Related papers: The geometry of proper quaternion random variables
Over the past few years, the applications of dual-quaternions have not only developed in many different directions but has also evolved in exciting ways in several areas. As dual-quaternions offer an efficient and compact symbolic form with…
This paper revisits the topic of rotation estimation through the lens of special unitary matrices. We begin by reformulating Wahba's problem using $SU(2)$ to derive multiple solutions that yield linear constraints on corresponding…
We study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries (considering each quaternion as a $2\times 2$ complex matrix). We define the ensemble by the matrix probability distribution function that is…
Many phenomena are described by bivariate signals or bidimensional vectors in applications ranging from radar to EEG, optics and oceanography. The time-frequency analysis of bivariate signals is usually carried out by analyzing two separate…
Although regular conditional distributions (r.c.d.) are well-defined and widely used measure-theoretic objects, they can violate our intuition from the classical definition of a conditional probability given an event. For that purpose, the…
A novel approach towards the spectral analysis of stationary random bivariate signals is proposed. Using the Quaternion Fourier Transform, we introduce a quaternion-valued spectral representation of random bivariate signals seen as…
We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a…
The spectral density of random matrices is studied through a quaternionic generalisation of the Green's function, which precisely describes the mean spectral density of a given matrix under a particular type of random perturbation. Exact…
Dual quaternions have gained significant attention due to their wide applications in areas such as multi-agent formation control, 3D motion modeling, and robotics. A fundamental aspect in dual quaternion research involves the projection…
The interior structure of arbitrary sets of quaternion units is analyzed using general methods of the theory of matrices. It is shown that the units are composed of quadratic combinations of fundamental objects having a dual mathematical…
Quaternions are important for a wide variety of rotation-related problems in computer graphics, machine vision, and robotics. We study the nontrivial geometry of the relationship between quaternions and rotation matrices by exploiting the…
The immediate purpose of the paper was neither to review the basic definitions of percolation theory nor to rehearse the general physical notions of universality and renormalization (an important technique to be described in Part Two). It…
The presence of second-order smoothness for objective functions of optimization problems can provide valuable information about their stability properties and help us design efficient numerical algorithms for solving these problems. Such…
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…
Objects' rigid motions in 3D space are described by rotations and translations of a highly-correlated set of points, each with associated $x,y,z$ coordinates that real-valued networks consider as separate entities, losing information.…
We introduce real second-order freeness in second-order noncommutative probability spaces. We demonstrate that under this definition, three real models of random matrices, namely real Ginibre matrices, Gaussian orthogonal matrices, and real…
We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The…
In recursive state estimation, numerical error can play a major role in an algorithm's overall performance and reliability. Roundoff errors due to finite precision arithmetic can violate theoretical guarantees, leading to asymmetric and…
Solving dual quaternion equations is an important issue in many fields such as scientific computing and engineering applications. In this paper, we first introduce a new metric function for dual quaternion matrices. Then, we reformulate…
Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear…