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Xref: bloom-picayune.mit.edu comp.compression:4949 news.answers:3737 Path: bloom-picayune.mit.edu!snorkelwacker.mit.edu!news.media.mit.edu!micro-heart-of-gold.mit.edu!wupost!darwin.sura.net!paladin.american.edu!news.univie.ac.at!hp4at!mcsun!corton!chorus!chorus.fr From: jloup@chorus.fr (Jean-loup Gailly) Newsgroups: comp.compression,news.answers Subject: comp.compression Frequently Asked Questions (part 2/2) Summary: *** READ THIS BEFORE POSTING *** Keywords: data compression, FAQ Message-ID: <compr2_29oct92@chorus.fr> Date: 30 Oct 92 13:11:19 GMT Expires: 10 Dec 92 16:17:20 GMT References: <compr1_29oct92@chorus.fr> Sender: news@chorus.chorus.fr Reply-To: jloup@chorus.fr Followup-To: comp.compression Lines: 800 Approved: news-answers-request@MIT.Edu Supersedes: <compr2_2oct92@chorus.fr> Archive-name: compression-faq/part2 Last-modified: Oct 2nd, 1992 This file is part 2 of a set of Frequently Asked Questions for the groups comp.compression and comp.compression.research. If you don't want to see this FAQ regularly, please add the subject line to your kill file. If you have corrections or suggestions for this FAQ, send them to Jean-loup Gailly <jloup@chorus.fr>. Thank you. Contents ======== (Long) introductions to data compression techniques [70] Introduction to data compression (long) Huffman and Related Compression Techniques Arithmetic Coding Substitutional Compressors The LZ78 family of compressors The LZ77 family of compressors [71] Introduction to MPEG (long) What is MPEG? Does it have anything to do with JPEG? Then what's JBIG and MHEG? What has MPEG accomplished? So how does MPEG I work? What about the audio compression? So how much does it compress? What's phase II? When will all this be finished? How do I join MPEG? How do I get the documents, like the MPEG I draft? [72] What is wavelet theory? [73] What is the theoretical compression limit? [74] Introduction to JBIG [99] Acknowledgments Search for "Subject: [#]" to get to question number # quickly. Some news readers can also take advantage of the message digest format used here. ------------------------------------------------------------------------------ Subject: [70] Introduction to data compression (long) Written by Peter Gutmann <pgut1@cs.aukuni.ac.nz>. Huffman and Related Compression Techniques ------------------------------------------ *Huffman compression* is a statistical data compression technique which gives a reduction in the average code length used to represent the symbols of a alphabet. The Huffman code is an example of a code which is optimal in the case where all symbols probabilities are integral powers of 1/2. A Huffman code can be built in the following manner: (1) Rank all symbols in order of probability of occurrence. (2) Successively combine the two symbols of the lowest probability to form a new composite symbol; eventually we will build a binary tree where each node is the probability of all nodes beneath it. (3) Trace a path to each leaf, noticing the direction at each node. For a given frequency distribution, there are many possible Huffman codes, but the total compressed length will be the same. It is possible to define a 'canonical' Huffman tree, that is, pick one of these alternative trees. Such a canonical tree can then be represented very compactly, by transmitting only the bit length of each code. This technique is used in most archivers (pkzip, lha, zoo, arj, ...). A technique related to Huffman coding is *Shannon-Fano coding*, which was suggested by Shannon and Weaver in 1949 and modified by Fano in 1961. It works as follows: (1) Divide the set of symbols into two equal or almost equal subsets based on the probability of occurrence of characters in each subset. The first subset is assigned a binary zero, the second a binary one. (2) Repeat step (1) until all subsets have a single element. The algorithm used to create the Huffman codes is bottom-up, and the one for the Shannon-Fano codes is top-down. Huffman encoding always generates optimal codes, Shannon-Fano sometimes uses a few more bits. Arithmetic Coding ----------------- It would appear that Huffman or Shannon-Fano coding is the perfect means of compressing data. However, this is *not* the case. As mentioned above, these coding methods are optimal when and only when the symbol probabilities are integral powers of 1/2, which is usually not the case. The technique of *arithmetic coding* does not have this restriction: It achieves the same effect as treating the message as one single unit (a technique which would, for Huffman coding, require enumeration of every single possible message), and thus attains the theoretical entropy bound to compression efficiency for any source. Arithmetic coding works by representing a number by an interval of real numbers between 0 and 1. As the message becomes longer, the interval needed to represent it becomes smaller and smaller, and the number of bits needed to specify that interval increases. Successive symbols in the message reduce this interval in accordance with the probability of that symbol. The more likely symbols reduce the range by less, and thus add fewer bits to the message. 