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Compression Frequently Asked Questions (part 2/3)

Please note this FAQ is rather old and may not contain all the latest information. Also some (for this site) irrelevant information might have been removed.


From: jloup@gzip.OmitThis.org (Jean-loup Gailly)
Newsgroups: comp.compression
Subject: comp.compression Frequently Asked Questions (part 2/3)
Date: 5 Sep 1999 22:36:25 +0200

Archive-name: compression-faq/part2
Last-modified: Sep 5th, 1999

his file is part 2 of a set of Frequently Asked Questions for the
groups comp.compression and comp.compression.research.

If you did not get part 1 or 3, you can get them at
http://www.faqs.org/faqs/compression-faq/part1/preamble.html or
or ftp://rtfm.mit.edu/pub/usenet/news.answers/compression-faq/

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 .  Thank you.

Contents
========

Part 2: (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
[75] Introduction to JPEG
[76] What is Vector Quantization?
[77] Introduction to Fractal compression
[78] The Burrows-Wheeler block sorting algorithm (long)

Part 3: (Long) list of image compression hardware

[85] Image compression hardware
[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 .

 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
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.

[See also "Practical Huffman coding" http://www.compressconsult.com/huffman/ ]


 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.



  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
                   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 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]




  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.


[A tutorial on some compression algorithms is available at
http://www.cs.sfu.ca/cs/CC/365/li/squeeze/ ]



Subject: [71] Introduction to MPEG (long)


For MPEG players, see item 15 in part 1 of the FAQ.  Frank Gadegast
 also posts a FAQ specialized in MPEG, available in
ftp://ftp.cs.tu-berlin.de/pub/msdos/dos/graphics/ mpegfa*.zip and
http://www.powerweb.de/mpeg/mpegfaq/

The site ftp://ftp.crs4.it/mpeg/ dedicated to the MPEG compression standard.
Another MPEG FAQ is available in http://www.vol.it/MPEG/

See also http://www.mpeg.org and http://www-plateau.cs.berkeley.edu/mpeg

A description of MPEG can be found in: "MPEG: A Video Compression
Standard for Multimedia Applications" Didier Le Gall, Communications
of the ACM, April 1991, Vol 34. No.4, pp.46-58.

Several books on MPEG have been published, see the list in
http://www.mpeg.org/MPEG/starting-points.html#books

MPEG-2 bitstreams are available on wuarchive.wustl.edu in directory
/graphics/x3l3/pub/bitstreams. MPEG-2 Demultiplexer source code is
in /graphics/x3l3/pub/bitstreams/systems/munsi_v13.tar.gz

Public C source encoder for all 3 layers for mpeg2 including mpeg1 is in
ftp://ftp.tnt.uni-hannover.de/pub/MPEG/audio/mpeg2/public_software/
technical_report/dist08.tar.gz


Introduction to MPEG originally written by Mark Adler
 around January 1992; modified and updated by
Harald Popp  in March 94:

Q: What is MPEG, exactly?

A: MPEG is the "Moving Picture Experts Group", working under the
   joint direction of the International Standards Organization (ISO)
   and the International Electro-Technical Commission (IEC). This
   group works on standards for the coding of moving pictures and
   associated audio.

Q: What is the status of MPEG's work, then? What's about MPEG-1, -2,
   and so on?

A: MPEG approaches the growing need for multimedia standards step-by-
   step. Today, three "phases" are defined:

   MPEG-1: "Coding of Moving Pictures and Associated Audio for
           Digital Storage Media at up to about 1.5 MBit/s"

   Status: International Standard IS-11172, completed in 10.92

   MPEG-2: "Generic Coding of Moving Pictures and Associated Audio"

   Status: Comittee Draft CD 13818 as found in documents MPEG93 /
           N601, N602, N603 (11.93)

   MPEG-3: no longer exists (has been merged into MPEG-2)

   MPEG-4: "Very Low Bitrate Audio-Visual Coding"

   Status: Call for Proposals 11.94, Working Draft in 11.96

Q: MPEG-1 is ready-for-use. How does the standard look like?

A: MPEG-1 consists of 4 parts:

   IS 11172-1: System
   describes synchronization and multiplexing of video and audio

   IS 11172-2: Video
   describes compression of non-interlaced video signals

   IS 11172-3: Audio
   describes compression of audio signals

   CD 11172-4: Compliance Testing
   describes procedures for determining the characteristics of coded
   bitstreams and the decoding porcess and for testing compliance
   with the requirements stated in the other parts

Q. Does MPEG 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. So how does MPEG-1 work? Tell me about video coding!

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 intracode 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. What will MPEG-2 do for video coding?

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 MPEG-2 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-1?

A. The main difficulty is the interlacing.  The simplest way to extend
   MPEG-1 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-2 is to figure out what works well.
   MPEG-2 is not limited to consider only derivations of MPEG-1.
   There were several non-MPEG-1-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-1 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: Talking about MPEG audio coding, I heard a lot about "Layer 1, 2
   and 3". What does it mean, exactly?

A: MPEG-1, IS 11172-3, describes the compression of audio signals
   using high performance perceptual coding schemes. It specifies a
   family of three audio coding schemes, simply called Layer-1,-2,-3,
   with increasing encoder complexity and performance (sound quality
   per bitrate). The three codecs are compatible in a hierarchical
   way, i.e. a Layer-N decoder is able to decode bitstream data
   encoded in Layer-N and all Layers below N (e.g., a Layer-3
   decoder may accept Layer-1,-2 and -3, whereas a Layer-2 decoder
   may accept only Layer-1 and -2.)

Q: So we have a family of three audio coding schemes. What does the
   MPEG standard define, exactly?

A: For each Layer, the standard specifies the bitstream format and
   the decoder. To allow for future improvements, it does *not*
   specify the encoder , but an informative chapter gives an example
   for an encoder for each Layer.

Q: What have the three audio Layers in common?

A: All Layers use the same basic structure. The coding scheme can be
   described as "perceptual noise shaping" or "perceptual subband /
   transform coding".

   The encoder analyzes the spectral components of the audio signal
   by calculating a filterbank or transform and applies a
   psychoacoustic model to estimate the just noticeable noise-
   level. In its quantization and coding stage, the encoder tries
   to allocate the available number of data bits in a way to meet
   both the bitrate and masking requirements.

   The decoder is much less complex. Its only task is to synthesize
   an audio signal out of the coded spectral components.

   All Layers use the same analysis filterbank (polyphase with 32
   subbands). Layer-3 adds a MDCT transform to increase the frequency
   resolution.

   All Layers use the same "header information" in their bitstream,
   to support the hierarchical structure of the standard.

   All Layers use a bitstream structure that contains parts that are
   more sensitive to biterrors ("header", "bit allocation",
   "scalefactors", "side information") and parts that are less
   sensitive ("data of spectral components").

   All Layers may use 32, 44.1 or 48 kHz sampling frequency.

   All Layers are allowed to work with similar bitrates:
   Layer-1: from 32 kbps to 448 kbps
   Layer-2: from 32 kbps to 384 kbps
   Layer-3: from 32 kbps to 320 kbps

Q: What are the main differences between the three Layers, from a
   global view?

A: From Layer-1 to Layer-3,
   complexity increases (mainly true for the encoder),
   overall codec delay increases, and
   performance increases (sound quality per bitrate).

Q: Which Layer should I use for my application?

A: Good Question. Of course, it depends on all your requirements. But
   as a first approach, you should consider the available bitrate of
   your application as the Layers have been designed to support
   certain areas of bitrates most efficiently, i.e. with a minimum
   drop of sound quality.

   Let us look a little closer at the strong domains of each Layer.

   Layer-1: Its ISO target bitrate is 192 kbps per audio channel.

