chapter7 image compression - ustc

Chapter7 Image Compression

• Preview7 1 I d i• 7.1 Introduction

• 7.2 Fundamental concepts and theories i di h d• 7.3 Basic coding methods

• 7.4 Predictive coding• 7.5 Transform coding• 7.6 Introduction to international standards

Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang

1

7.1 Introduction7.1.1 why code data?

• To reduce storage volume7.1.2 Classification

To reduce storage volume• To reduce transmission

time

Lossless compression: reversible

Lossy compression: irreversibletime– One colour image

• 760 by 580 pixels

Lossy compression: irreversible

• 3 channels, each 8 bits• 1.3 Mbyte

Vid d t– Video data• same resolution• 25 frames per secondp• 33 Mbyte/second


2

7.1 Introduction7.1.3 Compression methods

•Entropy coding: Huffman coding arithmetic coding•Entropy coding: Huffman coding, arithmetic coding•Run length coding: G3, G4

di j( ) i ( i d ) ( )•LZW coding: arj(DOS), zip(Windows), compress( UNIX)•Predictive coding: linear prediction, non-linear predictionT f di DCT KLT W l h T WT( l )•Transform coding: DCT, KLT, Walsh-T, WT(wavelet)

•Others: vector quantization (VQ), Fractal codingthe second generation coding


3

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.1 Data redundancyIf n1denote the original data number, and n2 denote the compressed data number, then the data redundancy can be defined as

11DR

RC

= −R

where CR, commonly called the compression ratio, is

1

2R

nCn

=

F lFor example n2 =n1 RD=0

2 1n n , 1R DC R→ ∞ →


4

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.1 Data redundancy

•Other expression of compression ratio

2

1R

nCn

= or 2 8 // , Rn bits pixelL bits pixel C

M N L= =

×

•Three basic image redundancy

Coding redundancy

Interpixle redundancye p e edu d cy

psychovisual redundancy


5

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.1 Data redundancy: coding redundancy

If rk represents the gray levels of an image and each rk occurswith probability pr(rk), then

( ) , 0,1, 1kr k

np r k Ln

= = −

where L is the number of gray levels, nk is number of times that the kth gray level appears in the image and n is the total numberthe kth gray level appears in the image, and n is the total numberof pixels in the image


6


If the number of bits used to represent each value of rk is l(rk),thenthe average number of bits required to represent each pixel is

1L 1

0

( ) ( )L

avg k r kk

L l r p r−

=

= ∑For example

rk Pr(rk) Code1 l1(rk) Code2 l2(rk)r0 0.19 000 3 11 20

r1 0.25 001 3 01 2r2 0.21 010 3 10 2r3 0.16 011 3 001 3r4 0.08 100 3 0001 4r5 0.06 101 3 00001 5R6 0.03 110 3 000001 6


7r7 0.02 111 3 000000 7


7

20

( ) ( )avg k r kk

L l r p r=

=∑2(0.19) 2(0.25) 2(0.21) 3(0.16)

4(0.08) 5(0.06) 6(0.03) 6(0.02)= + + ++ + + +2.7bits=

3/ 2 7 1 11C 3/ 2.7 1.11RC = =

11 0 099R = =1 0.0991.11DR = − =


8

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.1 Data redundancy: interpixel redundancy


9


Autocorrelation coefficients

( )( )(0)

A nnA

γ ΔΔ =

11 N n− −Δ

where

1

0

1( ) ( , ) ( , )N n

y

A n f x y f x y nN n

Δ

=

Δ = + Δ−Δ ∑


10



11

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.1 Data redundancy: psychovisual redundancyPsychovisual redundancy is associated with real or quantifiable.Psychovisual redundancy is associated with real or quantifiable visual information. The information itself is not essential for normal visual processing

8bits/pixel 4bits/pixel 2bits/pixel


12

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.2 Fidelity criteria: objective L t t i t i d d t th( )f x y ^

( )fLet represent an input image and denote the Decompressed image. For any value x and y, the error

( , )f x y ( , )f x y

^

e(x,y) bewteen and can be defined as( , )f x y^( , )f x y

^( , ) ( , ) ( , )e x y f x y f x y= −

So that the total error between the two images isSo that the total error between the two images is