1 Codewords +-----------+-----------+-----------+ /-----\ | |8/9 YY | Detail |<- 31/32 .11111 | +-----------+-----------+<- 15/16 .1111 | Y | | too small |<- 14/16 .1110 |2/3 | YX | for text |<- 6/8 .110 +-----------+-----------+-----------+ | | |16/27 XYY |<- 10/16 .1010 | | +-----------+ | | XY | | | | | XYX |<- 4/8 .100 | |4/9 | | | +-----------+-----------+ | | | | | X | | XXY |<- 3/8 .011 | | |8/27 | | | +-----------+ | | XX | | | | | |<- 1/4 .01 | | | XXX | | | | | |0 | | | +-----------+-----------+-----------+ As an example of arithmetic coding, lets consider the example of two symbols X and Y, of probabilities 0.66 and 0.33. To encode this message, we examine the first symbol: If it is a X, we choose the lower partition; if it is a Y, we choose the upper partition. Continuing in this manner for three symbols, we get the codewords shown to the right of the diagram above - they can be found by simply taking an appropriate location in the interval for that particular set of symbols and turning it into a binary fraction. In practice, it is also necessary to add a special end-of-data symbol, which is not represented in this simpe example. In this case the arithmetic code is not completely efficient, which is due to the shortness of the message - with longer messages the coding efficiency does indeed approach 100%. Now that we have an efficient encoding technique, what can we do with it? What we need is a technique for building a model of the data which we can then use with the encoder. The simplest model is a fixed one, for example a table of standard letter frequencies for English text which we can then use to get letter probabilities. An improvement on this technique is to use an *adaptive model*, in other words a model which adjusts itself to the data which is being compressed as the data is compressed. We can convert the fixed model into an adaptive one by adjusting the symbol frequencies after each new symbol is encoded, allowing the model to track the data being transmitted. However, we can do much better than that. Using the symbol probabilities by themselves is not a particularly good estimate of the true entropy of the data: We can take into account intersymbol probabilities as well. The best compressors available today take this approach: DMC (Dynamic Markov Coding) starts with a zero-order Markov model and gradually extends this initial model as compression progresses; PPM (Prediction by Partial Matching) looks for a match of the text to be compressed in an order-n context. If no match is found, it drops to an order n-1 context, until it reaches order 0. Both these techniques thus obtain a much better model of the data to be compressed, which, combined with the use of arithmetic coding, results in superior compression performance. So if arithmetic coding-based compressors are so powerful, why are they not used universally? Apart from the fact that they are relatively new and haven't come into general use too much yet, there is also one major concern: The fact that they consume rather large amounts of computing resources, both in terms of CPU power and memory. The building of sophisticated models for the compression can chew through a fair amount of memory (especially in the case of DMC, where the model can grow without bounds); and the arithmetic coding itself involves a fair amount of number crunching. There is however an alternative approach, a class of compressors generally referred to as *substitutional* or *dictionary-based compressors*. Substitutional Compressors -------------------------- The basic idea behind a substitutional compressor is to replace an occurrence of a particular phrase or group of bytes in a piece of data with a reference to a previous occurrence of that phrase. There are two main classes of schemes, named after Jakob Ziv and Abraham Lempel, who first proposed them in 1977 and 1978. <The LZ78 family of compressors> LZ78-based schemes work by entering phrases into a *dictionary* and then, when a repeat occurrence of that particular phrase is found, outputting the dictionary index instead of the phrase. There exist several compression algorithms based on this principle, differing mainly in the manner in which they manage the dictionary. The most well-known scheme (in fact the most well-known of all the Lempel-Ziv compressors, the one which is generally (and mistakenly) referred