   Layer-1 is a simplified version of Layer-2. It is most useful for
   bitrates around the "high" bitrates around or above 192 kbps. A
   version of Layer-1 is used as "PASC" with the DCC recorder.

   Layer-2: Its ISO target bitrate is 128 kbps per audio channel.

   Layer-2 is identical with MUSICAM. It has been designed as trade-
   off between sound quality per bitrate and encoder complexity. It
   is most useful for bitrates around the "medium" bitrates of 128 or
   even 96 kbps per audio channel. The DAB (EU 147) proponents have
   decided to use Layer-2 in the future Digital Audio Broadcasting
   network.

   Layer-3: Its ISO target bitrate is 64 kbps per audio channel.

   Layer-3 merges the best ideas of MUSICAM and ASPEC. It has been
   designed for best performance at "low" bitrates around 64 kbps or
   even below. The Layer-3 format specifies a set of advanced
   features that all address one goal: to preserve as much sound
   quality as possible even at rather low bitrates. Today, Layer-3 is
   already in use in various telecommunication networks (ISDN,
   satellite links, and so on) and speech announcement systems.

Q: Tell me more about sound quality. How do you assess that?

A: Today, there is no alternative to expensive listening tests.
   During the ISO-MPEG-1 process, 3 international listening tests
   have been performed, with a lot of trained listeners, supervised
   by Swedish Radio. They took place in 7.90, 3.91 and 11.91. Another
   international listening test was performed by CCIR, now ITU-R, in
   92.

   All these tests used the "triple stimulus, hidden reference"
   method and the CCIR impairment scale to assess the audio quality.
   The listening sequence is "ABC", with A = original, BC = pair of
   original / coded signal with random sequence, and the listener has
   to evaluate both B and C with a number between 1.0 and 5.0. The
   meaning of these values is:

   5.0 = transparent (this should be the original signal)
   4.0 = perceptible, but not annoying (first differences noticable)
   3.0 = slightly annoying
   2.0 = annoying
   1.0 = very annoying

   With perceptual codecs (like MPEG audio), all traditional
   parameters (like SNR, THD+N, bandwidth) are especially useless.
   Fraunhofer-IIS works on objective quality assessment tools, like
   the NMR meter (Noise-to-Mask-Ratio), too. BTW: If you need more
   informations about NMR, please contact nmr@iis.fhg.de.

Q: Now that I know how to assess quality, come on, tell me the
   results of these tests.

A: Well, for low bitrates, the main result is that at 60 or 64 kbps
   per channel), Layer-2 scored always between 2.1 and 2.6, whereas
   Layer-3 scored between 3.6 and 3.8. This is a significant increase
   in sound quality, indeed! Furthermore, the selection process for
   critical sound material showed that it was rather difficult to
   find worst-case material for Layer-3 whereas it was not so hard to
   find such items for Layer-2.

Q: OK, a Layer-2 codec at low bitrates may sound poor today, but
   couldn't that be improved in the future? I guess you just told me
   before that the encoder is not fixed in the standard.

A: Good thinking! As the sound quality mainly depends on the encoder
   implementation, it is true that there is no such thing as a "Layer-
   N"- quality. So we definitely only know the performance of the
   reference codecs during the international tests. Who knows what
   will happen in the future? What we do know now, is:

   Today, Layer-3 already provides a sound quality that comes very
   near to CD quality at 64 kbps per channel. Layer-2 is far away
   from that.

   Tomorrow, both Layers may improve. Layer-2 has been designed as a
   trade-off between quality and complexity, so the bitstream format
   allows only limited innovations. In contrast, even the current
   reference Layer-3-codec exploits only a small part of the powerful
   mechanisms inside the Layer-3 bitstream format.

Q: All in all, you sound as if anybody should use Layer-3 for low
   bitrates. Why on earth do some vendors still offer only Layer-2
   equipment for these applications?

A: Well, maybe because they started to design and develop their
   system rather early, e.g. in 1990. As Layer-2 is identical with
   MUSICAM, it has been available since summer of 90, at latest. In
   that year, Layer-3 development started and could be successfully
   finished in spring 92. So, for a certain time, vendors could only
   exploit the existing part of the new MPEG standard.

   Now the situation has changed. All Layers are available, the
   standard is completed, and new systems need not limit themselves,
   but may capitalize on the full features of MPEG audio.

Q: How do I get the MPEG documents?

A: You may order it from your national standards body.

   E.g., in Germany, please contact:
   DIN-Beuth Verlag, Auslandsnormen
   Mrs. Niehoff, Burggrafenstr. 6, D-10772 Berlin, Germany
   Phone: 030-2601-2757, Fax: 030-2601-1231

   E.g., in USA, you may order it from ANSI [phone (212) 642-4900] or
   buy it from companies like OMNICOM phone +44 438 742424
                                      FAX   +44 438 740154

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.)



Subject: [72] What is wavelet theory?


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
M-band 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 .

A wavelet transform coder construction kit is available at
http://www.cs.dartmouth.edu/~gdavis/wavelet/wavelet.html
Contact: Geoff Davis 

A matlab toolbox for constructing multi-scale image representations,
including Laplacian pyramids, QMFs, wavelets, and steerable pyramids,
is available at ftp://ftp.cis.upenn.edu/pub/eero/
Contact: Eero Simoncelli .

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 wavelet-based HARC-C 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 :

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.)



Subject: [73] What is the theoretical compression limit?


This question can be understood in two different ways:

(a) For a given compressor/decompressor, what is the best possible lossless
    compression for an arbitrary string (byte sequence) given as input?

(b) For a given string, what is the best possible lossless
    compressor/decompressor?

For case (a), the question is generally meaningless, because a specific
compressor may compress one very large input file down to a single bit, and
enlarge all other files by only one bit.  There is no lossless 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 item 9).  In case (a), the size of the decompressor is
not taken into account for the determination of the compression ratio since the
decompressor is fixed and it may decompress an arbitrary number of files of
arbitrary length.

For case (b), it is of course necessary to take into account the size of the
decompressor. The problem may be restated as "What is the shortest program P
which, when executed, produces the string S?".  The size of this program
is known as the Kolmogorov complexity of the string S.  Some (actually most)
strings are not compressible at all, by any program: the smallest
representation of the string is the string itself.  On the other hand, the
output of a pseudo-random number generator can be extremely compressible, since
it is sufficient to know the parameters and seed of the generator to reproduce
an arbitrary long sequence.

   Ming Li and Paul Vitanyi, 2nd edition, Springer-Verlag, ISBN 0-387-94868-6
   http://www.cwi.nl/~paulv/kolmogorov.html

If you don't want to read a whole book, I recommend the excellent lecture
"Randomness & Complexity in Pure Mathematics" by G. J. Chaitin:
  http://www.cs.auckland.ac.nz/CDMTCS/chaitin/ijbc.html
The decimal and binary expansions of Chaitin's number Omega are examples of
uncompressible strings. There are more papers on
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/



Subject: [74] Introduction to JBIG


JBIG software and the JBIG specification are available in
ftp://nic.funet.fi/pub/graphics/misc/test-images/jbig.tar.gz

The ISO JBIG committee's home page is http://www.jpeg.org/public/welcome.htm


A short introduction to JBIG, written by Mark Adler :

  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.


Another introduction to JBIG, written by Hank van Bekkem :

  The following description of the JBIG algorithm is derived from
  experiences with a software implementation I wrote following the
  specifications in the revision 4.1 draft of September 16, 1991. The
  source will not be made available in the public domain, as parts of
  JBIG are patented.

