Data Compression

1
Data Compression
2
Terminology

• Physical versus logical
• Physical
• Performed on data regardless of what information
  it contains
• Translates a series of bits to another series of
  bits
• Logical
• Knowledge-based
• Change United Kingdom to UK

3
Terminology

• Symmetric
• Compression and decompression roughly use the
  same techniques and take just as long
• Data transmission which requires compression and
  decompression on-the-fly will require these types
  of algorithms

4
Terminology

• Asymmetric
• Most common is where compression takes a lot more
  time than decompression
• In an image database, each image will be
  compressed once and decompressed many times
• Less common is where decompression takes a lot
  more time than compression
• Creating many backup files which will hardly ever
  be read

5
Terminology

• Non-adaptive
• Contain a static dictionary of predefined
  substrings to encode which are known to occur
  with high frequency
• Adaptive
• Dictionary is built from scratch

6
Terminology

• Semi-adaptive
• In pass 1, an optimal dictionary is constructed
• In pass 2, the actual compression occurs

7
Terminology

• Lossless
• decompress(compress(data)) data
• Lossy
• decompress(compress(data)) ? data
• A small change in pixel values may be invisible,
  however

8
Pixel Packing
9
Run-Length Encoding

• Repeating string of characters, called a run, is
  coded into two bytes
• First byte contains the run count, one less than
  the number of repetitions
• Second byte contains the run value, the character
  being repeated

10
Run-Length Encoding

• 77777zzzyyyyyyV becomes 472z5y0V
• 15 byte string becomes 8 bytes long
• Compression ratio of almost 2 to 1
• Some strings become twice as long
• 7fu5JLY9jhYIujG

11
(No Transcript)
12
Lempel-Ziv-Welch (LZW)

• Lossless
• GIF, TIFF, V.42bis modem compression standard,
  PostScript Level 2
• Substitutional or dictionary-based
• Algorithm builds a data dictionary
• Code emitted if pattern found in dictionary,
  while if not already in dictionary, it is added
• Not necessary to have dictionary to do
  decompression

13
Lempel-Ziv-Welch (LZW)

• History
• 1977
• Abraham Lempel and Jakob Ziv published a paper on
  a universal data compression algorithm
• Called LZ77
• 1978
• Lempel and Ziv formulated an improved,
  dictionary-based data compression algorithm
• Called LZ78

14
Lempel-Ziv-Welch (LZW)

• History
• 1981
• While working for Sperry, Lempel and Ziv, with
  some other researchers filed for a patent for
  LZ78
• Granted in 1984
• 1984
• While working for Sperry, Terry Welch modified
  LZ78
• Result was LZW algorithm
• Published in IEEE Computer

15
Lempel-Ziv-Welch (LZW)

• History
• 1985
• Sperry granted a patent for Welchs modification
  and for implementation of LZW
• 1986
• Sperry and Burroughs merged to form Unisys
• Ownership of Sperry patent transferred to Unisys

16
Lempel-Ziv-Welch (LZW)

• History
• 1987
• CompuServe created GIF file format
• Required use of LZW algorithm
• Didnt check patents for LZW
• Unisys also didnt realize GIF used LZW 1988
• Aldus released Revision 5.0 of TIFF file format
• Used LZW algorithm
• 1990
• Unisys licensed Adobe for use of LZW patent for
  PostScript

17
Lempel-Ziv-Welch (LZW)

• History
• 1991
• Unisys licensed Aldus for use of LZW patent in
  TIFF
• 1993
• Unisys became aware the GIF file format used LZW
• Negotiations began with CompuServe

18
Lempel-Ziv-Welch (LZW)

• History
• 1994
• Unisys and CompuServe came to an understanding
  that LZW algorithm by CompuServe would be
  licensed for the application of the GIF file
  format in software used primarily to access the
  CompuServe Information Service
• 1995
• America Online and Prodigy also entered into
  license agreements with Unisys for LZW

