Lossy Compression Iii_1

8/10/2019 Lossy Compression Iii_1

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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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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

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Lossy Compression Iii_1