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From: eero@media-lab.media.mit.edu (Eero Simoncelli)
Newsgroups: comp.dsp,comp.compression
Subject: Re: Wavelet Transforms?
Message-ID: <1991Jun17.190324.5903@news.media.mit.edu>
Date: 17 Jun 91 19:03:24 GMT
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John Moore (john@qip.UUCP) writes:

> Can anyone post a summary of the concept of wavelet transforms? I have
> read that these are the "hot new thing."
>
> .... With wavelets, presumably someone found a nifty
> set of orthogonal, spanning functions that resemble solitons or
> sinx/x functions or something else that is new and marvelous.

This is a reasonable description, but most people would add a few
additional properties:

	1) The basis functions form an octave subband (constant Q)
	transform (when viewed in the frequency domain).

	2) The transform is defined (and typically implemented)
	recursively.

In addition, people are often interested in the convergence properties
of the recursive construction.  If the process converges to a smooth
function, the wavelet set is called "regular". One useful mathematical
review that hasn't been mentioned is the following:

@ARTICLE{Strang89,
	AUTHOR = "G. T. Strang",
	TITLE = "Wavelets and dilation equations: A brief introduction",
	JOURNAL = "{SIAM} Review",
	VOLUME = 31,
	NUMBER = 4,
	MONTH = "December",
	YEAR = 1989,
	PAGES = "614--627"
	}

Also, researchers in the speech coding community have been using a
version of discrete wavelet transforms (under the name of Quadrature
Mirror Filter banks) since 1976.  They've also been used in image
coding.  Here are some references:

@INPROCEEDINGS{Croisier76,
	AUTHOR = "A. Croisier and D. Esteban and C. Galand",
	TITLE = "Perfect channel splitting by use of 
	interpolation/decimation/tree decomposition techniques",
	BOOKTITLE = "International Conference on Information Sciences 
	and Systems",
	YEAR = 1976,
	MONTH = "August",
	ADDRESS = "Patras",
	PAGES = "443--446"
	}

@INPROCEEDINGS{Esteban77,
	AUTHOR = "D. Esteban and C. Galand",
	TITLE = "Application of quadrature mirror filters to split band 
	voice coding schemes",
	BOOKTITLE = "Proceedings ICASSP",
	YEAR = "1977",
	PAGES = "191--195"
	}

@ARTICLE{Vetterli84,
	AUTHOR = "M. Vetterli",
	TITLE = "Multi-dimensional sub-band coding: 
	Some theory and algorithms",
	JOURNAL = "Signal Processing",
	PUBLISHER = "Elsevier Science Publishers~B.V. (North-Holland)",
	MONTH = "February",
	YEAR = "1984",
	VOLUME = "6",
	NUMBER = "2",
	PAGES = "97--112"
	}

@ARTICLE{Woods86,
	AUTHOR = "J. W. Woods and S. D. O'Neil",
	TITLE = "Subband coding of images",
	JOURNAL = "IEEE Trans. ASSP",
	MONTH = "October",
	YEAR = "1986",
	VOLUME = "ASSP-34",
	NUMBER = "5",
	PAGES = "1278--1288"
	}

@INPROCEEDINGS{Gharavi86,
	AUTHOR = "Gharavi, H. and Tabatabai, A.",
	TITLE = "Sub-band coding of digital images using two-dimensional 
	quadrature mirror filtering",
	BOOKTITLE = "Proceedings of SPIE",
	YEAR = "1986",
	VOLUME = "707",
	PAGES = "51--61"
	}

@INPROCEEDINGS{Adelson87,
	AUTHOR = "E. H. Adelson and E. Simoncelli and R. Hingorani",
	TITLE = "Orthogonal pyramid transforms for image coding",
	BOOKTITLE = "Proceedings of SPIE",
	ADDRESS = "Cambridge, MA",
	VOLUME = 845,
	MONTH = "October",
	PAGES = "50--58",
	YEAR = 1987
	}

@ARTICLE{Simoncelli89,
	AUTHOR = "E. Simoncelli and E. H. Adelson",
	TITLE = "Non-separable Extensions of Quadrature Mirror Filters 
	to Multiple Dimensions",
	JOURNAL = "Proceedings of the IEEE: Special Issue on
	Multi-dimensional Signal Processing",
	MONTH = "April",
	YEAR = 1990
	}

@BOOK{Woods90,
	EDITOR = "J. W. Woods",
	BOOKTITLE = "Subband Image Coding",
	PUBLISHER = "Kluwer Academic Publishers",
	ADDRESS = "Norwell, MA",
	YEAR = 1990
	}

Finally, two interesting related transforms are the Laplacian pyramid
and the Lapped Orthogonal Transform:

@ARTICLE{Burt83,
	AUTHOR = "P. J. Burt and E. H. Adelson",
	TITLE = "The {L}aplacian pyramid as a compact image code",
	JOURNAL = "IEEE Trans. Communications",
	MONTH = "April",
	YEAR = "1983",
	VOLUME = "COM-31",
	NUMBER = "4",
	PAGES = "532--540"
	}

@ARTICLE{Malvar89,
	AUTHOR = "H. S. Malvar and D. H. Staelin",
	TITLE = "The {LOT}: Transform Coding Without Blocking Effects",
	JOURNAL = "IEEE Trans. ASSP",
	VOLUME = "ASSP-37",
	NUMBER = 4,
	PAGES = "553--559",
	MONTH = "April",
	YEAR = 1989
	}

Hope you find these useful,

Eero.
-- 
Eero Simoncelli			 	eero@media-lab.media.mit.edu
MIT Media Laboratory, E15-385, 20 Ames Street, Cambridge, MA  02139