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Ant Colony Optimization ACOFractal Image Compression

鄭志宏義守大學 資工系 高雄縣大樹鄉

J. H. Jeng

Department of Information Engineering

I-Shou University, Kaohsiung County

22

Outline

Fractal Image Compression (FIC) Encoder and Decoder Transform Method Evolutionary Computation Methods Ant Colony Optimization () ACO for FIC

33

Multimedia vs 心經 眼耳鼻舌身意 色聲香味觸法 眼： Text, Graphics, Image, Animation, Video 耳： Midi, Speech, Audio 鼻：電子鼻 , 機車廢氣檢測 舌：成份分析儀 , 血糖機 , Terminator III 身：壓力 , 溫度感測器 , 高分子壓電薄膜 意： Demolition Man

7-th “Sensor”

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Digital Image Compression

Finite Set• a, b, c, … ASCII

• 你 , 我 , … Big 5 Geometric Pattern

• Circle --- (x,y,r)

• Spline --- control points and polynomials Fractal Image

• Procedure, Iteration Natural Image

• JPEG, GIF

55

Fractal Image –having details in every scale

66

Fractal Image

77

321

3

2

1

0

2/1

2/10

02/1

2/1

0

2/10

02/1

2/10

02/1

wwwW

y

x

y

xw

y

x

y

xw

y

x

y

xw

Affine Transformations

88

Local Self-Similarity

99

Fractal Image Compression Proposed by Barnsley in 1985, Realized by Jacquin

in1992 Partitioned Iterated Function System (PIFS) Explore Self-similarity Property in Natural Image Lossy Compression Advantage:

• High compressed ratio

• High retrieved image quality

• Zoom invariant

Drawback:• Time consuming in encoding

1010

Domain Pool (D) Range Pool (R)

0r 1r

1922d

6538d

…….

Original Image

…….

……

.

Search for Best Match

1111

Expanded Codebook

Search Every Vector in the Domain Pool

For Each Search Entry:• Eight orientations• Contrast adjustment• Brightness adjustment

1212

The Best Match

: range block to be encoded

: search entry in the Domain Pool

: eight orientations,

})),(({min)(2

,,,,vqjiupv k

qpkji

v

),( jiu

),( jiuk 81 k

1313

Eight Orientations (Dihedral Group)

87654321 ,,,,,,, ttttttttT

1 2

4 3

3 4

2 1

4 1

3 2

1 4

2 3

2 1

3 4

3 2

4 1

4 3

1 2

2 3

1 41t 2t 4t3t

5t 6t 8t7t

90

flip

1 2

34

1414

210

0 21 : 1 case

0 21

21 0 :6 case

0 21

21 0 :7 case

0 21

21 0 :8 case

0 21

21 0 :5 case

210

0 21 : 2 case

210

0 21 : 3 case

210

0 21 : 4 case

Rotate 0º

Rotate 90º

Rotate 270º

Rotate 180º

Flip of case 1

Flip of case 6

Flip of case 7

Flip of case 4

Matrix Representations

1515

])),((,[

]),(),(,[

21

0

1

0

2

1

0

1

0

1

0

1

0

2

N

i

N

jkkk

N

i

N

j

N

i

N

jkk

k

jiuuuN

jivjiuvuN

p

1

0

1

0

1

0

1

02

),(),(1 N

i

N

jkk

N

i

N

jk jiupjiv

Nq

Contrast and Brightness

})),(({min),(2

8..1vqjiupji kkkk

k

1616

Affine Transform and Coding Format

q

j

i

z

y

x

p

dc

ba

z

y

x

W kk

kk

00

0

0

kkkk dcba ,,,p : contrast scale q : intensity offset

z : The gray level of a pixel

yx, : The position of a pixel

ji, : dihedral group: position

) 7 , 5 , 3 ,8 ,8(

) ,, , ,( qpTji k

1717

De-Compression

Make up all the Affine Transformations Choose any Initial Image Perform the Transformation to Obtain a New

Image and Proceed Recursively Stop According to Some Criterions

1818

The Decoding Iterations

Init Image Iteration=1 Iteration=2

Iteration=3 Iteration=4 Iteration=8

1919

Original 256256 Lena image Encoding time = 22.4667 minutes PSNR=28.515 dB

Full Search Coder

2020

58081116256 2

Domain block=1616 down to 8*8

#Domain blocks =

#MSE= 580818 = 464648

Contrast and Brightness Adjustment

Domain Pool (D) Range Pool (R)

0r 1r

1922d

6538d

…….

