# 0.7 Generalizations of the basic multiresolution wavelet system  (Page 14/28)

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## Support

In general, the finite length of $H\left(k\right)$ and $G\left(k\right)$ ensure the finite support of $\text{Φ}\left(t\right)$ and $\Psi \left(t\right)$ . However, there are no straightforward relations between the support length and the number of nonzero coefficients in $H\left(k\right)$ and $G\left(k\right)$ . An explanation is the existence of nilpotent matrices [link] . A method to estimate the support is developed in [link] .

## Orthogonality

For these scaling functions and wavelets to be orthogonal to each other and orthogonal to their translations, we need [link]

$\mathbf{H}\left(\text{ω}\right){\mathbf{H}}^{†}\left(\text{ω}\right)+\mathbf{H}\left(\text{ω}+\pi \right){\mathbf{H}}^{†}\left(\text{ω}+\pi \right)={I}_{R},$
$\mathbf{G}\left(\text{ω}\right){\mathbf{G}}^{†}\left(\text{ω}\right)+\mathbf{G}\left(\text{ω}+\pi \right){\mathbf{G}}^{†}\left(\text{ω}+\pi \right)={I}_{R},$
$\mathbf{H}\left(\text{ω}\right){\mathbf{G}}^{†}\left(\text{ω}\right)+\mathbf{H}\left(\text{ω}+\pi \right){\mathbf{G}}^{†}\left(\text{ω}+\pi \right)={0}_{R},$

where $†$ denotes the complex conjugate transpose, ${I}_{R}$ and ${0}_{R}$ are the $R×R$ identity and zero matrix respectively. These are the matrix versions of [link] and [link] . In the scalar case, [link] can be easily satisfied if we choose the wavelet filter by time-reversing the scaling filter and changing the signsof every other coefficients. However, for the matrix case here, since matrices do notcommute in general, we cannot derive the $G\left(k\right)$ 's from $H\left(k\right)$ 's so straightforwardly. This presents some difficulty in finding the wavelets from the scaling functions; however, this also gives usflexibility to design different wavelets even if the scaling functions are fixed [link] .

The conditions in [link][link] are necessary but notsufficient. Generalization of Lawton's sufficient condition (Theorem  Theorem 14 in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients ) has been developed in [link] , [link] , [link] .

## Implementation of multiwavelet transform

Let the expansion coefficients of multiscaling functions and multiwavelets be

${c}_{i,j}\left(k\right)=〈f,\left(t\right),,,{\phi }_{i,j,k},\left(t\right)〉,$
${d}_{i,j}\left(k\right)=〈f,\left(t\right),,,{\psi }_{i,j,k},\left(t\right)〉.$

We create vectors by

${C}_{j}\left(k\right)={\left[{c}_{1,j}\left(k\right),...,{c}_{R,j}\left(k\right)\right]}^{T},$
${D}_{j}\left(k\right)={\left[{d}_{1,j}\left(k\right),...,{d}_{R,j}\left(k\right)\right]}^{T}.$

For $f\left(t\right)$ in ${V}_{0}$ , it can be written as linear combinations of scaling functions and wavelets,

$f\left(t\right)=\sum _{k}{C}_{{j}_{0}}{\left(k\right)}^{T}{\text{Φ}}_{{J}_{0},k}\left(t\right)+\sum _{j={j}_{0}}^{\infty }\sum _{k}{D}_{j}{\left(k\right)}^{T}{\Psi }_{j,k}\left(t\right).$

Using [link] and [link] , we have

${C}_{j-1}\left(k\right)=\sqrt{2}\sum _{n}H\left(n\right){C}_{j}\left(2k+n\right)$

and

${D}_{j-1}\left(k\right)=\sqrt{2}\sum _{n}G\left(n\right){C}_{j}\left(2k+n\right).$ Discrete Multiwavelet Transform

Moreover,

${C}_{j}\left(k\right)=\sqrt{2}\sum _{k}\left(H,{\left(k\right)}^{†},{C}_{j-1},\left(2k+n\right),+,G,{\left(k\right)}^{†},{D}_{j-1},\left(2k+n\right)\right).$

These are the vector forms of [link] , [link] , and [link] . Thus the synthesis and analysis filter banks for multiwavelet transforms have similar structures as the scalar case. Thedifference is that the filter banks operate on blocks of $R$ inputs and the filtering and rate-changing are all done in terms of blocks of inputs.

To start the multiwavelet transform, we need to get the scaling coefficients at high resolution. Recall that in the scalar case, thescaling functions are close to delta functions at very high resolution, so the samples of the function are used as the scaling coefficients. However, for multiwavelets we need the expansion coefficients for $R$ scaling functions. Simply using nearby samples as the scaling coefficients is abad choice. Data samples need to be preprocessed ( prefiltered ) to produce reasonable values of the expansion coefficients for multi-scalingfunction at the highest scale. Prefilters have been designed based on interpolation [link] , approximation [link] , and orthogonal projection [link] .

## Examples

Because of the larger degree of freedom, many methods for constructing multiwavelets have been developed.

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