0.7 Generalizations of the basic multiresolution wavelet system  (Page 13/28)

Multiwavelets

In Chapter: A multiresolution formulation of Wavelet Systems , we introduced the multiresolution analysis for the space of $L^2$ functions, where we have a set of nesting subspaces

where each subspace is spanned by translations of scaled versions of a single scaling function $\phi$ ; e.g.,

The direct difference between nesting subspaces are spanned by translations of asingle wavelet at the corresponding scale; e.g.,

There are several limitations of this construction. For example, nontrivial orthogonal wavelets can not be symmetric. To avoid this problem, we generalizedthe basic construction, and introduced multiplicity- $M$ ( $M$ -band) scaling functions and wavelets in [link] , where the difference spaces are spanned by translationsof $M-1$ wavelets. The scaling is in terms of the power of $M$ ; i.e.,

In general, there are more degrees of freedom to design the M-band wavelets. However, the nested $V$ spaces are still spanned by translations of a single scaling function.It is the multiwavelets that removes the above restriction, thus allowing multiple scaling functions to span the nested $V$ spaces [link] , [link] , [link] . Although it is possible to construct $M$ -band multiwavelets, here we only present results on the two-band case, as most of the researches in theliterature do.

Construction of two-band multiwavelets

Assume that $V_0$ is spanned by translations of $R$ different scaling functions ${\phi }_i\left(t\right)$ , $i=1,...,R$ . For a two-band system, we define the scaling and translation of these functions by

The multiresolution formulation implies

We next construct a vector scaling function by

Since $V_0\subset V_1$ , we have

where $H\left(k\right)$ is a $R\times R$ matrix for each $k\in \mathbf{Z}$ . This is a matrix version of the scalar recursive equation [link] . The first and simplest multiscaling functions probably appear in [link] , and they are shown in [link] .

The first scaling function ${\phi }_1\left(t\right)$ is nothing but the Haar scaling function,and it is the sum of two time-compressed and shifted versions of itself, as shown in [link] (a). The second scaling function can be easilydecomposed into linear combinations of time-compressed and shifted versions of the Haar scaling function and itself, as

This is shown in [link]

Putting the two scaling functions together, we have

Further assume $R$ wavelets span the difference spaces; i.e.,

Since $W_0\subset V_1$ for the stacked wavelets $\Psi \left(t\right)$ there must exist a sequence of $R\times R$ matrices $G\left(k\right)$ , such that

These are vector versions of the two scale recursive equations [link] and [link] .

We can also define the discrete-time Fourier transform of $H\left(k\right)$ and $G\left(k\right)$ as

Properties of multiwavelets

Approximation, regularity and smoothness

Recall from Chapter: Regularity, Moments, and Wavelet System Design that the key to regularity and smoothness is having enough number of zeros at $\pi$ for $H\left(\text{ω}\right)$ . For multiwavelets, it has been shown that polynomials can be exactly reproduced by translatesof $\text{Φ}\left(t\right)$ if and only if $\mathbf{H}\left(\text{ω}\right)$ can be factored in special form [link] , [link] , [link] . The factorization is used to study the regularity and convergence of refinablefunction vectors [link] , and to construct multi-scaling functions with approximation and symmetry [link] . Approximation and smoothness of multiple refinable functions are also studied in [link] , [link] , [link] .