第1页 / 共405页
第2页 / 共405页
第3页 / 共405页
第4页 / 共405页
第5页 / 共405页
第6页 / 共405页
第7页 / 共405页
第8页 / 共405页
Quaternion and Octonion Color Image Processing With Matlab Grigoryan 2018.pdf
PM279 Cover
Terms of Use
Copyright
Table of Contents
Preface
Chapter 1 Complex and Hypercomplex Numbers
1.1 Complex and Hypercomplex Numbers
1.2 Vector Space and Pure Quaternions
1.3 Quaternion Multiplication and Rotation
1.4 The Quaternion Exponential Function
1.5 Quaternion Trigonometric and Hyperbolic Functions
1.6 Quaternion-Type Numbers
Chapter 2
Octonion Numbers
2.1 Octonion Arithmetic
2.2 Functions on Octonions
Chapter 3 Quaternions and Color Images
3.1 Digital Grayscale Images
3.2 Model 1: (2×2)-Mapping
3.3 Color Models
3.4 Image Models with Octonions
Chapter 4 Color Images as 2-D Grayscale Images
4.1 2-D DFT of the Image
4.2 The (3x2) Color-to-Gray Model
4.3 The (3x1) Color-to-Gray Model
4.4 The (1x3) Color-to-Gray Model
4.5 Model of Color-to-Octonion Image
4.6 Color images and Quaternion Multiplication
Chapter 5 1-D Quaternion and Octonion Discrete Fourier Transforms
5.1 Discrete Fourier Transform
5.2 1-D Right-Side QDFT
5.3 1-D Left-Side QDFT
5.4 Octonion Discrete Fourier Transform
Chapter 6 2-D Quaternion and Octonion Discrete Fourier Transforms
6.1 Tensor Representation of the Image
6.2 Mapping of 2-D DFTs in Quaternion Algebra
6.3 2-D Two-Side QDFT
6.4 2-D Right-Side QDFT
6.5 Method of Symplectic Decomposition
6.6 Tensor Representation of the 2-D Right-Side QDFT
6.7 Tensor Representation of Grayscale and Color Images
6.8 Direction Components of Quaternion Images
6.9 2-D Left-Side QDFT and Tensor Transform
6.10 2-D QDFT on the Hexagonal Lattice
6.11 2-D Two-Side Octonion DFT
6.12 2-D Right-Side ODFT
6.13 2-D Left-Side ODFT
Chapter 7 Color Image Enhancement and QDFT
7.1 Transform-Based Image Enhancement
7.2 Quantitative Measure of Image Enhancement
7.3 New Color Image Quality Measure
7.4 Enhancement of Images by Colors
7.5 2-D Quaternion DFT in Image Enhancement
Chapter 8 Gradients, Face Recognition, Visualization, and Quaternions
8.1 Correlation Function and Gradients
8.2 Laplacian Gradient
8.3 Weber-Fechner Visibility Images
8.4 Image VIsualization by the Michelson Contrast
8.5 EME-Type Measures and Visibility
8.6 Multiplicative Visibility Images
8.7 Two-Window Gradient Operators
8.8 Facial Image Representation
8.9 Color Visibility Images
8.10 Quaternion Image Gradients
8.11 Color Facial Image Representation
Chapter 9 Color Image Restoration and QDFT
9.1 Problem of Image Restoration
9.2 Classical Model of Image Restoration
9.3 Optimal Filtration of Color Images
Index
Author Biographies
Color image processing has attracted much interest in recent years. The use of color
in image processing is motivated by the facts that (1) the human eyes can discern
thousands of colors, and image processing is used both for human interaction and
computer interpretation; (2) a color image comprises more information than a grayscale
image; (3) color features are robust to several image-processing procedures (for
example, to the translation and rotation of the regions of interest); (4) color features are
efficiently used in many vision tasks, including object recognition and tracking, image
segmentation and retrieval, image registration, etc.; and (5) color is necessary in many
real-life applications such as visual communications, multimedia systems, fashion and
food industries, computer vision, entertainment, consumer electronics, production
printing and proofing, digital photography, biometrics, digital artwork reproduction,
industrial inspection, and biomedical applications. Finally, the enormous number of color
images that are constantly uploaded to the Internet require new approaches to visual
media creation, retrieval, processing, and applications. This also gives us new
opportunities to create a number of large, visual data-driven applications. The main goal
of this book is to provide the mathematics of quaternions and octonions and to show
how they can be used in emerging areas of color image processing.
