Quantization Heejune AHN Embedded Communications Laboratory Seoul National Univ. of Technology Fall 2013 Last updated 2013. 9. 24.

Quantization

Heejune AHNEmbedded Communications Laboratory

Seoul National Univ. of TechnologyFall 2013

Last updated 2013. 9. 24

Heejune AHN: Image and Video Compression p. 2

Outline

Quantization Concept Quantizer Design Quantizer in Video Coding Vector Quantizer Rate-distortion optimization Concept

Lagrangian Method

Appendix Detail for Lloyd-max Quantization Algorithm


1. Quantization

Reduce representation set Nlarge points ([0 ~ Nlarge]) to Nsmall points ([0, … Nsmall]) Inevitable when Analog to Digital Conversion Not Reversible !

Original

(digital/analog)

ADC/Quant

Quanitzed

Levels

ReScale


Scalar Quantizer

Quantize a single value (single input & single output) In contrast to VQ (Vector Quantizer, to be discussed later)

Mapping function {[xi-1, xi) => yi} Decision levels to

reconstruction level

Quantizer

input

Quantizer

output

xi-1 xi

yi

Output : {yi}

Input : x

Q q

Q-1

x

q x̂


Quantizer’s Effects

+ Bit Rate Reduction Bits required to represent the values: (E1bits=> E2 bits)

• Nlarge (= 2E1)points ([0 ~ N]) to Nsmall (= 2E2)points ([0, … Nsmall])

• Bit reduction = E2 - E1 = log2 (NLarge/Nsmall)

- Distortion e = yi - x (in [xi-1, xi)) MSE (mean squared error)

Trade-off problem

1 2

1 2

1 12

1 2 1 20 01 2

1ˆMSE= , ,

N N

n n

x n n x n nN N


2. Quantization Design

Distortion Quantization error = difference original (x) and quantized (y i) Distortion Measure

Quantizer design parameters Step size (N) decision level (xi)

reconstruction level (yi) Distortion depends on Input signal distribution (f(x))

Optimization Problem How to set the ranges and reconstruction values, i.e., {yi & [xi-1,

xi)} for i =0, … N, so that the resulting MSE is minimized.

N

i

x

x i

i

iX

dxxfyxD1

2

1

)()(


Optimal Quantizer

Solutions Note performance depends on the input statistics.

Simpler design • Uniform (Fixed quant step) quantizer • With/Without dead-zone near zero

Lloyd-Max algorithm• a optimal quantizer design algorithm• More levels for dense region, Less levels for sparse region.

(non-linear quantizer)


Uniform Quantizers

Properties xi and yi are spaced evenly (so, uniform), with space Δ (step size) Mid-tread (odd # of recon. levels) vs Mid-rise (no zero recon. ) mid-tread is more popular than mid-rise.

∆ 2∆ 3∆ Input

-3∆ -2∆ -∆

Reconstruction3.5∆

2.5∆

1.5∆0.5 ∆

-0.5∆

-1.5∆

-2.5∆-3.5∆

Uniform Midrise Quantizer

-2.5∆ -1.5∆ -0.5∆

Reconstruction3∆

2∆

∆

-∆

-2∆-3∆

Uniform Midtread Quantizer

0.5∆ 1.5∆ 2.5∆ Input

What will be the mathematical representation for quant/inverse?



Uniform Quantizer Performance

xA A

A21

)(xf X

For Uniform dist. Input, Uniform quantizer is optimal Can you prove this?

Uniform Input X ~ U(-A,A)

Error signal e = y – x ~ U(-Δ/2, Δ/2)


Signal Power (variance)

MSE

SNR

Note If one bit added, Δ is halved, noise variance reduces to 1/4, SNR

increases by 6 dB. SNR does not depends on input range, MSE does.

2 , 2 )2(3

12

12

1

1222

2

222r

xr

rN

A

N

AD

bitdB

dBrD

dBSNR rx

/6

)( 02.6 )2( log 10 log 10)( 22

3

22 Ax


Non-uniform Quantizer

Issue Many data are non-uniformly distributed and even unbounded

Concept General Intuition: Dense Levels for Dense Input region Boundness: Use saturation Post filter for unbounded input Generally no closed-form expression for MSE.

