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

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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

$\left\{0\right\}\subset \cdots \subset {V}_{-2}\subset {V}_{-1}\subset {V}_{0}\subset {V}_{1}\subset {V}_{2}\subset \cdots \subset {L}^{2},$

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

${V}_{j}=\underset{k}{\mathrm{Span}}\left\{{2}^{j/2}\phi \left({2}^{j}t-k\right)\right\}.$

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

${W}_{j}={V}_{j+1}\ominus {V}_{j}=\underset{k}{\mathrm{Span}}\left\{{2}^{j/2}\psi \left({2}^{j}t-k\right)\right\}.$

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 $M\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}1$ wavelets. The scaling is in terms of the power of $M$ ; i.e.,

${\phi }_{j,k}\left(t\right)={M}^{j/2}\phi \left({M}^{j}t-k\right).$

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

${\phi }_{i,j,k}\left(t\right)={2}^{j/2}{\phi }_{i}\left({2}^{j}t-k\right).$

The multiresolution formulation implies

${V}_{j}=\underset{k}{\mathrm{Span}}\left\{{\phi }_{i,j,k}\left(t\right):i=1,...,R\right\}.$

We next construct a vector scaling function by

$\text{Φ}\left(t\right)={\left[{\phi }_{1},\left(t\right),,,...,,,{\phi }_{R},\left(t\right)\right]}^{T}.$

Since ${V}_{0}\subset {V}_{1}$ , we have

$\phantom{\rule{0.277778em}{0ex}}\text{Φ}\left(t\right)=\sqrt{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\sum _{n}\phantom{\rule{0.166667em}{0ex}}H\left(n\right)\phantom{\rule{0.166667em}{0ex}}\text{Φ}\left(2t-n\right)$

where $H\left(k\right)$ is a $R×R$ 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

${\phi }_{2}\left(t\right)=\frac{\sqrt{3}}{2}{\phi }_{1}\left(2t\right)+\frac{1}{2}{\phi }_{2}\left(2t\right)-\frac{\sqrt{3}}{2}{\phi }_{1}\left(2t-1\right)+\frac{1}{2}{\phi }_{2}\left(2t-1\right).$

This is shown in [link]

Putting the two scaling functions together, we have

$\left[\begin{array}{c}{\phi }_{1}\left(t\right)\\ {\phi }_{2}\left(t\right)\end{array}\right]=\left[\begin{array}{cc}1& 0\\ \sqrt{3}/2& 1/2\end{array}\right]\left[\begin{array}{c}{\phi }_{1}\left(2t\right)\\ {\phi }_{2}\left(2t\right)\end{array}\right]+\left[\begin{array}{cc}1& 0\\ -\sqrt{3}/2& 1/2\end{array}\right]\left[\begin{array}{c}{\phi }_{1}\left(2t-1\right)\\ {\phi }_{2}\left(2t-1\right)\end{array}\right].$

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

${W}_{j}={V}_{j+1}\ominus {V}_{j}=\underset{k}{\mathrm{Span}}\left\{{\psi }_{i,j,k}\left(t\right):i=1,...,R\right\}.$

Since ${W}_{0}\subset {V}_{1}$ for the stacked wavelets $\Psi \left(t\right)$ there must exist a sequence of $R×R$ matrices $G\left(k\right)$ , such that

$\phantom{\rule{0.277778em}{0ex}}\Psi \left(t\right)=\sqrt{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\sum _{k}\phantom{\rule{0.166667em}{0ex}}G\left(k\right)\phantom{\rule{0.166667em}{0ex}}\text{Φ}\left(2t-k\right)$

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

$\mathbf{H}\left(\text{ω}\right)=\sum _{k}H\left(k\right){e}^{i\text{ω}k},\phantom{\rule{2.em}{0ex}}\mathbf{G}\left(\text{ω}\right)=\sum _{k}G\left(k\right){e}^{i\text{ω}k}.$

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] .

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