to as "Lempel-Ziv Compression"), is Terry Welch's LZW scheme, which he designed in 1984 for implementation in hardware for high- performance disk controllers. Input string: /WED/WE/WEE/WEB Character input: Code output: New code value and associated string: /W / 256 = /W E W 257 = WE D E 258 = ED / D 259 = D/ WE 256 260 = /WE / E 261 = E/ WEE 260 262 = /WEE /W 261 263 = E/W EB 257 264 = WEB <END> B LZW starts with a 4K dictionary, of which entries 0-255 refer to individual bytes, and entries 256-4095 refer to substrings. Each time a new code is generated it means a new string has been parsed. New strings are generated by appending the current character K to the end of an existing string w. The algorithm for LZW compression is as follows: set w = NIL loop read a character K if wK exists is in the dictionary w = wK else output the code for w add wK to the string table w = K endloop A sample run of LZW over a (highly redundant) input string can be seen in the diagram above. The strings are built up character-by-character starting with a code value of 256. LZW decompression takes the stream of codes and uses it to exactly recreate the original input data. Just like the compression algorithm, the decompressor adds a new string to the dictionary each time it reads in a new code. All it needs to do in addition is to translate each incoming code into a string and send it to the output. A sample run of the LZW decompressor is shown in below. Input code: /WED<256>E<260><261><257>B Input code: Output string: New code value and associated string: / / W W 256 = /W E E 257 = WE D D 258 = ED 256 /W 259 = D/ E E 260 = /WE 260 /WE 261 = E/ 261 E/ 262 = /WEE 257 WE 263 = E/W B B 264 = WEB The most remarkable feature of this type of compression is that the entire dictionary has been transmitted to the decoder without actually explicitly transmitting the dictionary. At the end of the run, the decoder will have a dictionary identical to the one the encoder has, built up entirely as part of the decoding process. LZW is more commonly encountered today in a variant known as LZC, after its use in the UNIX "compress" program. In this variant, pointers do not have a fixed length. Rather, they start with a length of 9 bits, and then slowly grow to their maximum possible length once all the pointers of a particular size have been used up. Furthermore, the dictionary is not frozen once it is full as for LZW - the program continually monitors compression performance, and once this starts decreasing the entire dictionary is discarded and rebuilt from scratch. More recent schemes use some sort of least-recently-used algorithm to discard little-used phrases once the dictionary becomes full rather than throwing away the entire dictionary. Finally, not all schemes build up the dictionary by adding a single new character to the end of the current phrase. An alternative technique is to concatenate the previous two phrases (LZMW), which results in a faster buildup of longer phrases than the character-by-character buildup of the other methods. The disadvantage of this method is that a more sophisticated data structure is needed to handle the dictionary. [A good introduction to LZW, MW, AP and Y coding is given in the yabba package. For ftp information, see question 2 in part one, file type .Y] <The LZ77 family of compressors> LZ77-based schemes keep track of the last n bytes of data seen, and when a phrase is encountered that has already been seen, they output a pair of values corresponding to the position of the phrase in the previously-seen buffer of data, and the length of the phrase. In effect the compressor moves a fixed-size *window* over the data (generally referred to as a *sliding window*), with the position part of the (position, length) pair referring to the position of the phrase within the window. The most commonly used algorithms are derived from the LZSS scheme described by James Storer and Thomas Szymanski in 1982. In this the compressor maintains a window of size N bytes and a *lookahead buffer* the contents of which it tries to find a match for in the window: while( lookAheadBuffer not empty ) { get a pointer ( position, match ) to the longest match in the window for the lookahead buffer; if( length > MINIMUM_MATCH_LENGTH ) { output a ( position, length ) pair; shift the window length characters along; } else { output the first character in the lookahead buffer; shift the window 1 character along; } } Decompression is simple and fast: Whenever a ( position, length ) pair is encountered, go to that ( position ) in the window and copy ( length ) bytes to the output. Sliding-window-based schemes can be simplified by numbering the input text characters mod N, in effect creating a circular buffer. The sliding window approach automatically creates