lecture 6 scalar and vector quantization -...

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 2013

Wen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2013 1 / 61

Lecture 6 Scalar and Vector Quantization Introduction

Quantization

Lossy compression method

Reduce distinct output values to a 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. ....

Objectives1 Minimize distortion s.t. number of reconstruction levels2 Minimize distortion s.t. entropy rate constraint (or vice versa)



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



Midrise vs. Midtread Quantizers

Midrise 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

Midrise 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 )

ZBld(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 usually of zero mean (as we shallshow later)

1Depend on distortion measure d(�) and source distribution p(f )Wen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2013 8 / 61


Scalar Quantizer


Lecture 6 Scalar and Vector Quantization Uniform Scalar Quantizer

Uniform Scalar Quantizer

Equal distances betweenadjacent boundaries andreconstruction values

bl � bl�1 = gl � gl�1 = q

where

bl = fmin + l � q

gl =bl + bl�1

2 Q(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

Quantizer distortion

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 =

BL, 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�12(f � g �l )p(f )df = 0

Nearest-neighbor condition

(b�l � g �l )2 = (b�l � g �l+1)2 (1)

) b�l =g �l + g

�l+1

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

Centroid condition

g �l =1

P(Bl )

Z b�l

b�l�1f p(f )df = E (FjF 2 Bl )| {z }

Conditional Mean

(2)

Eqs. (1) & (2) generally DO NOT have a closed-form solutionWen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2013 13 / 61


Properties of MMSE Scalar Quantizer

SymbolsF : quantizer inputG: quantizer outputQ = F � G: quantization error

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

G is orthogonal to (and uncorrelated with) Q, whereas F and Q arecorrelated

E (GQ) = 0,E (FQ) 6= 0Reduced signal variance

σ2G = σ2F � σ2Q

Equalized error contribution

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



MMSE Scalar Quantizers of Various Sources

All sources are with zero mean and unit variance2

Quantizer Design (y j= g j , xj = bj ) SNRU: Uniform, G: Gaussian, L: Laplacian, Γ: Gamma

2See Appendix for arbitrary PDFWen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2013 15 / 61

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

MMSE Scalar Quantizer: High Rate Approximation

High rate approx.: 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(b�, g�)∂gl

= 0)Z b�l

b�l�1(f � g �l )df = 0) g �l =

b�l + b�l�1

2| {z }High Rate





Dq ' ∑l2L

P(Bl )∆l

Z b�l

b�l�1(f � g �l )2df = ∑

l2LP(Bl )

∆2l12|{z}Dq,l

where ∆l = bl � bl�1Equalized error contribution

P(Bl )Dq,l =112P(Bl )"

∆2l#= constant

Special case: uniform quantizer and uniform source

P(Bl ) = P(Bl 0),∆l = ∆l 0 = ∆,Dq =∆2

12




Alternative expression (more useful)

Dq ' ∑l2LP(Bl )

∆2l12' ∑

l2L

pF (g �l )∆3l

12' σ2f 2

�2R ε2 (See Appendix)

where

ε2 =112

�Z fmax/σf

fmin/σf

3qpF 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



MMSE Scalar Quantizer of Various Sources

∆SNR = 6.02R (Uniform Source and Quantizer)�maxfSNRgSolid (PDF-opt. non-uniform quantizer) vs. Dashed (PDF-opt.uniform quantizer)

Uniform quant. becomes increasingly ine¢ cient with increasing RNon-uniform quant. attains a PDF-speci�c asymptote with increasingR



Companding: Compressing and Expanding

A technique for analyzingquantizers at high rate

Companding law c(x) de�nes Bl a

Interval intensity

dc(gl )dx

� 2xmaxL∆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)dx

dx = c(xmax)� c(xmin) = 2xmax



MMSE Scalar Quantizer Design Using Companding

MMSE scalar quantizer

minx2max3L2

Z xmax

xminp(x)

�dc(x)dx

��2dx| {z }

Dq

s.t.Z xmax

xmin

dc(x)dx

dx = 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 xmax

03pp(x)dx

3qp(x)

D�q '23L2

�Z xmax

0

3qp(x)dx

�3= σ2f 2

�2R ε2


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 g �l

b�l�1p(f )df �

Z b�l

g �lp(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�1p(f )df =

Z b�l

g �lp(f )df


Lecture 6 Scalar and Vector Quantization Optimal Scalar Quantizer

Optimal Scalar Quantizer

Nearest-neighbor condition (same as MMSE case and obvious)

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 (distortion measure dependent) 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 2013 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 iteratively1 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.)

