DynaMap project

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DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF
JOENSUU JOENSUU, FINLAND
Image Compression Lecture 8 Scalar
Quantization Alexander Kolesnikov
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Quantization
Any analog quantity that is to be processed by a
digital computer or digital system must be
converted to an integer number proportional to
its amplitude. The conversion process between
analog samples and discrete-valued samples is
called quantization.
Q
Quantizer
Input signal
Quantized signal
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Quantization application areas
Analog-to-digital conversion of signals
(audio,video,etc.)
Quantization of transform coefficients (JPEG,
JPEG2000)
Binarization and multilevel thresholding of
digital images
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Quantization Rounding-off
Any real number x can be rounded-off to the
nearest integer value q(x) round(x)
If k?0.5 ? x ltk 0.5 then q(x)k
5 5.5 6
6.5 7
x6.4
q(x)6.0
x5.6
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Uniform quantizer M8 levels
Input-output characteristic of uniform quantizer
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Nonuniform quantizer as line partition
yi ? Reproduction (reconstructed)
levels ai ? Decision levels (thresholds) Si
ai-1, ai) ? Cell ai -ai-1 ? Cell width
a0 -?, a6 ?.
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Nonuniform quantizer M 8 levels
Input-output characteristic of nonuniform
quantizer
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Quantization error
Input signal x
Quantized signal q(x)
Quantization error e(x) x?q(x)
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Distortion measure
Probability density function (pdf) of x is
p(x) Quantization error e(x) x ? q (x) Mean
(average value) ? of quantization error
Variance ?2 of quantization error as distortion
measure
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Optimal quantization problem
Given a signal x, with probability density
function (or histogram) p(x), find a quantizer
q(x) of x, which minimizes the quantization error
variance ?2
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Formulation of quantization problem (contd)
Optimization problem Find decision aj and
representation levels yj to minimize variance
?2.
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Max-Lloyd solution
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Max-Lloyd Conditions for optimal quantizer
Decision levels ai are midpoints
Representation levels yi are centroids
yj1
aj
yj-1
yj
aj-1
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How to construct optimal quantizer?

• If we have some set of levels, with Max-Lloyd
  equations
• we can check do these levels provide minimum
  of
• error variance.
• But these equations dont tell us how to find
  the
• optimal levels.

?
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Max-Lloyd Iterative algorithm
0. Guess initial set of decision levels aj 1.
Calculate representation levels (centroids)
yj
?
2. Calculate decision levels aj
3. Repeat 1. and 2. until no further variance ?2
reduction
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Iterative algorithm Discrete case
0. Guess initial set of decision levels aj 1.
Calculate representation levels
(centroids) yj
?
2. Calculate decision levels aj
3. Repeat 1. and 2. until no further variance ?2
reduction
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Anyway, how to construct optimal quantizer?

• Iterative algorithm of Lloyd cannot guarante
  global
• minimum of quantizaton error variance.
• Other heuristic approaches merge, genetic
  algorithm,
• reinforcement learning, etc.
• How to achieve the global minimum
• of the quantization error?

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Summary
1) Uniform and nonuniform quantizers 2) Mean and
variance of quantization error 3) Formulation of
optimal quantization problem 4) Max-Lloyd
Conditions for optimal scalar quantizer 5)
Max-Lloyd Iterative algorithm for scalar
quantizizatoin