• Range and Reconstruction Level is cross-related


Lloyd-Max Algorithm

Algorithm Iteration algorithm with decision level (xi) and recon. level (yi) Run until code book does not change any more Step 1: find Near Region Ri((xi-1 , xi])

• Ri = {x | d(x,yi) <= d(x,yj), for all j =\= I }

Step 2: find level yi

• yi = E[X| x in Ri ]

최근린 분할 무게중심 계산Cm Cm+1

{Ri}mcentroid

calculation range

update


L

k

t

t x

k

k

k dxxfyxyxE1

22 1

)()(])'[(

pdfinput on the depending ]|[)(

)(

2

)(

1

1

1

kt

t x

t

t x

k

kkk

xxEdxxf

dxuxfy

yyt

k

k

k

k

where Tk is the k th interval [tk,tk+1)

0

kk yt

Lloyd-Max Algorithm Minimize mean squared error.


Example

Quantization and Quality (Lloyd-Max applied) Original (8bpp) vs 6 bpp (spatial domain)


Original (8bpp) vs 4 bpp (spatial domain)


Original (8bpp) vs 2 bpp (spatial domain)


High Density Quantization

,x

iR

conditional pdf of

in the interval iii RRyx 1in )(

0][][ 1 Nrr RPRP

1212)(

1)(

])[()(])ˆ[(])ˆ[(

22/

2/

22

222

ii

ii

iiii

RPdeeRP

RyXERPRXXEEXXED

2,

2x

)(xf X

High Quality Condition Observation

• Slow change (Constant) over the quantization region

• Approximate with Uniform distribution


Quantization in Video Coding

UTQ (uniform Threshold Quantization) often is used Difficult in adaptive design (a LM quantizer design) Video quantizers are high quality ones

Quantization in VC Input signal: quantize the (DCT) coefficients Parameters

• q (Δ): quantization scale factor ( 2 ~ 62, usu. q = 2QUANT, 1 ~31 )

• Index (Level) : index for the reconstruction values

• Note: index, not reconstruction values are transmitted E.g.

• Index: I (n,m) = floor(F(n,m)/q) = floor(F(n,m)/2QUANT)– i.e. [i-1, i) *q => i

• Reconstruction: F’(n.m) = (I(n,m) + ½) x q = (2*I(n,m) +1)xQUANT – i.e. i => (i +1/2) q (i.e, middle of range [i -1, i)*q)

• Note: with QUANT, no 1/2 in expression.


Uniform Threshold Quantizer

Threshold (dead zone) for reducing dct coeff. Intra DC (zero threshold), Intra AC, inter DC, AC (non-zero) 2*Q(UANT) = q

Quantizer input

Quantizer output

q/2

-q/2

q 2q

-q-2q

-q

-2q

-3q

2q

3q

q

3q

-3qQuantizer

input

Quantizer output

2Q 3Q-2Q

-3Q

-Q

-3Q

-5Q

3Q

5Q

q

5Q

-5Q

AC (With Threshold)DC (without Threshold, q = 8 )

dead zone


3. Quantization in Video Coding

Applied on the Transform Coefficients Key Throttle for Bit-Rate and Quality

Very Important Trade-off in Video coding

Estimation

(MC, IntraP)Xorig

Transform

(e.g. DCT)

e = Xorig - Xest Y = T(e)

Quantization

Entrophy

Coding

Yq = Q(Y) VLC(Yq)


Example #1

Example #2: Comparison with Coarse and Fine Quant See Textbook Table 7.4

185 3 1 1 -3 2 -1 0

1 1 -1 0 -1 0 0 1

0 0 1 0 -1 0 0 0

1 1 0 -1 0 0 0 -1

0 0 1 0 0 0 -1 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

Q(8)

1480 26.0 9.5 8.9 -26.4 15.1 -8.1 0.3

11.0 8.3 -8.2 3.8 -8.4 -6.0 -2.8 10.6

-5.5 4.5 9.0 5.3 -8.0 4.0 -5.1 4.9

10.7 9.8 4.9 -8.3 -2.1 -1.9 2.8 -8.1

1.6 1.4 8.2 4.3 3.4 4.1 -7.9 1.0

-4.5 -5.0 -6.4 4.1 -4.4 1.8 -3.2 2.1

5.9 5.8 2.4 2.8 -2.0 5.9 3.2 1.1

-3.0 2.5 -1.0 0.7 4.1 -6.1 6.0 5.7

Zig-zag scanTransformed 8x8 block


DCT Coef. Quantization

Human visual weight coarse quant to High frequency components NDZ-UTQ to Intra DC, DZ-UTQ to all others Intra block Case

• I (n,m) = round(F(n,m)/(q*W(n,n))/8 )

• F’(n.m) = (I(n,m) + ½) x q x W(n,m)/8

8 16 19 22 26 27 29 34

16 16 22 24 27 29 34 37

19 22 26 27 29 34 34 38

22 22 26 27 29 34 37 40

22 26 27 29 32 35 40 48

26 27 29 32 35 40 48 58

26 27 29 34 38 46 56 69

27 29 35 38 46 56 69 83

intra

16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16

16 16 16 16 16 16 16 16

inter


Quantizer Example

MPEG-4 example Inverse Quantization only Defined

if LEVEL = 0, REC = 0Else If QUANT is ODD

|REC| = QUANT *(2 |LEVEL| +1)Else

|REC| = QUANT *(2|LEVEL| +1) -1

• + 1 : Dead-zone • (sorry, I cannot explain why odd/even)

Quantization is not defined • Why?