Parameters to be determined: L and dc (x )dx

Assumptions � (1) high rate, (2) the value of L is arbitrary




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)dx

dx

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

minZ xmax

xminp(x) log2

dc(x)dx

dx + λ

Z xmax

xminp(x)

�dc(x)dx

��2dx �D 0

!Here we have assumed that L takes on its best value




Uniform quantizer attains minimum output entropy regardless ofPDF.

dc�(x)dx

=p2λ ln 2 = 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 amount at quantizer inputlog2 ∆ = h(Q) : average information loss due to quantization (oraverage information conveyed by quantization error)



Comparison with Rate-Distortion Bound

Memoryless Non-Gaussian Sources (D(R) has NO closed-form)

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 quantization ) vectorquantization can be bene�cial even for memoryless sources



Comparison with MMSE Scalar Quantizer

For a given distortion, maximum bit rate reduction

maxf∆Rg = 12log2

σ2fDq

ε2 �H�Q (x) =12log2

�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(MMSE Scalar Quantizer L=8)

For a given L, HQ (entropy of quantizer output symbols) can bereduced at the cost of σ2q . How?For a given σ2q , the entropy can be reduced by increasing L; likewise,for a given output entropy HQ , σ2q can be reduced by increasing L.



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 to occur than others

Parameters

Quantization levels LReconstruction vectors (or 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 criterionComplexity 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 ) =1N ∑N

n=1(fn � gl ;n)2


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 PointsLattice 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

Dq = E (dN (�!F ,Q(�!F )))

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

=L

∑l=1

P(Bl )Dq,l

whereDq,l =

ZBlpN (f j

�!F 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 V1 =�1 00 1

�vs. Hexagonal V2 =

� p3/2 01/2 1

�

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�Z

dN (f,Q(f))p(f)df�

= argmin�

∑P(Bl )ZBldN (f,Q(f))p(f j

�!F 2 Bl )df�

Optimal Solution1 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

�ZdN (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

ZB�ldN (f, g)p(f j

�!F 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


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 2013 47 / 61


Gaussian-Markov Source + 8-D VQ

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



Memoryless Laplacian + 8-D 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 an 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 (and uncorrelated with) Q.

E (GQ) = E (G(F � G))= EGEF jG((G(F � G)))= EG(GEF jG(F )� G2)= EG(G2 � G2)= 0

Note that E (FQ) = E (F (F � G)) = E (F 2)� E (FG) =E (F 2)� E (G2) 6= 0




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)




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 )

) ddf 0CF 0(f

0) =ddf 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 2013 56 / 61




Dq ' ∑l2L

P(Bl )∆2l12

= ∑l2L

p(g �l )∆3l

12= ∑

l2L

α3l12

whereαl =

3qp(g �l )∆l

Observe that

∑l2L

αl = ∑l2L

3qp(g �l )∆l '

Z fmax

fmin

3qp(f )df = constant

MMSE Scalar Quantizer

min ∑l2L

α3l12s.t. ∑

l2Lαl = C

) α�l = constant =1L

Z fmax

fmin

3qp(f )df



MMSE Scalar Quantizer High Rate Approximation

Given

α�l =1L

Z fmax

fmin

3qp(f )df


Dq ' ∑l2L

(α�l )3

12= L

(α�l )3

12

=1L2112

�Z fmax

fmin

3qp(f )df

�3= 2�2R

112

�Z fmax

fmin

3qp(f )df

�3where L = 2R



MMSE Scalar Quantizer High Rate Approximation

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)

) ddf 0CF 0(f

0) =ddf 0CF (σF f

0)

) pF 0(f0) = σf pF (σf f

0) or pF (f ) =1

σfpF 0(

fσf)


Dq ' 2�2R

12

�Z fmax

fmin

3qpF (f )df

�3=2�2R

12

Z fmax

fmin

3

s1

σfpF 0(

fσf)df

!3= σ2f 2

�2R ε2

where ε2 = 112

�R fmax/σffmin/σf

3ppF 0(f )df

�3Wen-Hsiao Peng, Ph.D (NCTU CS) MAPL May 2013 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

= glZ 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 Communications2 Jayant, N. S., et. al - Digital Coding of Waveforms3 B. Girod, http://www.stanford.edu/class/ee368b/Handouts/


lecture 6 scalar and vector quantization -...

Documents