Data Compression

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Data Compression

• Rejean Lau

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Data Compression

• Data transmission and storage cost money
• It is advantageous to represent data files in the
  most compact form during either transmission or
  storage

Storage
Player decompessor
Native Format
Compressor
Encoded Files (video, audio)
MPEG, MP3
End User
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Compression

• A compression program is used to convert data
  from a native format to one optimized for
  compactness.
• An uncompression program returns the compressed
  format to a usable form which may or may not be a
  replication of the original depending on
  whether the compression is lossy or not

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Compression

• A lossless compression algorithm means that the
  restored data is identical to the original
• Necessary for many applications executable code,
  ascii files etc.
• A lossy compression algorithm, allows for some
  degradation
• Ok for many types of signal applications audio,
  video, graphics

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Lossy versus Lossless
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Input and Output Group Size

• Most compression algorithms operate by taking a
  group of data from the original (native) file,
  and writing a compressed group to the output
  file.
• Some algorithms have a fixed-input fixed output
  scheme. That is, a fixed number of bits are read
  from the input file and a smaller fixed number of
  bits are written to the output file.
• CSQ reads 24 bits of an audio file, and outputs
  8 bits, resulting in a 31 compression ratio.
• Others allow for a variable number of bits to be
  read or written

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Input and Output Group Size
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Run-Length Encoding

• Digitized signals can have runs of the same value
  over and over again.
• An image of the night sky could contain long runs
  of the same signal representing the black
  background.
• Digitized albums on CD might have a long run of
  zeroes between tracks.

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Run length Encoding
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PackBits

• Created for Macintosh users
• Each byte (8 bits) from the input file is
  replaced by 9 bits in the compressed file.
• The added bit is interpreted as the sign of the
  number
• Each character read from the input file is
  between 0 to 255 (ASCII).
• Each character added to the encoded file is
  between -255 and 255.

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Packbits

• Consider the input file 1,2,3,4,2,2,2,2,4 and
  the compressed file generated by PackBits
  1,2,3,4,2,-3,4
• The compressed file replaced 2,2,2,2 with 2,-3.
• The 2 represents the character run. The -3
  indicates the number of characters, found by
  taking the absolute value and incrementing one.
• Essentially, the negative indicates that this is
  to be interpreted as the number of character runs
  of the previous character.
• Ex. 4,-2 means 4,4,4 21,-4 means
  21,21,21,21,21

Why is the secondary value, one less then the
character run? Why not just use the exact number?
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Huffman Encoding

• Let us examine the histogram of the byte values
  from a sample ASCII file.

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Huffman Encoding

• 96 of the sample text consists of only 31
  characters the lower case letters, the space,
  comma, period, and the return.

The observation can be used to make an
appropriate compression scheme for the file.
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Huffman Encoding

• Assign each of the 31 common characters a five
  bit binary code 00000 a, 00001 b, 00010
  c, etc.
• This allows 96 of the file to be reduced in size
  by 5/8
• Let 11111 be a flag indicating that the character
  being transmitted is not one of the 31 common
  characters and add 8 bits according to the
  ASCII code
• This results in 4 of the characters to be
  represented by 5813 bits

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Huffman Encoding

• The principle is to assign the vast majority of
  characters fewer bits and rare characters more
  bits
• Huffman encoding takes the principle to the
  extreme
• Characters that occur most often, such as the
  space and period, may be assigned one or two
  bits.
• Characters that occur most often, such as the
  space and period, may require a dozen or more
  bits.
• The optimal solution is reached when the number
  of bits used for each character is proportional
  to the logarithm of the characters probability
  of occurrence
• Huffman encoding is a fixed input, variable
  output scheme, however the output is regrouped as
  fixed output bytes for transmission and storage

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Huffman Encoding
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Huffman Encoding

• Figure 27-3 shows a simplified Huffman encoding
  scheme the characters A through G occur in the
  original data stream with the probabilities
  shown.
• Since the character A is the most common, it will
  be represented with a single bit, the code 1.
• The next most common character, B, receives two
  bits, the code 01.
• The least frequent character, G, is assigned six
  bits, 000011

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Huffman Encoding

• The variable length codes are resorted into eight
  bit groups or bytes
• The uncompression program looks at the stream of
  ones and zeros until a valid code is formed,and
  then looks for the next character.
• The way that codes are formed insures that no
  ambiguity exists in the separation.
• Note that there is no need for delimiters between
  the codes, if the code is chosen correctly

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Delta Encoding

• Delta encoding refers to several techniques that
  store data as the difference between successive
  samples (or characters)

• The first value in the delta encoded file is the
  same as the first value in the original value
• All the following values in the encoded file are
  equal to the delta (difference) between the
  corresponding value in the input file, and the
  previous value in the input file.

