[HN Gopher] Wavelets Allow Researchers to Transform - and Unders...
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Wavelets Allow Researchers to Transform - and Understand - Data
Author : theafh
Score : 72 points
Date : 2021-10-13 16:59 UTC (6 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| TimMurnaghan wrote:
| Nice to see some love for wavelets. They're great for many more
| things. Robust volatility measurement, galerkin methods for
| numerical solutions to differential equations in finance. Just
| please don't use a Haar wavelet for anything other than teaching.
| eutectic wrote:
| I though Haar wavelets were good for text and other hard-edged
| images?
| nickff wrote:
| Hear wavelets can be extremely useful when the signal of
| interest is square-shaped, or fast processing is required.
| ur-whale wrote:
| >Nice to see some love for wavelets.
|
| Yeah, I agree. Wavelets were all the rage in the late 80's and
| 90's and they seem to have fallen out of fashion.
|
| As a matter of fact, It's kind of strange that applied math
| techniques be subject to fashion.
|
| I think there is quite a lot to be done looking at deep nets in
| terms of wavelets for example.
| nrr wrote:
| Mathematicians are human just the same and are wont to get
| their hands on shiny toys too from time to time. There's some
| value in seeking out novelty.
|
| Apropos deep nets, with the explosion in machine learning in
| the past few years, I've been seeing a lot of research
| interest statements change to meet that. In particular,
| there's an awful lot of numerical linear algebra being done
| now.
|
| I suspect that things will come full-circle soon enough, and
| those tools developed in numerical linear algebra (via their
| connections to functional analysis) will make their way to
| harmonic analysis.
|
| (This is notwithstanding the fact that compressed sensing is
| picking up a little momentum as a research area in applied
| mathematics and other disciplines that study signal
| processing. Wavelets, curvelets, shearlets, chirplets, etc.
| will likely see some action there too.)
| kragen wrote:
| It's rational to seek out novelty. What's more likely: that
| you'll be the first to discover a momentous consequence of
| a theorem published last week, or that you'll be the first
| to discover a momentous consequence of a theorem published
| by Euler?
| mensetmanusman wrote:
| Wavelets are like localized Fourier transforms. Super useful in
| some compression algorithms.
| nickff wrote:
| I agree that the discrete wavelet transform is analogous to the
| discrete Fourier transform, but they are calculated
| differently, and present very different information in a very
| different way. I think most people find DWT results much more
| confusing than FFT results.
|
| If you're very familiar with both of them, it can be easy to
| underestimate how confusing each is to a newcomer.
| graycat wrote:
| The referenced article has:
|
| "Wavelets came about as a kind of update to an enormously useful
| mathematical technique known as the Fourier transform. In 1807,
| Joseph Fourier discovered that any periodic function -- an
| equation whose values repeat cyclically -- could be expressed as
| the sum of trigonometric functions like sine and cosine."
|
| Due to the "periodic", that math is Fourier series, not the
| Fourier transform. Given almost any periodic function, the
| Fourier transform won't exist.
|
| The math of Fourier theory is done carefully in two of the W.
| Rudin texts: Fourier series is in his _Principles of Mathematical
| Analysis_ , and Fourier transforms is in his _Real and Complex
| Analysis_. For more, his _Functional Analysis_ covers the related
| _distributions_. See also the "celebrated" Riesz-Fischer
| theorem, IIRC in his _Real and Complex Analysis_.
|
| As I recall, wavelets have some good _completeness_ properties.
|
| Power spectra are of interest, and for that see the Wiener-
| Khinchin theorem. For the relevant statistics of estimating power
| spectra, see the work of Blackman and Tukey.
|
| The referenced article also has
|
| "That's because Fourier transforms have a major limitation: They
| only supply information about the frequencies present in a
| signal, saying nothing about their timing or quantity."
|
| Phase gives some information on timing, and the power spectra, on
| quantity.
|
| The article also has
|
| "Whenever you have a particularly good match, a mathematical
| operation between them known as the dot product becomes zero, or
| very close to it."
