Top Document: comp.compression Frequently Asked Questions (part 2/3) Previous Document: [71] Introduction to MPEG (long) Next Document: [73] What is the theoretical compression limit? See reader questions & answers on this topic!  Help others by sharing your knowledge Preprints and software are available by anonymous ftp from the Yale Mathematics Department computer ftp://ceres.math.yale.edu/pub/wavelets/ and /pub/software/ . For source code of several wavelet coders, see item 15 in part one of this FAQ. A list of pointers, covering theory, papers, books, implementations, resources and more can be found at http://www.amara.com/current/wavelet.html#Wavelinks Bill Press of Harvard/CfA has made some things available on ftp://cfata4.harvard.edu/pub/ There is a short TeX article on wavelet theory (wavelet.tex, to be included in a future edition of Numerical Recipes), some sample wavelet code (wavelet.f, in FORTRAN  sigh), and a beta version of an astronomical image compression program which he is currently developing (FITS format data files only, in fitspress08.tar.Z). The Rice Wavelet Toolbox Release 2.0 is available in ftp://cml.rice.edu/pub/dsp/software/ and /pub/dsp/papers/ . This is a collection of MATLAB of "mfiles" and "mex" files for twoband and Mband filter bank/wavelet analysis from the DSP group and Computational Mathematics Laboratory (CML) at Rice University, Houston, TX. This release includes application code for Synthetic Aperture Radar despeckling and for deblocking of JPEG decompressed Images. Contact: Ramesh Gopinath <ramesh@rice.edu>. A wavelet transform coder construction kit is available at http://www.cs.dartmouth.edu/~gdavis/wavelet/wavelet.html Contact: Geoff Davis <gdavis@cs.dartmouth.edu> A matlab toolbox for constructing multiscale image representations, including Laplacian pyramids, QMFs, wavelets, and steerable pyramids, is available at ftp://ftp.cis.upenn.edu/pub/eero/ Contact: Eero Simoncelli <eero.simoncelli@nyu.edu>. A mailing list dedicated to research on wavelets has been set up at the University of South Carolina. To subscribe to this mailing list, send a message with "subscribe" as the subject to wavelet@math.sc.edu. For back issues and other information, check the Wavelet Digest home page at http://www.wavelet.org/ A tutorial by M. Hilton, B. Jawerth, and A. Sengupta, entitled "Compressing Still and Moving Images with Wavelets" is available in ftp://ftp.math.sc.edu/pub/wavelet/papers/varia/tutorial/ . The files are "tutorial.ps.Z" and "fig8.ps.Z". fig8 is a comparison of JPEG and wavelet compressed images and could take several hours to print. The tutorial is also available at http://www.mathsoft.com/wavelets.html A page on waveletbased HARCC compression technology is available at http://www.harc.edu/HARCC.html Commercial wavelet image compression software: http://www.aware.com http://www.summus.com http://www.infinop.com Details of the wavelet transform can be found in ftp://ftp.isds.duke.edu/pub/brani/papers/ ftp://ftp.isds.duke.edu/pub/brani/papers/ A 5 minute course in wavelet transforms, by Richard Kirk <rak@crosfield.co.uk>: Do you know what a Haar transform is? Its a transform to another orthonormal space (like the DFT), but the basis functions are a set of square wave bursts like this... ++ ++ +  + +  + ++ ++ ++ ++ +  + +  + ++ ++ ++ ++ +  + +  + ++ ++ ++ ++ +  + + + ++ This is the set of functions for an 8element 1D Haar transform. You can probably see how to extend this to higher orders and higher dimensions yourself. This is dead easy to calculate, but it is not what is usually understood by a wavelet transform. If you look at the eight Haar functions you see we have four functions that code the highest resolution detail, two functions that code the coarser detail, one function that codes the coarser detail still, and the top function that codes the average value for the whole `image'. Haar function can be used to code images instead of the DFT. With bilevel images (such as text) the result can look better, and it is quicker to code. Flattish regions, textures, and soft edges in scanned images get a nasty `blocking' feel to them. This is obvious on hardcopy, but can be disguised on color CRTs by the effects of the shadow mask. The DCT gives more consistent results. This connects up with another bit of maths sometimes called Multispectral Image Analysis, sometimes called Image Pyramids. Suppose you want to produce a discretely sampled image from a continuous function. You would do this by effectively `scanning' the function using a sinc function [ sin(x)/x ] `aperture'. This was proved by Shannon in the `forties. You can do the same thing starting with a high resolution discretely sampled image. You can then get a whole set of images showing the edges at different resolutions by differencing the image at one resolution with another version at another resolution. If you have made this set of images properly they ought to all add together to give the original image. This is an expansion of data. Suppose you started off with a 1K*1K image. You now may have a 64*64 low resolution image plus difference images at 128*128 256*256, 512*512 and 1K*1K. Where has this extra data come from? If you look at the difference images you will see there is obviously some redundancy as most of the values are near zero. From the way we constructed the levels we know that locally the average must approach zero in all levels but the top. We could then construct a set of functions out of the sync functions at any level so that their total value at all higher levels is zero. This gives us an orthonormal set of basis functions for a transform. The transform resembles the Haar transform a bit, but has symmetric wave pulses that decay away continuously in either direction rather than square waves that cut off sharply. This transform is the wavelet transform ( got to the point at last!! ). These wavelet functions have been likened to the edge detecting functions believed to be present in the human retina. Loren I. Petrich <lip@s1.gov> adds that order 2 or 3 Daubechies discrete wavelet transforms have a speed comparable to DCT's, and usually achieve compression a factor of 2 better for the same image quality than the JPEG 8*8 DCT. (See item 25 in part 1 of this FAQ for references on fast DCT algorithms.) User Contributions:Comment about this article, ask questions, or add new information about this topic:Top Document: comp.compression Frequently Asked Questions (part 2/3) Previous Document: [71] Introduction to MPEG (long) Next Document: [73] What is the theoretical compression limit? Part1  Part2  Part3  Single Page [ Usenet FAQs  Web FAQs  Documents  RFC Index ] Send corrections/additions to the FAQ Maintainer: jloup@gzip.OmitThis.org
Last Update March 27 2014 @ 02:11 PM
