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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
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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
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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
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