|
| A dot product value of zero means orthogonality, that is, in the
| usual senses, neither signal is useful for saying any much about
| the other.
|
| Of course, in the case of stochastic processes, the orthogonality
| of a dot product ( _inner_ product) is not the same as
| probabilistic independence.
|
| For computations, of course, see the many versions of the fast
| Fourier transform, in response to a question from R. Garwin,
| _rediscovered_ by Tukey, first programmed by Cooley, later given
| various developments and many applications.
|
| At one point early in my career, these topics got the company I
| was working for "sole source" on a US Navy contract and got me a
| nice, new high end Camaro, a nice violin, a nice piano for my
| wife, some good French food, a sack full of nice Nikon camera
| equipment, and a nice stock market account! My annual salary was
| 6+ the cost of the Camaro.
| VikingCoder wrote:
| So, dumb question...
|
| If you were training some deep learning model...
|
| ...should you be trying a few Wavelet transforms on your inputs,
| and feeding those in to your model, too, to see if your model
| performs better with wavelet inputs?
| nickff wrote:
| It very much depends on your inputs, but it's likely worth a
| shot. Note that wavelet transforms are generally much slower
| than fast Fourier transforms.
| [deleted]
| actusual wrote:
| Instead of saying "I'm going to try this, maybe it will work",
| you should instead be asking if wavelet transforms are
| appropriate given the domain you are building a model for.
| Don't just transform data in the hopes that it will magically
| work.
| cinntaile wrote:
| I guess the overarching question is...
|
| How do you determine if they are good for your application
| and how do you choose which family of wavelets to apply?
| taneq wrote:
| Or if you do, write down the results and publish them even if
| they're negative. That's how science is _meant_ to work.
| mr_luc wrote:
| So if I'm understanding this right -- if I have a stream of
| numbers coming in forming a squiggly line, and I have a bucket of
| these wavelet shapes, I can pick up wavelets and stretch and
| squeeze and resize them and overlay them on my line, and ... use
| them to characterize that squiggly line? Working as feature
| recognition and also serving as a way to compress it? So instead
| of 20k data points, I have a sequence like 'mexican hat, mexican
| hat', and maybe elements in the sequence are different sizes and
| overlap?
|
| (a) if my intuition is super wrong I'd love for HN to correct it,
| heh. (b) long shot, but anyone have links to code? It's HN after
| all -- maybe some commenters in this thread are aware of
| cool/idiomatic/simple/etc uses of wavelet code in open source
| software.
|
| (I know of a bunch of cool _uses_ they 've been put to, from
| JPEGs to the spacex fluid dynamics presentation about using
| wavelet compression on gpu, but _I 've_ personally never used
| them as a tool for anything, and it'd be fun to learn about them
| with code!)
| 0x70dd wrote:
| Wavelets are used for pattern recognition in many iris
| recognition systems. First, the position of the iris in the
| input image is determined, then any eyelash and eyelid
| occlusions are removed and the iris is extracted by converting
| it to a polar coordinated image. The resulting image signal is
| convolved with wavelets of different shapes and sizes. The
| resulting signal is encoded using phase demodulation to produce
| an IrisCode. Two iris codes can be checked for a match by
| computing their hamming distance. [1] is the paper which
| describes the original system invented by Daugman. [2] is an
| open source implementation of that method.
|
| [1] https://www.robots.ox.ac.uk/~az/lectures/est/iris.pdf [2]
| http://iris.giannaros.org/
| maliker wrote:
| Yes.
|
| A mathematician explained it to me as an alternative to a
| fourier transform: instead of describing the function as a sum
| of sine waves, its a sum of mexican hats (or whatever basis
| function). And it turns out that's a simpler representation in
| the case of function with sharp discontinuities. It's also an
| alternative to a Taylor series, replacing sums of derivatives
| with the sum of the scaled basis function. Seemed like a pretty
| elegant explanation to me.
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