  JBIG (Joint Bi-level Image Experts Group) is an experts group of ISO,
  IEC and CCITT (JTC1/SC2/WG9 and SGVIII). Its job is to define a
  compression standard for lossless image coding ([1]). The main
  characteristics of the proposed algorithm are:
  - Compatible progressive/sequential coding. This means that a
    progressively coded image can be decoded sequentially, and the
    other way around.
  - JBIG will be a lossless image compression standard: all bits in
    your images before and after compression and decompression will be
    exactly the same.

  In the rest of this text I will first describe the JBIG algorithm in
  a short abstract of the draft. I will conclude by saying something
  about the value of JBIG.


  JBIG algorithm.
  --------------

  JBIG parameter P specifies the number of bits per pixel in the image.
  Its allowable range is 1 through 255, but starting at P=8 or so,
  compression will be more efficient using other algorithms. On the
  other hand, medical images such as chest X-rays are often stored with
  12 bits per pixel, while no distorsion is allowed, so JBIG can
  certainly be of use in this area. To limit the number of bit changes
  between adjacent decimal values (e.g. 127 and 128), it is wise to use
  Gray coding before compressing multi-level images with JBIG. JBIG
  then compresses the image on a bitplane basis, so the rest of this
  text assumes bi-level pixels.

  Progressive coding is a way to send an image gradually to a receiver
  instead of all at once. During sending, more detail is sent, and the
  receiver can build the image from low to high detail. JBIG uses
  discrete steps of detail by successively doubling the resolution. The
  sender computes a number of resolution layers D, and transmits these
  starting at the lowest resolution Dl. Resolution reduction uses
  pixels in the high resolution layer and some already computed low
  resolution pixels as an index into a lookup table. The contents of
  this table can be specified by the user.

  Compatibility between progressive and sequential coding is achieved
  by dividing an image into stripes. Each stripe is a horizontal bar
  with a user definable height. Each stripe is separately coded and
  transmitted, and the user can define in which order stripes,
  resolutions and bitplanes (if P>1) are intermixed in the coded data.
  A progressive coded image can be decoded sequentially by decoding
  each stripe, beginning by the one at the top of the image, to its
  full resolution, and then proceeding to the next stripe. Progressive
  decoding can be done by decoding only a specific resolution layer
  from all stripes.

  After dividing an image into bitplanes, resolution layers and
  stripes, eventually a number of small bi-level bitmaps are left to
  compress. Compression is done using a Q-coder. Reference [2]
  contains a full description, I will only outline the basic principles
  here.

  The Q-coder codes bi-level pixels as symbols using the probability of
  occurrence of these symbols in a certain context. JBIG defines two
  kinds of context, one for the lowest resolution layer (the base
  layer), and one for all other layers (differential layers).
  Differential layer contexts contain pixels in the layer to be coded,
  and in the corresponding lower resolution layer.

  For each combination of pixel values in a context, the probability
  distribution of black and white pixels can be different. In an all
  white context, the probability of coding a white pixel will be much
  greater than that of coding a black pixel. The Q-coder assigns, just
  like a Huffman coder, more bits to less probable symbols, and so
  achieves compression. The Q-coder can, unlike a Huffmann coder,
  assign one output codebit to more than one input symbol, and thus is
  able to compress bi-level pixels without explicit clustering, as
  would be necessary using a Huffman coder.

  Maximum compression will be achieved when all probabilities (one set
  for each combination of pixel values in the context) follow the
  probabilities of the pixels. The Q-coder therefore continuously
  adapts these probabilities to the symbols it sees.


  JBIG value.
  ----------

  In my opinion, JBIG can be regarded as two combined devices:
  - Providing the user the service of sending or storing multiple
    representations of images at different resolutions without any
    extra cost in storage. Differential layer contexts contain pixels
    in two resolution layers, and so enable the Q-coder to effectively
    code the difference in information between the two layers, instead
    of the information contained in every layer. This means that,
    within a margin of approximately 5%, the number of resolution
    layers doesn't effect the compression ratio.
  - Providing the user a very efficient compression algorithm, mainly
    for use with bi-level images. Compared to CCITT Group 4, JBIG is
    approximately 10% to 50% better on text and line art, and even
    better on halftones. JBIG is however, just like Group 4, somewhat
    sensitive to noise in images. This means that the compression ratio
    decreases when the amount of noise in your images increases.

  An example of an application would be browsing through an image
  database, e.g. an EDMS (engineering document management system).
  Large A0 size drawings at 300 dpi or so would be stored using five
  resolution layers. The lowest resolution layer would fit on a
  computer screen. Base layer compressed data would be stored at the
  beginning of the compressed file, thus making browsing through large
  numbers of compressed drawings possible by reading and decompressing
  just the first small part of all files. When the user stops browsing,
  the system could automatically start decompressing all remaining
  detail for printing at high resolution.

  [1] "Progressive Bi-level Image Compression, Revision 4.1", ISO/IEC
      JTC1/SC2/WG9, CD 11544, September 16, 1991
  [2] "An overview of the basic principles of the Q-coder adaptive
      binary arithmetic coder", W.B. Pennebaker, J.L. Mitchell, G.G.
      Langdon, R.B. Arps, IBM Journal of research and development,
      Vol.32, No.6, November 1988, pp. 771-726 (See also the other
      articles about the Q-coder in this issue)



Subject: [75] Introduction to JPEG

Here is a brief overview of the inner workings of JPEG, plus some references
for more detailed information, written by Tom Lane .
Please read item 19 in part 1 first.

JPEG works on either full-color or gray-scale images; it does not handle
bilevel (black and white) images, at least not well.  It doesn't handle
colormapped images either; you have to pre-expand those into an unmapped
full-color representation.  JPEG works best on "continuous tone" images.
Images with many sudden jumps in color values will not compress well.

There are a lot of parameters to the JPEG compression process.  By adjusting
the parameters, you can trade off compressed image size against reconstructed
image quality over a *very* wide range.  You can get image quality ranging
from op-art (at 100x smaller than the original 24-bit image) to quite
indistinguishable from the source (at about 3x smaller).  Usually the
threshold of visible difference from the source image is somewhere around 10x
to 20x smaller than the original, ie, 1 to 2 bits per pixel for color images.
Grayscale images do not compress as much.  In fact, for comparable visual
quality, a grayscale image needs perhaps 25% less space than a color image;
certainly not the 66% less that you might naively expect.

JPEG defines a "baseline" lossy algorithm, plus optional extensions for
progressive and hierarchical coding.  There is also a separate lossless
compression mode; this typically gives about 2:1 compression, ie, about 12
bits per color pixel.  Most currently available JPEG hardware and software
handles only the baseline mode.


Here's the outline of the baseline compression algorithm:

1. Transform the image into a suitable color space.  This is a no-op for
grayscale, but for color images you generally want to transform RGB into a
luminance/chrominance color space (YCbCr, YUV, etc).  The luminance component
is grayscale and the other two axes are color information.  The reason for
doing this is that you can afford to lose a lot more information in the
chrominance components than you can in the luminance component: the human eye
is not as sensitive to high-frequency chroma info as it is to high-frequency
luminance.  (See any TV system for precedents.)  You don't have to change the
color space if you don't want to, since the remainder of the algorithm works
on each color component independently, and doesn't care just what the data
is.  However, compression will be less since you will have to code all the
components at luminance quality.  Note that colorspace transformation is
slightly lossy due to roundoff error, but the amount of error is much smaller
than what we typically introduce later on.

2. (Optional) Downsample each component by averaging together groups of
pixels.  The luminance component is left at full resolution, while the chroma
components are often reduced 2:1 horizontally and either 2:1 or 1:1 (no
change) vertically.  In JPEG-speak these alternatives are usually called 2h2v
and 2h1v sampling, but you may also see the terms "411" and "422" sampling.
This step immediately reduces the data volume by one-half or one-third.
In numerical terms it is highly lossy, but for most images it has almost no
impact on perceived quality, because of the eye's poorer resolution for chroma
info.  Note that downsampling is not applicable to grayscale data; this is one
reason color images are more compressible than grayscale.

