Related papers: Representations of 3D Rotations: Mathematical Foun…
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…
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…
The attitude of a rigid-body in the three dimensional space has a unique and global definition on the Special Orthogonal Group SO (3). This paper gives an overview of the rotation matrix, attitude kinematics and parameterization. The four…
Many settings in machine learning require the selection of a rotation representation. However, choosing a suitable representation from the many available options is challenging. This paper acts as a survey and guide through rotation…
The attitude space has been parameterized in various ways for practical purposes. Different representations gain preferences over others based on their intuitive understanding, ease of implementation, formulaic simplicity, and physical as…
In this paper, we provide some careful analysis of certain pathological behavior of Euler angles and unit quaternions encountered in previous works related to rotation representation in neural networks. In particular, we show that for…
This paper presents an experimental study on the application of quaternions in several machine learning algorithms. Quaternion is a mathematical representation of rotation in three-dimensional space, which can be used to represent complex…
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…
Symmetric orthogonalization via SVD, and closely related procedures, are well-known techniques for projecting matrices onto $O(n)$ or $SO(n)$. These tools have long been used for applications in computer vision, for example optimal 3D…
Accurate rotation estimation is at the heart of robot perception tasks such as visual odometry and object pose estimation. Deep neural networks have provided a new way to perform these tasks, and the choice of rotation representation is an…
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.…
This article introduces yet another representation of rotations in 3-space. The rotations form a 3-dimensional projective space, which fact has not been exploited in Computer Science. We use the four affine patches of this projective space…
An arbitrary rigid transformation in $\mathbf{SE}(3)$ can be separated into two parts, namely, a translation and a rigid rotation. This technical report reviews, under a unifying viewpoint, three common alternatives to representing the…
The general 4D rotation matrix is specialised to the general 3D rotation matrix by equating its leftmost top element (a00) to 1. Its associate matrix of products of the left-hand and right-hand quaternion components is specialised…
It is shown that the groups of Euclidian rotations, rigid motions, proper, orthochronous Lorentz transformations, and the complex rigid motions can be represented by the groups of unit-norm elements in the algebras of real, dual, complex,…
This article is an exhaustive revision of concepts and formulas related to quaternions and rotations in 3D space, and their proper use in estimation engines such as the error-state Kalman filter. The paper includes an in-depth study of the…
3D frame fields are auxiliary for hexahedral mesh generation. There mainly exist three ways to represent 3D frames: combination of rotations, spherical harmonics and fourth order tensor. We propose here a representation carried out by the…
This paper analyzes the robustness of recent 3D shape descriptors to SO(3) rotations, something that is fundamental to shape modeling. Specifically, we formulate the task of rotated 3D object instance detection. To do so, we consider a…
The purpose of this effort is to investigate if the use of quaternion mathematics can be used to better model and simulate the electromagnetic fields that occur from moving electromagnetic charges. One observed deficiency with the commonly…
The parameterisation of rotations in three dimensional Euclidean space is an area of applied mathematics that has long been studied, dating back to the original works of Euler in the 18th century. As such, many ways of parameterising a…