Eckhard Hitzer

Quaternion Fourier Transform on Quaternion Fields and Generalizations

Abstract.We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear (GL) transformation behavior of the QFT with matrices, Clifford geometric algebra and with examples. We finally arrive at wide-ranging non-commutative multivector FT generalizations of the QFT. Examples given are new volume-time and spacetime algebra Fourier transformations.

Directional Uncertainty Principle for Quaternion Fourier Transform

Abstract.This paper derives a new directional uncertainty principle for quaternion valued functions subject to the quaternion Fourier transformation. This can be generalized to establish directional uncertainty principles in Clifford geometric algebras with quaternion subalgebras. We demonstrate this with the example of a directional spacetime algebra function uncertainty principle related to multivector wave packets.

Quaternion and Clifford Fourier Transforms and Wavelets

Quaternion and Clifford Fourier and wavelet transformations generalize the classical theory to higher dimensions and are becoming increasingly important in diverse areas of mathematics, physics, computer science and engineering. This edited volume presents the state of the art in these hypercomplex transformations. The Clifford algebras unify Hamiltons quaternions with Grassmann algebra. A Clifford algebra is a complete algebra of a vector space and all its subspaces including the measurement of volumes and dihedral angles between any pair of subspaces. Quaternion and Clifford algebras permit the systematic generalization of many known concepts. This book provides comprehensive insights into current developments and applications including their performance and evaluation. Mathematically, it indicates where further investigation is required. For instance, attention is drawn to the matrix isomorphisms for hypercomplex algebras, which will help readers to see that software implementations are within our grasp. It also contributes to a growing unification of ideas and notation across the expanding field of hypercomplex transforms and wavelets. The first chapter provides a historical background and an overview of the relevant literature, and shows how the contributions that follow relate to each other and to prior work. The book will be a valuable resource for graduate students as well as for scientists and engineers.

Square Roots of –1 in Real Clifford Algebras

It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Systematic research has been done [33] on the biquaternion roots of –1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cl 3,0 of ℝ3. Further research on general algebras Cl p,q has explicitly derived the geometric roots of –1for p + q≤4 [20]. The current research abandons this dimension limit and uses the Clifford algebra to matrix algebra isomorphisms in order to algebraically characterize the continuous manifolds of square roots of –1f ound in the different types of Clifford algebras, depending on the type of associated ring (ℝ,ℍ,ℝ2,ℍ2, or ℂ). At the end of the chapter explicit computer generated tables of representative square roots of –1 are given for all Clifford algebras with n = 5,7, and s = 3 (mod 4) with the associated ring ℂ. This includes, e.g., Cl 0,5 important in Clifford analysis, and Cl 4,1 which in applications is at the foundation of conformal geometric algebra. All these roots of –1 are immediately useful in the construction of new types of geometric Clifford–Fourier transformations.

A General Geometric Fourier Transform

The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straightforward definition of a general geometric Fourier transform covering most versions in the literature. We show which constraints are additionally necessary to obtain certain features such as linearity or a shift theorem. As a result, we provide guidelines for the target-oriented design of yet unconsidered transforms that fulfill requirements in a specific application context. Furthermore, the standard theorems do not need to be shown in a slightly different form every time a new geometric Fourier transform is developed since they are proved here once and for all.

The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations

The two-sided quaternionic Fourier transformation (QFT) was introduced in [2] for the analysis of 2D linear time-invariant partial-differential systems. In further theoretical investigations [4, 5] a special split of quaternions was introduced, then called ±split. In the current chapter we analyze this split further, interpret it geometrically as an orthogonal 2D planes split (OPS), and generalize it to a freely steerable split of H into two orthogonal 2D analysis planes. The new general form of the OPS split allows us to find new geometric interpretations for the action of the QFT on the signal. The second major result of this work is a variety of new steerable forms of the QFT, their geometric interpretation, and for each form, OPS split theorems, which allow fast and efficient numerical implementation with standard FFT software.

Optimal learning rates for clifford neurons

Neural computation in Clifford algebras, which include familiar complex numbers and quaternions as special cases, has recently become an active research field. As always, neurons are the atoms of computation. The paper provides a general notion for the Hessian matrix of Clifford neurons of an arbitrary algebra. This new result on the dynamics of Clifford neurons then allows the computation of optimal learning rates. A thorough discussion of error surfaces together with simulation results for different neurons is also provided. The presented contents should give rise to very efficient second-order training methods for Clifford Multilayer perceptrons in the future.

Multivector differential calculus

Universal geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. This paper treats the fundamentals of the multivector differential calculus part of geometric calculus. The multivector differential is introduced, followed by the multivector derivative and the adjoint of multivector functions. The basic rules of multivector differentiation are derived explicitly, as well as a variety of basic multivector derivatives. Finally factorization, which relates functions of vector variables and multivector variables is discussed, and the concepts of both simplicial variables and derivatives are explained. Everything is proven explicitly in a very elementary level step by step approach. The paper is thus intended to serve as reference material, providing a number of details, which are usually skipped in more advanced discussions of the subject matter. The arrangement of the material closely followschapter 2 of [3].

Moment Invariants for 2D Flow Fields Using Normalization

The analysis of 2D flow data is often guided by the search for characteristic structures with semantic meaning. One way to approach this question is to identify structures of interest by a human observer. The challenge then, is to find similar structures in the same or other datasets on different scales and orientations. In this paper, we propose to use moment invariants as pattern descriptors for flow fields. Moment invariants are one of the most popular techniques for the description of objects in the field of image recognition. They have recently also been applied to identify 2D vector patterns limited to the directional properties of flow fields. In contrast to previous work, we follow the intuitive approach of moment normalization, which results in a complete and independent set of translation, rotation, and scaling invariant flow field descriptors. They also allow to distinguish flow features with different velocity profiles. We apply the moment invariants in a pattern recognition algorithm to a real world dataset and show that the theoretic results can be extended to discrete functions in a robust way.

OPS‐QFTs: A New Type of Quaternion Fourier Transforms Based on the Orthogonal Planes Split with One or Two General Pure Quaternions

We explain the orthogonal planes split (OPS) of quaternions based on the arbitrary choice of one or two linearly independent pure unit quaternions f,g. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by f,g, or by only one pure unit quaternion f, comment on their geometric meaning, and establish inverse transformations.