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Scalar Quantization. CAP5015 Fall 2004. 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

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Scalar quantization l.jpg

Scalar Quantization

CAP5015

Fall 2004


Quantization l.jpg
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

  • 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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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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Probability Density Function so that

  • 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( X<= x ).

    • The corresponding cumulative distribution function CDF or F(x) requires that F(x) is non-decreasing for x[1] <= x[2]. 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-> 0 from the right positive abscissa.

      • In the discrete case the point probabilities of particular values of x[i] have a probability that is always greater or equal to 0, p[i] == Pr( X = x[i] ) >= 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: so that

    • – Let M be the number of reconstruction levels

      where the decision boundaries are

      and the reconstruction levels are


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Quantization Problem so that

  • 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 so that

  • 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 so that

  • 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 Quantizer so that

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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Image Compression so that

Original 8bits/pixel

3bits/pixel


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Image Compression so that

2bits/pixel

1bit/pixel