Original Image

…….

……

.

10248/256 2

Image Size = 256256

Range block = 88

#Range block =

Complexity

2121

Deterministic

Contrast and Brightness: Optimization The Dihedral Group: Transform Method

})),(({min),(2

8..1vqjiupji kk

k

)},({min,

jiji

2222

Non-Deterministic

Classification Method Correlation Method Soft Computing Method

})),(({min),(2

8..1vqjiupji kk

k

)},({min,

jiji

2323

Soft Computing

Machine Learning• ANN, FNN, RBFN, CNN

• Statistical Learning, SVM Global Optimization Techniques

• Branch and Bound, Tabu Search

• MSC, SA

• GA, PSO, ACO To infinity and beyond

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Global Optimization Techniques

Deterministic• Branch and Bound (Decision Tree)

Stochastic• Monte-Carlo Simulation

• Simulated Annealing (Physics) Heuristics

• Tabu Search

• Evolutionary Computation (Survival of the Fittest)

2525

Evolutionary Computation

Genotype and Phenotype• Genetic Algorithms (GA)

• Memetic Algorithm (MA)

• Genetic Programming (GP)

• Evolutionary Programming (EP)

• Evolution Strategy (ES) Social Behavior

• Particle Swarm Optimization (PSO)

• Ant Colony Optimization (ACO)

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Genetic Algorithm

Developed by John Holland in 1975 Mimicking the natural selection and natural

genetics Advantage:

• Global search technique

• Suited to rough landscape Drawback:

• Final solution usually not optimal

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Spatial Correlation Genetic Algorithm (1)

Two stage GA: 1. spatial correlation

1Dr Vr 2Dr

Hr jr

Hd

Vd1Dd

2Dd

HS

VS 1DS

2DS

W

L

27

2828

Particle Swarm Optimization (PSO) Particle Swarm Optimization

• Introduced in 1995 by Kennedy and Eberhart • Swarm Intelligence• Simulation of a social model• Population-based optimization• Evolutionary computation

Social Psychology Principles• Bird flocking• Fish schooling• Elephant Herding

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Edge-Property Adapted PSO for FIC

Hybrid Method vs Fused Methods

Visual-Salience Tracking Edge-type Classifier, 5 Edge Types Predict the Best k (Dihedral Transformation) Intuitively Direct the Swarm Velocity Direction

according to Edge Property

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Behavior of Ants

Secrete and Lay Pheromone Detect and Follow with High Probability Reinforce the Trail

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Ant Colony Optimization (ACO)

Proposed by Dorigo et al. (1996) Learn from real ants Pheromone

• Intensity

• Accumulation

• Communication

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Artificial Ants

E

A

B

C

D

H

t=0

30 ants

30 ants

E

A

B

C

D

H

t=1

10 ants

10 ants

20 ants

20 ants

30 ants

30 ants

E

A

B

C

D

H

d=1

d=1

d=0.5

d=0.5

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Ant system

Proposed by Dorigo et al. (1996) Characteristics of AS to solve TSP

• Choose the town with a probability• Town distance• Amount of trail (pheromone)

• Force the ant to make legal tours• Disallow visited towns until a tour is completed

• Lay trail on each edge visited when it completes a tour

),( ji

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TSP

Traveling Salesman Problem

Problem of finding a minimal length closed tour that visits each town once.

Parameters•

•

townsofset a :n

jidij and wnsbetween topath theoflength the:

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Probability of selecting town

•

• visibility ( )

• control the relative importance of trail versus visibility

Transition probability is a trade-off between visibility and trail intensity at time

ijij d

1

0)( allowed

][)]([

][)]([

kk ikik

ijij

tτ

tkij tp

kj allowed if

otherwise

}tabu{allowed kk N :ij:,

t

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Pheromone Accumulation

• the evaporation of trail ( )

• the intensity of trail on edge at time

• the sum of trail on edge by the ants

between time and

ijijij tnt )()(:1 :)(tij:ij

10 ),( ji

),( ji

t nt

m

k

kijij

1

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Global update

• constant

• the tour length of the kth ant

0

kLQ

kij

if kth ants uses edge (i,j) in its tour (between time t and t+n)

otherwise

:Q

:kL

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Local update

Ant-density model

•

Ant-quantity model

•

• Shorter edges are made more desirable

0

Qkij

if the kth ant goes from i and j between time t to t+1

otherwise

0

ijdQ

kij if the kth ant goes from i and j between time t to t+1

otherwise

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TSP (Traveling Salesman Problem)

