ITCT Lecture 9.1itct/slide/2019/ITCT Lecture 9.1.pdf · – Inevitably results in some distortion...

ITCT Lecture 9.1:Image Data Compression (1)

Prof. Ja-Ling Wu

Department of Computer Science

and Information Engineering

National Taiwan University

Information Theory 2

Contents

Lecture 9.1

I. Introduction

II. Predictive Techniques

Lecture 9.2

I. Image Coding in Visual Telephony

II. Coding of Two-Tone Images

III. References


Image Data Compression

I. Introduction :

Image data Compression is concerned with

minimizing the number of bits required to represent

an image.

Applications of data compression are primarily in

“Transmission” and “Storage” of information.

Application of data compression is also in the

development of “fast algorithms” where the number

of operations required to implement an algorithm is

reduced by working with the compressed data.

--- Compressed Domain Signal Processing


Image data Compression techniques

Pixel

Coding

Predictive

Coding

Transform

CodingOthers

• PCM/quantization

• Run-length coding

• Bit-plane coding

JPEG2000/DVC

• Delta modulation

• Line-by-line DPCM

• 2-D DPCM

• Intra-/Inter-frame

techniques

•Adaptive

• Zonal coding

• Threshold coding

•DCT/Waveform

•Real/integer/Lapped

• Multi-D techniques

•Adaptive

• Hybrid Coding

• Vector quantization

•Compressed sensing


Image data Compression methods fall into two

common categories :

A. Redundancy Coding :

– Redundancy reduction

– Information lossless

Predictive coding : DM, DPCM

B. Entropy Coding :

– Entropy reduction

– Inevitably results in some distortion

Transform coding

For digitized data, “Distortionless Compression”

techniques are possible.


Some methods for Entropy reduction:

Subsampling : reduce the sampling rate

Coarse Quantization : reduce the number of

quantization levels

Frame Repetition / Interlacing :

reduce the refresh rate (number of frames per second)

TV signals


II. Predictive Techniques :

Basic Principle :

: to remove mutual redundancy between successive

pixels and encode only the new information.

DPCM :

A Sampled sequence u(m), coded up to m=n-1. Let

be the value of the reproduced

(decoded) sequence.

,2~,1~ nunu


At m=n, when u(n) arrives, a quantity , an

estimate of u(n), is predicted from the previously

decoded samples , i.e.,

prediction error :

If is the quantized value of e(n), then the

reproduced value of u(n) is :

ne~

nenunu ~~~

"rule prediction:" ; ,2~,1~~ nununU

nu~

,2~,1~ nunu

nunune ~


DPCM CODEC

Quantizer Communication

Channel

Predictor

with delay

Predictor

with delay

+

+

+–

nu ne ne~ ne~ nu~

nu~

nu~ nu~

Encoder Reconstruction

filter/Decoder


Note :

Remarks:

1. The pointwise coding error in the input sequence is exactly

equal to q(n), the quantization error in e(n)

2. With a reasonable predictor the mean square value of the

differential signal e(n) is much smaller than that of u(n)

e(n)in error on Quantizati the:

