09: Scalar Quantization CSCI 6990 Data Compression Vassil Roussev 1 CSCI 6990.002: Data Compression 09: Scalar Quantization Vassil Roussev <vassil @ cs.uno.edu> UNIVERSITY of NEW ORLEANS DEPARTMENT OF COMPUTER SCIENCE 2 Quantization Definition: The process of representing a large—possibly infinite— set of values with a much smaller set. Example: Source: Real numbers in the [-10.0, 10.0] Quantization Q(x) = ⎣x+0.5⎦ [-10.0, -10.0] {-10, -9, …, -1, 0, 1, 2, …, 9, 10} Scalar vs. vector quantization Scalar: applied to scalars Vector: applied to vectors
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09: Scalar Quantization CSCI 6990 Data Compression
Vassil Roussev 1
CSCI 6990.002: Data Compression
09: Scalar Quantization
Vassil Roussev<vassil@ cs.uno.edu>
UNIVERSITY of NEW ORLEANSDEPARTMENT OF COMPUTER SCIENCE
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Quantization
Definition:The process of representing a large—possibly infinite—set of values with a much smaller set.Example:
Scalar vs. vector quantizationScalar: applied to scalarsVector: applied to vectors
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The Quantization Process
Two aspectsEncoder mapping
Map a range of values to a codewordIf source is analog A/D converterKnowledge of the source can help pick more appropriate ranges
Decoder mappingMap the codeword to a value in the rangeIf output is analog D/A converterKnowledge of the source distribution can help pick better approximations
Quantizer = encoder + decoder
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Quantization Example
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Quantizer Input-Output Map
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Quantization Problem Formulation
Input:X – random variablefX(x) – probability density function (pdf)
Output:{bi}i=0..M decision boundaries
{yi}i=1..M reconstruction levels
Discrete processes are often approximated by continuous distributions
E.g.: Laplacian model of pixel differenceIf source is unbounded, then first/last decision boundaries = ±∞
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Quantization Error
Mean squared quantization error
( )
( ) dxfyx
dxfxQx
X
M
i
b
b i
Xq
i
i∑∫
∫
=
∞
∞−
−
−=
−=
1
2
22
1
)(σ
iii bxbyxQ ≤<= −1iff)(
Quantization error is a.k.a.Quantization noiseQuantizer distortion
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Quantization Problem Formulation (2)
Bit rates w/ fixed-length codewordsR = ⎡log2M⎤E.g.: M = 8 R = 3
Quantizer design problemGiven:
input pdf fX(x) & MFind:
decision boundaries {bi} and
Reconstruction levels {yi}Such that:
MSQE is minimized
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Quantization Problem Formulation (3)
Bit rates w/ variable-length codewordsR depends on boundary selectionExample
∑ ==
M
i ii yPlR1
)(
∫−
=i
i
b
b Xi dxxfyP1
)()(
∑ ∫=−
=M
i
b
b Xii
i
dxxflR1
1
)(
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Quantization Problem Formulation (4)
Rate-optimization formulation
Given:
Distortion constraint σq2 ≤ D*
Find:{bi},{yi}, binary codes
Such that:R is minimized
Distortion-optimization formulation
Given:Rate constraint R ≤ R*
Find:{bi},{yi}, binary codes
Such that:
σq2 is minimized
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Uniform Quantizer
All intervals of the same sizeI.e., boundaries are evenly spaced (∆)Outer intervals may be an exception
ReconstructionUsually the midpoint is selected
Midrise quantizerZero is not an output level
Midtread quantizer Zero is an output level
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Midrise vs. Midtread Quantizer
Midrise Midtread
09: Scalar Quantization CSCI 6990 Data Compression