Quaternion Fourier Transforms for Signal and Image Processing

Todd A. Ell, Engineering Fellow, Goodrich Sensors and Integrated Systems, Burnsville, MN, USA. Nicolas Le Bihan, Researcher, CNRS, Grenoble, France. Stephen J. Sangwine, Senior Lecturer, University of Essex, United Kingdom.

• Nomenclature ixPreface xiIntroduction xiiiChapter 1. Quaternion algebra 11.1. Definitions 11.2. Properties 21.3. Exponential and logarithm of a quaternion 71.3.1. Exponential of a pure quaternion 71.3.2. Exponential of a full quaternion 91.3.3. Logarithm of a quaternion 101.4. Representations 111.4.1. Polar forms 111.4.2. The ℂj -pair notation 151.4.3. ℝ and ℂ matrix representations 171.5. Powers of a quaternion 181.6. Subfields 18Chapter 2. Geometric applications 212.1. Euclidean geometry (3D and 4D) 212.1.1. 3D reflections 222.1.2. 3D rotations 222.1.3. 3D shears 242.1.4. 3D dilations 242.1.5. 4D reflections 252.1.6. 4D rotations 252.2. Spherical geometry 262.3. Projective space (3D) 282.3.1. Systems of linear quaternion functions 312.3.2. Projective transformations 33Chapter 3. Quaternion fourier transforms 353.1. 1D quaternion Fourier transforms 383.1.1. Definitions 383.1.2. Basic transform pairs 403.1.3. Decompositions 423.1.4. Inter-relationships between definitions 453.1.5. Convolution and correlation theorems 473.2. 2D quaternion Fourier transforms 483.2.1. Definitions 483.2.2. Basic transform pairs 523.2.3. Decompositions 543.2.4. Inter-relationships between definitions 553.3. Computational aspects 573.3.1. Coding 573.3.2. Verification 623.3.3. Verification of transforms 62Chapter 4. Signal and image processing 674.1. Generalized convolution 674.1.1. Classical grayscale image convolution filters 674.1.2. Color images as quaternion arrays 704.1.3. Quaternion convolution 704.1.4. Quaternion image spectrum 734.2. Generalized correlation 794.2.1. Classical correlation and phase correlation 814.2.2. Quaternion correlation 864.2.3. Quaternion phase correlation 884.3. Instantaneous phase and amplitude of complex signals 914.3.1. Important properties of 1D QFT of a complex signal z(t) 914.3.2. Hilbert transform and right-sided quaternion spectrum 964.3.3. The quaternion signal associated with a complex signal 984.3.4. Instantaneous amplitude and phase 1014.3.5. The instantaneous frequency of a complex signal 1024.3.6. Examples 1044.3.7. The quaternion Wigner-Ville distribution of a complex signal 1094.3.8. Time marginal 1134.3.9. The mean frequency formula 113Bibliography 117Index 123

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