the LRU effect which must be done explicitly in LZ78 schemes. Variants of this method apply additional compression to the output of the LZSS compressor, which include a simple variable-length code (LZB), dynamic Huffman coding (LZH), and Shannon-Fano coding (ZIP 1.x)), all of which result in a certain degree of improvement over the basic scheme, especially when the data are rather random and the LZSS compressor has little effect. Recently an algorithm was developed which combines the ideas behind LZ77 and LZ78 to produce a hybrid called LZFG. LZFG uses the standard sliding window, but stores the data in a modified trie data structure and produces as output the position of the text in the trie. Since LZFG only inserts complete *phrases* into the dictionary, it should run faster than other LZ77-based compressors. All popular archivers (arj, lha, zip, zoo) are variations on the LZ77 theme. ------------------------------------------------------------------------------ Subject: [71] Introduction to MPEG (long) Written by Mark Adler <madler@cco.caltech.edu>. Q. What is MPEG? A. MPEG is a group of people that meet under ISO (the International Standards Organization) to generate standards for digital video (sequences of images in time) and audio compression. In particular, they define a compressed bit stream, which implicitly defines a decompressor. However, the compression algorithms are up to the individual manufacturers, and that is where proprietary advantage is obtained within the scope of a publicly available international standard. MPEG meets roughly four times a year for roughly a week each time. In between meetings, a great deal of work is done by the members, so it doesn't all happen at the meetings. The work is organized and planned at the meetings. Q. So what does MPEG stand for? A. Moving Pictures Experts Group. Q. Does it have anything to do with JPEG? A. Well, it sounds the same, and they are part of the same subcommittee of ISO along with JBIG and MHEG, and they usually meet at the same place at the same time. However, they are different sets of people with few or no common individual members, and they have different charters and requirements. JPEG is for still image compression. Q. Then what's JBIG and MHEG? A. Sorry I mentioned them. Ok, I'll simply say that JBIG is for binary image compression (like faxes), and MHEG is for multi-media data standards (like integrating stills, video, audio, text, etc.). For an introduction to JBIG, see question 74 below. Q. Ok, I'll stick to MPEG. What has MPEG accomplished? A. So far (as of January 1992), they have completed the "Committee Draft" of MPEG phase I, colloquially called MPEG I. It defines a bit stream for compressed video and audio optimized to fit into a bandwidth (data rate) of 1.5 Mbits/s. This rate is special because it is the data rate of (uncompressed) audio CD's and DAT's. The draft is in three parts, video, audio, and systems, where the last part gives the integration of the audio and video streams with the proper timestamping to allow synchronization of the two. They have also gotten well into MPEG phase II, whose task is to define a bitstream for video and audio coded at around 3 to 10 Mbits/s. Q. So how does MPEG I work? A. First off, it starts with a relatively low resolution video sequence (possibly decimated from the original) of about 352 by 240 frames by 30 frames/s (US--different numbers for Europe), but original high (CD) quality audio. The images are in color, but converted to YUV space, and the two chrominance channels (U and V) are decimated further to 176 by 120 pixels. It turns out that you can get away with a lot less resolution in those channels and not notice it, at least in "natural" (not computer generated) images. The basic scheme is to predict motion from frame to frame in the temporal direction, and then to use DCT's (discrete cosine transforms) to organize the redundancy in the spatial directions. The DCT's are done on 8x8 blocks, and the motion prediction is done in the luminance (Y) channel on 16x16 blocks. In other words, given the 16x16 block in the current frame that you are trying to code, you look for a close match to that block in a previous or future frame (there are backward prediction modes where later frames are sent first to allow interpolating between frames). The DCT coefficients (of either the actual data, or the difference between this block and the close match) are "quantized", which means that you divide them by some value to drop bits off the bottom end. Hopefully, many of the coefficients will then end up being zero. The quantization can change for every "macroblock" (a macroblock is 16x16 of Y and the corresponding 8x8's in both U and V). The results of all of this, which include the DCT coefficients, the