Quaternion and Clifford Fourier Transforms and Wavelets

Quaternion and Clifford Fourier and Wavelet Transformations as refinement and generalization of the one-dimensional complex Fourier and Wavelet theory play an increasing role in different areas of mathematics, physics, computer science, engineering and beyond. This edited volume presents and discusses the state of the art of these hypercomplex Fourier and Wavelet Transformations, both from the side of theoretical investigations and in the context of applications. A historical introduction gives a representative overview of the relevant literature in the field of hypercomplex signal processing, hypercomplex algebra and analysis, complex numbers and quaternions, and abstract mathematics. The book aims at providing comprehensive insight into the wide variety of current developments and of concrete implementations including their merits, performance and evaluation. On the mathematical side, it indicates where further systematic mathematical investigation is required and helps in communicating new results to practitioners. For instance, attention is drawn to the matrix isomorphisms for hypercomplex algebras which will help to see that software implementations are within easy reach. Finally, the effort to unify the notation will make the book suitable for graduate students as well as researchers and engineers.