Scalar Quantization

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Scalar Quantization

• CAP5015
• Fall 2004

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Quantization

• Definition
• Quantization a process of representing a large
  possibly infinite set of values with a much
  smaller set.
• Scalar quantization a mapping of an input value
  x into a finite number of output values, y
• Q x y
• One of the simplest and most general idea in
  lossy compression.

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Scalar Quantization

• Many of the fundamental ideas of quantization and
  compression are easily introduced in the simple
  context of scalar quantization.
• An example any real number x can be rounded off
  to the nearest integer, say
• q(x) round(x)
• Maps the real line R (a continuous space) into a
  discrete space.

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Quantizer

• The design of the quantizer has a significant
  impact on the amount of compression obtained and
  loss incurred in a lossy compression scheme.
• Quantizer encoder mapping and decode mapping.
• Encoder mapping
• The encoder divides the range of source into a
  number of intervals
• Each interval is represented by a distinct
  codeword
• Decoder mapping
• For each received codeword, the decoder
  generates a reconstruct value

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Components of a Quantizer

1. Encoder mapping Divides the range of values that
   the source generates into a number of intervals.
   Each interval is then mapped to a codeword. It
   is a many-to-one irreversible mapping. The code
   word only identifies the interval, not the
   original value. If the source or sample value
   comes from a analog source, it is called a A/D
   converter.

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Mapping of a 3-bit Encoder
Codes

000 001 010 011 100 101 110
111
-3.0 -2.0 -1.0 0 1.0 2.0
3.0 input
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Mapping of a 3-bit D/A Converter
Input Codes Output
000 -3.5
001 -2.5
010 -1.5
011 -0.5
100 0.5
101 1.5
110 2.5
111 3.5
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Components of a Quantizer
2. Decoder Given the code word, the decoder
gives a an estimated value that the source might
have generated. Usually, it is the midpoint of
the interval but a more accurate estimate will
depend on the distribution of the values in the
interval. In estimating the value, the decoder
might generate some errors. (Give Table 8.1 and
explain)
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Digitizing a Sine Wave
t 4cos(2Pit) A/D Output D/A Output Error
0.1 3.8 111 3.5 0.3
0.1 3.2 111 3.5 -0
0.2 2.4 110 2.5 -0
0.2 1.2 101 1.5 -0
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Step Encoder
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(No Transcript)
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• resulting quantization error (noise)
  so that

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Probability Density Function

• A probability density function f(x) of the random
  variable x is said to meet the following
  criterion
• Probability associated with a value of x in its
  domain X is given by Pr( Xlt x ).
• The corresponding cumulative distribution
  function CDF or F(x) requires that F(x) is
  non-decreasing for x1 lt x2. When sampling
  occurs at discrete intervals then F(x) is said to
  be monotonically increasing.
• F(x) is said to be continuous from the right or
  that the limit of f(x e) exists when evaluated
  as e-gt 0 from the right positive abscissa.
• In the discrete case the point probabilities of
  particular values of xi have a probability that
  is always greater or equal to 0, pi Pr( X
  xi ) gt 0.
• CDF may be expressed as 
• In the continuous case, the CDF may be expressed
  as the following relationship 

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• Quantization operation
• Let M be the number of reconstruction levels
• where the decision boundaries are
• and the reconstruction levels are

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Quantization Problem

• MSQE (mean squared quantization error)
• If the quantization operation is Q
• Suppose the input is modeled by a random variable
  X with pdf fX(x).
• The MSQE is

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Quantization Problem

• Rate of the quantizer
• The average number of bits required to represent
  a single quantizer output
• For fixed-length coding, the rate R is
• For variable-length coding, the rate will depend
  on the probability of occurrence of the outputs

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Quantization Problem

• Quantizer design problem
• Fixed -length coding
• Variable-length coding
• If li is the length of the codeword corresponding
  to the output yi, and the probability of
  occurrence of yi is
• The rate is given by

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Uniform Quantization
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Uniform Quantizer
Zero is one of the output levels M is odd
Zero is not one of the output levels M is even
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Uniform Quantization of A Uniformly Distributed
Source
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Uniform Quantization of A Uniformly Distributed
Source
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Uniform Quantization of A Non-uniformly
Distributed Source
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Image Compression
3bits/pixel
Original 8bits/pixel
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Image Compression
2bits/pixel
1bit/pixel