3. Group the pixel values for each component into 8x8 blocks.  Transform each
8x8 block through a discrete cosine transform (DCT).  The DCT is a relative of
the Fourier transform and likewise gives a frequency map, with 8x8 components.
Thus you now have numbers representing the average value in each block and
successively higher-frequency changes within the block.  The motivation for
doing this is that you can now throw away high-frequency information without
affecting low-frequency information.  (The DCT transform itself is reversible
except for roundoff error.)  See question 25 for fast DCT algorithms.

4. In each block, divide each of the 64 frequency components by a separate
"quantization coefficient", and round the results to integers.  This is the
fundamental information-losing step.  The larger the quantization
coefficients, the more data is discarded.  Note that even the minimum possible
quantization coefficient, 1, loses some info, because the exact DCT outputs
are typically not integers.  Higher frequencies are always quantized less
accurately (given larger coefficients) than lower, since they are less visible
to the eye.  Also, the luminance data is typically quantized more accurately
than the chroma data, by using separate 64-element quantization tables.
Tuning the quantization tables for best results is something of a black art,
and is an active research area.  Most existing encoders use simple linear
scaling of the example tables given in the JPEG standard, using a single
user-specified "quality" setting to determine the scaling multiplier.  This
works fairly well for midrange qualities (not too far from the sample tables
themselves) but is quite nonoptimal at very high or low quality settings.

5. Encode the reduced coefficients using either Huffman or arithmetic coding.
(Strictly speaking, baseline JPEG only allows Huffman coding; arithmetic
coding is an optional extension.)   Notice that this step is lossless, so it
doesn't affect image quality.  The arithmetic coding option uses Q-coding;
it is identical to the coder used in JBIG (see question 74).  Be aware that
Q-coding is patented.  Most existing implementations support only the Huffman
mode, so as to avoid license fees.  The arithmetic mode offers maybe 5 or 10%
better compression, which isn't enough to justify paying fees.

6. Tack on appropriate headers, etc, and output the result.  In a normal
"interchange" JPEG file, all of the compression parameters are included
in the headers so that the decompressor can reverse the process.  These
parameters include the quantization tables and the Huffman coding tables.
For specialized applications, the spec permits those tables to be omitted
from the file; this saves several hundred bytes of overhead, but it means
that the decompressor must know a-priori what tables the compressor used.
Omitting the tables is safe only in closed systems.


The decompression algorithm reverses this process.  The decompressor
multiplies the reduced coefficients by the quantization table entries to
produce approximate DCT coefficients.  Since these are only approximate,
the reconstructed pixel values are also approximate, but if the design
has done what it's supposed to do, the errors won't be highly visible.
A high-quality decompressor will typically add some smoothing steps to
reduce pixel-to-pixel discontinuities.

The JPEG standard does not specify the exact behavior of compressors and
decompressors, so there's some room for creative implementation.  In
particular, implementations can trade off speed against image quality by
choosing more accurate or faster-but-less-accurate approximations to the
DCT.  Similar tradeoffs exist for the downsampling/upsampling and colorspace
conversion steps.  (The spec does include some minimum accuracy requirements
for the DCT step, but these are widely ignored, and are not too meaningful
anyway in the absence of accuracy requirements for the other lossy steps.)


Extensions:

The progressive mode is intended to support real-time transmission of images.
It allows the DCT coefficients to be sent piecemeal in multiple "scans" of
the image.  With each scan, the decoder can produce a higher-quality
rendition of the image.  Thus a low-quality preview can be sent very quickly,
then refined as time allows.  The total space needed is roughly the same as
for a baseline JPEG image of the same final quality.  (In fact, it can be
somewhat *less* if a custom Huffman table is used for each scan, because the
Huffman codes can be optimized over a smaller, more uniform population of
data than appears in a baseline image's single scan.)  The decoder must do
essentially a full JPEG decode cycle for each scan: inverse DCT, upsample,
and color conversion must all be done again, not to mention any color
quantization for 8-bit displays.  So this scheme is useful only with fast
decoders or slow transmission lines.  Up until 1995, progressive JPEG was a
rare bird, but its use is now spreading as software decoders have become fast
enough to make it useful with modem-speed data transmission.

The hierarchical mode represents an image at multiple resolutions.  For
example, one could provide 512x512, 1024x1024, and 2048x2048 versions of the
image.  The higher-resolution images are coded as differences from the next
smaller image, and thus require many fewer bits than they would if stored
independently.  (However, the total number of bits will be greater than that
needed to store just the highest-resolution frame in baseline form.)
The individual frames in a hierarchical sequence can be coded progressively
if desired.  Hierarchical mode is not widely supported at present.

Part 3 of the JPEG standard, approved at the end of 1995, introduces several
new extensions.  The one most likely to become popular is variable
quantization, which allows the quantization table to be scaled to different
levels in different parts of the image.  In this way the "more critical"
parts of the image can be coded at higher quality than the "less critical"
parts.  A signaling code can be inserted at any DCT block boundary to set a
new scaling factor.

Another Part 3 extension is selective refinement.  This feature permits a
scan in a progressive sequence, or a refinement frame of a hierarchical
sequence, to cover only part of the total image area.  This is an
alternative way of solving the variable-quality problem.  My (tgl's) guess
is that this will not get widely implemented, with variable quantization
proving a more popular approach, but I've been wrong before.

The third major extension added by Part 3 is a "tiling" concept that allows
an image to be built up as a composite of JPEG frames, which may have
different sizes, resolutions, quality settings, even colorspaces.  (For
example, a color image that occupies a small part of a mostly-grayscale page
could be represented as a separate frame, without having to store the whole
page in color.)  Again, there's some overlap in functionality with variable
quantization and selective refinement.  The general case of arbitrary tiles
is rather complex and is unlikely to be widely implemented.  In the simplest
case all the tiles are the same size and use similar quality settings.
This case may become popular even if the general tiling mechanism doesn't,
because it surmounts the 64K-pixel-on-a-side image size limitation that was
(not very foresightedly) built into the basic JPEG standard.  The individual
frames are still restricted to 64K for compatibility reasons, but the total
size of a tiled JPEG image can be up to 2^32 pixels on a side.

Lossless JPEG:

The separate lossless mode does not use DCT, since roundoff errors prevent a
DCT calculation from being lossless.  For the same reason, one would not
normally use colorspace conversion or downsampling, although these are
permitted by the standard.  The lossless mode simply codes the difference
between each pixel and the "predicted" value for the pixel.  The predicted
value is a simple function of the already-transmitted pixels just above and
to the left of the current one (for example, their average; 8 different
predictor functions are permitted).  The sequence of differences is encoded
using the same back end (Huffman or arithmetic) used in the lossy mode.

Lossless JPEG with the Huffman back end is certainly not a state-of-the-art
lossless compression method, and wasn't even when it was introduced.  The
arithmetic-coding back end may make it competitive, but you're probably best
off looking at other methods if you need only lossless compression.

The main reason for providing a lossless option is that it makes a good
adjunct to the hierarchical mode: the final scan in a hierarchical sequence
can be a lossless coding of the remaining differences, to achieve overall
losslessness.  This isn't quite as useful as it may at first appear, because
exact losslessness is not guaranteed unless the encoder and decoder have
identical IDCT implementations (ie, identical roundoff errors).  And you
can't use downsampling or colorspace conversion either if you want true
losslessness.  But in some applications the combination is useful.


