Wavelets: Multiresolution Analysis (MRA)

By and large I try not to flee into cold, formal mathematics, not here. You can find all too many books that will give you just the mathematics (and some of them are invaluable references). On the other hand, sometimes I am way too vague. Let me try to give you a clear statement of what I’m going on about when I say “multiresolution analysis” (MRA).

(Don’t misunderstand me. When I’m studying something, I’m sitting there with collections of definitions, theorems, proofs, and examples – trying to make sense of them. I just think that the collection itself is not a substitute for understanding.)

There’s a lot to be said for having a clearly stated theorem. Working through this has had a large impact on my own grasp of the properties of wavelets.

The most lively summary of multiresolution analysis can be found in the first couple of pages of chapter 5 of Daubechies’ “Ten Lectures on Wavelets”. The most lively introduction can be found on pages 10-16.

I have more than a few, close to several, books that seem to present the following concepts in the same way. First, they use multi-resolution analysis to describe orthonormal wavelets; then they use filter banks to describe biorthogonal wavelets; finally, they explain how biorthogonal wavelets could be described by a modified multi-resolution analysis.
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Wavelets: semi-orthogonal, from linear splines

Edit: 13 July, just one remark added. see “edit” below.

How are things going? As I said yesterday in “happenings”, the simplest answer is: right now I’m just thrilled when I can reproduce a drawing in a book, even if I don’t understand why the method works or where the function came from.

This is post that I ended up with when I started out to write a “happenings” post yesterday. I promised you a picture. In fact, I’ll give you a mother wavelet for the linear spline scaling function. (That link occurs a few more times. What can I say? That’s what I’m building on, in more than one way.)

You may recall that I have shown you two ways to approximate a scaling function. The one I do not understand applies convolution and downsampling repeatedly. The one I do understand is the dyadic expansion, and I’ve been using it ever since I worked it out, in preference to the other method.

Well. I have now seen a scaling function which is infinite at all the dyadic points, so the dyadic expansion hasn’t got a prayer of working! But the convolution and downsampling algorithm gives me a drawing which seems to match the book.
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Happenings Jul 11

I sat down to write a happenings post this morning, and found myself looking at a wavelets post when I was finished with it.

Since the first draft turned into a very nice technical post, let me try this again. When I began, what else did I think I might write about?

I certainly do not mind that I ended up with a post different from what I intended. One of the secrets to writing is: don’t edit yourself while it’s happening. Of course, drafts need to be edited — but that is a different process. By giving myself free rein, I am often surprised at how well things turn out.
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