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The Chirplet Transform

#### The Chirplet Transform

In traditional signal processing, we use waves or wavelets. Waves are harmonic oscillations, such as sin(wt), where w is the frequency of the wave. "Wavelets" in the broadest sense are "pieces of waves", namely windowed waves. In a more strict use of the term, there are other mathematical restrictions such as the absence of a DC component.

A recently-proposed alternative to waves and wavelets are the so-called chirplets. The chirplet is to a wavelet as the chirp is to a wave, as shown in the figure below:

The chirplet is a windowed portion of a chirp. Four examples of chirplets (note for example, that a wave is a special case of a chirplet where the "chirprate" is zero and the window size is infinity).

A chirp is a would-be harmonic function but instead of having a fixed period (or frequency=1/period) the period changes with position (or time, or the like) along the function.

The most notable examples of chirps are the sounds made by birds where the resonant cavity changes size while oscillating. A slide-whistle can also make a chirping sound, by virtue of a cotton ball on a piece of wire that allows the user to change the length of the active portion of the wistle while blowing on it.

#### P-Chirps and P-Chirplets

Another notable example of chirping is the result of projective geometry acting on a periodic structure (e.g. a picture of a periodic structure), giving rise to periodicity-in-perspective, which may be regarded as being composed of a weighted sum of projected sinusiods that we call 'p-chirps', as depicted in the figure below:

Here is a picture (a) containing periodicity-in-perspective. If we plot a single raster across the center of the image, we observe that it is nearly periodic but that the period evolves (in this case, decreases) from left to right (or equivalently, the frequency increases from left to right). The most dominant p-chirp component is plotted in (c), below the plot of the raster. This dominant chirp component has a frequency that increases with the coordinate (e.g. time or space), and might therefore be called an "upchirp".

#### First published reference to chirplets

The first published reference to chirplets and the chirplet transform can be found here:
```@ARTICLE{mannVI91,
author = "Mann, Steve and Haykin, Simon",
title = "The Chirplet Transform: A Generalization of
{G}abor's Logon Transform",
journal = "Vision Interface '91",
publisher = "",
organization = "Canadian Image Processing and Patern Recognition Society",
month = "June 3-7",
pages = "205--212",
note = "ISSN 0843-803X",
year = "1991"
}
```
A scan of the 8 page article is here and you can also download a gzipped tarfile in html with jpeg files if you'd like to mirror this portion of the site.

Copies of the full conference proceedings are obtainable from:

```Canadian Information Processing Society
430 King St. West, Suite 205
Toronto, Ont.
M5V 1L5
Tel: (416) 593-4040
```

I an adaptive version of the chirplet tranform, a smaller number of chirplets are used to approximate an arbitrary signal, by selecting a small number of chirplets that, when added together, best describe the signal.

This work was also published in 1991:

```@ARTICLE{mannspieconference,
author = {Mann, Steve and Haykin, Simon},
journal = "SPIE, 36th Annual International Symposium on Optical and
Optoelectronic Applied Science and Engineering",
organization = "The International Society for Optical Engineering",
month = "21-26 July",
year = "1991"
}
```
The above paper describes Logon Expectation Maximization (LEM) which is a form of EM in the Time Frequency plane to form an optimal set of chirplet functions for the analysis of particular classes of signals.

There was also a 1992 paper on the Adaptive Chirplet Transform.

#### Additional publications on the Chirplet Transform

Further publications on the Chirplet Transform can be found at http://wearcam.org/chirplet/index.htm

#### Q-Chirps and Q-Chirplets

Quadratic chirps/chirplets are also of interest, especially in radar applications. You can Look at a PostScript paper on q-chirplets or you can download a PDF version with the actual pagination, etc., exactly as it appeared in the IEEE Transactions. (if you're using mosaic or glynx, a window should pop up to allow you to read this document right off the screen), or you can just take a a look at the html version below, if you don't like Adobe's file formats: A revised draft of this html document also appeared in the IEEE Transactions on Signal Processing, Vol 43, No. 11, November 1995.
```@ARTICLE{mannsp,
author    = "Steve Mann and Simon Haykin",
title     = "The Chirplet Transform: Physical Considerations",
journal = "{IEEE} Trans. Signal Processing",
year = "1995",
volume = 43,
number = 11,
pages = 2745--2761",
month = "November",
organization = "The Institute for Electrical and Electronics Engineers"}
# publisher = "{IEEE}",
# in above line, IEEE doesn't get included so i put it as part of the journal
```