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Thursday, December 29, 2005

Elementary Wavelet Analysis

This is the abstract for my MSci thesis on wavelet analysis. For the full article, see my main site.

Abstract: A discussion of orthonormal wavelet bases and wavelet
series, utilising the notion of multiresolution approximations. I start by discussing multiresolution approximations and show how to find an orthonormal basis of the generating space, consisting of translates of a `scaling function' φ. The notion of orthornormal wavelet bases is then introduced, and it is shown how one can go from a multiresolution approximation to a wavelet basis with a scaling function, and vice-versa. The article then turns to the description of wavelet bases of compact support and multi-dimensional wavelet bases (using the tensor product method). Finally, the regular wavelets of compact support are seen to form an unconditional basis of the space H1(Rn) of Stein and Weiss, this being an interesting subspace of L1(Rn).

This is my MSci project, which I undertook in the fourth year of my undergraduate degree at Imperial College, London. It is concerned with mathematical entities known as wavelets, and uses the wavelets to investigate a space called H1(Rn).

Though the title of the project is 'Elementary' wavelet analysis, you still do need to know some maths to understand it! In particular, you should be aquainted with undergraduate functional analysis. If you'd prefer something less technical, why don't you try The Engineers Ultimate Guide to Wavelet Analysis. This Internet article was written by a non-mathematician and designed for non-mathematicians, so anyone (well any scientist or engineer) should find it quite accessible. If you want to read more on the topic, then another not-so-technical resource is G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, Boston 1994. A good introductory volume for the more mathematically minded amongst you is C. K. Chui, An Introduction to Wavelets, Academic Press Inc., New York, 1992. But the book I referred to mostly whilst writing my project was Y. Meyer, Wavelets and operators, Cambridge University Press, Cambridge, 1992. It's a pretty tough going book and I wouldn't recommend it unless you're really up to scratch on your real analysis; nonetheless it is required reading if you seriously want to understand the deeper theoretical foundations of the wavelet theory.

In case you meet the above requirement and are interested in reading further, you might first like to know a little about what wavelets are and how they can be used.

A one-dimensional wavelet basis is a basis of L2(R) (depending upon the author, the basis may or may not be required to be orthonormal) and equally an n-dimensional wavelet basis is a basis of L2(Rn), in much the same manner the complex exponentials form an orthonormal basis of L2(0,2π). So indeed wavelet analysis shares some similarities with Fourier analysis. However if you read the article, you'll find that wavelets really kick arse, compared with Fourier analysis.


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