Hypercomplex number

Abstract: "Introduction to Theoretical Kinematics" provides a uniform presentation of the mathematical foundations required for studying the movement of a kinematic chain that makes up robot arms, mechanical hands, walking machines, and similar mechanisms It is a concise and readable introduction that takes a more modern approach than other kinematics texts and introduces several useful derivations that are new to the literature The author employees a unique format, highlighting the similarity of the mathematical results for planar, spherical, and spatial cases by studying them all in each chapter rather than as separate topics For the first time, he applies to kinematic theory two tools of modern mathematics--the theory of multivectors and the theory of Clifford algebras--that serve to clarify the seemingly arbitrary nature of the construction of screws and dual quaternions The first two chapters formulate the matrices that represent planar, spherical, and spatial displacements and examine a continuous set of displacements which define a continuous movement of a body, introducing the "tangent operator" Chapter 3 focuses on the tangent operators of spatial motion as they are reassembled into six-dimensional vectors or screws, placing these in the modern setting of multivector algebra Clifford algebras are used in chapter 4 to unify the construction of various hypercomplex "quaternion" numbers Chapter 5 presents the elementary formulas that compute the degrees of freedom, or mobility, of kinematic chains, and chapter 6 defines the structure equations of these chains in terms of matrix transformations The last chapter computes the quaternion form ofthe structure equations for ten specific mechanisms These equations define parameterized manifolds in the Clifford algebras, or "image spaces," associated with planar, spherical, and spatial displacements McCarthy reveals a particularly interesting result by showing that these parameters can be mathematically manipulated to yield hyperboloids or intersections of hyperboloids

Abstract: Hypercomplex Fourier transforms based on quaternions have been proposed by several groups for use in image processing, particularly of color images. So far, however, there has not been a coherent explanation of what the spectral domain coefficients produced by a hypercomplex Fourier transform represent and this paper attempts to present such an explanation for the first time making use of the polar form of a quaternion and a separation of a quaternion spectral coefficient into components parallel and perpendicular to the hypercomplex exponentials in the transform (the basis functions).

Abstract: In this work, we move beyond the traditional complex-valued representations, introducing more expressive hypercomplex representations to model entities and relations for knowledge graph embeddings. More specifically, quaternion embeddings, hypercomplex-valued embeddings with three imaginary components, are utilized to represent entities. Relations are modelled as rotations in the quaternion space. The advantages of the proposed approach are: (1) Latent inter-dependencies (between all components) are aptly captured with Hamilton product, encouraging a more compact interaction between entities and relations; (2) Quaternions enable expressive rotation in four-dimensional space and have more degree of freedom than rotation in complex plane; (3) The proposed framework is a generalization of ComplEx on hypercomplex space while offering better geometrical interpretations, concurrently satisfying the key desiderata of relational representation learning (i.e., modeling symmetry, anti-symmetry and inversion). Experimental results demonstrate that our method achieves state-of-the-art performance on four well-established knowledge graph completion benchmarks.

Abstract: The construction of Gabor's (1946) complex signal-which is also known as the analytic signal-provides direct access to a real one-dimensional (1-D) signal's local amplitude and phase. The complex signal is built from a real signal by adding its Hilbert transform-which is a phase-shifted version of the signal-as an imaginary part to the signal. Since its introduction, the complex signal has become an important tool in signal processing, with applications, for example, in narrowband communication. Different approaches to an n-D analytic or complex signal have been proposed in the past. We review these approaches and propose the hypercomplex signal as a novel extension of the complex signal to n-D. This extension leads to a new definition of local phase, which reveals information on the intrinsic dimensionality of the signal. The different approaches are unified by expressing all of them as combinations of the signal and its partial and total Hilbert transforms. Examples that clarify how the approaches differ in their definitions of local phase and amplitude are shown. An example is provided for the two-dimensional (2-D) hypercomplex signal, which shows how the novel phase concept can be used in texture segmentation.