特性• 規則簡單• 計算複雜

• 拜訪 42 個城市需走過 演算法比較

• 螞蟻演算法 (Ant Colony Optimization)

• 彈性網路 (Elastic Net)

• 基因演算法 (Genetic Algorithm)

• 人腦

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TSP result

演算法比較推銷員問題 彈性網絡 螞蟻王國 基因演算法 人腦（平均）

Att48 5.81% 2.86%( 875) 3.0%(3256) !4.41%(7)

Berlin52 6.90%1.52%(1388

)7.4%(3816) !5.18%(6)

Eil101 9.10%7.64%(1488

)14.2%(5000

)8.83%(6)

Eil51 3.37%4.41%(1115

)4.4%(5000) 8.98%(3)

St70 4.16% 3.42%( 283) 5.9%(4408) 7.03%(3)

Ulysses16 1.30% 0%(3289) -0.1%( 901) !1.05%(2)

Ulysses22 1.57% 0%(4562) 0.3%(1364) N/A

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TSP result 種子數為 10 ， 20 ，… 100 產生 30 個城市推銷員問題 彈性網絡 螞蟻王國

(1000)螞蟻王國(2000)

基因演算法

#1 4.442 4.597( 62)4.442(224

4)

#2 4.053 4.053(602)4.053(288

7)

#3 4.634 4.480(367)4.480(211

7)

#4 4.744 4.744(170) 4.480(1207)4.799(214

9)

#5 4.869 4.759(994) 4.737(1759)4.737(134

4)

#6 4.316 4.214(120)4.369(173

4)

#7 5.498 5.061(467) 5.049(1365)5.322(108

3)

#8 4.621 4.601(416)4.846(115

3)

#9 4.362 4.358(250)4.387(177

6)

#10 5.535 5.211(139)5.454(223

7)

Average 4.707 4.608(359) 4.601(629)4.689(197

2)

Variance 0.236 0.128 0.125 0.192

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ACO for FIC

Ant: range block • Secrete pheromone at cities instead of on the path

between two cities City: domain block Visibility: reciprocal of the MSE

• Between the agent (range block) and the city (domain block)

otherwise,0

)( if,))(())((

))(())((

)()(

tJitt

tt

tpg

tJuuu

ii

gi

g

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(a) Original image (b) Full search, 28.90 dB (c) ACO, 27.66 dB

Lena FIC-ACO

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(a) Original image (b) Full search, 30.40 dB (c) ACO, 28.78 dB

Pepper FIC-ACO

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Various pheromone evaporate rates

Pheromone evaporate

rate

Quality(PSNR)

Average(PSNR)

0.1 27.59 27.48 27.53 27.67 27.55 27.56

0.2 27.63 27.60 27.59 27.60 27.55 27.59

0.3 27.56 27.57 27.52 27.57 27.53 27.55

0.4 27.54 27.55 27.58 27.63 27.59 27.58

0.5 27.55 27.66 27.46 27.60 27.56 27.57

0.6 27.50 27.57 27.55 27.55 27.53 27.54

0.7 27.63 27.57 27.62 27.51 27.54 27.57

0.8 27.58 27.50 27.61 27.59 27.66 27.59

0.9 27.53 27.58 27.49 27.56 27.53 27.54

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Various parameters

Quality(PSNR)

Average(PSNR)

1 1 27.58 27.50 27.61 27.59 27.66 27.59

2 1 27.17 27.23 27.27 27.24 27.10 27.20

1 2 26.71 26.59 26.67 27.03 26.61 26.72

2 2 26.62 26.34 26.63 26.51 26.65 26.55

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Result on various images

Lena Baboon F16 Pepper

Full search method

Quality (PSNR)

28.90 20.13 26.09 30.41

Time 3620 3716 3684 3709

Proposed method

Quality (PSNR)

27.58 19.75 25.70 28.74

27.50 19.77 25.81 28.78

27.61 19.80 25.74 28.69

27.59 19.72 25.80 28.80

27.66 19.78 24.48 28.69

Average(PSNR)

27.59 19.76 25.52 28.74

Time 144 145 144 146

Speedup 25.1 25.6 25.8 25.2

4848

Thanks