Continuous and orthogonal wavelets

Jacques Lewalle

Syracuse University

If you are familiar with my old tutorial and/or wish to continue using it, click here. Some of my notations have evolved over ten years, but the original step-by-step calculation of wavelet transforms of simple functions still seems helpful.

Many books and websites are devoted to wavelets, and can be both useful and frustrating. In the text, I reference a number of sources, listed in a short introductory bibliography. Over the years, the applications I dealt with have led me to some changes in emphasis, reflected in these notes. Their purpose is two-fold:

Update the presentation of my old tutorial, including orthogonal wavelets, from the perspective that makes most sense to me at this time. This includes links to Matlab files.
Provide detailed proofs of why my presentation is mathematically correct – including links to Maple worksheets – in a way that is not generally possible in archival publications.

This project owes much to the preparation of the Springer Handbook of Experimental Fluid Mechanics section on wavelets, both for the content selection and for the justification of my perspective.

As a relative newcomer to orthogonal wavelets, I respect the elegance of the tool, its effectiveness at packaging the information for transmission, storage and simulation, and the quality of some results related to fluid mechanics and turbulence. I am also unconvinced about some marginal but published interpretations, to be addressed in these notes, and unenthusiastic about the frequency resolution when it comes to data interrogation. Among continuous wavelets, my experience centers on the Mexican hat and related wavelets, and on the Morlet wavelet:

For the latter, my departure from orthodoxy is three-fold. First, the parameter z₀, usually associated with the number of oscillations in a Gaussian envelope of unit duration, is used instead to determine the corresponding duration of the Gaussian envelope of an oscillation of unit frequency: this facilitates, in my view, the interpretation of the wavelet dilation as a wavelength, and is otherwise inconsequential. Also, absorbing the normalization factor c__ψ in the wavelet itself simplifies the formulae, which I view as an internally-consistent triplet of wavelet transform, inverse transform and Parseval/Plancherel relations. Finally (as for the Mexican hat, below), the use of the logarithmic frequency scale for integration (inverse transform, Parseval) leads to absorbing various factors in the definition of the wavelet itself.
For the Hermitian wavelets, I am adamant about the relation to Gaussian filtering; about the cautious use of the alternative inverse transform formula, consistent with the multiscale decomposition of the signal; etc. The arduous progression can be retraced from my archival publications, but the end-result is much simpler and is summarized here.
What is lost in my presentation is the reference to the `mother’ wavelet and the L² formulation; I trade them for simpler formulae and what I hope is a more transparent interpretation of the results. This is a matter of presentation, not of content.

To facilitate navigation, the notes are at two levels below this, corresponding to topical outlines and to extended footnotes. The annotated table of contents, below, is linked to each outline. There, links will be found to the pdf text and discussion, Maple worksheets with detailed derivations, and Matlab codes. All orthogonal wavelet results use the Wavelab library, which the user should obtain independently.

Data: discretized values, time series vs. spatial distribution (1 or n-dimensional), test signals

Fourier transforms and spectra. Basic formulae, FFT and linear frequency scale, spectra and their presentation.

Continuous wavelets: general definition and formulae. This is the orthodox presentation, including corresponding matlab code.

1. The Mexican hat wavelet: transform, inverse, spectra.

2. The Morlet wavelet: transform, inverse, spectra.

The part below is still being edited… stay tuned…

Discrete wavelets

1. Orthogonal discrete filters

2. Orthogonal wavelets and multiresolution

3. Spectra using orthogonal wavelets

Continuous vs. orthogonal: comparison and discussion.

Selected applications: outline.