Lecture 6 Scalar and Vector Quantizationmapl.nctu.edu.tw/course/VC_2009/files/06.Lecture 6...Lecture 6 Scalar and Vector Quantization MMSE Scalar Quantizer High Rate Approximation

Lecture 6 Scalar and Vector Quantization

Wen-Hsiao Peng, Ph.D

Multimedia Architecture and Processing Laboratory (MAPL)Department of Computer Science, National Chiao Tung University

May 2008

Wen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2008 1 / 61

Lecture 6 Scalar and Vector Quantization Introduction

Quantization

Lossy Compression Method

Reduce distinct output values to a much smaller setMap an input value/vector to an approximated value/vector

Approaches

Scalar Quant. - quantize each sample separately

Uniform vs. Non-uniformMSE vs. MAE vs. ....

Vector Quant. - quantize a group of samples jointly

Uniform (Lattice Quantizer) vs. Non-uniformMSE vs. MAE vs. ....

Design Objectives

1 Minimize Distortion s.t. Fixed Number of Reconstructions2 Minimize Entropy Rate s.t. Distortion Constraint (Minimize Distortions.t. Entropy Rate Constraint)



Example

Scalar Quantization

Quantization Levels L = 2R

Reconstruction Values gl , l = 1, 2, ..., LBoundary Values bl , l = 0, 1, ..., LQuantizer Mapping Function

Q(f ) = gl if f 2 Bl = [bl�1, bl )



Quantizer Mapping Function

Q(f ) = gl if f 2 Bl Q(f ) =jf�fminq

k�q+ q

2+f min

Non-uniform Quantizer Uniform Quantizer



Uniform vs. Non-uniform



Midright vs. Midtread Quantizers

Midright Uniform Quantizer (Even L)

"0" is a decision boundary

Midtread Uniform Quantizer (Odd L)

"0" is a reconstruction levelBetter for small L in video/image coding

Midright vs. Midtread (for Symmetric Distributions)



Quantizer Distortion

Dq = σ2q(granular )| {z }Our Focus

+ σ2q(overload)| {z }0 for bounded inputs




Quantizer Distortion Dq1

Dq = E (d(F ,Q(F )))

= ∑l2L

�Zf 2Bl

d(f ,Q(f ))p(f )df

�= ∑

l2LP(Bl )

Zf 2Bl

d(f ,Q(f ))p(f jf 2 Bl )df| {z }Dq,l

Example: d(F ,Q(F )) = (F �Q(F ))2

Dq = E (d(F ,Q(F )) = E�(F �Q(F ))2

�| {z }

MSE

= σ2q

where Quant. Error F �Q(F ) is of zero mean1Depend on distortion measure d(�) and source distribution p(f )



Scalar Quantizer


Lecture 6 Scalar and Vector Quantization Uniform Scalar Quantizer

Uniform Scalar Quantizer

Equal Quantization Stepsize

bl � bl�1 = gl � gl�1 = q

where

bl = fmin + l � q

gl =bl + bl�1

2Q(f ) =

�f � fminq

��q+q

2+f min| {z }

Closed�form


Lecture 6 Scalar and Vector Quantization Uniform Scalar Quantizer

Uniform Quantizer Optimized for Uniform Distribution

Uniform Distribution

p(f ) =

�1/B f 2 (fmin, fmax)0 otherwise

where B = fmax � fmin

Distortion Measure

d(F ,Q(F )) � (F �Q(F ))2


Dq = ∑l2LP(Bl )Dq,l = Dq,l =

q2

12= σ2f 2

�2R

SNR = 10 logσ2fDq

= (20 log 2)R = 6.02R

where

Dq,l =q2

12, q =

B

L, L = 2R , Signal Variance σ2f = B

2/12


Lecture 6 Scalar and Vector Quantization MMSE Scalar Quantizer

Minimum Mean Square Error Scalar Quantizer

Distortion Measure

d(F ,Q(F )) � (F �Q(F ))2

Objective

(b�, g�) = arg minfb,gg

∑l2L

Z bl

bl�1(f � gl )2p(f )df

!| {z }

J(b,g)

whereb = (b0, b1, ..., bL)

