A note on Hermite multiwavelets with polynomial and exponential vanishing moments
Introduction
It is well-known that multiwavelets generalize classical wavelets in the sense that the corresponding multiresolution analysis (for which we will often use the acronym MRA) is generated by translates and dilates of not just one but several functions. These functions can be assembled in a vector, also known as multi-scaling function, satisfying a vector refinement equation, whose coefficients are matrices rather than scalars (see [17] for an overview on the topic). Such generalization can result in some advantages connected to the possibility of constructing multiwavelet bases, for example, with short support and a high number of vanishing moments. In the scalar situation, the latter is a very crucial and desired property in applications since it reflects on the filters, in the sense that they are able to cancel polynomial discrete data up to a certain degree, thus assuring certain compression capabilities of the overall system associated to the discrete wavelet transform. In the vector setting, nevertheless, it cannot be exploited directly in practical implementations, because the vanishing moments associated to the multiwavelet function do not imply corresponding discrete cancellation properties on the filters side. This results in combining the discrete multiwavelet transform with computationally costly pre-processing and post-processing steps [2], [13], unless full-rank filters [12], [4], [5] or balanced multiwavelets [1], [18] are used. Also, except in these cases, no easy factorization of the symbol as in the scalar situation can be considered.
In this paper we are especially interested in multiwavelets of Hermite-type, representing a special instance of multiwavelets, where the sequences involved in the transform process are connected not to generic vector data, but to vectors consisting of function values and consecutive derivatives up to a certain order (in this respect, no pre-processing step is necessary). Such wavelet systems can find applications in many situations where Hermite data are available (for example in problems of motion control) and need to be processed (for example compressed or smoothed).
Our idea is to exploit the close connection between subdivision schemes and wavelet analysis in order to study and construct Hermite multiwavelets from Hermite subdivision schemes, so that the underlying multi-scaling function corresponds to the limit function of the subdivision process. Hermite schemes are a particular case of vector subdivision, as they act on vectors representing function values and derivatives (see, for example, [8], [9], [10], [15], [14], [20]).
In particular, we focus on Hermite multiwavelet filters which provide not only polynomial but also exponential data cancellation. We thus use a notion of vanishing moment which extends the one usually given, which refers just to polynomials. This generalized property assures compression capabilities of the wavelet system also in the case where the given data exhibit transcendental features. Wavelets possessing such property have already been studied for example in [24] in a scalar framework. The vector context offers the advantage of providing a higher number of vanishing moments together with a short support. Hermite-type multiwavelets allow, in addition, to express the cancellation property as the factorization of the wavelet filter in terms of the so-called annihilator or cancellation operator introduced in [6] in the context of the study of level-dependent Hermite subdivision schemes. In [6], [7] some conditions have been proved connected to the preservation of elements in the (polynomial and exponential) space spanned by with . In particular, the preservation property allows the factorization of the subdivision operator in terms of a minimal annihilator.
We show how, given a Hermite subdivision operator based on a level-dependent mask , satisfying the -spectral condition, in the sense specified later, it is always possible to complete it to a biorthogonal system, where the wavelet filter possesses the desired polynomial/exponential cancellation property. In general, when dealing either with matrix filters or the bivariate case, such completion can be involved, but, as we will show, it can be proved and carried out in a very simple and elegant way starting from subdivision schemes of interpolatory type (as, for example, in [5], [11]). Various families of such Hermite multiwavelets can be thus generated by interpolatory Hermite schemes, for example the one provided in [7] and described later in the paper. In this special case, the MRA is associated to level-dependent vector refinable functions which provide a generalization of the well-known Hermite (or finite element) multi-scaling functions proposed, for example, by Strang and Strela in [23]. Some other examples, as we will show, can be considered as well, all of them giving rise, in the limit (i.e. when the frequencies tend to zero), to Hermite multiwavelets with polynomial vanishing moments.
The paper is organized as follows. In Section 2 we fix the notation and present some basic facts about level-dependent (nonstationary) multiresolution analyses of and related discrete wavelet transforms. In Section 3 we provide some details and properties of Hermite subdivision schemes preserving exponential and polynomial data. A strategy for generating Hermite multiwavelets from such schemes is proposed in Section 4, and a factorization result is formulated. In particular, we focus on multiwavelet systems associated to subdivision schemes of interpolatory type. Finally, in Section 5 we give some examples of our construction, based on explicitly given families of interpolatory Hermite subdivision possessing preservation properties. Some conclusions are finally drawn.
Section snippets
Preliminaries and basic facts
Let and , respectively, denote the spaces of all vector-valued and matrix-valued sequences defined on .
A level-dependent MRA of is defined as the nested sequence of spaces each spanned by the dilates and translates of a finite set of functions, which differs from level to level, that is, for , Nonstationary MRAs, in the scalar case (), have been introduced, for example, in [3], [21].
For each , such
Hermite subdivision preserving exponentials and polynomials
Since our aim is to propose an MRA based on Hermite subdivision schemes, we recall some basic facts on such schemes, focusing on subdivision preserving exponential and polynomial data.
An Hermite subdivision scheme consists of the successive applications of level-dependent subdivision operators, acting on vector valued data, whose k-th component corresponds to a k-th derivative of some function f. At the generic iteration , the data correspond to such function evaluated at the
MRA based on Hermite subdivision
In this section we describe how to build multiresolution analyses associated to a convergent Hermite level-dependent scheme, where the subdivision operator plays the role of the reconstruction low-pass filter. If such operator has the polynomial-exponential preservation property, then the wavelet decomposition filter can be easily constructed in order to cancel elements in the space . In particular, we take into account interpolatory Hermite subdivision schemes because they provide a simple
Families of Hermite multiwavelets
In this section we propose some examples of biorthogonal multiwavelet filter banks based on the construction of Hermite subdivision schemes reproducing polynomial and exponential data. All the illustrated schemes are interpolatory and “two point”, which means that the scheme involves only two values at each step for generating a new one. This implies that the masks have support and makes them implementable at reasonable costs, even if the dimension of the spaces is not small.
The
Conclusion
In this paper we have presented Hermite-type multiwavelet systems satisfying the vanishing moment property with respect to elements in the space spanned by exponentials and polynomials. These systems naturally generate MRAs which differ from the classical ones, in the sense that they are of nonstationary type, and the decomposition/reconstruction rules change accordingly to the level. In addition, for such kind of Hermite multiwavelets some nice results connected to the factorization of the
Acknowledgements
The second author was partially supported by BayFOR, Grant BayIntAn_UPA_2016_34, and, further, by the German Science Foundation in the context of the Emmy Noether Junior Research Group KR 4512/1-1 and the collaborative research center TR-109.
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