motion vectors, and the quantization parameters (and other stuff) is Huffman coded using fixed tables. The DCT coefficients have a special Huffman table that is "two-dimensional" in that one code specifies a run-length of zeros and the non-zero value that ended the run. Also, the motion vectors and the DC DCT components are DPCM (subtracted from the last one) coded. Q. So is each frame predicted from the last frame? A. No. The scheme is a little more complicated than that. There are three types of coded frames. There are "I" or intra frames. They are simply a frame coded as a still image, not using any past history. You have to start somewhere. Then there are "P" or predicted frames. They are predicted from the most recently reconstructed I or P frame. (I'm describing this from the point of view of the decompressor.) Each macroblock in a P frame can either come with a vector and difference DCT coefficients for a close match in the last I or P, or it can just be "intra" coded (like in the I frames) if there was no good match. Lastly, there are "B" or bidirectional frames. They are predicted from the closest two I or P frames, one in the past and one in the future. You search for matching blocks in those frames, and try three different things to see which works best. (Now I have the point of view of the compressor, just to confuse you.) You try using the forward vector, the backward vector, and you try averaging the two blocks from the future and past frames, and subtracting that from the block being coded. If none of those work well, you can intra- code the block. The sequence of decoded frames usually goes like: IBBPBBPBBPBBIBBPBBPB... Where there are 12 frames from I to I (for US and Japan anyway.) This is based on a random access requirement that you need a starting point at least once every 0.4 seconds or so. The ratio of P's to B's is based on experience. Of course, for the decoder to work, you have to send that first P *before* the first two B's, so the compressed data stream ends up looking like: 0xx312645... where those are frame numbers. xx might be nothing (if this is the true starting point), or it might be the B's of frames -2 and -1 if we're in the middle of the stream somewhere. You have to decode the I, then decode the P, keep both of those in memory, and then decode the two B's. You probably display the I while you're decoding the P, and display the B's as you're decoding them, and then display the P as you're decoding the next P, and so on. Q. You've got to be kidding. A. No, really! Q. Hmm. Where did they get 352x240? A. That derives from the CCIR-601 digital television standard which is used by professional digital video equipment. It is (in the US) 720 by 243 by 60 fields (not frames) per second, where the fields are interlaced when displayed. (It is important to note though that fields are actually acquired and displayed a 60th of a second apart.) The chrominance channels are 360 by 243 by 60 fields a second, again interlaced. This degree of chrominance decimation (2:1 in the horizontal direction) is called 4:2:2. The source input format for MPEG I, called SIF, is CCIR-601 decimated by 2:1 in the horizontal direction, 2:1 in the time direction, and an additional 2:1 in the chrominance vertical direction. And some lines are cut off to make sure things divide by 8 or 16 where needed. Q. What if I'm in Europe? A. For 50 Hz display standards (PAL, SECAM) change the number of lines in a field from 243 or 240 to 288, and change the display rate to 50 fields/s or 25 frames/s. Similarly, change the 120 lines in the decimated chrominance channels to 144 lines. Since 288*50 is exactly equal to 240*60, the two formats have the same source data rate. Q. You didn't mention anything about the audio compression. A. Oh, right. Well, I don't know as much about the audio compression. Basically they use very carefully developed psychoacoustic models derived from experiments with the best obtainable listeners to pick out pieces of the sound that you can't hear. There are what are called "masking" effects where, for example, a large component at one frequency will prevent you from hearing lower energy parts at nearby frequencies, where the relative energy vs. frequency that is masked is described by some empirical curve. There are similar temporal masking effects, as well as some more complicated interactions where a temporal effect can unmask a frequency, and vice-versa. The sound is broken up into spectral chunks with a hybrid scheme that combines sine transforms with subband transforms, and the psychoacoustic model written in terms of those chunks. Whatever can be removed or reduced in precision is, and the remainder is sent. It's a little more complicated than that, since the bits have to be allocated across the