T , g = (g1, g2, ..., gL)T

Necessary Conditions

rJ(b�, g�) = 0) (1) ∂J(b�, g�)/∂bl = 0(2) ∂J(b�, g�)/∂gl = 0



MMSE Scalar Quantizer

Solution

(1) ∂J(b�, g�)/∂bl = (b�l � g �l )2 p(b�l )� (b�l � g �l+1)2p(b�l ) = 0

(2) ∂J(b�, g�)/∂gl = �Z b�l

b�l�1

2(f � g �l )p(f )df = 0

Nearest-Neighbor Condition

b�l =g �l + g

�l+1

2) Bl = ff : d(f , gl ) � d(f , gl 0), 8l 0 6= lg (1)

Centroid Condition

g �l =1

P(Bl )

Z b�l

b�l�1

f p(f )df = E (FjF 2 Bl )| {z }Conditional Mean

(2)

Eqs. (1) & (2) generally DO NOT have a closed-form solution



Properties of MMSE Scalar Quantizer

Symbols

F : original random variableG: quantized random variableQ = F � G: quantization error

G is unbiased estimator of FE (G) = E (F ),E (Q) = 0

G is orthogonal to Q, whereas F is correlated to QE (GQ) = 0,E (FQ) 6= 0

Reduced signal variance

σ2G = σ2F � σ2Q

Equalized error contribution

P(Bl )Dq,l =DqL, 8l 2 L



MMSE Scalar Quantizer of Various Sources

All sources are with zero mean and unit variance2

Quantizer Design (y j= g j , xj = bj ) SNR

U: Uniform, G: Gaussian, L: Laplacian, Γ: Gamma2See Appendix for arbitrary PDFWen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2008 15 / 61

Lecture 6 Scalar and Vector Quantization MMSE Scalar Quantizer High Rate Approximation

MMSE Scalar Quantizer High Rate Approximation

High Rate Approximation: p(f ) is approximately at in all intervals Bl

Dq = ∑l2L

Z bl

bl�1(f � gl )2 p(f )|{z} df

' ∑l2Lp(gl )| {z }

Z bl

bl�1(f � gl )2df

= ∑l2L

P(Bl )∆l| {z }

Z bl

bl�1(f � gl )2df , where ∆l = bl � bl�1

Centroid Condition

∂Dq∂gl

= 0)Z bl

bl�1(f � gl )df = 0) g �l =

bl + bl�12| {z }

High Rate



MMSE Quantizer Distortion High Rate Approximation


Dq ' ∑l2L

P(Bl )∆l

Z bl

bl�1(f � g �l )2df = ∑

l2LP(Bl )

∆2l12|{z}Dq,l

where ∆l = bl � bl�1Equalized Error Contribution

P(Bl )Dq,l =1

12P(Bl )"

∆2l#= constant

Special Case: Uniform Quantizer and Uniform Source

P(Bl ) = P(Bl 0),∆l = ∆l 0 = ∆,Dq =∆2

12



MMSE Quantizer Distortion High Rate Approximation

Alternative Expression (More Useful)

Dq ' ∑l2LP(Bl )

∆2l12' ∑

l2L

pF (g�l )∆

3l

12' σ2f 2

�2Rε2 (See Appendix)

where

ε2 =1

12

�Z fmax/σf

fmin/σf

3

qpF 0(f )df

�3pF 0(f ) = σf pF (σf f ) is PDF of an unit variance source

Uniform Quantizer/Source vs. Non-uniform Quantizer/Source

Uniform : SNR ' 10 log σ2fDq

= 10 log 22R = 6.02R

Non-uniform : SNR ' 10 log σ2fDq

= 6.02R � 20 log ε




∆SNR = 6.02R (Uniform Source and Quantizer)�maxfSNRgSolid (Non-uniform Quantizer) vs. Dashed (Uniform Quantizer)

Uniform quantizer becomes increasingly ine�cient with increasing R

Non-uniform quantizer attains an asymptote with increasing R



Companding: Compressing and Expanding

A technique for analyzing

quantizers at high rate

Companding Law c(x) de�nes Bl a

Interval Intensity

dc(gl )

dx� 2xmax

L∆l

where

c(xmax) = xmax,c(0) = 0,c(xmin) = xmin

aQuantizers designed by c(x) is nolonger common



Companding: Compressing and Expanding

Distortion Measure

d(F ,Q(F )) � (F �Q(F ))2

Quantizer Distortion (in terms of c(x))

Dq ' ∑l2LP(Bl )

∆2l12

' x2max3L2 ∑

l2Lp(gl )∆l

�dc(gl )

dx

��2' x2max

3L2

Z xmax

xminp(x)

