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The HR-Calculus: Enabling Information Processing with Quaternion Algebra

Danilo P. Mandic, Sayed Pouria Talebi, Clive Cheong Took, Yili Xia, Dongpo Xu, Min Xiang, Pauline Bourigault

TL;DR

The foundations of the HR-calculus are revised and the required tools for deriving adaptive learning techniques suitable for dealing with quaternion-valued signals, such as the gradient operator, chain and product derivative rules, and Taylor series expansion are presented.

Abstract

From their inception, quaternions and their division algebra have proven to be advantageous in modelling rotation/orientation in three-dimensional spaces and have seen use from the initial formulation of electromagnetic filed theory through to forming the basis of quantum filed theory. Despite their impressive versatility in modelling real-world phenomena, adaptive information processing techniques specifically designed for quaternion-valued signals have only recently come to the attention of the machine learning, signal processing, and control communities. The most important development in this direction is introduction of the HR-calculus, which provides the required mathematical foundation for deriving adaptive information processing techniques directly in the quaternion domain. In this article, the foundations of the HR-calculus are revised and the required tools for deriving adaptive learning techniques suitable for dealing with quaternion-valued signals, such as the gradient operator, chain and product derivative rules, and Taylor series expansion are presented. This serves to establish the most important applications of adaptive information processing in the quaternion domain for both single-node and multi-node formulations. The article is supported by Supplementary Material, which will be referred to as SM.

The HR-Calculus: Enabling Information Processing with Quaternion Algebra

TL;DR

The foundations of the HR-calculus are revised and the required tools for deriving adaptive learning techniques suitable for dealing with quaternion-valued signals, such as the gradient operator, chain and product derivative rules, and Taylor series expansion are presented.

Abstract

From their inception, quaternions and their division algebra have proven to be advantageous in modelling rotation/orientation in three-dimensional spaces and have seen use from the initial formulation of electromagnetic filed theory through to forming the basis of quantum filed theory. Despite their impressive versatility in modelling real-world phenomena, adaptive information processing techniques specifically designed for quaternion-valued signals have only recently come to the attention of the machine learning, signal processing, and control communities. The most important development in this direction is introduction of the HR-calculus, which provides the required mathematical foundation for deriving adaptive information processing techniques directly in the quaternion domain. In this article, the foundations of the HR-calculus are revised and the required tools for deriving adaptive learning techniques suitable for dealing with quaternion-valued signals, such as the gradient operator, chain and product derivative rules, and Taylor series expansion are presented. This serves to establish the most important applications of adaptive information processing in the quaternion domain for both single-node and multi-node formulations. The article is supported by Supplementary Material, which will be referred to as SM.
Paper Structure (13 sections, 125 equations, 11 figures)

This paper contains 13 sections, 125 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Overview of Quaternions
  3. Three-Dimensional Rotations
  4. Main Advantages of Quaternions
  5. Generalising Rotations: Quaternion Involution and the Augmented Representation
  6. The $\mathbb{HR}$-calculus
  7. Multiplication, Rotation, and Conjugation Rules in the $\mathbb{HR}$-Calculus
  8. Chain Derivative Rule in the $\mathbb{HR}$-Calculus
  9. Information Processing and Decision Making
  10. Quaternion-Valued Least Mean Square (QLMS)
  11. The Quaternion Kalman Filter
  12. Linear Dynamic Programming and Initial Elements of Reinforcement Learning
  13. Conclusion

Figures (11)

  • Figure 1: Schematic of a rotation around $\eta$ by an angle of $\theta$, with $q_{\text{pre}}$ and $q_{\text{post}}$ pointing to the pre- and post-rotation orientation of the object in question.
  • Figure 2: Setting of the inertial body motion sensor with the fixed coordinate system (blue), sensor coordinate system (red), and Euler angles (green). The "N" axis is used as a visual guide to indicate the yaw angle. Figure taken from PouriaPhD.
  • Figure 3: Absolute value of the phase of quaternion-valued body motion signal employing the QLMS and a quadrivariate real-valued LMS approach.
  • Figure 4: System voltage, $q_{\left[n\right]}$, positive sequence element, $q^{+}_{\left[n\right]}$, and negative sequence element, $q^{-}_{\left[n\right]}$, of an unbalanced three-phase system suffering from an 80% drop in the amplitude of one phase and $20$ degree shifts in the other phases. Note that p.u. stands for per unit, i.e. the variables are normalised to their nominal value.
  • Figure 5: Frequency estimation performance (bias and MSE) using the derived quaternion-valued model, quaternion frequency estimator (QFE), as compared to complex-valued linear (CLFE) and widely liner (CWLFE) techniques detailed in PouriaPhD. The results pertain to the unbalanced system in Figure \ref{['Fig:Plot2']}.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Remark 10
  • ...and 2 more