Examples• Q= 4

31

23

15

7

-7

-15

-23

-31

27

19

11

0

-11

-19

-27

3

Len = 14 > 8

34

26

18

10

-10

-18

-26

-34

27

19

11

0

-11

-19

-27

Len = 14 > 8

Un-biased biased


4. Vector Quantizater

Vector Quantizer Use statistical correlation of N dimension Input


Vector Quantizer Design Generalized Loyd-Max Algorithm

Why Not Vector Quantization in Video Standards Transform Coding utilizes the “Correlation”, the remaining signal does

not has Correlation much.



Performance of VQ


Embedded quantizers

Motivation: “scalability” – decoding of compressed bitstreams at different rates (with different qualities)

Nested quantization intervals

In general, only one quantizer can be optimum(exception: uniform quantizers)

Q0

Q1

Q2

x


Rate Distortion Theory

Rate vs Distortion Inverse relation between Information/Data Rate and Distortion Revisit to a uniform distributed r.v. with a uniform quantizer

Plot it !

2 , 2 )2(3

12

12

1

1222

2

222r

xr

rN

A

N

AD

Exponential decreasing

Proportional to Complexity of source in statistics


Rate Distortion Theory

Rate-distortion Function and Theroy Inverse relation between Information/Data Rate and Distortion Example with Gaussian R.V.

3.0

2.0

1.0

0.1 0.2 0.4 1

R DI( ) ( )

D / 2

0.1 0.5 1.0

)()( RD I

R

2

R D DD

D

I( ) ( )log ,

,

1

20

0

22

2


Rate-Distortion Optimization

Motivation Image coding

• Condition: we can use 10KBytes for a whole picture

• Each part has different complexity (i.e, simpler and complex, variance)

• Which parts I have to assign more bits?

Analogy to school score • Students are evaluated sum of Math and

English grade.

• I gets 90pts in Math, 50pts in Eng.

• Which subject I have to study?

• (natural assumption: 90 to 100 is much harder than 50 to 60)

simpler blocks

More complex blocks


Lagrange multiplier

Constrained Optimization Problem Minimize distortion with bits no larger than max bits

Cannot use partial differential for minima and maxima

Lagrange Multiplier technique Insert one more imaginary variable (called Lagrange) We have multivariable minimization problem

maxR R subject to )}(min{ RD

})(min{ RRDJ


Side note: why not partial differential

Simple example min subject to

Solution• By substitution• minimum ½ at x = y = 1/sqrt(2)

Wrong Solution With complex constraints, we cannot use substitution of variable

method We cannot change partial different with ordinary difference

• If we check,

• x = y = 0 ?

22 yx 1

122),( yxyxf 1),( yxyx

02)( 22

xx

yx 02)( 22

yy

yx


An Example: Lagrange Multiplier

Optimization Problem multiple independent Gaussian variables

Lagrange multiplier optimization

BbM

ii

)(1

i hλb

J

B b b DJ

ibi

i

M

ii

ii

22

1

2)2ln( 2 0

)()(

1 ,22)( ),()( 2222

1

hhhbDbDBD ii b

ib

iiii

M

iii

iChibi for ])2 ln 2[( 2 122


Solution Multiplying M equations, and take M-square

Lagrange value

Bit-allocation

• Large variance, then more bit allocation

• But the increment is logarithmic (not linear)

CbM

bMbM

ii

i

ii

222

2

12

1

2 222

M

bb i

M

ii

,

1

22

2

2

2 log2

1

0 i

i bb

bM

ii

b hh 2

1

222 22ln222ln2

Appendix

Lloyd-Max Algorithm Details


Lloyd-Max scalar quantizer

Problem : For a signal x with given PDF find a quantizer with

M representative levels such that

( )Xf x

Solution : Lloyd-Max quantizer [Lloyd,1957] [Max,1960]

M-1 decision thresholds exactly half-way between representative levels.

M representative levels in the centroid of the PDF between two successive decision thresholds.