References:

For a good technical introduction to JPEG, see:
	Wallace, Gregory K.  "The JPEG Still Picture Compression Standard",
	Communications of the ACM, April 1991 (vol. 34 no. 4), pp. 30-44.
(Adjacent articles in that issue discuss MPEG motion picture compression,
applications of JPEG, and related topics.)  If you don't have the CACM issue
handy, a PostScript file containing a revised version of this article is
available at ftp://ftp.uu.net/graphics/jpeg/wallace.ps.gz.  This file
(actually a preprint for a later article in IEEE Trans. Consum. Elect.)
omits the sample images that appeared in CACM, but it includes corrections
and some added material.  Note: the Wallace article is copyright ACM and
IEEE, and it may not be used for commercial purposes.

An alternative, more leisurely explanation of JPEG can be found in "The Data
Compression Book" by Mark Nelson ([Nel 1991], see question 7).  This book
provides excellent introductions to many data compression methods including
JPEG, plus sample source code in C.  The JPEG-related source code is far from
industrial-strength, but it's a pretty good learning tool.

An excellent textbook about JPEG is "JPEG Still Image Data Compression
Standard" by William B. Pennebaker and Joan L. Mitchell.  Published by Van
Nostrand Reinhold, 1993, ISBN 0-442-01272-1.  650 pages, price US$59.95.
(VNR will accept credit card orders at 800/842-3636, or get your local
bookstore to order it.)  This book includes the complete text of the ISO
JPEG standards, DIS 10918-1 and draft DIS 10918-2.  Review by Tom Lane:
"This is by far the most complete exposition of JPEG in existence.  It's
written by two people who know what they are talking about: both served on
the ISO JPEG standards committee.  If you want to know how JPEG works or
why it works that way, this is the book to have."

There are a number of errors in the first printing of the Pennebaker and
Mitchell book.  An errata list is available at
ftp://ftp.uu.net/graphics/jpeg/pm.errata.gz.  At last report, all known
errors were fixed in the second printing.

The official specification of JPEG is not currently available on-line, and
is not likely ever to be available for free because of ISO and ITU copyright
restrictions.  You can order it from your national standards agency as ISO
standards IS 10918-1, 10918-2, 10918-3, or as ITU-T standards T.81, T.83,
T.84.  See ftp://ftp.uu.net/graphics/jpeg/jpeg.documents.gz for more info.
NOTE: buying the Pennebaker and Mitchell textbook is a much better deal
than purchasing the standard directly: it's cheaper and includes a lot of
useful explanatory material along with the full draft text of the spec.
The book unfortunately doesn't include Part 3 of the spec, but if you need
Part 3, buy the book and just that part and you'll still be ahead.




Subject: [76] What is Vector Quantization?

Some vector quantization software for data analysis that is available
in the ftp://cochlea.hut.fi/pub/ directory.  One package is lvq_pak and
one is som_pak (som_pak generates Kohonen maps of data using lvq to
cluster it).

A VQ-based codec that is based on the Predictive Residual Vector
Quantization is in ftp://mozart.eng.buffalo.edu/pub/prvq_codec/

VQ software is also available in ftp://isdl.ee.washington.edu/pub/VQ/


For a book on Vector Quantization, see the reference (Gersho and Gray)
given in item 7 of this FAQ. For a review article: N. M. Nasrabadi and
R. A. King, "Image Coding Using Vector Quantization: A review",
IEEE Trans. on Communications, vol. COM-36, pp. 957-971, Aug. 1988.


A short introduction to Vector Quantization, written by Alex Zatsman
:

In Scalar  Quantization one represents  the values by  fixed subset of
representative values. For  examples, if you  have  16 bit  values and
send  only 8 most  signifcant bits, you  get an  approximation  of the
original data  at  the expense of  precision.  In this case  the fixed
subset is all the 16-bit numbers divisable by 256, i.e 0, 256, 512,...

In  Vector Quantization  you   represent   not individual values   but
(usually small) arrays of them. A typical  example  is a color  map: a
color picture can  be  represented by  a  2D array  of  triplets  (RGB
values). In most pictures  those triplets do  not cover the  whole RGB
space but  tend to   concetrate  in certain   areas. For  example, the
picture of a forest will typically have a lot of green. One can select
a relatively  small  subset (typically 256 elements) of representative
colors, i.e RGB  triplets,  and then approximate  each  triplet by the
representative of  that small set. In case  of 256 one  can use 1 byte
instead of 3 for each pixel.

One  can do   the  same   for  any large  data  sets,  especialy  when
consecutive points are correlated in some way. CELP speech compression
algorithms  use  those subsets  "codebooks" and   use them to quantize
exciation vectors  for linear prediction  -- hence the name CELP which
stands for Codebook Excited Linear Prediction.  (See item 26 in part 1
of this FAQ for more information about CELP.)

Note that Vector Quantization, just like Scalar Quantization, is a lossy
compression.



Subject: [77] Introduction to Fractal compression (long)


Written by John Kominek 

Seven things you should know about Fractal Image Compression (assuming that
you want to know about it).

   1. It is a promising new technology, arguably superior to JPEG --
      but only with an argument.
   2. It is a lossy compression method.
   3. The fractals in Fractal Image Compression are Iterated Function
      Systems.
   4. It is a form of Vector Quantization, one that employs a virtual
      codebook.
   5. Resolution enhancement is a powerful feature but is not some
      magical way of achieving 1000:1 compression.
   6. Compression is slow, decompression is fast.
   7. The technology is patented.

That's the scoop in condensed form. Now to elaborate, beginning with a little
background.


 A Brief History of Fractal Image Compression
 --------------------------------------------

The birth of fractal geometry (or rebirth, rather) is usually traced to IBM
mathematician Benoit B. Mandelbrot and the 1977 publication of his seminal
book The Fractal Geometry of Nature. The book put forth a powerful thesis:
traditional geometry with its straight lines and smooth surfaces does not
resemble the geometry of trees and clouds and mountains. Fractal geometry,
with its convoluted coastlines and detail ad infinitum, does.

This insight opened vast possibilities. Computer scientists, for one, found a
mathematics capable of generating artificial and yet realistic looking land-
scapes, and the trees that sprout from the soil. And mathematicians had at
their disposal a new world of geometric entities.

It was not long before mathematicians asked if there was a unity among this
diversity. There is, as John Hutchinson demonstrated in 1981, it is the branch
of mathematics now known as Iterated Function Theory. Later in the decade
Michael Barnsley, a leading researcher from Georgia Tech, wrote the popular
book Fractals Everywhere. The book presents the mathematics of Iterated Func-
tions Systems (IFS), and proves a result known as the Collage Theorem. The
Collage Theorem states what an Iterated Function System must be like in order
to represent an image.

This presented an intriguing possibility. If, in the forward direction, frac-
tal mathematics is good for generating natural looking images, then, in the
reverse direction, could it not serve to compress images? Going from a given
image to an Iterated Function System that can generate the original (or at
least closely resemble it), is known as the inverse problem. This problem
remains unsolved.

Barnsley, however, armed with his Collage Theorem, thought he had it solved.
He applied for and was granted a software patent and left academia to found
Iterated Systems Incorporated (US patent 4,941,193. Alan Sloan is the co-
grantee of the patent and co-founder of Iterated Systems.)  Barnsley announced
his success to the world in the January 1988 issue of BYTE magazine. This
article did not address the inverse problem but it did exhibit several images
purportedly compressed in excess of 10,000:1. Alas, it was not a breakthrough.
The images were given suggestive names such as "Black Forest" and "Monterey
Coast" and "Bolivian Girl" but they were all manually constructed. Barnsley's
patent has come to be derisively referred to as the "graduate student algo-
rithm."

Graduate Student Algorithm
   o Acquire a graduate student.
   o Give the student a picture.
   o And a room with a graphics workstation.
   o Lock the door.
   o Wait until the student has reverse engineered the picture.
   o Open the door.