�dc(x)

dx

��2dx

Constraint Z xmax

xmin

dc(x)

dxdx = c(xmax)� c(xmin) = 2xmax



MMSE Scalar Quantizer Design Using Companding


minx2max3L2

Z xmax

xminp(x)

�dc(x)

dx

��2dx| {z }

Dq

s.t.Z xmax

xmin

dc(x)

dxdx = 2xmax

Lagrange Multiplier (�nd dc�(x)/dx = g �)

g � = argmin

�Z xmax

xminp(x) (g)�2 dx + λ

�Z xmax

xmingdx � 2xmax

��Optimal Companding Law c�(x)

g � =dc�(x)

dx=

xmaxR xmax0

3pp(x)dx

3

qp(x)

D�q '2

3L2

�Z xmax

0

3

qp(x)dx

�2= σ2f 2

�2Rε2


Lecture 6 Scalar and Vector Quantization MMAE Scalar Quantizer

Minimum Mean Absolute Error Quantizer

Distortion Measure

d(F ,Q(F )) � jF �Q(F )j

Objective

(b�, g�) = arg minfb,gg

∑l2L

Z bl

bl�1jf � gl j p(f )df

!| {z }

J(b,g)

whereb = (b0, b1, ..., bL)

T , g = (g1, g2, ..., gL)T

Necessary Conditions

rJ(b�, g�) = 0) (1) ∂J(b�, g�)/∂bl = 0(2) ∂J(b�, g�)/∂gl = 0


Lecture 6 Scalar and Vector Quantization MMAE Scalar Quantizer

Minimum Mean Absolute Error Quantizer

Solution

(1) ∂J(b�, g�)/∂bl = jb�l � g �l j p(b�l )� jb�l � g �l+1j p(b�l ) = 0

(2) ∂J(b�, g�)/∂gl =Z gl

bl�1p(f )df �

Z bl

glp(f )df = 0 (Appendix)

Nearest Neighbor Condition (Same as MMSE Quantizer)

jb�l � g �l j p(b�l )� jb�l � g �l+1j p(b�l ) = 0) b�l =g �l + g

�l+1

2

Generalized Centroid ConditionZ g �l

b�l�1

p(f )df =Z b�l

g �l

p(f )df


Lecture 6 Scalar and Vector Quantization Optimal Scalar Quantizer

Optimal Scalar Quantizer

Nearest-Neighbor Condition (same as MMSE case)

B�l = ff : d(f , gl ) � d(f , gl 0), 8l 0 6= lg

Generalized Centroid Condition3

g �l = argminglE (d(F , g)jF 2 Bl )

Remarks

Nearest-Neighbor condition is NOT a�ectedCentroid SHALL be adapted to Distortion MeasureSource distribution MUST be knownReconstruction level is indexed by a �xed-length code

3Solution depends on distortion measureWen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2008 25 / 61


Lloyd-Max Algorithm

Optimal quantizer design based on training data

Applicable when source distribution is unknownUse sample averages to replace expectations

Update reconstruction and boundary values iteratively

1 Initialization

Choose initial reconstruction valuesCalculate initial boundary values (Nearest Neighbor)Calculate initial distortion

2 Iterations

Find new reconstructions (Centroid, Sample Mean)Find new boundaries (Nearest Neighbor)Calculate new distortionRepeat till distortion converges



Lloyd-Max Algorithm


Lecture 6 Scalar and Vector Quantization Entropy Constrained Optimal Scalar Quantizer

Entropy Constrained Optimal Scalar Quantizer

Minimize Quantizer Output Entropy s.t. Quantizer Distortion in MSE

minHQ = �∑l2LP(Bl ) log2 P(Bl )| {z }

Quantizer Output Entropy

s.t. Dq =x2max3L2

Z xmax

xminp(x)

�dc(x)

dx

��2dx| {z } = D

Quantizer Distortion with L Levels (High Rate Approx.)