1 1 ^[ ( ) ( )]

M N

f x y f x y− −

−∑ ∑0 0

[ ( , ) ( , )]x y

f x y f x y= =∑ ∑


13

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.2 Fidelity criteria: objective

The root-mean-square error, eems, between and is( , )f x y^( , )f x y

1 1 ^1 M N1 1 ^2 1/ 2

0 0

1{ [ ( , ) ( , )] }M N

rmsx y

e f x y f x yMN

− −

= =

= −∑ ∑

The mean-square signal noise ratio is defined as 1 1 ^

2( )M N

f− −

∑ ∑ 2

0 01 1 ^

( , )10 lg x y

rms M N

f x ySNR = =

− −=∑ ∑

2

0 0

[ ( , ) ( , )]x y

f x y f x y= =

−∑ ∑


14

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.2 Fidelity criteria: objective

The signal noise ratio is defined as 21 1 ^

( ) ( )M N

f x y f x y− − ⎡ ⎤−⎢ ⎥∑ ∑0 0

1 1 ^2

( , ) ( , )10 lg

[ ( ) ( )]

x yM N

f x y f x ySNR

f f

= =− −

−⎢ ⎥⎣ ⎦=∑ ∑

∑ ∑ 2

0 0

[ ( , ) ( , )]x y

f x y f x y= =

−∑ ∑The peak signal noise ratio is defined asThe peak signal noise ratio is defined as

1 12

max 2

M N

ff

− −

∑ ∑0 0 max

1 1 2^2

10 lg 10lg[ ( , ) ( , )]

x yM N

rms

fPSNRef x y f x y

= =− −= =

−∑ ∑Digital Image Processing

Prof.Zhengkai Liu Dr.Rong Zhang15

0 0

( ) ( )x y

f y f y= =∑ ∑

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.2 Fidelity criteria: subjective

Side-by-side comparisonsSide by side comparisons


16

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.3 image compression models

Source encoder model

( , )f x y decorrelation quantizer Symbolencoder

^( , )f x y

R RRemoveInterpixel

redundancy

Removepsychovisualredundancy

Removecodingredundancy redundancy

redundancy


17

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.4 Elements of information theory: terms

1948, Cloude Shannon, A mathematical theory of communication

News: dataNews: datainformation: contentsInformation source: symbols

Memoryless source

Information source: symbols

Markov source


18

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.4 Elements of information theory: Self-information

A random event E that occurs with probability is said to contain units of information.

( )P E

( )I E

1( ) log log ( )( )

I E P EP E

= = −

( )

( )I E = ∞

( )P E

For example ( ) 0,P E =

( ) 1,P E = ( ) 0I E =

Unit: Base=2, bits ( binary digits)U :Base=e, nats ( nature digits)Base=10, Hartly


19

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.4 Elements of information theory: Entropy of the source

definition

Suppose X={x0 , x1,… xN-1 }is a discrete random variable, andthe probability of xj is P(xj),namely

0 1 1

0 1 1( ) ( ) ( )( )Nx x xX

p x p x p xP X−⎡ ⎤⎡ ⎤

= ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦0 1 1( ) ( ) ( )( ) p p p⎣ ⎦ ⎣ ⎦

then 1 1

0 0

1( ) ( ) log ( ) log ( )( )

N N

j j jj j

H X p x p x p xp x

− −

= = −∑ ∑0 0( )j jjp x= =


20

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.4 Elements of information theory: Entropy of the source

properties

( ) 0H X ≥(1) ( ) 0H X ≥(1)

( ) (1,0, 0) (0,1, 0) (0,0, 1) 0H X H H H= = = = =(2)

Wh P( ) 1/N f ll j H(X) l N

0 1 1( ) ( , ) logNH X H x x x N−= ≤(3)

When P(xj)=1/N for all j,, H(X)=logN

example X={1 2 3 4 5 6 7 8 } p =1/8 for each jexample X={1,2,3,4,5,6,7,8 }, pj=1/8 for each j8

21( ) log 8 3 /8

H X bits symbol= =∑Digital Image Processing


1 8j=

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.4 Elements of information theory: Conditional entropy