bands. And, of course, what is sent is entropy coded. Q. So how much does it compress? A. As I mentioned before, audio CD data rates are about 1.5 Mbits/s. You can compress the same stereo program down to 256 Kbits/s with no loss in discernable quality. (So they say. For the most part it's true, but every once in a while a weird thing might happen that you'll notice. However the effect is very small, and it takes a listener trained to notice these particular types of effects.) That's about 6:1 compression. So, a CD MPEG I stream would have about 1.25 MBits/s left for video. The number I usually see though is 1.15 MBits/s (maybe you need the rest for the system data stream). You can then calculate the video compression ratio from the numbers here to be about 26:1. If you step back and think about that, it's little short of a miracle. Of course, it's lossy compression, but it can be pretty hard sometimes to see the loss, if you're comparing the SIF original to the SIF decompressed. There is, however, a very noticeable loss if you're coming from CCIR-601 and have to decimate to SIF, but that's another matter. I'm not counting that in the 26:1. The standard also provides for other bit rates ranging from 32Kbits/s for a single channel, up to 448 Kbits/s for stereo. Q. What's phase II? A. As I said, there is a considerable loss of quality in going from CCIR-601 to SIF resolution. For entertainment video, it's simply not acceptable. You want to use more bits and code all or almost all the CCIR-601 data. From subjective testing at the Japan meeting in November 1991, it seems that 4 MBits/s can give very good quality compared to the original CCIR-601 material. The objective of phase II is to define a bit stream optimized for these resolutions and bit rates. Q. Why not just scale up what you're doing with MPEG I? A. The main difficulty is the interlacing. The simplest way to extend MPEG I to interlaced material is to put the fields together into frames (720x486x30/s). This results in bad motion artifacts that stem from the fact that moving objects are in different places in the two fields, and so don't line up in the frames. Compressing and decompressing without taking that into account somehow tends to muddle the objects in the two different fields. The other thing you might try is to code the even and odd field streams separately. This avoids the motion artifacts, but as you might imagine, doesn't get very good compression since you are not using the redundancy between the even and odd fields where there is not much motion (which is typically most of image). Or you can code it as a single stream of fields. Or you can interpolate lines. Or, etc. etc. There are many things you can try, and the point of MPEG II is to figure out what works well. MPEG II is not limited to consider only derivations of MPEG I. There were several non-MPEG I-like schemes in the competition in November, and some aspects of those algorithms may or may not make it into the final standard for entertainment video compression. Q. So what works? A. Basically, derivations of MPEG I worked quite well, with one that used wavelet subband coding instead of DCT's that also worked very well. Also among the worked-very-well's was a scheme that did not use B frames at all, just I and P's. All of them, except maybe one, did some sort of adaptive frame/field coding, where a decision is made on a macroblock basis as to whether to code that one as one frame macroblock or as two field macroblocks. Some other aspects are how to code I-frames--some suggest predicting the even field from the odd field. Or you can predict evens from evens and odds or odds from evens and odds or any field from any other field, etc. Q. So what works? A. Ok, we're not really sure what works best yet. The next step is to define a "test model" to start from, that incorporates most of the salient features of the worked-very-well proposals in a simple way. Then experiments will be done on that test model, making a mod at a time, and seeing what makes it better and what makes it worse. Example experiments are, B's or no B's, DCT vs. wavelets, various field prediction modes, etc. The requirements, such as implementation cost, quality, random access, etc. will all feed into this process as well. Q. When will all this be finished? A. I don't know. I'd have to hope in about a year or less. Q. How do I join MPEG? A. You don't join MPEG. You have to participate in ISO as part of a national delegation. How you get to be part of the national delegation is up to each nation. I only know the U.S., where you have to attend the corresponding ANSI meetings to be able to attend the ISO meetings. Your company or institution has to be willing to sink some bucks into travel since, naturally, these meetings are held all