## on Wavelet Transforms

January 2nd, 2008In the October 2007 issue of Physics Today beginning on page 78 there is an article on Wavelets by Ivan Selesnick, son of my friend Stephen Selesnick. He must be a very proud father.

I read the article. I wish I understood what I read. What does sparse mean?

Here is my picture of the story: A fourier transform tells us to what extent the signal is periodic; it displays the frequencies embedded in the signal. Signal polution ‘has no frequency’ – white noise – yielding small amplitude at all frequencies. Thus it can, in the transform, be distinguished from what we are looking for – periodicity. A spectrum shows us what frequencies are present.

But signals may have distinguishing features other than frequency for which we want to search. The idea is to create a transform to display the distinguishing feature we wish to favor and to blunt what we wish to devalue. Wavelets are the transforms that do this: enhance what we favor, blunt what we devalue.

Is this a valid overview? Have I got it right?

If so, how wonderful! And Selesnick writes that there are “data-adaptive transforms that can automatically find a good transform for a specific signal”. This seems miraculous! Doesn’t each signal have different attributes that might be of interest? Doesn’t a person decide? Doesn’t the whole scheme depend upon knowing in advance what we are looking for? Seems like one would need a second input, besides the data, to guide the adaptation!

I would love to understand this. Especially in terms of how the chosen transform function embodies the search criteria. Here’s what I mean using fourier. The transform function, exp ikx, expresses the irreducible representations for the the group of altered scrutinies of translation along x. These functions embody sameness against various amounts of translation – i.e. they have periodicity. And the irreducible representation label, k, labels this periodicity.

How are other visual criteria encoded into a transform? What is the philosophy for doing it? Surely there is some group theoretic overview. And it must yield labels – irrep labels – for marking the symmetries embedded in the criteria used in deciding what one seeks in the signal.

I’ve written to Ivan’s dad because I am holding him accountable for his son. I wrote:” Stephen, if you know the answer do tell me. I beg you, though, don’t tell me where I can find the answer. I am already looking, on my own, for too many answers. Either give me a sweet answer or let us cast the problem adrift to float in the sea with all the other problems that I have cast adrift. And wish your son well in his endeavors.”

As you point out, Fourier analysis exploits the underlying group structure of the various domains: and any compact (including all finite) groups, and locally compact abelian groups, can – and have been – used. (And some non-compact non-abelian ones, like the Poincare group, which have a well-behaved representation theory). There are certain limitations and rigidities in the exploitation of the group structure in this kind of analysis, called harmonic, very subtle as a rule. Consider the case of ordinary Fourier transforms on the real line (considered as a locally compact group under addition): if a function has compact support its transform cannot have compact support, and conversely. This complicates all sorts of things, like the usual proof of the sampling theorem. Also there arise things like the Gibbs phenomenon. Wavelets do not participate so much in the group structures and can be used more flexibly as overdetermined bases to expand functions in terms of, instead of the exp ikx’s. Their effective use was discovered by Ingrid Daubechies, now at Princeton.

- Stephen @ 11 January 2008I would recommend the somewhat expanded version of Ivan’s QuickStudy, to be found at:

http://taco.poly.edu/selesi/pubs/WaveletQuickStudy_expanded.pdf