1∑ ∑

definitions0

1( ) ( ) log( )i

i i

H X p xp x

=∑

1 11

1( ) ( ) ( | ) log( )i i i

i i i i

H X p x p x xp x x−

−

=∑ ∑1( ) ( ) ( | ) logH X p x p x x x∑ ∑2 1 2

1 2

( ) ( ) ( | ) log( )i i i i

i i i i i

H X p x p x x xp x x x− −

− −

=∑ ∑

11

1

1( ) ( ) ( | ) log( )m i i i i m

i i i i i m

H X p x p x x xp x x x− −

− −

=∑ ∑

tproperty1 2 1 0m mH H H H H H∞ −= = < < < < <


22

7.2 Fundamental concepts and theories7.2.4 Elements of information theory: using information theory

A simple 8 bits image:

21 21 21 95 169 243 243 24321 21 21 95 169 243 243 243 1∑

A simple 8-bits image:Zero-order entropy:

21 21 21 95 169 243 243 24321 21 21 95 169 243 243 24321 21 21 95 169 243 243 243

01( ) ( ) log( )i

i i

H X p xp x

=∑

1 81 /bits pixel

Gray-levelGa Count probability

1.81 /bits pixel=Probability of the source symbols:

Gray levelGa Count probability21 12 3/895 4 1/8169 4 1/8169 4 1/8243 12 3/8


23

7.2 Fundamental concepts and theories7.2.4 Elements of information theory: using information theory

Relative frequency of pairs of pixels:Gray-level

PairaCount probability

(21,21) 8 2/8(21 95) 4 1/8(21,95) 4 1/8(95,169) 4 1/8(169,243) 4 1/8(243,243) 8 2/8(243,21) 4 1/8

Fi t d t

1 11( ) ( ) ( ) log

( )i i iH X p x p x xp x x−=∑ ∑

First-order entropy:

1( )i i i ip x x −∑ ∑1.25 /bits pixel=


24

7.2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.4 Elements of information theory: Noiseless coding theorem

Also called Shannon first theorem. Defines the minimum averagecode word length per source symbol that can be achieved

( ) avgH X L≤

For memoryless (zero-memory) source:

1( ) ( ) log( )i

i i

H X p xp x

=∑ ( )i ip

But, the image data usually are Markov source, how to code?


25

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.4 Elements of information theory: Noiseless coding theorem

Coding steps decorrelationX Ymarcov memoryless

entropycoding

entropycoding

LX

g g

LY

H(X)=Hm(X)<H0(X), LX= H0(X)

X Y

H(Y)=H0(Y), LY= H0(Y)

H(X)=H(Y) LY= Hm(X)<LX


26

( ) ( ) Y m( ) X

7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.4 Elements of information theory: Noise coding theorem

For a given rate distortion D, the rate distortion function R(D)is less than H(X), and R(D) is the minimum of coding length Lavg


27

7 3 Basic coding methods7.3 Basic coding methods7.3.1 Huffman coding

Huffman, 1952

• code the low probabilities symbols with long wordcode the high probabilities symbols with short word

• smallest possible number of code symbols per source symbol

• the source symbols must be coded one at a time


28


step1:create a series of source reduction by ordering the probabilitiesstep2:combine the lowest probability symbols into a single symbol

h l h i h d ithat replaces them in the next source reductionstep3:repeat step2 until the lowest probability is 1

For example Symbol Probability 21 43

ABCD

0.40.30.10 1

0.40.30.10 1

0.40.30.20 1

0.40.30.3

0.60.4

DEF

0.10.060.04

0.10.1

0.1


29


t 4 d h d d ith 0 d 1 t i ith thstep4:code each reduced source with 0 and 1, staring with the smallest source and working back to the original source Symbol Probability 21 43Symbol Probability 21 43

AB

0.4 1 0.4 1 0.4 10 3 00

0.4 10 3 00

0.6 00 4 1B

CDE

0.3 000.1 0110.1 01000 06 01010

0.3 000.1 0110.1 01000 1 0101

0.3 000.2 0100.1 011

0.3 000.3 01

0.4 1

0.4 1 0.3 2 0.1 3 0.1 4avgL = × + × + × + ×6

01

( ) ( ) log ( )i ii

H A p a p a=

= −∑EF

0.06 010100.04 01011

0.1 0101

0.06 5 0.04 5avg

+ × + ×2.2 /bits symbol=

1i=

2.14 /bits symbol=

( )L H X>Digital Image Processing


0 ( )avgL H X>

7.3 Basic coding methodsg7.3.2 Arithmetic coding•An entire sequence of source symbols( or message) is assigned as a single arithmetic code word• the code word itself defines an interval of real numbersbetween 0 and 1