Necessary (but not sufficient) conditions

1

1

1

1ˆ ˆ 1, 2, , -1

2

( )

ˆ 0,1, , -1

( )

q

q

q

q

q q q

t

X

t

q t

X

t

t x x q M

x f x dx

x q M

f x dx

1

1

1

1ˆ ˆ 1, 2, , -1

2

( )

ˆ 0,1, , -1

( )

q

q

q

q

q q q

t

X

t

q t

X

t

t x x q M

x f x dx

x q M

f x dx

2ˆ min .d MSE E X X 2ˆ min .d MSE E X X


Iterative Lloyd-Max quantizer design

1. Guess initial set of representative levels

2. Calculate decision thresholds

3. Calculate new representative levels

4. Repeat 2. and 3. until no further distortion reduction

ˆ 0,1,2, , -1qx q M

1

1

( )

ˆ 0,1, , -1

( )

q

q

q

q

t

X

t

q t

X

t

x f x dx

x q M

f x dx

1

1

( )

ˆ 0,1, , -1

( )

q

q

q

q

t

X

t

q t

X

t

x f x dx

x q M

f x dx

1

1ˆ ˆ 1, 2, , -1

2q q qt x x q M 1

1ˆ ˆ 1, 2, , -1

2q q qt x x q M


Example of use of the Lloyd algorithm (I)

X zero-mean, unit-variance Gaussian r.v. Design scalar quantizer with 4 quantization indices with

minimum expected distortion D* Optimum quantizer, obtained with the Lloyd algorithm

Decision thresholds -0.98, 0, 0.98 Representative levels –1.51, -0.45, 0.45, 1.51 D*=0.12=9.30 dB

Boundary

Reconstruction


Example of use of the Lloyd algorithm (II)

Convergence Initial quantizer A:

decision thresholds –3, 0 3

Initial quantizer B:decision thresholds –½, 0, ½

After 6 iterations, in both cases, (D-D*)/D*<1%

5 10 15-6

-4

-2

0

2

4

6

Qu

an

tiz

ati

on

Fu

nc

tio

n

Iteration Number

0 5 10 150

0.2

0.4

D

0 5 10 150

5

10

SN

RO

UT

[dB

]

Iteration Number

SNROUT

final

= 9.2978

5 10 15-6

-4

-2

0

2

4

6

Qu

an

tiz

ati

on

Fu

nc

tio

n

Iteration Number

0 5 10 150.1

0.15

0.2

D

0 5 10 156

8

10S

NR

OU

T [d

B]

Iteration Number

SNROUT

final

= 9.298


Example of use of the Lloyd algorithm (III)

X zero-mean, unit-variance Laplacian r.v. Design scalar quantizer with 4 quantization indices with

minimum expected distortion D* Optimum quantizer, obtained with the Lloyd algorithm

Decision thresholds -1.13, 0, 1.13 Representative levels -1.83, -0.42, 0.42, 1.83 D*=0.18=7.54 dB

Threshold

Representative


Example of use of the Lloyd algorithm (IV)

Convergence Initial quantizer A,

decision thresholds –3, 0 3

Initial quantizer B,decision thresholds –½, 0, ½

After 6 iterations, in both cases, (D-D*)/D*<1%

0 5 10 150

0.2

0.4

D

0 5 10 154

6

8

SN

R [

dB

]

Iteration Number

SNRfinal

= 7.5415

2 4 6 8 10 12 14 16-10

-5

0

5

10

Qu

an

tiz

ati

on

Fu

nc

tio

n

Iteration Number2 4 6 8 10 12 14 16

-10

-5

0

5

10

Qu

an

tiz

ati

on

Fu

nc

tio

n

Iteration Number

0 5 10 150

0.2

0.4

D

0 5 10 154

6

8S

NR

[d

B]

Iteration Number

SNRfinal

= 7.5415


Lloyd algorithm with training data

1. Guess initial set of representative levels

2. Assign each sample xi in training set T to closest representative

3. Calculate new representative levels

4. Repeat 2. and 3. until no further distortion reduction

ˆ ; 0,1,2, , -1qx q M

x

1ˆ 0,1, , -1

q

qBq

x x q MB

x

1ˆ 0,1, , -1

q

qBq

x x q MB

ˆqx

: 0,1,2, , -1qB x Q x q q M T : 0,1,2, , -1qB x Q x q q M T


Lloyd-Max quantizer properties

Zero-mean quantization error

Quantization error and reconstruction decorrelated

Variance subtraction property

ˆ 0E X X

ˆ ˆ 0E X X X

22 2ˆ

ˆXX

E X X

Quantization Heejune AHN Embedded Communications Laboratory Seoul National Univ. of Technology Fall 2013 Last updated 2013. 9. 24.

Documents

video compressionp

problem heejune ahn

uniform input x

qheejune ahn

uniform quantizer performance

input range

crossrelated heejune

input statistics