over the world. (For example, Paris, Santa Clara, Kurihama Japan, Singapore, Haifa Israel, Rio de Janeiro, London, etc.) Q. Well, then how do I get the documents, like the MPEG I draft? A. MPEG is a draft ISO standard. It's exact name is ISO CD 11172. The draft consists of three parts: System, Video, and Audio. The System part (11172-1) deals with synchronization and multiplexing of audio-visual information, while the Video (11172-2) and Audio part (11172-3) address the video and the audio compression techniques respectively. You may order it from your national standards body (e.g. ANSI in the USA) or buy it from companies like OMNICOM phone +44 438 742424 FAX +44 438 740154 ------------------------------------------------------------------------------ Subject: [72] What is wavelet theory? Preprints and software are available by anonymous ftp from the Yale Mathematics Department computer ceres.math.yale.edu[130.132.23.22], in pub/wavelets and pub/software. epic is a pyramid wavelet coder. (For source code, see item 15 in part one). Bill Press of Harvard/CfA has made some things available for anonymous ftp on cfata4.harvard.edu [128.103.40.79] 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). An experimental wavelet decomposition program is available in linc.cis.upenn.edu:/pub/grasp/wavelet.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 wavelet@math.scarolina.edu. 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 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 <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.) ------------------------------------------------------------------------------ Subject: [73] What is the theoretical compression limit? There is no compressor that is guaranteed to compress all possible input files. If it compresses some files, then it must enlarge some others. This can be proven by a simple counting argument (see question 9). As an extreme example, the following algorithm achieves 100% compression for one special input file and enlarges all other files by only one bit: - if the input data is <insert your favorite one here>, output an empty file. - otherwise output one bit (zero or one) followed by the input data. The concept of theoretical compression limit is meaningful only if you have a model for your input data. See question 70 above for some examples of data models. ------------------------------------------------------------------------------ Subject: [74] Introduction to JBIG Written by Mark Adler <madler@cco.caltech.edu>. JBIG losslessly compresses binary (one-bit/pixel) images. (The B stands for bi-level.) Basically it models the redundancy in the image as the correlations of the pixel currently being coded with a set of nearby pixels called the template. An example template might be the two pixels preceding this one on the same line, and the five pixels centered above this pixel on the previous line. Note that this choice only involves pixels that have already been seen from a scanner. The current pixel is then arithmetically coded based on the eight-bit (including the pixel being coded) state so formed. So there are (in this case) 256 contexts to be coded. The arithmetic coder and probability estimator for the contexts are actually IBM's (patented) Q-coder. The Q-coder uses low precision, rapidly adaptable (those two are related) probability estimation combined with a multiply-less arithmetic coder. The probability estimation is intimately tied to the interval calculations necessary for the arithmetic coding. JBIG actually goes beyond this and has adaptive templates, and probably some other bells and whistles I don't know about. You can find a description of the Q-coder as well as the ancestor of JBIG in the Nov 88 issue of the IBM Journal of Research and Development. This is a very complete and well written set of five articles that describe the Q-coder and a bi-level image coder that uses the Q-coder. You can use JBIG on grey-scale or even color images by simply applying the algorithm one bit-plane at a time. You would want to recode the grey or color levels first though, so that adjacent levels differ in only one bit (called Gray-coding). I hear that this works well up to about six bits per pixel, beyond which JPEG's lossless mode works better. You need to use the Q-coder with JPEG also to get this performance. Actually no lossless mode works well beyond six bits per pixel, since those low bits tend to be noise, which doesn't compress at all. Anyway, the intent of JBIG is to replace the current, less effective group 3 and 4 fax algorithms. ------------------------------------------------------------------------------ Subject: [99] Acknowledgments There are too many people to cite. Thanks to all people who directly or indirectly contributed to this FAQ.