The encoding process of an arithmetic coding can be explainedthrough the following example


31

7.3 Basic coding methodsg7.3.2 Arithmetic coding: example

encoding

•Let us assume that the source symbols are {a1 a2 a3a4} and theprobabilities of these symbols are {0 2 0 2 0 4 0 2}probabilities of these symbols are {0.2, 0.2, 0.4, 0.2}

•To encode a message of a sequence: a1 a2 a3a3a4

h i l [0 1) b di id d f b i l•The interval [0,1) can be divided as four sub-intervals:[0.0, 0.2), [0.2, 0.4), [0.4, 0.8), [0.8, 1.0),

symbols probabilites Initial intervalsa1 0.2 [0.0,0.2)a2 0.2 [0.2,0.4)a3 0.4 [0.4,0.8)a 0 2 [0 8 1 0)


32

a4 0.2 [0.8,1.0)

7.3 Basic coding methodsg7.3.2 Arithmetic coding: example

•We take the first symbol a1 from the message and find its encoding range is [0.0, 0.2).

•The second symbol a2is encodedby taking the 20th-40th of y y ginterval [0.0, 0.2) as the new interval [0.02,0.04).

•And so on Visually we can use the figure:

Fi ll h b

And so on. Visually, we can use the figure:

•Finally, choose a number from the interval of [0 0688 0 06752) as the[0.0688,0.06752) as theoutput: 0.06800


33

7.3 Basic coding methodsg7.3.2 Arithmetic coding : example

0< 0 068800 <0 2 a1

decoding

0< 0.068800 <0.2 a1

0.04< 0.068800 <0.08 a2

0 056< 0 068800 <0 072 a0.056< 0.068800 <0.072 a3

0.0624< 0.068800 <0.0688 a3

0.06752< 0.068800 <0.0688 a4


34

7 3 Basic coding methods7.3 Basic coding methods7.3.4 Binary image coding: constant area coding

all white

all blackall black

mixed intensity

•The most probable category is coded with 1-bit code word 0p g y

•The other two category is coded with 2-bit code word 10 and 11

•The code assigned to the mixed intensity category is used as a prefix


35

7 3 Basic coding methods7.3 Basic coding methods7.3.4 Binary image coding: run length coding

•Represent each row of a image by a sequence of lengths thatdescribe runs of black and white pixels

• the standard compression approach in facsimile (FAX) coding. CCITT, G3,G4.

(1,8) (0,3) (1,14) (0,4) (1, 28) (0,3) (1,23) (0,3) (1,29) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )(0, 3) (1,28) (0,3) (1,31) (0,4) (1,25) (0,4) (1,38) (0,4)

8 3 14 4 28 3 23 3 29 3 28 9 3 31 4 25 4 38 4


36

7 3 Basic coding methods7.3 Basic coding methods7.3.4 Binary image coding: contour tracing and coding

• is the difference between the starting coordinates of the front contours adjacent lines

'Δ

i h diff b h f b k l h• is the difference between the front-to-back contour lengths contours adjacent lines

''Δ

• code: new start and'Δ ''Δ• code: new start, and Δ Δ


37

But for gray image

162 161 159 161 162 160 158 156 156 161162 161 159 161 162 160 158 156 156 161

But for gray image…

162 161 159 161 162 160 158 156 156 161163 158 159 159 160 158 155 155 156 158159 157 159 156 159 159 154 152 155 153155 157 156 156 158 157 156 155 154 156156 157 155 151 157 156 155 156 156 154157 156 156 156 156 156 154 156 155 155158 157 155 155 156 155 155 155 155 155156 155 156 153 156 155 156 155 154 156155 155 157 154 157 155 157 158 158 158


