
Wavelet Analysis for Dummies - kevinskii
http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html
======
mbenjaminsmith
Does anyone here know signal theory well enough to explain wavelet transform
vis a vis audio data compression? I used to follow the talk around audio
compression algorithms and I remember wavelets were often described as a
(potential) holy grail.

Any DSP gurus here?

~~~
btilly
I can answer very broadly.

Compressing data is the art of taking a large amount of data, and turning it
into a smaller amount of data saying much the same thing. Compression comes in
two fundamental forms, _lossless_ and _lossy_. Lossy compression is just a
fancy name for selecting some of the data for the great bit bucket in the sky.
Lossless compression is about finding a compact description of the data.
Wavelets are useful for both.

The simplest way to describe a signal is as a bunch of data points. This is
perfectly localized in space, but not in frequency. With solid blocks it gives
you an easily compressed description, but a smoothly varying curve takes a
tremendous amount of data to describe this way.

The next simplest approach is to describe it as a superposition of pure
frequencies. This is what the Fourier transform does. A smooth curve can be
described very efficiently this way, but if there is a discontinuity in it
anywhere, then it takes a tremendous amount of data to represent it.

Wavelets are little waves that are somewhat localized in both space and
frequency. A wavelet can be moved around or stretched to provide many
different trade-offs. This is good for efficient description of mixed signals,
such as pictures with smooth regions and sudden jumps, or the sound of talking
with boundaries between different phonemes. A lot of audio and visual data
looks like this.

This is good for lossless compression because wavelets tend to wind up fairly
efficiently describing each part of the signal in an appropriate way for that
part, which is overall very efficient.

But wavelets give you another trick. To specify a lot of detail takes a lot of
wavelets, but you can track how much "energy" each wavelet has. This lets you
prioritize the signal and figure out which parts you can get rid of without
changing the signal too noticeably. This is great for lossy compression.

But, and this is a big but, wavelets were very, very popular in the mid-90s,
lots of research was done, and then it was all locked away with patents. So
there is a lot of good stuff out there that nobody wants to use until the
patents expire. Which has stifled research, and makes a lot of this stuff
we'll all ignore for a few years. :-(

(The legal questions are one of the reasons that JPEG 2000 has not been widely
adopted. See
[http://upload.wikimedia.org/wikipedia/commons/7/78/JPEG_2000...](http://upload.wikimedia.org/wikipedia/commons/7/78/JPEG_2000_Artifacts_Demonstration.png)
for a visual demonstration of how much of an image remains even after you've
thrown away 99% of the data.)

~~~
snikolov
Another interesting side note is that wavelets (Gabor wavelets in particular)
have been shown to look very much like receptive fields in the primary visual
cortex. The suggestion is that such receptive fields evolved as an efficient
way to encode images of natural scenes. A couple of nice references are

David. J Field. Wavelets, vision, and the statistics of natural scenes. (1999)
[http://redwood.psych.cornell.edu/papers/field_wavelets_1999....](http://redwood.psych.cornell.edu/papers/field_wavelets_1999.pdf)

David J. Field. What is the goal of sensory coding? (1994)

[http://redwood.psych.cornell.edu/courses/psych527fall05/pape...](http://redwood.psych.cornell.edu/courses/psych527fall05/papers/Field1994_Sensory.pdf)

------
slig
Best use so far for Readability.

~~~
billmcneale
Funny I was about to post exactly that. Have an upvote.