Attempts to automate this process have met little success. As Barnsley admit-
ted in 1988: "Complex color images require about 100 hours each to encode and
30 minutes to decode on the Masscomp [dual processor workstation]." That's 100
hours with a _person_ guiding the process.

Ironically, it was one of Barnsley's PhD students that made the graduate
student algorithm obsolete. In March 1988, according to Barnsley, he arrived
at a modified scheme for representing images called Partitioned Iterated
Function Systems (PIFS). Barnsley applied for and was granted a second patent
on an algorithm that can automatically convert an image into a Partitioned
Iterated Function System, compressing the image in the process. (US patent
5,065,447. Granted on Nov. 12 1991.) For his PhD thesis, Arnaud Jacquin imple-
mented the algorithm in software, a description of which appears in his land-
mark paper "Image Coding Based on a Fractal Theory of Iterated Contractive
Image Transformations." The algorithm was not sophisticated, and not speedy,
but it was fully automatic. This came at price: gone was the promise of
10,000:1 compression. A 24-bit color image could typically be compressed from
8:1 to 50:1 while still looking "pretty good." Nonetheless, all contemporary
fractal image compression programs are based upon Jacquin's paper.

That is not to say there are many fractal compression programs available.
There are not. Iterated Systems sell the only commercial compressor/decompres-
sor, an MS-Windows program called "Images Incorporated." There are also an
increasing number of academic programs being made freely available. Unfor-
tunately, these programs are -- how should I put it? -- of merely academic
quality.

This scarcity has much to do with Iterated Systems' tight lipped policy about
their compression technology. They do, however, sell a Windows DLL for pro-
grammers. In conjunction with independent development by researchers else-
where, therefore, fractal compression will gradually become more pervasive.
Whether it becomes all-pervasive remains to be seen.

Historical Highlights:
   1977 -- Benoit Mandelbrot finishes the first edition of The Fractal
           Geometry of Nature.
   1981 -- John Hutchinson publishes "Fractals and Self-Similarity."
   1983 -- Revised edition of The Fractal Geometry of Nature is
           published.
   1985 -- Michael Barnsley and Stephen Demko introduce Iterated
           Function Theory in "Iterated Function Systems and the Global
           Construction of Fractals."
   1987 -- Iterated Systems Incorporated is founded.
   1988 -- Barnsley publishes the book Fractals Everywhere.
   1990 -- Barnsley's first patent is granted.
   1991 -- Barnsley's second patent is granted.
   1992 -- Arnaud Jacquin publishes an article that describes the first
           practical fractal image compression method.
   1993 -- The book Fractal Image Compression by Michael Barnsley and Lyman
           Hurd is published.
        -- The Iterated Systems' product line matures.
   1994 -- Put your name here.


 On the Inside
 -------------

The fractals that lurk within fractal image compression are not those of the
complex plane (Mandelbrot Set, Julia sets), but of Iterated Function Theory.
When lecturing to lay audiences, the mathematician Heinz-Otto Peitgen intro-
duces the notion of Iterated Function Systems with the alluring metaphor of a
Multiple Reduction Copying Machine. A MRCM is imagined to be a regular copying
machine except that:

  1. There are multiple lens arrangements to create multiple overlapping
     copies of the original.
  2. Each lens arrangement reduces the size of the original.
  3. The copier operates in a feedback loop, with the output of one
     stage the input to the next. The initial input may be anything.

The first point is what makes an IFS a system. The third is what makes it
iterative. As for the second, it is implicitly understood that the functions
of an Iterated Function Systems are contractive.

An IFS, then, is a set of contractive transformations that map from a defined
rectangle of the real plane to smaller portions of that rectangle. Almost
invariably, affine transformations are used. Affine transformations act to
translate, scale, shear, and rotate points in the plane. Here is a simple
example:


     |---------------|              |-----|
     |x              |              |1    |
     |               |              |     |
     |               |         |---------------|
     |               |         |2      |3      |
     |               |         |       |       |
     |---------------|         |---------------|

         Before                      After

     Figure 1. IFS for generating Sierpinski's Triangle.

This IFS contains three component transformations (three separate lens ar-
rangements in the MRCM metaphor). Each one shrinks the original by a factor of
2, and then translates the result to a new location. It may optionally scale
and shift the luminance values of the rectangle, in a manner similar to the
contrast and brightness knobs on a TV.

The amazing property of an IFS is that when the set is evaluated by iteration,
(i.e. when the copy machine is run), a unique image emerges. This latent image
is called the fixed point or attractor of the IFS. As guaranteed by a result
known as the Contraction Theorem, it is completely independent of the initial
image. Two famous examples are Sierpinski's Triangle and Barnsley's Fern.
Because these IFSs are contractive, self-similar detail is created at every
resolution down to the infinitesimal. That is why the images are fractal.

The promise of using fractals for image encoding rests on two suppositions: 1.
many natural scenes possess this detail within detail structure (e.g. clouds),
and 2. an IFS can be found that generates a close approximation of a scene
using only a few transformations. Barnsley's fern, for example, needs but
four. Because only a few numbers are required to describe each transformation,
an image can be represented very compactly. Given an image to encode, finding
the optimal IFS from all those possible is known as the inverse problem.

The inverse problem -- as mentioned above -- remains unsolved. Even if it
were, it may be to no avail. Everyday scenes are very diverse in subject
matter; on whole, they do not obey fractal geometry. Real ferns do not branch
down to infinity. They are distorted, discolored, perforated and torn. And the
ground on which they grow looks very much different.

To capture the diversity of real images, then, Partitioned IFSs are employed.
In a PIFS, the transformations do not map from the whole image to the parts,
but from larger parts to smaller parts. An image may vary qualitatively from
one area to the next (e.g. clouds then sky then clouds again). A PIFS relates
those areas of the original image that are similar in appearance. Using Jac-
quin's notation, the big areas are called domain blocks and the small areas
are called range blocks. It is necessary that every pixel of the original
image belong to (at least) one range block. The pattern of range blocks is
called the partitioning of an image.

Because this system of mappings is still contractive, when iterated it will
quickly converge to its latent fixed point image. Constructing a PIFS amounts
to pairing each range block to the domain block that it most closely resembles
under some to-be-determined affine transformation. Done properly, the PIFS
encoding of an image will be much smaller than the original, while still
resembling it closely.

Therefore, a fractal compressed image is an encoding that describes:
   1. The grid partitioning (the range blocks).
   2. The affine transforms (one per range block).

The decompression process begins with a flat gray background. Then the set of
transformations is repeatedly applied. After about four iterations the attrac-
tor stabilizes. The result will not (usually) be an exact replica of the
original, but reasonably close.


 Scalelessnes and Resolution Enhancement
 ---------------------------------------

When an image is captured by an acquisition device, such as a camera or scan-
ner, it acquires a scale determined by the sampling resolution of that device.
If software is used to zoom in on the image, beyond a certain point you don't
see additional detail, just bigger pixels.

A fractal image is different. Because the affine transformations are spatially
contractive, detail is created at finer and finer resolutions with each itera-
tion. In the limit, self-similar detail is created at all levels of resolu-
tion, down the infinitesimal. Because there is no level that 'bottoms out'
fractal images are considered to be scaleless.

What this means in practice is that as you zoom in on a fractal image, it will
still look 'as it should' without the staircase effect of pixel replication.
The significance of this is cause of some misconception, so here is the right
spot for a public service announcement.

                        /--- READER BEWARE ---\

Iterated Systems is fond of the following argument. Take a portrait that is,
let us say, a grayscale image 250x250 pixels in size, 1 byte per pixel. You
run it through their software and get a 2500 byte file (compression ratio =
25:1). Now zoom in on the person's hair at 4x magnification. What do you see?
A texture that still looks like hair. Well then, it's as if you had an image
1000x1000 pixels in size. So your _effective_ compression ratio is 25x16=400.

