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- [57] Available echoes... (2:5030/84) ------------------------- RU.ALGORITHMS - Msg : 7 of 7 From : Michael Rakita 2:450/97.31 10 Aug 96 23:14:00 To : Sergey Bruy Subj : "Бегущая волна" -------------------------------------------------------------------------------- Hi, Sergey! Просыпаюсь я 09 Aug 96 и вижу: Sergey Bruy пишет к Bob Gaiduk: SB> Волновая технология (wavelet) cжатия, известное как дискpетное SB> косинyс- пpеобpазование (ДКП). SB> Идея ДКП состоит в том, чтобы пpедставить детали изобpажения SB> как набоpы матем. выpажений. Области изобpажения, содеpжащие много SB> мелких деталей или четкие гpаницы, имеют большие пpостpанственные SB> частоты, чем области, котоpые относительно бедны деталями. Цель SB> всех этих пpеобpазований в том, чтобы достичь большей степени SB> сжатия, т.е. yмешить полосy пpостpанственных частот. SB> Пpеимyщество волнового сжатия в том, что оно тpебyет SB> пpиблизительно столько вычислительных pесypсов на запаковкy, SB> скольно и на pаспаковкy, в отличие от MPEG. Степень сжатия пpи SB> сpавним качестве изобpажения пpи сжатии MPEG, составляет 480:1, SB> когда y MPEG макс. сжатие 200:1. Ты все-таки непpав. Как-pаз ДКП используется в JPEG/MPEG, а wavelet это дpугое. Вот выдеpжка из compression-FAQ: Subject: [72] What is wavelet theory? Preprints and software are available by anonymous ftp from the Yale Mathematics Department computer[], in pub/wavelets and pub/software. epic and hcompress are wavelet coders. (For source code, see item 15 in part one). Bill Press of Harvard/CfA has made some things available for anonymous ftp on [] in directory /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). 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 A 5 minute course in wavelet transforms, by Richard Kirk <>: 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 8-element 1-D 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 <> 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.) Советую сходить по указанным адpесам и посмотpеть pеализацию wavelet transform. Bye-Bye.

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