
Wavelets (1994) [pdf] - rubenbe
http://cybertester.com/data/wavelet.pdf
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IIAOPSW
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 (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.

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

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liuyao
These kind of heuristics are supposedly a prerequisite to the formal study,
but sadly most textbooks don't spend the time talking about it (since
logically it's not necessary). Instead of reading up on functional analysis
(Banach spaces and Hilbert spaces), read bits of its history to get a taste.
Here's one that's not too long [http://courses.mai.liu.se/GU/TATM85/FA-
history.pdf](http://courses.mai.liu.se/GU/TATM85/FA-history.pdf)

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patentatt
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!

~~~
ad404b8a372f2b9
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](https://edouardoyallon.github.io/thesis.pdf)

~~~
ska
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.

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jstewartmobile
Gilbert Strang is a delightful writer.

" _What do you say to a thesis student you don 't remember? In that position I
suggest something very short: 'Tell me more.' The most amazing part was his
thesis topic. 'I am designing the filter bank for MIT's entry in the HDTV
competition.' Some days you can't lose, even if you deserve to._"

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misiti3780
i 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/Contourlet)

[https://en.wikipedia.org/wiki/Shearlet](https://en.wikipedia.org/wiki/Shearlet)

~~~
ska
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.

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high_pathetic
My thesis defense was about wavelets and it was in 1995! Had to implement my
own FTT and rendered 3D charts in Excel...

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gsmethells
JPEG2000 uses Wavelets
[https://en.wikipedia.org/wiki/JPEG_2000](https://en.wikipedia.org/wiki/JPEG_2000)

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woliveirajr
I love how you can decompose one image in wavelets and edit just some layers
of it (in GIMP), and those retouches end up as subtle improvements of the
image.

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FredrikMeyer
The 2017 Abel Prize was awarded to Yves Meyer (no relation) for his
contribution to the theory of wavelets. This is worth a read:
[https://www.quantamagazine.org/yves-meyer-wavelet-expert-
win...](https://www.quantamagazine.org/yves-meyer-wavelet-expert-wins-abel-
prize-20170321/)

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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.

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CamperBob2
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?

~~~
person_of_color
Probably 500 Hz. Haha

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ohazi
Worth reading, but boy is that font painful. Looks like one of the older
bitmapped latex CM font?

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gdy
Implemented FDWT as part of a coursework project many many years ago when it
was hip :)

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ncmncm
Who gets taught wavelets now?

Are they supposed to displace Fourier decompositions, or did that fizzle?

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WillDaSilva
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.

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mrcactu5
how does this compare to Fast Fourier transform ?

[https://en.wikipedia.org/wiki/Fast_Fourier_transform](https://en.wikipedia.org/wiki/Fast_Fourier_transform)

Or the Discrete Cosine Transform ?

[https://en.wikipedia.org/wiki/Discrete_cosine_transform](https://en.wikipedia.org/wiki/Discrete_cosine_transform)

Or the Gabor Transform ?

[https://en.wikipedia.org/wiki/Gabor_transform](https://en.wikipedia.org/wiki/Gabor_transform)

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planteen
A wavelet naturally gives a spectrogram. The Fourier transform taken on a
signal will give you only frequency content, but no idea in time where this
frequency occurred. A wavelet naturally gives both time and frequency
information. Of course, if you take the Fourier transform at discrete time
intervals in blocks, you can also construct a spectrogram.

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mjfl
Anyone know good sources on shift invariant wavelets?

~~~
mirsadm
The dual tree complex wavelet transform is approximately shift invariant
[https://en.wikipedia.org/wiki/Complex_wavelet_transform#Dual...](https://en.wikipedia.org/wiki/Complex_wavelet_transform#Dual-
tree_complex_wavelet_transform)