Assumptions

High Rate ApproximationAn arbitrary number of quantization levels L




Quantizer Output Entropy (in terms of c(x))

HQ = �∑l2LP(Bl ) log2 P(Bl ) (3)

' �∑l2Lp(gl )∆l log2 (p(gl )∆l )

' �Z xmax

xminp(x) log2 p(x)dx| {z }

h(X )=E (� log2 p(x)) di�erential entropy

+ log2L

2xmax+Z xmax

xminp(x) log2

dc(x)

dxdx

Optimal quantizer is equivalent to �nding g � = dc�(x)/dx such that

minZ xmax

xminp(x) log2

dc(x)

dxdx + λ

Z xmax

xminp(x)

�dc(x)

dx

��2dx �D 0

!Wen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2008 29 / 61



Uniform Quantizer attains minimum output entropy regardless of PDF

dc�(x)

dx=p2λ = Constant

c�(x) = x given c(xmax) = xmax, c(0) = 0 (4)

Optimal Output Entropy (From Eqs. (3) & (4))

H�Q(x) = h(X )� log∆ = h(X )� 12log2 (12Dq)

Information-theoretical Interpretation

h(X ) average information at quantizer inputlog2 ∆ = h(Q) average information loss due to quantization



Comparison with Rate-Distortion Bound

Memoryless Non-Gaussian Sources (No closed-form D(R))

LD(R)| {z }Lower Bound

� D(R) � D(R)G| {z }Upper Bound

where distortion D is in MSE and

LD(R) = (2πe)�12�2[R�h(X )],

LR(D) = h(X )� 12log2 2πeD

For a given distortion

H�Q(x)�L R(D) = 0.255bits

Lower e�ciency is caused by scalar (or memoryless) quantization



Comparison with MMSE Scalar Quantizer

For a given distortion, maximum bit rate reduction

maxf∆Rg = 1

2log2

σ2fDq

ε2 �H�Q(x) =1

2log2

�12σ2f ε2

�� h(X )

For a given rate, maximum SNR gain

maxf∆SNRg = SNREC � SNRMMSE = 6.02maxf∆Rg dB




maxf∆RgRate reduction with entropy constrained quantizerA higher number of quantization levels

∆Rpdf�optRate reduction by entropy encoding MMSE quantizer outputsA �xed number of quantization levels




Gaussian Source Even L vs. Odd L (Entropy Constrained)

(MMSE Scalar Quantizer L=8)



Summary

∆Rpdf�optNOT the best value achievable by entropy coding

maxf∆Rg = (3/2)∆Rpdf�optAchieved with uniform quantizer and more quantization levelsAdditional quantization levels are used for outer part of PDFFall short of R-D bound by 0.255bits (1.53dB) at high ratesApproach R-D bound at low rates



Vector Quantizer


Lecture 6 Scalar and Vector Quantization Vector Quantization

Vector Quantization

Why Vector Quantization?

Samples in a block are correlatedSome block patterns are more likely than others

Parameters

Quantization Levels LReconstruction Vectors (Codewords) gl , l = 1, 2, ..., LPartition Regions Bl , l = 1, 2, ..., LQuantizer Mapping Function

gl = Q(f) if f 2 Bl

wheref = [f1, f2, ..., fN ]

T , gl = [gl ;1, gl ;2, ..., gl ;N ]T

Bits/Sample R = 1N dlog2 Le


Lecture 6 Scalar and Vector Quantization Nearest-Neighbor Vector Quantizer

Nearest-Neighbor Vector Quantizer

Quantized vector gl determined by Nearest Neighbor criterion

Complexity (increases exponentially with vector size N)

Operations NL (or N2NR ), storage NL (or N2NR )

Bl = ff 2 RN : dN(f, gl ) � dN(f, g0l ), 8l 0 6= lg

dN(f, gl ) =1

N ∑N

n=1(fn � gl ;n)


Lecture 6 Scalar and Vector Quantization Lattice Vector Quantizer

Lattice Vector Quantizer

Realization of Uniform Vector Quantizer

All partitions have same shape and size (good for uniform source)Closed-form quantizer mapping function (low complexity)

Reconstruction Vectors gl are Lattice Points (or Lattice Coset)

Lattice Λ (de�ned by a set of basis vectors fvng)

gl =N

∑n=1

ml ;nvn = [V]ml ,

whereml 2 Zn

[V] = [v1, v2, ..., vN ] is Lattice Generating Matrix

Determination of Quantized Vector

1 m =[V]�1f, where m 2 Rn is a real index vector2 Evaluate distortions of bm = dme or bmc, where bm 2 Zn




Vector Quantizer Distortion (Same as Scalar Quantizer)

Dq = E (dN(�!F ,Q(�!F )))

