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Wavelets (1994) [pdf]

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Re: Wavelets (1994) [pdf]

#21

Post by ska »

ad404b8a372f2b9 wrote: post #16
patentatt wrote: post #3

Wavelets 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…

I'm not familiar with this particular research direction.

On the more general area Mallat has a good and approachable (not very technical) book on the area: "A Wavelet Tour of Signal Processing". Worth a look for anyone intrigued by wavelets.

Re: Wavelets (1994) [pdf]

#22

Post by wallyowen »

We explored using fast wavelet transforms to compress EKG samples for an ambulatory cardiac monitoring device, but the risks of regulators not understanding the math behind the compression meant we walked away from a ~3:1 compression of the sampled waveforms and attendant bandwidth savings.

Re: Wavelets (1994) [pdf]

#23

Post by laretluval »

IIAOPSW wrote: post #19

Something 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(-2 dx), f(-dx), f(0), f(dx), f(2 dx)...]. 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 (che…

What should I google to read more about this kind of thing?

Re: Wavelets (1994) [pdf]

#24

Post by ska »

IIAOPSW wrote: post #19

Something 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(-2 dx), f(-dx), f(0), f(dx), f(2 dx)...]. 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 (che…

It's a bit weirder than this suggests. You are right as long as you have a complete basis you can transform code your function by a weighted sum of bases vectors, where the weights are inner products of the function against he dual vectors of that basis, and the basis vector form a complete orthonormal basis (we can relax that last bit).

So that's great, but what do the basis vectors look like? You need a family of functions so that scalings and translations of them have inner product 0 against each other. So this is easy if the shared support is 0, or if the +ve and -ve parts cancel out. It doesn't take too much playing about to see how this works with Haar basis, or sin/cos (i.e. FFT), so it looks pretty intuitive.

It also seem kind of intuitive that these might be the only way to do this... but that's wrong.

It turns out you can construct other families that work, unlike the nicely behaved sin/cos, or step functions, they are not smooth, not symmetric (at least for orthonormal basis) - quite odd. They don't have closed forms, so if you want to see what one looks like you'll have to realized it as a fixed point or by some other approximation method: e.g. https://en.wikipedia.org/wiki/Daubechies_wavelet#/media/File...

Re: Wavelets (1994) [pdf]

#25

Post by mlevental »

laretluval wrote: post #23
IIAOPSW wrote: post #19

Something 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(-2 dx), f(-dx), f(0), f(dx), f(2 dx)...]. 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 (che…

What should I google to read more about this kind of thing?

https://en.wikipedia.org/wiki/Functional_analysis

Re: Wavelets (1994) [pdf]

#26

Post by heyitsguay »

ad404b8a372f2b9 wrote: post #16
patentatt wrote: post #3

Wavelets 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…

My old research group was working on stuff relating to scattering network theory and application. The tl;dr is it's got nice math but isn't practically useful for much. The network construction also isn't as similar to, e.g., convnets as some of the scattering literature suggested.

Re: Wavelets (1994) [pdf]

#27

Post by userbinator »

zyxzevn wrote: post #18
lp251 wrote: post #9

Earlier quoted context omitted.

JPEG2000, I believe.

Blocked by Patent-bullshit

I believe either the patents have expired or the basic functionality was always royalty-free; but what really killed JPEG2000 was the immense complexity and increased processing power required. It still has applications in some niche areas, and probably most people's contact with it will be in the form of images embedded in PDFs (many of archive.org's ebooks use JPEG2000), where the noticeably slower rendering of pages is a big turn-off.

Re: Wavelets (1994) [pdf]

#28

Post by CamperBob2 »

wallyowen wrote: post #22

We explored using fast wavelet transforms to compress EKG samples for an ambulatory cardiac monitoring device, but the risks of regulators not understanding the math behind the compression meant we walked away from a ~3:1 compression of the sampled waveforms and attendant bandwidth savings.

I'm surprised bandwidth was a consideration at all given the slow signals involved. What's a typical Nyquist rate used when sampling an EKG?

Re: Wavelets (1994) [pdf]

#29

Post by person_of_color »

CamperBob2 wrote: post #28
wallyowen wrote: post #22

We explored using fast wavelet transforms to compress EKG samples for an ambulatory cardiac monitoring device, but the risks of regulators not understanding the math behind the compression meant we walked away from a ~3:1 compression of the sampled waveforms and attendant bandwidth savings.

I'm surprised bandwidth was a consideration at all given the slow signals involved. What's a typical Nyquist rate used when sampling an EKG?

Probably 500 Hz. Haha
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