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Wavelets (1994) [pdf]

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Re: Wavelets (1994) [pdf]

#41

Post by planteen »

mrcactu5 wrote: post #35

how does this compare to Fast Fourier transform ? https://en.wikipedia.org/wiki/Fast_Fourier_transform Or the Discrete Cosine Transform ? https://en.wikipedia.org/wiki/Discrete_cosine_transform Or the Gabor Transform ? https://en.wikipedia.org/wiki/Gabor_transform

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.

Re: Wavelets (1994) [pdf]

#42

Post by goalieca »

pavlov wrote: post #32
goalieca wrote: post #14

Earlier quoted context omitted.

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.

There's no wavelet-based video codec in widespread use (unless one counts DCP, a file format used for physical deliveries of movies to theaters, where the picture track is encoded as JPEG2000 frames).

H.264 is pretty widespread.

Re: Wavelets (1994) [pdf]

#43

Post by liuyao »

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?

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

Re: Wavelets (1994) [pdf]

#44

Post by jacobolus »

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…

> 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 be warned that the polynomial you get depends on where you center your Taylor series. There are multiple different Taylor-series approximations for the same function, and each one only converges within a disk in the complex plane which does not include any poles.

If you try to fit the Runge function 1 / (1 + 25x^2) with a Taylor series centered at 0 you’re going to have a bad time.

https://www.desmos.com/calculator/uyjyjug07m

Re: Wavelets (1994) [pdf]

#45

Post by ksec »

goalieca wrote: post #42
pavlov wrote: post #32

Earlier quoted context omitted.

There's no wavelet-based video codec in widespread use (unless one counts DCP, a file format used for physical deliveries of movies to theaters, where the picture track is encoded as JPEG2000 frames).

H.264 is pretty widespread.

H.264 is not a wavelet-based video codec.

Re: Wavelets (1994) [pdf]

#46

Post by ksec »

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!

The problem is that the economical model of Wavelet, or TCO of Wavelet codec doesn't do well. And like everything in out world it is only the cost and values on offers that matters. The total energy ( CPU cycles, or hardware implementation ) it requires to encode and decode far outweighs the benefits it offers. Not to mention the current block based still has many tricks. VVC ( H.266 ) seems to be doing well. ( Patents issues aside )

Re: Wavelets (1994) [pdf]

#47

Post by TooSmugToFail »

ksec wrote: post #46
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!

The problem is that the economical model of Wavelet, or TCO of Wavelet codec doesn't do well. And like everything in out world it is only the cost and values on offers that matters. The total energy ( CPU cycles, or hardware implementation ) it requires to encode and decode far outweighs the benefits it offers. Not to mention the current block based still has many tricks. VVC ( H.266 ) seems to be doing well. ( Paten…

Do you happen to know what's the encoding time for a DWT based encoder vs H.264? Assuming the same hardware, of course...

Are we talking about roughly the same encoding time, or the DWT is slower by an order of magnitude?

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