From: Ben Tilly Date: 2001-01-22T13:14:16+09:00 Subject: [ruby-talk:9698] Re: 101 Misconceptions About Dynamic Languages I realize that this conversation (while interesting to me) has drifted radically off-topic. Should we take it to private email or are people still following? "Christoph Rippel" wrote: > > -----Original Message----- > > From: Ben Tilly [mailto:ben_tilly@hotmail.com] > > Sent: Sunday, January 21, 2001 08:34 AM [...] >Ben (sorry about the Till) I was kidding you - I forgot to mention that >these funny >objects for whatever reason seem to model the real world but this real >world meaning >is IMO attached to them by physicist not mathematicians. Well there is some question about how "real-world" it is, anyone who is interested in the topic should wander over to http://math.ucr.edu/home/baez/ and browse for a bit. > > Now you named encryption as a contribution of algebra. > > Well to name but one relatively recent advance from > > analysis, consider the theory of wavelets. This provides > > entire classes of ways to break data in way that tends to > > extract and segment overall smooth data and interesting > > boundaries. Much real-world data shows this pattern. As > > a result this is applicable in compression, speech > > recognition, etc. >Sure wavelets are important and the same goes for the more mundane FFTs but >they are >useful in large part because of their formal properties (which translates >into easy >computations) not because of their inherent meaning - which is properly why >they >where not invented by signal processing engineers. I think that saying who invented them is a little hard, the subject was a convergence of too many ideas. But certainly they must be poorly understood by anyone who doesn't realize that Ingrid Daubechies was important. And she is (IMO at least) an analyst. :-) Anyways I disagree that the formal properties allowing quick computation (O(n), but with constants bad enough that FFT is faster in practice) are the key to why wavelets are so important. Rather it is the fact that they allow breaking signals into components which can trade off locality in frequency with locality in space. (Subject to the Heisenberg Uncertainty principle of course. *) Real world features tend to be localized in both, so wavelets model real-world features much better than alternatives. This is seen directly in the fact that wavelets can avoid the Gibbs effect. Therefore the bulk of the energy from your signal is concentrated in fewer terms. A few examples may make the point about why this matters. In the MRI example that I gave the computational effort was a non-issue. Having patients tie up very expensive machines for 8 hours was. Locality of reference and avoidance of Gibbs ringing made taking measurements based on a wavelet basis require far fewer measurements to get a better resolution image with fewer artifacts. (As a bonus they could also parallelize measurements to some extent.) When wavelet transforms are compared to JPEG, the computational effort required for wavelets is much greater. Again, the fact that with fewer terms you get images with fewer artifacts is why people are interested in them. (CPU power is more available than bandwidth.) Adaptive wavelet transforms (eg wavelet packets) are of interest for their ability to pick out features (eg phoenemes in speech, camouflaged objects from a photo) and to model real signals with even better compression. The ability to produce these algorithms depends upon being able to combine a conceptual understanding of what wavelets mean with existing ideas about information and entropy from CS. Adaptive transforms are harder to compute than non-adaptive ones, yet are worthwhile because they allow problems to be tackled which could not be tackled without. >Going off on a tangent the reason why analysis is often spectacularly more >successfully is IMO that its two most powerful abstractions - limits and >averaging - >are much more natural in its realm. The same ideas (formalisms?) have been >applied >to algebra but to use them successfully tends to be much harder (abstract) >and this >expalins IMO why you won't find them applied too often to the bred and >butter >discreet world most software engineers encounter every day. How dare you go off on a tangent when we are on one already? :-) From what I have encountered of limits in category theory, I think that formalism is a far more accurate term than idea. :-) Incidentally I don't happen to believe that limits are a particularly good conceptual basis for calculus. IMHO but for an accident of history, big-O, little-o would have been much better. I am not alone in that belief, I was interested to find out that Knuth came up with the same ideas: http://www-cs-faculty.stanford.edu/~knuth/ocalc.tex Cheers, Ben * No joke. The Heisenberg Uncertainty Principle is a theorem about Fourier Transforms, you cannot localize in both space and frequency at the same time. Thus (for instance) it is impossible for a clap to have a well-defined pitch. _________________________________________________________________ Get your FREE download of MSN Explorer at http://explorer.msn.com