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Quaternion contains one real part and three imaginary parts, which provided a more expressive hypercomplex space for learning knowledge graph. Existing quaternion embedding models measure the plausibility of a triplet either through…
The field of deep learning has seen significant advancement in recent years. However, much of the existing work has been focused on real-valued numbers. Recent work has shown that a deep learning system using the complex numbers can be…
The use of quaternions as a novel tool for color image representation has yielded impressive results in color image processing. By considering the color image as a unified entity rather than separate color space components, quaternions can…
Since their first applications, Convolutional Neural Networks (CNNs) have solved problems that have advanced the state-of-the-art in several domains. CNNs represent information using real numbers. Despite encouraging results, theoretical…
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…
A formal description of quaternions by means of exterior calculus is presented. Considering a three-dimensional space-time characterized by three time-like coordinates, we have been able to consistently recover a suitable formulation of…
Machine Learning algorithms have had a profound impact on the field of computer science over the past few decades. These algorithms performance is greatly influenced by the representations that are derived from the data in the learning…
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…
This paper revisits the little-known Gibbs-Rodrigues representation of rotations in a three-dimensional space and demonstrates a set of algorithms for handling it. In this representation the rotation is itself represented as a…
Imitation learning techniques have been used as a way to transfer skills to robots. Among them, dynamic movement primitives (DMPs) have been widely exploited as an effective and an efficient technique to learn and reproduce complex discrete…
Recently, the connectionist temporal classification (CTC) model coupled with recurrent (RNN) or convolutional neural networks (CNN), made it easier to train speech recognition systems in an end-to-end fashion. However in real-valued models,…
A novel single-frame quaternion estimator processing two vector observations is introduced. The singular cases are examined, and appropriate rotational solutions are provided. Additionally, an alternative method involving sequential…
A systematic theory is introduced for calculating the derivatives of quaternion matrix function with respect to quaternion matrix variables. The proposed methodology is equipped with the matrix product rule and chain rule and it is able to…
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…
Quaternions provide a unified algebraic and geometric framework for representing three-dimensional rotations without the singularities that afflict Euler-angle parametrisations. This article develops a pedagogical and conceptual analysis of…
We introduce motions as real six-dimensional vectors. A motion means a rotation and a translation. We define a motion operator which maps unit dual quaternions to motions, and a UDQ operator which maps motions to unit dual quaternions. By…
In neural networks, it is often desirable to work with various representations of the same space. For example, 3D rotations can be represented with quaternions or Euler angles. In this paper, we advance a definition of a continuous…
Over the last decades quaternions have become a crucial and very successful tool for attitude representation in robotics and aerospace. However, there is a major problem that is continuously causing trouble in practice when it comes to…
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…
Learning from data in the quaternion domain enables us to exploit internal dependencies of 4D signals and treating them as a single entity. One of the models that perfectly suits with quaternion-valued data processing is represented by 3D…