Post on 02-Jun-2018
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LOSSY COMPRESSION III
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IntroductionCompression in all the lossy schemes is achievedthrough quantization.
The process of representing a large possiblyinfinite set of values with a much smaller set iscalled quantization
Example: Source generates numbers between -10.0and +10.0 Simple scheme is to represent eachoutput of the source with the integer value closer toit.
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IntroductionTwo types of quantization
Scalar Quantization.Vector Quantization.
The design of the quantizer has a significantimpact on the amount of compression (i.e.,rate) obtained and loss (distortion) incurred ina lossy compression scheme
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Scalar QuantizationMany of the fundamental ideas of quantization andcompression are easily introduced in the simplecontext of scalar quantization.
Any real number x can be rounded off to the nearestinteger, say
Q(x) = round(x)
Maps the real line R (a continuous space) into adiscrete space.
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Scalar Quantization Quantizer: encoder mapping and decodermapping.
Encoder mappingThe encoder divides the range of source into a numberof intervals Each interval is represented by a distinct codeword
Decoder mappingFor each received codeword, the decoder generates ar econstr uct value
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Scalar Quantization Encoder mapping: Divides the range of values that thesource generates into a number of intervals. Each interval isthen mapped to a codeword. It is a many-to-one irreversiblemapping. The code word only identifies the interval, not theoriginal value.
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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Scalar Quantization Decoder: Given the code word, the decodergives 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 thedistribution of the values in the interval.
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Mapping of a 3-bit Decoder
Input Codes Output000 -3.5001 -2.5010 -1.5011 -0.5
100 0.5101 1.5110 2.5111 3.5
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Encoder Decoder Example
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Scalar Quantization
Quantization operation: Let M be the number of reconstruction levels
where the decision boundaries are
and the reconstruction levels are
ii j b xbiff y xQ 1)(
M ibi 0
M ii y 1
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Scalar Quantization
MSQE (mean squared quantization error)If the quantization operation is Q
Suppose the input is modeled by a random variable Xwith pdf f X ( x). The MSQE is
ii j b xbiff y xQ 1)(
dx x f y xdx x f xQ x X
M
i
b
bi X q
i
i
)()()())((1
222
1
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Scalar Quantization
Rate of the quantizerThe average number of bits required to represent a
single quantizer outputFor fixed-length coding, the rate R is:
For variable-length coding, the rate will depend on the probability of occurrence of the outputs
M
i
b
b
X i
i
i
dx x f l R1 1
)(
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Scalar Quantization
Quantizer Design Problem :Given an input pdf f X (x) and the number of levels M in
the quantizer, find the decision boundaries {b i} and thereconstruction levels {y i} so as to minimize the MSQE(Mean Square Quantization Error)
dx x f y x
dx x f xQ x
X
M
i
b
bi
X q
i
i
)()(
)())((
1
2
22
1
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Scalar Quantization
Find the optimum partitions, codes and representationlevels
Given a distortion constraint, find the decision boundaries,reconstruction levels, and binary codes that minimize therate, while satisfying the distortion constraint given above.
Given a rate constraint find the decision boundaries,reconstruction levels, and binary codes that minimize thedistortion.
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Uniform Quantization of a UniformlyDistributed Source
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Uniform Quantization of a UniformlyDistributed Source
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Uniform Quantization of a UniformlyDistributed Source
Summary:If the distortion constraint is given as D*, then step size can
be calculated directly, since
D* =
M = (x max xmin)/
If the rate constraint is given as R *, then M can be calculated,hence can be calculated.
Then distortion is D =
12
2
12
2
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Example Image compression
Assume Image pixels are uniformly distributed between 0& 255.
1 bit/pixel [0,255] is divided into two intervals [0,127]and [128,255]
Reconstruction levels midpoints of intervals {64, 196}.
2 bits/pixel Four intervals [0,64,128,196,255] boundaries
Reconstruction levels {32,96,160,224}
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UNIFORM QUANTIZATION EXAMPLE
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UNIFORM QUANTIZATION EXAMPLE