P.O. Box 10
Bellingham, WA 98227-0010
ISBN: 9781510611368
SPIE Vol. No.: PM279
Library of Congress Cataloging-in-Publication Data
Names: Grigoryan, Artyom M., author. | Agaian, S. S., author.
Title: Quaternion and octonion color image processing with MATLAB / Artyom
M. Grigoryan and Sos S. Agaian.
Description: Bellingham, Washington : SPIE Press, [2018] | Includes bibliographical
references and index.
Identifiers: LCCN 2017013782 | ISBN 9781510611351 (print : alk. paper) | ISBN
1510611355 (print : alk. paper) | ISBN 9781510611368 (PDF) | ISBN 1510611363
(PDF)
ISBN
9781510611382 (Kindle/Mobi) | ISBN 151061138X (Kindle/Mobi)
ISBN 9781510611375 (ePub)
|
|
ISBN 1510611371 (ePub)
|
Subjects: LCSH: Image processing Digital techniques Mathematics. | Quaternions. |
Color vision. | Color. | MATLAB.
Classification: LCC TA1637.5 .G75 2017 | DDC 621.36/7028553 dc23 LC record
available at https://lccn.loc.gov/2017013782
Published by
SPIE
P.O. Box 10
Bellingham, Washington 98227-0010 USA
Phone: +1 360.676.3290
Fax: +1 360.647.1445
Email: books@spie.org
Web: http://spie.org
Copyright © 2018 Society of Photo-Optical Instrumentation Engineers (SPIE)
All rights reserved. No part of this publication may be reproduced or distributed in
any form or by any means without written permission of the publisher.
The content of this book reflects the work and thought of the authors. Every effort has
been made to publish reliable and accurate information herein, but the publisher is not
responsible for the validity of the information or for any outcomes resulting from
reliance thereon.
Printed in the United States of America.
First Printing.
For updates to this book, visit http://spie.org and type “PM279” in the search field.
Contents
Preface
1 Complex and Hypercomplex Numbers
1.1 Complex and Hypercomplex Numbers
1.1.1 Complex arithmetic
1.1.1.1 Geometry of complex numbers
Basic operations
Properties of multiplication of quaternions
1.1.2 Quaternion numbers
1.1.3 Rules for multiplication
1.1.4
1.1.5
Vector Space and Pure Quaternions
Inner product (or dot product)
1.2.1
1.2.2
Vector product
1.2
1.3 Quaternion Multiplication and Rotation
The 2nd rotation: Multiplication by a perpendicular vector
1.3.1 Multiplication and sum of elementary rotations
1.3.2 Rotation: Multiplication by a perpendicular vector
1.3.3
1.3.4 Multiplication: Rotation of any vector
1.3.5 Matrix representation of rotation
1.3.6
The Quaternion Exponential Function
Addition of rotations in 3-D space
1.4
1.5 Quaternion Trigonometric and Hyperbolic Functions
1.6 Quaternion-Type Numbers
References
2 Octonion Numbers
2.1 Octonion Arithmetic
2.2
Properties of the octonion multiplication
2.1.1
Functions on Octonions
2.2.1 Octonion exponential function
2.2.2 Octonion power and logarithm
2.2.3
2.2.4 Octonion logarithm
Power function
v
ix
1
2
2
6
12
13
16
19
35
36
39
41
41
48
52
54
55
60
65
77
80
82
85
85
88
99
99
102
102
104
vi
Contents
2.2.5 Octonion trigonometric functions
References
3 Quaternions and Color Images
3.1 Digital Grayscale Images
3.2 Model 1: (2 2)-Mapping
3.3 Color Models
Family of RGB models
3.3.1.1
3.3.1
3.3.2 Model 2: (3 3)-mapping
Script for the 2 2-mapping
Script for the 3 3-mapping
3.3.2.1
3.4
3.3.3 RGBY model
