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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
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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
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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
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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→ ∞ →
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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
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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
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7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.1 Data redundancy: coding redundancy
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
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7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.1 Data redundancy: coding redundancy
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 = − =
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7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.1 Data redundancy: interpixel redundancy
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7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.1 Data redundancy: interpixel redundancy
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 ∑
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7 2 Fundamental concepts and theories7.2 Fundamental concepts and theories7.2.1 Data redundancy: interpixel redundancy
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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
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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= =∑ ∑
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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= =
−∑ ∑
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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
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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
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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
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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
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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
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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= =
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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
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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∞ −= = < < < < <
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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
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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=
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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?
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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
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( ) ( ) 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
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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
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7 3 Basic coding methods7.3 Basic coding methods7.3.1 Huffman coding
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
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7 3 Basic coding methods7.3 Basic coding methods7.3.1 Huffman coding
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
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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
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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)
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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
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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
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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
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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
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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
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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
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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= +
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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=
⎡ ⎤= −⎢ ⎥⎣ ⎦
∑
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7 4 Predictive coding
[ ]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)
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(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
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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
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7 4 Predictive coding
ˆ
7.4 Predictive coding7.4.2 lossy coding: model
encoder n n nf e f= +
decoder
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7 4 Predictive coding
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 −=
⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭∑
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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)
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7 4 Predictive coding7.4 Predictive coding7.4.2 lossy coding: Optimal predictors
Experiment
original
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7 4 Predictive coding7.4 Predictive coding7.4.2 lossy coding: Optimal predictors
C l i d h d f h di iDigital Image Processing
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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
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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?
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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:
Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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
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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
Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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,
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
Prof.Zhengkai Liu Dr.Rong Zhang54
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
Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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7 5 Transform Coding7.5 Transform Coding7.5.5 bit allocation: threshold coding
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
Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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7 5 Transform Coding7.5 Transform Coding7.5.5 bit allocation: threshold coding
Z(u,v) is assigned a particular value c
Z(u v) used in JPEG stardardZ(u,v) used in JPEG stardard
Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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7 5 Transform Coding7.5 Transform Coding7.5.5 bit allocation: threshold coding
Experiment results
Left column: quantize with Z(u,v)q ( , )
Right column: quantize with 4*Z(u,v)
Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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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?
Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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7 5 Transform Coding7.5 Transform Coding7.5.3 JEPG 2000 lossy compression standard
Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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2003 Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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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
Prof.Zhengkai Liu Dr.Rong Zhang63
Video compression standardsVideo compression standards
2003 Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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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。
2003 Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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