~

~~~

~

~

nq

nene

nenunenu

nununu

nenunu


Conclusion:

For the same mean square quantization error, e(n)

requires fewer quantization bits than u(n).

The number of bits required for transmission has

been reduced while the quantization error is kept the

same.


Feedback Versus Feedforward Prediction

An important aspect of DPCM is that the prediction is

based on the output — the quantized samples — rather

than the input — the unquantized samples.

This results in the predictor being in the “feedback loop”

around the quantizer, so that the quantization error at a

given step is fed back to the quantizer input at the next

step. This has a “stabling effect” that prevents DC drift

and accumulation of error in the reconstructed signal

. nu~


If the prediction rule is based on the past input, the signal

reconstruction error would depend on all the past and

present quantization errors in the feedforward prediction-

error sequence (n).

Generally, the MSE of feedforward reconstruction will be

greater than that in DPCM.

Quantizer

PredictorPredictorEntropy

coder/decoder

+

+

+

–

nu n nu~

Feedforward coding

nu


Example

The sequence 100, 102, 120, 120, 120, 118, 116, is to be

predictively coded using the prediction rule:

Assume a 2-bit quantizer, as shown below, is used,

Except the first sample is quantized separately by a 7-bit uniform

quantizer, given

coder. predictive dfeedforwar for the 1

DPCMfor 1~~

nunu

nunu

-6 -4 -2 2 4 6

5

1

-1

-5

.10000~ uu


Input DPCM Feedforward Predictive Coder

N u(n) u(n) u(n)

0 100 — — — 100 0 — — — 100 0

1 102 100 2 1 101 1 100 2 1 101 1

2 120 101 19 5 106 14 102 18 5 106 14

3 120 106 14 5 111 9 120 0 -1 105 15

4 120 111 9 5 116 4 120 0 -1 104 16

5 118 116 2 1 117 1 120 -2 -5 99 19

nu~ ne ne~ nu~ nu nε nε~ nu~


Delta Modulation : (DM)

Predictor : one-step delay function

Quantizer : 1-bit quantizer

1~

1~~

nunune

nunu

Unit Delay

Unit Delay

+

–

+

–

nu

nu~

nu~

ne ne~

+

+

Integrator

+

+

ne~ nu~

nu~


Slope

overload

nu nu~ Granularity

Primary Limitation of DM :

1) Slope overload : large jump region

Max. slope = (step size) (sampling freq.)

2) Granularity Noise : almost constant region

3) Instability to channel Noise

Step size effect :

Step Size (i) slope overload

(sampling frequency ) (ii) granular Noise


Adaptive Delta Modulation

This adaptive approach simultaneously minimizes the

effects of both slope overload and granular noise.

Unit Delay

+1

–1

+

–

1kS1KE

+

+

Adaptive

Function

kX 1kX1k

stored

,, min kk E

11

min1min

min21

1

1

11

if

if

sgn

KKK

KK

KKKK

K

KKK

XX

E

EE

XSE


DPCM Design

There are two components to design in a DPCM

system :

i. The predictor

ii. The quantizer

Ideally, the predictor and quantizer would be optimized

together using a linear or Nonlinear technique. In

practice, a suboptimum design approach is adopted :

i. Linear predictor

ii. Zero-memory quantizer

Remark : For this approach, the number of quantizing

levels, M, must be relatively large (M8) to achieve

good performance.


Design of linear predictor

iniiii

n

niiii

ninii

inniii

ii

jiij

i

inn

inn

i

nn

i

nn

RRRRa

a

a

a

RRRR

RaRaRa

SSaSSaSSaER

SSESSE

SSER

SSSE

SSaSaSaSE

SSaSaSaSE

a

SaSaSaSE

a

SSE

SSe

SaSaSaS

0

1

21

2

1

210

2211

22110

00

00

22110

22110

2

22110

2

00

000

2211

,,,

,,,

ˆ

n1,2,i ,0ˆ

0

n1,2,i , 0

2

ˆ

ˆ

0ˆ


When comprises these optimized coefficients, ai, then the

mean square error signal is :

0S

signal original theof variance the:

signal difference theof variance the: σ

ˆˆσ

principle) l(orthogona 0ˆˆBut

ˆˆˆ

ˆσ

00

2

e

002201100

00

2

0000

2

e

000

000000

2

00

2

e

R

RaRaRaR

SSESESSSE

SSSE

SSSESSSE

SSE

nn

The variance of the error signal is less than the variance of the

original signal.


Remarks:

1. The complexity of the predictor depends on “n”.