But there is a catch. Detail has not been retained, but generated. With a
little luck it will look as it should, but don't count on it. Zooming in on a
person's face will not reveal the pores.

Objectively, what fractal image compression offers is an advanced form of
interpolation. This is a useful and attractive property. Useful to graphic
artists, for example, or for printing on a high resolution device. But it does
not bestow fantastically high compression ratios.

                        \--- READER BEWARE ---/

That said, what is resolution enhancement? It is the process of compressing an
image, expanding it to a higher resolution, saving it, then discarding the
iterated function system. In other words, the compressed fractal image is the
means to an end, not the end itself.


 The Speed Problem
 -----------------

The essence of the compression process is the pairing of each range block to a
domain block such that the difference between the two, under an affine trans-
formation, is minimal. This involves a lot of searching.

In fact, there is nothing that says the blocks have to be squares or even
rectangles. That is just an imposition made to keep the problem tractable.

More generally, the method of finding a good PIFS for any given image involves
five main issues:
   1. Partitioning the image into range blocks.
   2. Forming the set of domain blocks.
   3. Choosing type of transformations that will be considered.
   4. Selecting a distance metric between blocks.
   5. Specifying a method for pairing range blocks to domain blocks.

Many possibilities exist for each of these. The choices that Jacquin offered
in his paper are:
   1. A two-level regular square grid with 8x8 pixels for the large
      range blocks and 4x4 for the small ones.
   2. Domain blocks are 16x16 and 8x8 pixels in size with a subsampling
      step size of four. The 8 isometric symmetries (four rotations,
      four mirror flips) expand the domain pool to a virtual domain
      pool eight times larger.
   3. The choices in the last point imply a shrinkage by two in each
      direction, with a possible rotation or flip, and then a trans-
      lation in the image plane.
   4. Mean squared error is used.
   5. The blocks are categorized as of type smooth, midrange, simple
      edge, and complex edge. For a given range block the respective
      category is searched for the best match.

The importance of categorization can be seen by calculating the size of the
total domain pool. Suppose the image is partitioned into 4x4 range blocks. A
256x256 image contains a total of (256-8+1)^2  = 62,001 different 8x8 domain
blocks. Including the 8 isometric symmetries increases this total to 496,008.
There are (256-4+1)^2 = 64,009 4x4 range blocks, which makes for a maximum of
31,748,976,072 possible pairings to test. Even on a fast workstation an ex-
haustive search is prohibitively slow. You can start the program before de-
parting work Friday afternoon; Monday morning, it will still be churning away.

Increasing the search speed is the main challenge facing fractal image com-
pression.


 Similarity to Vector Quantization
 ---------------------------------

To the VQ community, a "vector" is a small rectangular block of pixels. The
premise of vector quantization is that some patterns occur much more frequent-
ly than others. So the clever idea is to store only a few of these common
patterns in a separate file called the codebook. Some codebook vectors are
flat, some are sloping, some contain tight texture, some sharp edges, and so
on -- there is a whole corpus on how to construct a codebook. Each codebook
entry (each domain block) is assigned an index number. A given image, then, is
partitioned into a regular grid array. Each grid element (each range block) is
represented by an index into the codebook. Decompressing a VQ file involves
assembling an image out of the codebook entries. Brick by brick, so to speak.

The similarity to fractal image compression is apparent, with some notable
differences.
   1. In VQ the range blocks and domain blocks are the same size; in an
      IFS the domain blocks are always larger.
   2. In VQ the domain blocks are copied straight; in an IFS each domain
      block undergoes a luminance scaling and offset.
   3. In VQ the codebook is stored apart from the image being coded; in
      an IFS the codebook is not explicitly stored. It is comprised of
      portions of the attractor as it emerges during iteration. For that
      reason it is called a "virtual codebook." It has no existence
      independent of the affine transformations that define an IFS.
   4. In VQ the codebook is shared among many images; in an IFS the
      virtual codebook is specific to each image.

There is a more refined version of VQ called gain-shape vector quantization in
which a luminance scaling and offset is also allowed. This makes the similari-
ty to fractal image compression as close as can be.


 Compression Ratios
 ------------------

Exaggerated claims not withstanding, compression ratios typically range from
4:1 to 100:1. All other things equal, color images can be compressed to a
greater extent than grayscale images.

The size of a fractal image file is largely determined by the number of trans-
formations of the PIFS. For the sake of simplicity, and for the sake of com-
parison to JPEG, assume that a 256x256x8 image is partitioned into a regular
partitioning of 8x8 blocks. There are 1024 range blocks and thus 1024 trans-
formations to store. How many bits are required for each?

In most implementations the domain blocks are twice the size of the range
blocks. So the spatial contraction is constant and can be hard coded into the
decompression program. What needs to be stored are:

   x position of domain block        8     6
   y position of domain block        8     6
   luminance scaling                 8     5
   luminance offset                  8     6
   symmetry indicator                3     3
                                    --    --
                                    35    26 bits

In the first scheme, a byte is allocated to each number except for the symme-
try indicator. The upper bound on the compression ratio is thus (8x8x8)/35 =
14.63. In the second scheme, domain blocks are restricted to coordinates
modulo 4. Plus, experiments have revealed that 5 bits per scale factor and 6
bits per offset still give good visual results. So the compression ratio limit
is now 19.69. Respectable but not outstanding.

There are other, more complicated, schemes to reduce the bit rate further. The
most common is to use a three or four level quadtree structure for the range
partitioning. That way, smooth areas can be represented with large range
blocks (high compression), while smaller blocks are used as necessary to
capture the details. In addition, entropy coding can be applied as a back-end
step to gain an extra 20% or so.


 Quality: Fractal vs. JPEG
 -------------------------

The greatest irony of the coding community is that great pains are taken to
precisely measure and quantify the error present in a compressed image, and
great effort is expended toward minimizing an error measure that most often is
-- let us be gentle -- of dubious value. These measure include signal-to-noise
ratio, root mean square error, and mean absolute error. A simple example is
systematic shift: add a value of 10 to every pixel. Standard error measures
indicate a large distortion, but the image has merely been brightened.

With respect to those dubious error measures, and at the risk of over-sim-
plification, the results of tests reveal the following: for low compression
ratios JPEG is better, for high compression ratios fractal encoding is better.
The crossover point varies but is often around 40:1. This figure bodes well
for JPEG since beyond the crossover point images are so severely distorted
that they are seldom worth using.

Proponents of fractal compression counter that signal-to-noise is not a good
error measure and that the distortions present are much more 'natural looking'
than the blockiness of JPEG, at both low and high bit rates. This is a valid
point but is by no means universally accepted.

What the coding community desperately needs is an easy to compute error meas-
ure that accurately captures subjective impression of human viewers. Until
then, your eyes are the best judge.


 Finding Out More
 ----------------

 Please refer to item 17 in part 1 of this FAQ for a list of references,
available software, and ftp sites concerning fractal compression.



Subject: [78] The Burrows-Wheeler block sorting algorithm (long)


A high-quality implementation of the Burrows-Wheeler
block-sorting-based lossless compression algorithm is available at
http://www.cs.man.ac.uk/arch/people/j-seward/bzip-0.21.tar.gz

Mark Nelson wrote an excellent article "Data Compression with the
Burrows-Wheeler Transform" for Dr. Dobb's Journal, September 1996.  A copy
of the article is at http://www.dogma.net/markn/articles/bwt/bwt.htm

Another introduction written by Sampo Syreeni :

    The Burrows-Wheeler block sorting compression algorithm is described in
"A Block-sorting Lossless Data Compression Algorithm" by M. Burrows and D.J.
Wheeler, dated in May 10, 1994. A postscript copy of this paper has been made
available by Digital on the Systems Research Center (SRC) FTP site at
ftp://ftp.digital.com/pub/DEC/SRC/research-reports/SRC-124.ps.Z

    The method was originally discovered by one of the authors (Wheeler) back
in 1983, but has not been published before. As such, the method is fairly new
and hasn't yet gained popularity.