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Delta Encoding

• Delta encoding is effective when the values in
  the original data is smooth that is, there is
  very little change between adjacent values
• not the case for ASCII text or executable code,
  but very true for signals

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Delta Encoding

• The essential feature is that the delta encoded
  signal has a much lower amplitude than the
  original signal.
• If the original signal is not changing, or is
  changing in a straight line, delta encoding will
  result in runs of samples having the same value
• Delta encoding followed by Huffman and/or
  run-length encoding is a common strategy for
  compressing signals

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JPEG

• Belongs to a family of lossy compression
  techniques called transform compression
• Transform compression is based on the following
  premise when the signal is passed through the
  Fourier (or other) transform, the resulting data
  values will be unequal in their information
  carrying capacity
• The low frequency components of a signal are more
  important than the high frequency components
• For example, removing 50 of the bits from the
  high frequency components might remove 5 of the
  of the encoded information

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JPEG

• JPEG compression starts by breaking the image
  into 88 pixel groups.
• Each pixel is a single byte, a greyscale value
  between 0 and 255.
• Each group is represented by 64 bytes.
• After transforming and removing data, each group
  is represented by 2 to 20 bytes.
• During uncompression, an inverse transform
  creates an approximation of the original 88
  group.

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JPEG

• The transform used for JPEG compression, is
  called the Discrete Cosine Transform (DCT)
• Like the discrete fourier transform (DFT), the
  DCT provides information about the signal in the
  frequency domain.
• Unlike the DFT, the DCT of a real signal is real
  valued.

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JPEG

• The transform is linear, so it can be expressed
  in a matrix-vector form

where xN is the N-vector describing the
signal, XCN is the N-vector describing the
result of the transform, and CN is a square
nonsingular NxN matrix describing the transform
itself. The matrix CN is real valued.
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JPEG

• The DCT is a transform that maps a block of pixel
  color values in the spatial domain to values in
  the frequency domain.
• The DCT can operate mathematically in any
  dimension, however an image is a two-dimensional
  (2-D) surface so the 2-D DCT transform is used.

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JPEG

• The 2-D DCT is given by

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JPEG

• The 2-D DCT is applied to each block so that an
  8x8 spectrum (8 x 8 matrix of DCT coefficients)
  is produced for each 8 x 8 block.
• 64 individual pixels are transformed into 64 real
  numbers representing frequency components of the
  entire block.

DC
The further an an AC component from the DC
component, the higher the frequency represented

• The top left component of the DCT matrix is
  called the discrete cosine (DC) coefficient and
  can be interpreted as the component responsible
  for the average background color of the block
• The remaining 63 components of the DCT matrix are
  the AC components, representing the frequency
  components of the image

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JPEG

• The 64 frequency components are described by 64
  basis functions, and the value in the DCT matrix
  is the amplitude of its corresponding 2D basis
  function

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JPEG
The one-half cycle basis function
(0,1) represents brightness gradually
changing over a region
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JPEG

• The original image block is recovered from the
  DCT coefficients by applying the inverse DCT
  (IDCT)

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JPEG

• Most of the signal amplitude concentrates in the
  low frequency components (upper left portion of
  the DCT matrix)
• This means the highest frequency components can
  be eliminated, while only degrading the signal a
  small amount

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JPEG
Reconstruction of the eye image with reduced
spectrum coefficients
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Quantization Tables

• Each value in the DCT spectrum is divided by the
  corresponding value in the
• quantization table, and the result rounded to
  the nearest integer
• For example, the lower right-hand value of a) is
  16, meaning that the original
• range of -127 to 127 is reduced to -7 to 7,
  correspondingly the value has been
• reduced in precision from 8 bits to 4 bits.

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Zig-Zag Pattern

• A zig-zag pattern places all the high frequency
  components together
• at the end of the linear sequence.
• This groups the zeros from the eliminated
  components into long runs
• Run-length encoding compresses the zeros
• Resulting sequence is encoded by Huffman or
  arithmetic encoding

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Compression Ratios