=ZBpN(f)dN(f,Q(f))df

=L

∑l=1

P(Bl )Dq,l

where

Dq,l =ZBlpN(f jf 2 Bl )dN(f, gl )df

Example: Lattice Quantizer for Uniform Distribution

Dq =L

∑l=1

P(Bl )Dq,l = Dq,l and Dq,l =1

jdet[V]j

ZBldN(f, 0)df



Lattice Vector Quantizer

Rectangular vs. Hexagonal

Better space packing makes VQ outperform SQ even with i.i.d source

dN (f, gl ) = maxf2Bl

dN (f, gl )

Rectangular Lattice: 2 uniform scalar quantizers


Lecture 6 Scalar and Vector Quantization Optimal Vector Quantizer

Optimal Vector Quantizer

Objective

(B�1 ,B�2 , ...,B�L; g�1, g�2, ..., g�L)= argminE (d(

�!F ,Q(�!F )))

= argmin

�ZBdN(f,Q(f))p(f)df

�= argmin

�∑P(Bl )

ZBldN(f,Q(f))p(f jf 2 Bl )df

�Optimal Solution

1 Given�g�1, g

�2, ..., g

�L

�, what would be optimal (B1,B2, ...,BL)?

2 Given�B�1,B�2, ...,B�L

�, what would be optimal (g1, g2, ..., gL)?




Given (g�1, g�2, ..., g

�L), what would be optimal (B1,B2, ...,BL)?

Q�(f) = arg minQ(f)=fg�1 ,g�2 ,...,g�Lg

�ZBdN(f,Q(f))p(f)df

�= arg min

Q(f)=fg�1 ,g�2 ,...,g�LgdN(f,Q(f))

B�l = ff : dN(f, g�l ) � dN(f, g�l 0), 8l 0 6= lg (Nearest Neighbor)

Given (B�1 ,B�2 , ...,B�L), what would be optimal (g1, g2, ..., gL)?

g�l = argming

�∑P(B�l )

ZB�ldN(f, g)p(f jf 2 B�l )df

�= argmin

g

ZBldN(f, g)p(f jf 2 B�l )df

= argmingE (dN(

�!F , g)j�!F 2 B�l ) (Centroid)




Nearest-Neighbor and Centroid are Necessary Conditions (Candidates)



Lloyd-Max Algorithm



Lloyd-Max Algorithm


Lecture 6 Scalar and Vector Quantization Entropy Constrained Vector Quantizer

Entropy Constrained Vector Quantizer

ObjectiveminDq s.t. �∑

l2LP(Bl ) log2 P(Bl )| {z }Entropy Rate

= RN

Solution (Necessary Conditions)

Lagrangian

(B�1,B�2, ...,B�L; g�1, g�2, ..., g�L)

= argmin

Dq + λ�

�∑l2LP(Bl ) log2 P(Bl )� RN

!!| {z }

J(B1,B2,...,BL;g1,g2,...,gL)4

λ� must be chosen such that

�∑l2LP(B�l ) log2 P(B�l ) = RN

4Lloyd-Max Algorithm is still applicable by replacing Dq with J(�)Wen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2008 47 / 61


Gaussian-Markov Source + 8D VQ

http://www.stanford.edu/class/ee368b/Handouts/06-Quantization.pdf



Memoryless Laplacian + 8D VQ

http://www.stanford.edu/class/ee368b/Handouts/06-Quantization.pdf


Lecture 6 Scalar and Vector Quantization Appendix

Appendix



Properties of MMSE Quantizer

G is unbiased estimator of F

E (G) = ∑l

P(Bl )gl

= ∑l

P(Bl )�Z

Blf p(f jf 2 Bl )df

�= ∑

l

ZBlf p(f )df

=ZBf p(f )df = E (F )

Q = F � G is of zero mean




G is orthogonal to Q

E (GQ) = E (G(F � G)) = ∑l

P(Bl )ZBlgl (f � gl ) p(f jf 2 Bl )df| {z }

=0

= 0

For an MMSE quantizer

gl = E (f jf 2 Bl ) =ZBlf p(f jf 2 Bl )df

ThusZBlgl (f � gl ) p(f jf 2 Bl )df = glE (f jf 2 Bl )| {z }

=gl

� g2l = 0, 8l 2 L

Note that E (F (F � G)) = E (F2)� E (FG) = E (F2)� E (G2) 6= 0Wen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2008 52 / 61



Reduced signal variance

σ2G = σ2F � σ2Q

By de�nition

σ2Q = E ((F � G)2)= E (

�(F�µF )�

�G�µG

��2)

= σ2F + σ2G � 2E ((F�µF )�G�µG

�)