19
Lempel-Ziv-Welch (LZW)

• GIF is not in public domain
• Some people were suspicious regarding the
  announcement of CompuServe that it was getting a
  license from Unisys
• In programming community it was known for many
  years prior to this that GIF used LZW and that
  LZW was patented by Unisys

20
Lempel-Ziv-Welch (LZW)

• Some people were suspicious regarding the
  announcement of CompuServe that it was getting a
  license from Unisys
• Unisys claimed that CompuServe only found out
  rather late that this was the case
• GIF was becoming an integral part of WWW for
  exchanging low-resolution graphics

21
Lempel-Ziv-Welch (LZW)

• Eventually, Unisys LZW patent and licensing
  agreements held
• Unisys reduced license fees after 1995
• Unisys wouldnt charge anything for inadvertent
  infringement by GIF software products delivered
  prior to 1995
• License fees still required for updates delivered
  after 1995

22
Lempel-Ziv-Welch (LZW)

• Not illegal to own, transmit, or receive GIF
  files, just to compress or decompress them
  without a license

23
Lempel-Ziv-Welch (LZW)
24
Lempel-Ziv-Welch (LZW)
25
Lempel-Ziv-Welch (LZW)
26
JPEG

• Joint Photographic Experts Group
• 1982
• ISO (International Standard Organization) formed
  Photographic Experts Group (PEG)
• Develop methods of transmitting video, images and
  text over ISDN (Integrated Services Digital
  Network) lines

27
JPEG

• 1986
• Subgroup of CCITT (International Telegraph and
  Telephone Consultative Committee) began to look
  at methods of compressing color and gray-scale
  data for fax transmission
• Methods for this were similar to those being
  considered by PEG

28
JPEG

• 1987
• Two groups combined into JPEG
• Most previous compression methods did poor job of
  compressing continuous-tone image data

29
JPEG

• Very few file formats can support 24-bit raster
  images
• GIF only works for 256 colors
• LZW doesnt work well on scanned image data
• TIFF and BMP didnt compress this type of image
  data very well

30
JPEG

• JPEG compresses continuous tone image data with a
  pixel depth of 6-24 bits with good efficiency
• JPEG itself doesnt define standard file format

31
JPEG

• Toolkit of methods with quality-compression
  trade-off
• Lossy
• Discards information that human eye cannot easily
  see
• Slight changes in color not perceived well
• Slight changes in intensity are well perceived

32
JPEG

• Works well with color or gray-scale continuous
  tone images photographs, video stills, complex
  graphics which resemble natural objects
• Doesnt work well for animations, ray tracing,
  line art, black-and-white documents, and typical
  vector graphics

33
JPEG

• End-user can tune quality of JPEG encoder through
  use of Q-factor, which ranges from 1-100
• Q-factor 1 produces smallest, worst quality
  images
• Q-factor 100 produces largest, best quality
  images
• Optimal value of Q-factor is image dependent

34
JPEG

• JPEG introduces artifacts in images containing
  large areas of a single color
• JPEG is slow if implemented in software
• Baseline JPEG
• Minimal subset of JPEG which all JPEG-aware
  applications are required to support

35
JPEG
36
JPEG

• Color transform
• Encodes each component in a color model
  separately
• Is independent of any color space model

37
JPEG

• Color transform
• Best compression ratios result if a luminance
  (gray scale)/chrominance (color) color space,
  such as YUV, is used
• Human eyes more sensitive to luminance
  information (Y) than to chrominance information
  (U, V)
• The other models spread human sensitive
  information across each of their 3 components

38
JPEG

• Down-sampling
• Average groups of pixels together
• To exploit humans lesser sensitivity to
  chrominance information, we use fewer pixels for
  the chrominance channels
• In an image of 1000 ? 1000 pixels, we might use
  1000 ? 1000 luminance pixels, but only
• 500 ? 500 chrominance pixels
• Each chrominance pixel covers the same area as a
• 2 ? 2 block of luminance pixels

