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 |  موضوع: كتاب Quaternion and Octonion Color Image Processing with MATLAB الأربعاء 03 نوفمبر 2021, 12:36 am | |
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أخواني في الله
أحضرت لكم كتاب
Quaternion and Octonion Color Image Processing with MATLAB
Artyom M. Grigoryan and Sos S. Agaian
و المحتوى كما يلي :
Contents
Preface ix
1 Complex and Hypercomplex Numbers 1
1.1 Complex and Hypercomplex Numbers 2
1.1.1 Complex arithmetic 2
1.1.1.1 Geometry of complex numbers 6
1.1.2 Quaternion numbers 12
1.1.3 Rules for multiplication 13
1.1.4 Basic operations 16
1.1.5 Properties of multiplication of quaternions 19
1.2 Vector Space and Pure Quaternions 35
1.2.1 Inner product (or dot product) 36
1.2.2 Vector product 39
1.3 Quaternion Multiplication and Rotation 41
1.3.1 Multiplication and sum of elementary rotations 41
1.3.2 Rotation: Multiplication by a perpendicular vector 48
1.3.3 The 2nd rotation: Multiplication by a perpendicular vector 52
1.3.4 Multiplication: Rotation of any vector 54
1.3.5 Matrix representation of rotation 55
1.3.6 Addition of rotations in 3-D space 60
1.4 The Quaternion Exponential Function 65
1.5 Quaternion Trigonometric and Hyperbolic Functions 77
1.6 Quaternion-Type Numbers 80
References 82
2 Octonion Numbers 85
2.1 Octonion Arithmetic 85
2.1.1 Properties of the octonion multiplication 88
2.2 Functions on Octonions 99
2.2.1 Octonion exponential function 99
2.2.2 Octonion power and logarithm 102
2.2.3 Power function 102
2.2.4 Octonion logarithm 104
v2.2.5 Octonion trigonometric functions 108
References 110
3 Quaternions and Color Images 111
3.1 Digital Grayscale Images 111
3.2 Model 1: (2 2)-Mapping 112
3.3 Color Models 113
3.3.1 Family of RGB models 114
3.3.1.1 Script for the 2 2-mapping 117
3.3.2 Model 2: (3 3)-mapping 118
3.3.2.1 Script for the 3 3-mapping 120
3.3.3 RGBY model 122
3.3.3.1 Other RGB models 123
3.3.4 CMY model 124
3.3.5 XYZ model 124
3.3.6 HSI color model 126
3.3.7 HSV color model 130
3.3.8 YCbCr color model 131
3.4 Image Models with Octonions 133
3.4.1 Model 1 with the 3 3 cross window 133
3.4.2 Model 2 with the 2 3 window 134
3.4.3 Model 3 with the 3 2 window 135
3.4.4 Model 4 with the 2 4 window 135
3.4.5 Model 5 with the 4 2 window 136
3.4.6 Model 6 with the 3 3 window 136
3.4.7 Model 7 with the 3 4 full hexagon 137
3.4.8 Models with a sequence of images or videos 138
References 139
4 Color Images as 2-D Grayscale Images 141
4.1 2-D DFT of the Image 143
4.2 The (3 2) Color-to-Gray Model 146
4.3 The (3 1) Color-to-Gray Model 148
4.4 The (1 3) Color-to-Gray Model 149
4.5 Model of Color-to-Octonion Image 149
4.5.1 Octonion image with RGB color model 150
4.5.2 Octonion image with RGBY color model 152
4.5.3 Octonion image with CMYK color model 153
4.5.4 Model with two color images 153
4.6 Color images and Quaternion Multiplication 154
4.6.1 Quaternion multiplication of images 155
4.6.1.1 Script for the quaternion image multiplication 164
References 166
vi Contents5 1-D Quaternion and Octonion Discrete Fourier Transforms 169
5.1 Discrete Fourier Transform 170
5.1.1 Fast DFT, or FFT 172
5.1.2 Inverse FFT, or inverse radix-2 FFT 174
5.1.3 Decimation-in-frequency FFT 175
5.1.4 Paired FFT 177
5.2 1-D Right-Side QDFT 184
5.2.1 Special cases 188
5.2.2 Codes for the 1-D right-side QDFTs 189
5.3 1-D Left-Side QDFT 193
5.4 Octonion Discrete Fourier Transform 197
5.4.1 1-D right-side ODFT 200
5.4.2 1-D left-side ODFT 206
References 211
6 2-D Quaternion and Octonion Discrete Fourier Transforms 215
6.1 Tensor Representation of the Image 215
6.1.1 Tensor transform and direction images 226
6.2 Mapping of 2-D DFTs in Quaternion Algebra 230
6.3 2-D Two-Side QDFT 231
6.3.1 Column-row algorithm of the two-side QDFT 235
6.3.2 Fast algorithms for the 2-D QDFT 238
6.4 2-D Right-Side QDFT 243
6.4.1 2-D QDFT with column-row algorithm 244
6.5 Method of Symplectic Decomposition 246
6.6 Tensor Representation of the 2-D Right-Side QDFT 252
6.6.1 Program for tensor transform-based 2-D QDFT 256
6.7 Tensor Representation of Grayscale and Color Images 258
6.8 Direction Components of Quaternion Images 259
6.9 2-D Left-Side QDFT and Tensor Transform 265
6.9.1 Tensor representation of the 2-D left-side QDFT 266
6.10 2-D QDFT on the Hexagonal Lattice 274
6.10.1 2-D Hexagonal DFT 275
6.10.2 2-D Quaternion HDFT 276
6.11 2-D Two-Side Octonion DFT 280
6.12 2-D Right-Side ODFT 284
6.12.1 Tensor transform-based 2-D rs-ODFT 285
6.13 2-D Left-Side ODFT 288
6.13.1 Tensor transform and the 2-D ls-ODFT 290
References 292
7 Color Image Enhancement and QDFT 297
7.1 Transform-Based Image Enhancement 298
7.2 Quantitative Measure of Image Enhancement 299