3.3.3.1 Other RGB models
YCbCr color model
3.3.4 CMY model
3.3.5
XYZ model
3.3.6 HSI color model
3.3.7 HSV color model
3.3.8
Image Models with Octonions
3.4.1 Model 1 with the 3 3 cross window
3.4.2 Model 2 with the 2 3 window
3.4.3 Model 3 with the 3 2 window
3.4.4 Model 4 with the 2 4 window
3.4.5 Model 5 with the 4 2 window
3.4.6 Model 6 with the 3 3 window
3.4.7 Model 7 with the 3 4 full hexagon
3.4.8 Models with a sequence of images or videos
References
4 Color Images as 2-D Grayscale Images
2-D DFT of the Image
4.1
The (3 2) Color-to-Gray Model
4.2
The (3 1) Color-to-Gray Model
4.3
The (1 3) Color-to-Gray Model
4.4
4.5 Model of Color-to-Octonion Image
4.5.1 Octonion image with RGB color model
4.5.2 Octonion image with RGBY color model
4.5.3 Octonion image with CMYK color model
4.5.4 Model with two color images
4.6 Color images and Quaternion Multiplication
4.6.1 Quaternion multiplication of images
4.6.1.1
Script for the quaternion image multiplication
References
108
110
111
111
112
113
114
117
118
120
122
123
124
124
126
130
131
133
133
134
135
135
136
136
137
138
139
141
143
146
148
149
149
150
152
153
153
154
155
164
166
Contents
5 1-D Quaternion and Octonion Discrete Fourier Transforms
5.1 Discrete Fourier Transform
Fast DFT, or FFT
Inverse FFT, or inverse radix-2 FFT
5.1.1
5.1.2
5.1.3 Decimation-in-frequency FFT
5.1.4
1-D Right-Side QDFT
5.2.1
Special cases
5.2.2 Codes for the 1-D right-side QDFTs
1-D Left-Side QDFT
5.3
5.4 Octonion Discrete Fourier Transform
Paired FFT
5.2
5.4.1
5.4.2
References
1-D right-side ODFT
1-D left-side ODFT
6 2-D Quaternion and Octonion Discrete Fourier Transforms
6.1
Tensor Representation of the Image
6.1.1
Tensor transform and direction images
6.2 Mapping of 2-D DFTs in Quaternion Algebra
6.3
2-D Two-Side QDFT
6.3.1 Column-row algorithm of the two-side QDFT
6.3.2
2-D Right-Side QDFT
6.4.1
2-D QDFT with column-row algorithm
Fast algorithms for the 2-D QDFT
6.4
6.5 Method of Symplectic Decomposition
6.6
Tensor Representation of the 2-D Right-Side QDFT
6.6.1
Program for tensor transform-based 2-D QDFT
Tensor Representation of Grayscale and Color Images
6.7
6.8 Direction Components of Quaternion Images
6.9
2-D Left-Side QDFT and Tensor Transform
6.9.1
Tensor representation of the 2-D left-side QDFT
6.10 2-D QDFT on the Hexagonal Lattice
6.10.1 2-D Hexagonal DFT
6.10.2 2-D Quaternion HDFT
6.11 2-D Two-Side Octonion DFT
6.12 2-D Right-Side ODFT
6.12.1 Tensor transform-based 2-D rs-ODFT
6.13 2-D Left-Side ODFT
6.13.1 Tensor transform and the 2-D ls-ODFT
References
7 Color Image Enhancement and QDFT
Transform-Based Image Enhancement
7.1
7.2 Quantitative Measure of Image Enhancement
vii
169
170
172
174
175
177
184
188
189
193
197
200
206
211
215
215
226
230
231
235
238
243
244
246
252
256
258
259
265
266
274
275
276
280
284
285
288
290
292
297
298
299
相关推荐
2023年江西萍乡中考道德与法治真题及答案.doc
2012年重庆南川中考生物真题及答案.doc
2013年江西师范大学地理学综合及文艺理论基础考研真题.doc
2020年四川甘孜小升初语文真题及答案I卷.doc
2020年注册岩土工程师专业基础考试真题及答案.doc
2023-2024学年福建省厦门市九年级上学期数学月考试题及答案.doc
2021-2022学年辽宁省沈阳市大东区九年级上学期语文期末试题及答案.doc
2022-2023学年北京东城区初三第一学期物理期末试卷及答案.doc
2018上半年江西教师资格初中地理学科知识与教学能力真题及答案.doc
2012年河北国家公务员申论考试真题及答案-省级.doc
2020-2021学年江苏省扬州市江都区邵樊片九年级上学期数学第一次质量检测试题及答案.doc
2022下半年黑龙江教师资格证中学综合素质真题及答案.doc