2. “n” depends on the covariance properties of the

original signal.


Design of the DPCM Quantizer

Review of uniform Quantizer:Quantization: 1. Zero-Memory quantization

2. Block quantization

3. Sequential quantization

Quantization Error : QE = y(X) – X

Average Distortion :

SNR :

Y

X

output

input

Midtread quantizer Midriser quantizer

dXXPXXyD

2

input x theof variance the:σ where

dBin log10

2

σ10

2

DSNR


Uniform quantizer : p(x) is constant within each interval

x1 x2 x3 x4 x5 x6

Granular RegionLower overload

regionupper overload

region

Y

X

Quantization Error for Midtread Quantizer


Output level yi always be in the midpoint of the input interval =xi-xi-1.

Assume p(x) is constant in the interval =xi-xi-1 and equal to p(xi)

Lower overload region : =x0-x1, x1 >> x0

Granular region : =xi-xi-1, 2 i M-1

upper overload region : =xM-xM-1, xM >> xM-1

y4

y3

y2

y1

y8

y7

y6

y5

x4x3x2x1 x8x7x6x5 x9X

Y

Uniform Midtread Quantizer M=9

1

2

3

1

2

1

1

3~

M

i

x

x

ii

M

i

i

x

xi

i

i

i

i

xxyxp

dxxpxxyD

Where we assume the contribution of the overload

region is negligible ; i.e. p(x1)=p(xM)=0


Since

V

V

V

M

i

i

M

i

i

ii

ii

dxx

xpx

VxV

D

xp

xpD

xyx

xyx

32V12

22

2V1

12

1

2

1

2

3

121

21

2

2

2

is anceinput vari the

)( p(x) is pdf theif

~

1But

~

Quantizer

characteristics

(Source Model)


Quantizerfor PCM only valid-

dB) (in 62log20

quantizer)bit -(n 2 if

log20log10

2for But

3

12log10

log10

Then

10

10

2

10

2

2

2

10

2

10

nnSNR

M

MMSNR

M

V

DSNR

n

MV


B. DPCM Quantizer

The pdf of the input signal to the DPCM quantizer is

not at all uniform. Since a “good” predictor would be

expected to result in many zero difference between

the predicted values and the actual values. A typical

shape for this distribution is a highly peaked, around

zero, distribution.

– +

pdf : p(d)

(Ex., Laplacian)

: Non-uniform Quantizer is required.


Compressor

Uniform

Quantizer

Expander

xL

x

dx

xdC

max2

C

Q

1C

xCQ

xC

xCQCY 1

X

Non-uniform Quantizer compressor + uniform Quantizer + Expander


Non-uniform Quantizer Expander

-Xmax

Xmax

Xmax

C(X) C(X)

y C-1(X)

Xmax-Xmax

x

uniform QuantizerCompressor


For this model, the mean-square distortion can be

approximately represented as :

functionnonlinear theof slope theis '

rangequantizer theis

'

12

1

12

12

22

2

1

xC

LL

LLxC

x

where

dxx

xp

MD

L

L


Lloyd-Max Quantizer : the most popular one.

1. Each interval limit should be midway between the

neighboring levels,

2. Each level should be at the centroid of the input prob.

Density function over the interval for that level, that is

2

1 ii yy

ix

01

i

i

x

xi dxxpyx

Logarithmic Quantizer : (log PCM)

-law

: US. Canada, Japan

1log

1log 1V

xVxy

1

KXdx

xdC

A-law

: Europe

Vx

xxy

AV

A

VV

AV

AAx

VAx

,

0,

log1

log

log1


If a Laplacian function is used to model p(e),

then the variance of the quantization error is:

the SNR for the non-uniform quantizer in DPCM becomes :

eep

ee

2exp

2

1

Input pdf of the DPCM Quantizer

V as

exp

2

2

312

2

92

3

0 3

2

2

1

3

22

Mg

V

Mg

e

ee

dee

2

2

2

2

2

22

2

2

10

10

n

9

210

10

log10

by SNR theimproves DPCM

65.6

: gives PCM pdf, same For the

log1065.6

2M Since

log10

log10

e

e

e

g

nSNR

nSNR

SNR

M


ADPCM :

i. Adaptive prediction

ii. Adaptive Quantization

DPCM for Image Coding :Each scan line of the image is coded independently by the

DPCM techniques. For every slow time-varying image (=0.95)

and a Laplacian-pdf Quantizer,

8 to 10 dB SNR improvement over PCM can be expected :

that is

The SNR of 6-bit PCM can be achieved by 4-bit line-by-line

DPCM for =0.97.

Two-Dimensional DPCM : two-D predictor

Ex :

1,11,1

1,,1,,

43

2

nmuanmua

nmuanmuanmu

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