Contents vii7.2.1 Quaternion transform-based image enhancement 301
7.3 New Color Image Quality Measure 302
7.4 Enhancement of Images by Colors 305
7.5 2-D Quaternion DFT in Image Enhancement 309
7.5.1 Examples and experimental results 312
References 315
8 Gradients, Face Recognition, Visualization, and Quaternions 319
8.1 Correlation Function and Gradients 320
8.2 Laplacian Gradient 322
8.2.1 Discrete gradient operators 323
8.3 Weber-Fechner Visibility Images 326
8.4 Image Visualization by the Michelson Contrast 333
8.5 EME-Type Measures and Visibility 335
8.6 Multiplicative Visibility Images 339
8.7 Two-Window Gradient Operators 340
8.8 Facial Image Representation 344
8.8.1 Facial image representation with the uniform LBP 349
8.9 Color Visibility Images 352
8.9.1 Other multiplicative visibility color images 354
8.10 Quaternion Image Gradients 356
8.10.1 1-in-4 Gradients 357
8.10.2 4-in-4 Gradients 359
8.11 Color Facial Image Representation 363
References 365
9 Color Image Restoration and QDFT 371
9.1 Problem of Image Restoration 371
9.2 Classical Model of Image Restoration 372
9.2.1 Optimal filter 373
9.3 Optimal Filtration of Color Images 378
References 384
Index 38
Index
2-D octonion DFT, 280, 284, 288
A
Agaian-Michelson enhancement
measure, 338
alpha-rooting, 298, 303
C
center-symmetric local binary
patterns, 349
color cube, 116
color direction images, 259
color enhancement measure, 302
color image enhancement, 305, 308
color image, 150
color models, 113, 123, 163
color splitting-signal, 254, 258
color visibility image, 352
color-in-quaternion image, 269
color-to-gray model, 146, 148
color-to-gray transform, 143
compass gradients, 360
complex conjugate, 5
complex exponential function, 8
complex number, 3
contrast, 333
convolution, 323
D
DFT, 143
direction images, 226, 262, 273
discrete Fourier transform, 143
division of quaternions, 29
E
edge, 322
EME entropy measure, 338
EME measure, 299, 312, 326
EME visibility operator, 336
EMEC measure, 302, 314
enhancement measure, 299
exponential function, 144
F
facial color image recognition, 363
facial image processing, 344
facial image representation, 318
facial image, 346
G
Gaussian function, 322, 344
generator of the signal, 268
golden ratio, 23
gradient, 322–323
Grigoryan-Agaian enhancement
measure, 337
H
hexagonal lattice, 274
hexagonal QDFT, 274
hue component, 126, 129
I
image brightness, 140
image correlation, 320
image enhancement, 296
image restoration, 369
389imaginary unit, 3, 12–13, 200
impulse response, 380
inner product, 36
intensity component, 129
inverse 2-D QDFT, 312
L
Laplacian gradient, 322, 324, 343
LBP image, 346, 349
local binary patterns, 318
M
mean operator, 327
mean-square-root measure, 339
Michelson visibility color
image, 355
Michelson visibility image, 333–334
Michelson visibility measure, 337
Michelson visibility operator, 339
models of mapping, 116
module of quaternion, 28
module, 5
multiplication of complex
numbers, 4, 10
multiplication of quaternions,
14, 19, 31, 154
multiplicative visibility image, 339
N
non-uniform pattern, 350
O
octonion DFT, 197, 200, 206, 210
octonion exponential function, 197
octonion image, 149, 152, 283
octonion signal, 203
octonion, 1
optimal filter, 381
P
paired representation, 264
paired splitting-signals, 264
Panetta-Agaian measure, 338
perpendicular vectors, 37
phase of the DFT, 145
pixel, 110
polar form, 8
Prewitt compass gradients, 361
principle of superposition, 273
pure quaternion image, 118
pure quaternion, 17, 19, 21
pure unit quaternion, 31
Q
quaternion conjugate, 27
quaternion DFT, 213, 232, 236,
240, 243
quaternion exponent, 75
quaternion gradient image, 357
quaternion gradient, 318
quaternion hyperbolic sine, 79
quaternion image enhancement, 301
quaternion image gradient, 356
quaternion image, 113, 116,
264, 276
quaternion logarithm, 74, 76
quaternion numbers, 1, 13, 16
quaternion sine, 77
quaternion Sobel gradient, 357
quaternion splitting-signals,
278, 286
R
retinex, 303, 314
Robinson compass gradients, 361
S
sampling points, 345
saturation component, 126
sine-Michelson visibility image, 334
Sobel operators, 324
splitting-signal, 216, 219,
222, 229, 254
square of the quaternion, 19
symplectic decomposition, 246
390 IndexT
Table of multiplications, 15
tensor representation, 213, 216, 252,
266, 276, 285
tensor transform, 272, 290
two-window gradient, 340
U
uniform LBP image, 349, 365
uniform LBP table, 351
uniform pattern, 349
V
value component, 130
vector product, 39
W
Weber ratio, 339
Weber visibility color image, 354
Weber visibility image, 328
Weber visibility operators, 330, 332
Weber-Fechner law, 326
Wiener filtration, 369, 373
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