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Multiwavelets

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Computational Complexity

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Article Outline

Glossary

Definition of the Subject

Introduction

Refinable Function Vectors

Multiresolution Approximation and Discrete Multiwavelet Transform

Moments and Approximation Order

The Discrete Multiwavelet Transform (DMWT) Algorithm

Pre- and Postprocessing and Balanced Multiwavelets

Boundary Handling

Applications

Polyphase Factorization

Lifting

Two‐Scale Similarity Transform (TST)

Examples and Software

Future Directions

Bibliography

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Abbreviations

Balanced multiwavelet:

A multiwavelet for which the expansion coefficients for polynomials up to a certain degree are polynomial sequences. Balanced multiwavelets do not require pre- or postprocessing.

Discrete multiwavelet transform (DMWT):

The algorithm which decomposes a signal into a coarse approximation and fine detail at several levels. A direct generalization of the Discrete Wavelet Transform (DWT) for scalar wavelets.

Lifting:

A method for modifying an existing multiwavelet, or building one from scratch. Lifting can be used to impose approximation order, balancing, or symmetry.

Modulation matrix:

The modulation matrix

$$ M(\xi) = \begin{pmatrix} H(\xi) & H(\xi+\pi) \\ G(\xi) & G(\xi+\pi) \end{pmatrix} \:, $$

can be used to describe the action of the DMWT. It is also used in the construction of multiwavelets by lifting or TST.

Multiresolution, multiresolution approximation (MRA):

Multiresolution is the fundamental concept underlying everything related to any kind of wavelet. A signal is decomposed into a low resolution approximation, plus fine detail at one or more levels of resolution. An MRA is a nested chain of subspaces of L 2 which describes this concept mathematically.

Multiscaling function, multiwavelet function,wavelet:

A multiwavelet of multiplicity r consists of a multiscaling function and a multiwavelet function. Both are functions from ℝ to ℂr, that is, function vectors. Multiwavelets are not the same as multivariate wavelets, which are functions from \( { \mathbb{R}^{n} } \) to ℂ.

Polyphase decomposition, polyphase matrix:

The polyphase decomposition separates a coefficient sequence into two sequences, by even and odd subscripts. It can be used to describe or implement the DMWT.The polyphase matrix

$$ P(z) = \begin{pmatrix} H^{(0)}(z) & H^{(1)}(z) \\ G^{(0)}(z) & G^{(1)}(z) \end{pmatrix}\:, $$

is useful in the construction of multiwavelets, especially orthogonal multiwavelets.

Pre- and postprocessing:

Before a DMWT can be applied to a signal, the signal needs to be written as a multiscaling function series. The DMWT works on the coefficients of the series expansion, not on the values of the signal.For scalar wavelets the distinction can usually be ignored, but not for multiwavelets. Having a multiwavelet of approximation order p does not mean that the coefficients of a polynomial up to order \( { p-1 } \) form a polynomial sequence.Preprocessing converts point samples into expansion coefficients. Postprocessing does the opposite.

Refinable function vector, refinement equation:

A refinable function vector of multiplicity r is a function from ℝ to \( { \mathbb{C}^{r} } \) which satisfies a refinement equation of the form

$$ \boldsymbol{\phi}(x) = \sqrt{2} \sum_{k=k_{0}}^{k_{1}} H_{k} \, \boldsymbol{\phi}(2x-k) $$

with recursion coefficients H k which are \( { r\! \times\! r } \) matrices.

Symbol:

Given any sequence \( { \mathbf{a} = \{a_{k} \} } \), the symbol of \( { \mathbf{a} } \) is defined as

$$ \begin{aligned} a(\xi) &= \sum_{k} a_{k} \text{e}^{-i\xi} \quad \text{or} \\[1mm] a(z) &= \sum_{k} a_{k} z^{k}\:,\quad z = \text{e}^{-i\xi}\:, \end{aligned}$$

possibly with a normalizing factor. The sequence may represent point samples of a signal, the recursion coefficients of a two‐scale refinement equation, or other quantities.Both the trigonometric and polynomial notations are useful, depending on the setting; it is trivial to switch back and forth between them.

Two‐scale similarity transform (TST):

The TST is a way of moving approximation orders in a biorthogonal multiwavelet pair from one side to the other. For scalar wavelets it corresponds to moving a factor of \( { (1+\text{e}^{-i\xi})/2 } \) back and forth.The TST can be used to modify an existing multiwavelet, or to build multiwavelets from scratch. It can also be used to impose or characterize symmetry, approximation order and balancing.

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Authors and Affiliations

  1. Iowa State University, Ames, USA

    Fritz Keinert

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Keinert, F. (2012). Multiwavelets . In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_127

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