= σ2F + σ2G � 2(E (FG)�µFµG)

= σ2F + σ2G � 2(E (G2)�µ2G)

= σ2F � σ2G

where µF = µG = E (G) and E (FG) =E (G2)





Dq ' ∑l2L

P(Bl )∆2l12

= ∑l2L

p(g �l )∆3l

12= ∑

l2L

α3l12

where

αl =3

qp(g �l )∆l

Observe that

∑l2L

αl = ∑l2L

3

qp(g �l )∆l '

Z fmax

fmin

3

qp(f )df = constant


min ∑l2L

α3l12s.t. ∑

l2Lαl = C

) α�l = constant =1

L

Z fmax

fmin

3

qp(f )df




Given

α�l =1

L

Z fmax

fmin

3

qp(f )df


Dq ' ∑l2L

(α�l )3

12= L

(α�l )3

12

=1

L21

12

�Z fmax

fmin

3

qp(f )df

�3= 2�2R

1

12

�Z fmax

fmin

3

qp(f )df

�3where L = 2R




Given F 0 = F/σf is an unit variance source

P(F 0 � f 0)| {z }CF0 (f

0)

= P(F � σf f0)| {z }

CF (σf f 0)

) d

df 0CF 0(f

0) =d

df 0CF (σF f

0)

) pF 0(f0) = σf pF (σf f

0) or pF (f ) =1

σfpF 0(

f

σf)


Dq ' 2�2R

12

�Z fmax

fmin

3

qpF (f )df

�3=2�2R

12

Z fmax

fmin

3

s1

σfpF 0(

f

σf)df

!3= σ2f 2

�2Rε2

where ε2 = 112

�R fmax/σffmin/σf

3ppF 0(f )df

�3Wen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2008 56 / 61



Given a source F 0= (F � µF ) /σF of zero mean and unit variance

P(F 0 � f 0)| {z }CF0 (f

0)

= P(F � σF f0 + µF )| {z }

CF (σF f 0+µF )

) d

df 0CF 0(f

0) =d

df 0CF (σF f

0 + µF )

) pF 0(f0) = σFpF (σF f

0 + µF ) or pF (f ) =1

σFpF 0(

f � µFσF

)

where

pF 0(f0) and pF (f ) are PDF of F 0 and F , respectively




Optimal Scalar Quantizer for F 0

(1) ebl = egl+egl+12

(2) egl = E (F 0jF 0 2 eBl ) = R eblebl�1 f 0 pF0 (f 0)df 0R eblebl�1 pF0 (f 0)df 0Optimal Scalar Quantizer for F

(1) bl =gl+gl+1

2 = σFebl + µF

(2) gl = E (FjF 2 Bl ) =R blbl�1

f pF (f )dfR blbl�1

pF (f )df= σFegl + µF




Let

f 0 =f � µF

σFAssume ebl = bl � µF

σFThen optimal gl for F can be obtained as follows:

gl = E (FjF 2 Bl ) =R blbl�1

f pF (f )dfR blbl�1

pF (f )df=

R blbl�1

f 1σFpF 0(

f�µFσF

)dfR blbl�1

1σFpF 0(

f�µFσF

)df

=

R eblebl�1 (σF f 0 + µF ) pF 0(f0)df 0R eblebl�1 pF 0(f 0)df 0 = σFegl + µF (5)

From Eq. (5), the assumption ebl = (bl � µF ) /σF is justi�ed

bl = (gl + gl+1) /2 = σFebl + µFWen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2008 59 / 61


MMAE Scalar Quantizer

Observe thatZ bl


=Z gl

bl�1(gl � f ) p(f )df +

Z bl

gl(f � gl ) p(f )df

= gl

Z gl

bl�1p(f )df �

Z gl

bl�1f p(f )df +

Z bl

glf p(f )df � gl

Z bl

glp(f )df

Then

∂

∂glJ(b�, g�) =

∂

∂gl

�Z bl


�=

Z gl

bl�1p(f )df �

Z bl

glp(f )df



References

1 Y. Wang, et. al - Video Processing and Communications

2 Jayant, N. S., et. al - Digital Coding of Waveforms

3 B. Giord, http://www.stanford.edu/class/ee368b/Handouts/


Documents

Lecture 6 Scalar and Vector Quantizationmapl.nctu.edu.tw/course/VC_2009/files/06.Lecture 6...Lecture 6 Scalar and Vector Quantization MMSE Scalar Quantizer High Rate Approximation