38

7 4 Predictive coding7.4 Predictive coding7.4.1 lossless coding

encoder

Huffman coding or

ˆ

d d

Arithmetic codingn n ne f f= −

decoder

ˆn n nf e f= +


39

n n nf f

7 4 Predictive coding

In most case the prediction is formed by a linear combination

7.4 Predictive coding7.4.1 lossless coding

In most case, the prediction is formed by a linear combination of m previous pixels. That is,

ˆm⎡ ⎤

1n̂ i n i

if round fα −

=

⎡ ⎤= ⎢ ⎥⎣ ⎦∑

h i th d f th li di twhere m is the order of the linear predictorαi are prediction coefficients

For example, 1-D linear predictive can be written

ˆ ( ) ( )m

f d f⎡ ⎤∑1

( , ) ( , )ii

f x y round f x y iα=

⎡ ⎤= −⎢ ⎥⎣ ⎦

∑


40


[ ]ˆ

7.4 Predictive coding7.4.1 lossless coding: Experimental result

m=1,α=1 [ ]( , ) ( , 1)f x y round f x y= −

Original image Residual image(128 represents 0)


41

(128 represents 0)

7 4 Predictive coding2

1( ) e

e

p e e σ−

=

7.4 Predictive coding7.4.1 lossless coding: Experimental result

m=1, α=1 ( )

2ee

p e eσ

Histogram of original image Histogram of residual image


42

Histogram of original image Histogram of residual image

7 4 Predictive coding7.4 Predictive coding7.4.1 lossless coding: JPEG lossless compression standard

P di ti P di ti

Prediction schemes

c b

a x

Prediction Scheme

Predictionvalue

0 Null1 a1 a2 b3 c

Location of pixels

4 a+b-c5 a+(b-c)/26 b+(a-c)/27 (a+b)/2


43


ˆ

7.4 Predictive coding7.4.2 lossy coding: model

encoder n n nf e f= +

decoder


44


diff ti l l d d l ti ( DPCM)

7.4 Predictive coding7.4.2 lossy coding: Optimal predictors

differential pulse code modulation( DPCM)•Minimizes the encoder’s mean-square prediction error

2 2ˆ2 2ˆ{ } {[ ] }n n nE e E f f= −Subject to the constraint thatj

ˆm

n i n if fα −=∑1i=

22

1{ }m

n n i nE e E f fα⎧ ⎫⎡ ⎤⎪ ⎪= −⎨ ⎬⎢ ⎥⎣ ⎦

∑ 11

{ }n n i ni

f f −=

⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭∑


45

7 4 Predictive coding7.4 Predictive coding7.4.2 lossy coding: Optimal predictors

Experiment: four predictors:

ˆ ( , ) 0.97 ( , 1)f x y f x y= −(1) ( , ) 0.97 ( , 1)f x y f x y(1)

ˆ ( , ) 0.5 ( , 1) 0.5 ( 1, )f x y f x y f x y= − + −(2)ˆ̂ ( , ) 0.75 ( , 1) 0.75 ( 1, ) 0.5 ( 1, 1)f x y f x y f x y f x y= − + − − − −(3)

0.97 ( , 1) ifˆ f x y h v− Δ ≤ Δ⎧⎨

0.97 ( , 1) if ( , )

0.97 ( 1, ) otherwisef x y h v

f x yf x y

Δ ≤ Δ⎧= ⎨ −⎩

(4)

wherewhere ( 1, ) ( 1, 1)h f x y f x yΔ = − − − −

( , 1) ( 1, 1)v f x y f x yΔ = − − − −

f(x-1,y-1) f(x-1,y)

f(x,y-1)

f(x-1,y+1)

f(x,y)


46


Experiment

original


47


C l i d h d f h di iDigital Image Processing


Conclusion:Error decreases as the order of the predictor increases

7 4 Predictive coding7.4 Predictive coding7.4.2 lossy coding: Optimal quantization

The staircase quantization function t=q(s) is shown as


49

7 5 Transform Coding7.5 Transform Coding7.5.1 Review of image transform: definitions

X YT

1 1

, , ( , , , )M N

m n i jy x i j m n− −

= Φ∑∑Y TX=Direct transform0 0i j= =

1X T Y−=Inverse transformpolynomial expression

Vector expression

Question: why transformation can compress data?


