[HN Gopher] Wavelets Allow Researchers to Transform - and Unders... ___________________________________________________________________ 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. ___________________________________________________________________ (page generated 2021-10-13 23:00 UTC)