Worth reading, but boy is that font painful. Looks like one of the older bitmapped latex CM font?
Wavelets (1994) [pdf]
11–20 of 47 posts
Re: Wavelets (1994) [pdf]
#12Wavelets were supposed to be a big thing like 15-20 years ago ... what happened? Do any mainstream modern codecs use wavelets? I’m not aware of any Edit: looks like the Dirac video codec is based on wavelets. Good to know all of that research wasn’t for nothing!
ImMix Cube products maybe? My memory fades.
Re: Wavelets (1994) [pdf]
#13Anyone know good sources on shift invariant wavelets?
Re: Wavelets (1994) [pdf]
#14Wavelets were supposed to be a big thing like 15-20 years ago ... what happened? Do any mainstream modern codecs use wavelets? I’m not aware of any Edit: looks like the Dirac video codec is based on wavelets. Good to know all of that research wasn’t for nothing!
Wavelets are the basis of modern video codecs. People also use them to watermark videos. I never found much advanced application in image processing which was my field of research.
Re: Wavelets (1994) [pdf]
#15Who gets taught wavelets now? Are they supposed to displace Fourier decompositions, or did that fizzle?
They accomplish different things. Fourier decompositions break signals down into a frequency based representation, whereas wavelet decomposition yields a sort of hybrid frequency/temporal representation. Wavelets have their uses, but I don't know that anyone studying them thought they would replace Fourier decompositions.
Re: Wavelets (1994) [pdf]
#16Wavelets were supposed to be a big thing like 15-20 years ago ... what happened? Do any mainstream modern codecs use wavelets? I’m not aware of any Edit: looks like the Dirac video codec is based on wavelets. Good to know all of that research wasn’t for nothing!
I attended a talk by Stephane Mallat a few months back about the work his lab is doing on wavelet scattering transforms.
It seems they're making great progress in developing deep neural networks based on them that don't need to be learned.
It's not as hyped as mainstream deep learning methods but I think it holds a lot of promise since it cuts down on learning time, is mostly unsupervised, gives you control over the network's features and is a more theoretically principled way to build networks than just defining the loss and crossing your fingers while it's optimizing as we're doing right now (bit of an hyperbole).
I could go on, it's absolutely fascinating. More info (in english & french): https://edouardoyallon.github.io/thesis.pdf
Re: Wavelets (1994) [pdf]
#17Who gets taught wavelets now? Are they supposed to displace Fourier decompositions, or did that fizzle?
It's a pretty standard signal processing topic. It doesn't replace a FFT, but is used similarly to windowed FFT.
Definitely has its uses. Definitely isn't a panacea.
Re: Wavelets (1994) [pdf]
#18Wavelets were supposed to be a big thing like 15-20 years ago ... what happened? Do any mainstream modern codecs use wavelets? I’m not aware of any Edit: looks like the Dirac video codec is based on wavelets. Good to know all of that research wasn’t for nothing!
JPEG2000, I believe.
Blocked by Patent-bullshit
Re: Wavelets (1994) [pdf]
#19Something that blew my mind the first time I learned it:
You can think of a function f(x) as the limit of an infinitely big vector where the entries index the infinitesimal. Eg f(x) = [...f(-2dx), f(-dx), f(0), f(dx), f(2dx)...]. The dot product of two functions (f,g) is still the usual sum[f(x)*g(x)] but the sum is replaced with an integral. Sin and Cos of integer frequencies happen to have a dot product of 0 (check it) which means doing a change of basis into those functions (aka the Fourier transform) happens to work out really nicely. Other than that Sin and Cos are not really privileged. For example you can do a transform into the basis of polynomial functions if you wanted to (aka the Taylor series). Any basis you can cook up would do just as well so long as its a complete basis. Just like in linear algebra, you have a complete basis if you can use linear combinations to construct the vectors ... [...1,0,0...], [...0,1,0...], [...0,0,1...] ... (aka the dirac delta functions). Differentiation is a matrix with dx on the off diagonal and -dx on the diagonal.
Re: Wavelets (1994) [pdf]
#20i wrote my master thesis on wavelets! a few years back they morphed into shearlets and contourlets. https://en.wikipedia.org/wiki/Contourlet https://en.wikipedia.org/wiki/Shearlet
They didn't really morph, these were approaches to deal with a fundamental issue in signal representation in higher dimensions that comes up when you use tensor product representations. Simply put, wavelets are not good at capturing edges in signals of dimension > 1 (although they are great at it for dimension 1)
The fundamental (functional analysis) research in this area split in a few directions. One as you mention, mostly driven by imaging research, but also frame theory and non tensor-product basis, etc.