50

7 5 Transform Coding7.5 Transform Coding7.5.1 review of image transform: properties

• Entropy keeping: H(X)=H(Y)• Energy keeping: 1 1 1 1

2 2( , ) ( , )M N M N

X m n Y k l− − − −

=∑∑ ∑∑0 0 0 0m n k l= = = =∑∑ ∑∑

• decorrelation:1 2 1 0m mH H H H H H∞ −= = < < < < <

• Energy re-assigned:

H(Y)=H0(Y) H(X)=Hm(X) H0(X)>H0(Y)

Energy re assigned:


51original DFT DCT

7 5 Transform Coding7.5 Transform Coding7.5.2 Transform coding system

( , )f x y Symbolencoder

quantizerConstruct

n*nsubimage

Forwardtransform

Bitsstream

subimage

encoder

Bitsstream

Symbolencoder

Inverse transform

Merge n*n

bi

^( , )f x ystream encoder transform subimage

decoder


52

7 5 Transform Coding7.5 Transform Coding7.5.3 Transform selection

Information packing ability:Computational complexity: WHT<DCT<DFT<KLT

KLT>DCT>DFT>WHT

7.5.4 sub-image size selection

•Computational complexityComputational complexity increase as the subimage size increase

•Correlation decrease as thesubimage size increase

•The most popular subimagesize are 8*8, and 16*16


53

,

7 5 Transform Coding7.5 Transform Coding7.5.5 bit allocation: zonal coding

• transform coefficients of maximum variance carry the mostinformation and should be retained

1 1

1 1

1

1

1 1

1 0 0

0

0

0

0

0 8 7

7 6

6

5

4 3

4 3 2

2

1

1

0

0

1

11

1

1

1

000

0

00

0

00

0

00

0

00

0

00

6

43

5

4

4

333

3

32

3

21

1

11

1

00

0

001

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

3

2

1

3

2

1

3

1

1

2

1

0

1

1

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

Zonal mask Bit allocationDigital Image Processing


7 5 Transform Coding7.5 Transform Coding7.5.5 bit allocation: threshold coding

1 1 0 1 1 0 0 0

1

1

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Zig-zag1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

g g

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

threshold mask


55


There are three basic ways to threshold a transformed coefficients

•A single global threshold can be applied to all subimagesg g pp g•A different threshold can be used for each subimage•The threshold can be varied as a function of the location of each fcoefficient within the subimage

( , )ˆ( ) T u vT u v round⎡ ⎤

= ⎢ ⎥( , )( , )

T u v roundZ u v

= ⎢ ⎥⎣ ⎦

where T(u,v): transform coefficients

Z(u,v): quantization matrix


56


Z(u,v) is assigned a particular value c

Z(u v) used in JPEG stardardZ(u,v) used in JPEG stardard


57


Experiment results

Left column: quantize with Z(u,v)q ( , )

Right column: quantize with 4*Z(u,v)


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7 5 Transform Coding7.5 Transform Coding7.5.3 JEPG lossy compression standard

( , )f x y Huffmancodingquantizer

Construct8*8

subimageDCT

BitsstreamZig-zag

subimage

encoderQuestion: why there are mosaics in JEPG images?Question: why there are mosaics in JEPG images?


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7 5 Transform Coding7.5 Transform Coding7.5.3 JEPG 2000 lossy compression standard


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2003 Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang

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7 6 Introduction to international7.6 Introduction to international standards

• Image compression standards: -JPEG,JPEG2000-JPEG2000

• Video compression standards:-MPEG-1,MPEG-2,MPEG-4, H 261 H 263 H 263+ H 264-H.261,H.263,H.263+,H.264

-H.264/AVC2003 Digital Image Processing


Video compression standardsVideo compression standards


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Homework

6.7(11-14) ( )(1)请说明是否能用变长变码法压缩1幅已直方图均衡化的具有

2n级灰度的图？这样的图像中包含像素间冗余吗(2) 这样的图像中包含像素间冗余吗？

6 8(11 15)6.8(11-15) (1) 对一个具有3个符号的信源，有多少唯一的Huffman码？

(2)构造这些码。( )构

6.11(11-22)已知符号的出现概率分别是已知符号a,e,i,o,u,?的出现概率分别是0.2，0.3，0.1，0.2，0.1，

0.1，0.23355进行解码，解码长度为6。


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chapter7 image compression - ustc

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