermite-type multiwavelets satisfying the vanishing moment property with respect to elements in the space spanned by exponentials and polynomials. Such functions satisfy a two-scale relation which is level-dependent as well as the corresponding multiresolution analysis. An important feature of the associated filters is the possibility of factorizing their symbols in terms of the so-called cancellation operator. A family of biorthogonal multiwavelet system possessing the above property and obtained from a Hermite subdivision scheme reproducing polynomial and exponential data is finally introduced.

No abstract

In this paper, multiwavelets are considered in the context of image compression and two orthonormal multiwavelet bases are experimented, each used in connection with its proper prefilter. For evaluating the effectiveness of multiwavelet transform for coding images at low bit-rates, an efficient embedded coding of multiwavelet coefficients has been realized. The performance of this multiwavelet-based coder is compared with the results obtained for scalar wavelets.