    The method described in the original paper is really a composite of three
different algorithms: the block sorting main engine (a lossless, very slightly
expansive preprocessor), the move-to-front coder (a byte-for-byte simple,
fast, locally adaptive noncompressive coder) and a simple statistical
compressor (first order Huffman is mentioned as a candidate) eventually doing
the compression. Of these three methods only the first two are discussed here
as they are what constitutes the heart of the algorithm. These two algorithms
combined form a completely reversible (lossless) transformation that - with
typical input - skews the first order symbol distributions to make the data
more compressible with simple methods. Intuitively speaking, the method
transforms slack in the higher order probabilities of the input block (thus
making them more even, whitening them) to slack in the lower order statistics.
This effect is what is seen in the histogram of the resulting symbol data.

    The block sorting preprocessor operates purely on a block basis. One way
to understand the idea is to think of the input block arranged as a circular
array where, for every symbol, the succeeding symbols are used as a predictor.
This predictor is then used to group the symbols with similar right neighbors
together. This predictor is realized (conceptually) as a two phase process.
The first phase forms all cyclic shifts of the input block whose size is
usually a power of two. Note here that the original string is always present
intact on some row of the resulting matrix. If the block length is n then
there exist n unique rotations of the original string (to the left). These
rotations are now viewed as the rows of an N x N matrix of symbols. The second
phase consists of sorting this resulting conceptual matrix. This phase results
in the rows coming into order based on their first few symbols. If there is
some commonly repeated string in the input block (the original paper gives
"the" as an example), the sorting phase brings all those rotations that have a
part of this string as the row start very close to each other. The preceding
symbol in this common string is then found in the last column of the sorted
matrix. This way common strings result in short bursts of just a few distinct
characters being formed in the last column of the matrix. The last column is
what is then output from the second phase. One further bit of information is
derived from the input data. This is an integer with enough bits to tell the
size of the input string (that is, log_2(n)). The number is used to note the
row position into which the original input block got in the sorting algorithm.
This integer always results in expansion of the data, but is necessary for us
to be able successfully decompress the string. The absolute amount of overhead
increases as the logarithm of the input block size so its percentage of the
output data becomes negligible with useful block sizes anyway.

    The characteristics of the transformation process make the output from the
sort ideal for certain kinds of further manipulation. The extreme local
fluctuations in the first order statistics of the output string lead one to
use a transformation that boosts and flattens the local fluttering of the
statistics. The best example (and, of course, the one given in the original
paper) is move-to-front coding. This coder codes a symbol as the number of
distinct symbols seen since the symbol's last occurrence. Basically this means
that the coder outputs the index of an input symbol in a dynamic LIFO stack
and then updates the stack by moving the symbol to the top. This is easy and
efficient to implement and results in fast local adaptation. As just a few
common symbols will (locally) govern the input to the coder, these symbols
will be kept on the top of the stack and thus the output will mainly consist
of low numbers. This makes it highly susceptible to first order statistical
compression methods which are, in case, easy and efficient to implement.

    The transform matrix described above would require enormous amounts of
storage space and would not result in a usable algorithm as such. The method
can, however, be realized very efficiently by suffix and quick sort methods.
Thus the whole transformation together with the eventual simple compression
engine is extremely fast but still achieves impressive compression on typical
input data. When implemented well, the speeds achieved can be in the order of
pure LZ and the compression ratios can still approach state-of-the-art Markov
modeling coders. The engine also responds well to increasing block sizes -
the longer the input block, the more space there is for the patterns to form
and the more similar input strings there will be in it. This results in almost
monotonously increasing compression ratios even as the block length goes well
into the megabyte range.
    The decompression cascade is basically just the compression cascade
backwards. More logic is needed to reverse the main sorting stage, however.
This logic involves reasoning around the order of the first the last column of
the conceptual coding matrix. The reader is referred to the original paper for
an in depth treatment of the subject. The original paper also contains a more
thorough discussion of why the method works and how to implement it.

    And now a little demonstration. The original block to be compressed is
chosen to be the (rather pathological) string "good, jolly good". This was
taken as an example because it has high redundancy and it is exactly 16 bytes
long. The first picture shows the cyclic shifts (rotations) of the input
string. The second shows the matrix after sorting. Note that the last column
now has many double characters in it. Note also that the original string has
been placed into the 6th row now. The third picture shows the output for this
input block. The index integer has been packed to a full byte although 4 bits
would suffice in this case (log_2(16)=4). The fourth and fifth pictures show
the transformed string after move-to-front-coding. The sixth picture shows the
statistical distribution of the characters in the output string. Notice the
disproportionately large amount of ones and zeros, even with a very short
string like this. This is the output that is then routed through the simple
statistical encoder. It should compress very well, as the distribution of the
characters in the input block is now very uneven.


      0 1 2 3 4 5 6 7 8 9 A B C D E F        0 1 2 3 4 5 6 7 8 9 A B C D E F
      -------------------------------        -------------------------------
  0 | g o o d ,   j o l l y   g o o d    0 |   g o o d g o o d ,   j o l l y
  1 | o o d ,   j o l l y   g o o d g    1 |   j o l l y   g o o d g o o d ,
  2 | o d ,   j o l l y   g o o d g o    2 | ,   j o l l y   g o o d g o o d
  3 | d ,   j o l l y   g o o d g o o    3 | d ,   j o l l y   g o o d g o o
  4 | ,   j o l l y   g o o d g o o d    4 | d g o o d ,   j o l l y   g o o
  5 |   j o l l y   g o o d g o o d ,    5 | g o o d ,   j o l l y   g o o d
  6 | j o l l y   g o o d g o o d ,      6 | g o o d g o o d ,   j o l l y
  7 | o l l y   g o o d g o o d ,   j    7 | j o l l y   g o o d g o o d ,
  8 | l l y   g o o d g o o d ,   j o    8 | l l y   g o o d g o o d ,   j o
  9 | l y   g o o d g o o d ,   j o l    9 | l y   g o o d g o o d ,   j o l
  A | y   g o o d g o o d ,   j o l l    A | o d ,   j o l l y   g o o d g o
  B |   g o o d g o o d ,   j o l l y    B | o d g o o d ,   j o l l y   g o
  C | g o o d g o o d ,   j o l l y      C | o l l y   g o o d g o o d ,   j
  D | o o d g o o d ,   j o l l y   g    D | o o d ,   j o l l y   g o o d g
  E | o d g o o d ,   j o l l y   g o    E | o o d g o o d ,   j o l l y   g
  F | d g o o d ,   j o l l y   g o o    F | y   g o o d g o o d ,   j o l l

             1. The shifts                    2. In lexicographic order


                                                121,45,102,114,0,1,36,0,
         "y,dood  oloojggl",5                   1,113,1,0,112,110,0,3,5

     3. The output from block sort          4. After move-to-front-coding


                                             00: 4;  01: 3;  03: 1;  05: 1;
          79,2D,66,72,0,1,24,0,              24: 1;  2D: 1;  66: 1;  6E: 1;
          1,71,1,0,70,6E,0,3,5               70: 1;  71: 1;  72: 1;  79: 1

           5. In hexadecimal                      6. The statistics

------------------------------------------------------------------------------


 === End of part 2 of the comp.compression faq ===



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