39
JPEG

• Down-sampling
• For each 2 ? 2 block, we can store 6 pixel values
  
• 4 luminance values and 2 chrominance values 1
  for each of 2 channels
• instead of 12
• 4 pixel values for each of 3 channels
• This 50 reduction in data has almost no
  perceivable effect

40
JPEG

• Discrete cosine transform
• For each color channel, the image data is divided
  into 8 ? 8 blocks
• DCT applied to each block
• Low-order, or DC, term represents average value
  in the block
• Successive higher-order, or AC, terms represent
  the strength of more rapid changes across the
  block

41
JPEG

• Discrete cosine transform
• Can discard high-frequency data
• DCT is lossless except for roundoff errors
• DCT is most costly step in JPEG

42
JPEG

• Scan-order of each 8 ? 8 block of pixels for DCT

43
JPEG

• An 8 ? 8 block from an 8 bit image

44
JPEG

• The DCT coefficients corresponding to the
  previous 8 ? 8 block

45
JPEG

• Quantization
• Divide DCT output by a quantization coefficient
  and round result to integer
• The larger the coefficient, the more data is lost
• Each of the 64 positions of the DCT output block
  has its own coefficient
• Higher order terms have a larger coefficient
• Different coefficients for luminance and
  chrominance channels

46
JPEG

• Quantization
• This is the step controlled by the quality-factor
• Selecting quantization coefficients is an art

47
JPEG

• Sample quantization table
• Coefficients based on human perception

48
JPEG

• Labels
• Label labij corresponding to the quantized value
  of the transform coefficient cij is
• where Qij is the (i,j)th element of the
  quantization table

49
JPEG

• Quantizer labels corresponding to the previous 8
  ? 8 block

50
Encoding

• Huffman compress resulting coefficients
• Can use arithmetic coding as well

51
Huffman Coding

• Lossless

52
Huffman Coding
53
Huffman Coding
54
Arithmetic Coding

• Lossless

String cadd
55
Arithmetic Coding
56
JPEG Extensions

• Progressive
• For applications that need to receive JPEG data
  streams and display them on the fly
• Baseline JPEG image can be displayed only after
  all of the image data has been received

57
JPEG Extensions

• Progressive
• Instead of interlacing, where a majority of the
  image must be sent to be able to tell what it is,
  we send successively better resolution images
• Lossless JPEG

58
Fractal Compression

• Suppose we have a linear, non-identity, function
  of one variable, g, having xf as a fixed point
• g(xf) xf
• We can compute the fixed point by the
  approximation x, g(x), g(g(x)), g(g(g(x))),
  , where x is any initial approximation

59
Fractal Compression

• Example
• f(x) ax b
• Fixed point is solution to xf axf b or
• For a 0.5, b 1, we have that xf 2

60
Fractal Compression

• Example
• To calculate the fixed point by the previous
  approximation, use the initial guess 1 and
  calculate g(1), g(g(1)), g(g(g(1))), , where
  g(x) x/2 1
• The approximations are 1.5, 1.75, 1.875, 1.9375,
  , which converges to 2, the fixed point

61
Fractal Compression

• Given an image I, treated as an array of
  integers, suppose we have a non-identity function
  g(I) I
• If it was cheaper to encode g than to encode I,
  we could communicate g and reconstruct I by the
  sequence of approximations I0, g(I0), g(g(I0)),
  g(g(g(I0))), , where I0 is the all zero image

62
Fractal Compression

• Partition image into equal size range blocks
• For each range block, Rk, find a domain block,
  Dk, twice the size of a range block, and a
  function gk such that

63
Fractal Compression

• Consider the function
• This function has a fixed point If g(If), where
• This function has a fixed point If g(If), where

64
Fractal Compression

• gk is a composition of a geometric transformation
  followed by a massic transformation
• Geometric transformation
• Moves domain block
• Changes the size of the domain block
• Massic transformation
• Adjusts intensity and orientation of pixels

65
Fractal Compression