Something that blew my mind the first time I learned it:<p>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<i>dx), f(-dx), f(0), f(dx), f(2</i>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.
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<p>Edit: looks like the Dirac video codec is based on wavelets. Good to know all of that research wasn’t for nothing!
Gilbert Strang is a delightful writer.<p>"<i>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.</i>"
i wrote my master thesis on wavelets! a few years back they morphed into shearlets and contourlets.<p><a href="https://en.wikipedia.org/wiki/Contourlet" rel="nofollow">https://en.wikipedia.org/wiki/Contourlet</a><p><a href="https://en.wikipedia.org/wiki/Shearlet" rel="nofollow">https://en.wikipedia.org/wiki/Shearlet</a>
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.
The 2017 Abel Prize was awarded to Yves Meyer (no relation) for his contribution to the theory of wavelets. This is worth a read: <a href="https://www.quantamagazine.org/yves-meyer-wavelet-expert-wins-abel-prize-20170321/" rel="nofollow">https://www.quantamagazine.org/yves-meyer-wavelet-expert-win...</a>
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.
how does this compare to Fast Fourier transform ?<p><a href="https://en.wikipedia.org/wiki/Fast_Fourier_transform" rel="nofollow">https://en.wikipedia.org/wiki/Fast_Fourier_transform</a><p>Or the Discrete Cosine Transform ?<p><a href="https://en.wikipedia.org/wiki/Discrete_cosine_transform" rel="nofollow">https://en.wikipedia.org/wiki/Discrete_cosine_transform</a><p>Or the Gabor Transform ?<p><a href="https://en.wikipedia.org/wiki/Gabor_transform" rel="nofollow">https://en.wikipedia.org/wiki/Gabor_transform</a>