900 research outputs found
The Quaternion-Based Spatial Coordinate and Orientation Frame Alignment Problems
We review the general problem of finding a global rotation that transforms a
given set of points and/or coordinate frames (the "test" data) into the best
possible alignment with a corresponding set (the "reference" data). For 3D
point data, this "orthogonal Procrustes problem" is often phrased in terms of
minimizing a root-mean-square deviation or RMSD corresponding to a Euclidean
distance measure relating the two sets of matched coordinates. We focus on
quaternion eigensystem methods that have been exploited to solve this problem
for at least five decades in several different bodies of scientific literature
where they were discovered independently. While numerical methods for the
eigenvalue solutions dominate much of this literature, it has long been
realized that the quaternion-based RMSD optimization problem can also be solved
using exact algebraic expressions based on the form of the quartic equation
solution published by Cardano in 1545; we focus on these exact solutions to
expose the structure of the entire eigensystem for the traditional 3D spatial
alignment problem. We then explore the structure of the less-studied
orientation data context, investigating how quaternion methods can be extended
to solve the corresponding 3D quaternion orientation frame alignment (QFA)
problem, noting the interesting equivalence of this problem to the
rotation-averaging problem, which also has been the subject of independent
literature threads. We conclude with a brief discussion of the combined 3D
translation-orientation data alignment problem. Appendices are devoted to a
tutorial on quaternion frames, a related quaternion technique for extracting
quaternions from rotation matrices, and a review of quaternion
rotation-averaging methods relevant to the orientation-frame alignment problem.
Supplementary Material covers extensions of quaternion methods to the 4D
problem.Comment: This replaces an early draft that lacked a number of important
references to previous work. There are also additional graphics elements. The
extensions to 4D data and additional details are worked out in the
Supplementary Material appended to the main tex
The geometry of proper quaternion random variables
Second order circularity, also called properness, for complex random
variables is a well known and studied concept. In the case of quaternion random
variables, some extensions have been proposed, leading to applications in
quaternion signal processing (detection, filtering, estimation). Just like in
the complex case, circularity for a quaternion-valued random variable is
related to the symmetries of its probability density function. As a
consequence, properness of quaternion random variables should be defined with
respect to the most general isometries in , i.e. rotations from .
Based on this idea, we propose a new definition of properness, namely the
-properness, for quaternion random variables using invariance
property under the action of the rotation group . This new definition
generalizes previously introduced properness concepts for quaternion random
variables. A second order study is conducted and symmetry properties of the
covariance matrix of -proper quaternion random variables are
presented. Comparisons with previous definitions are given and simulations
illustrate in a geometric manner the newly introduced concept.Comment: 14 pages, 3 figure
On quaternion based parametrization of orientation in computer vision and robotics
The problem of orientation parameterization for applications in computer vision and robotics is examined in detail herein.
The necessary intuition and formulas are provided for direct practical use in any existing algorithm that seeks to
minimize a cost function in an iterative fashion. Two distinct schemes of parameterization are analyzed: The first scheme
concerns the traditional axis-angle approach, while the second employs stereographic projection from unit quaternion
sphere to the 3D real projective space. Performance measurements are taken and a comparison is made between the two
approaches. Results suggests that there exist several benefits in the use of stereographic projection that include rational
expressions in the rotation matrix derivatives, improved accuracy, robustness to random starting points and accelerated
convergence
Solving the nearest rotation matrix problem in three and four dimensions with applications in robotics
Aplicat embargament des de la data de defensa fins ei 31/5/2022Since the map from quaternions to rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is sometimes erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception was clarified when we found a new division-free conversion method. This result triggered the research work presented in this thesis. At first glance, the matrix to quaternion conversion does not seem to be a relevant problem. Actually, most researchers consider it as a well-solved problem whose revision is not likely to provide any new insight in any area of practical interest. Nevertheless, we show in this thesis how solving the nearest rotation matrix problem in Frobenius norm can be reduced to a matrix to quaternion conversion. Many problems, such as hand-eye calibration, camera pose estimation, location recognition, image stitching etc. require finding the nearest proper orthogonal matrix to a given matrix. Thus, the matrix to quaternion conversion becomes of paramount importance. While a rotation in 3D can be represented using a quaternion, a rotation in 4D can be represented using a double quaternion. As a consequence, the computation of the nearest rotation matrix in 4D, using our approach, essentially follow the same steps as in the 3D case. Although the 4D case might seem of theoretical interest only, we show in this thesis its practical relevance thanks to a little known mapping between 3D displacements and 4D rotations. In this thesis we focus our attention in obtaining closed-form solutions, in particular those that only require the four basic arithmetic operations because they can easily be implemented on microcomputers with limited computational resources. Moreover, closed-form methods are preferable for at least two reasons: they provide the most meaningful answer because they permit analyzing the influence of each variable on the result; and their computational cost, in terms of arithmetic operations, is fixed and assessable beforehand. We have actually derived closed-form methods specifically tailored for solving the hand-eye calibration and the pointcloud registration problems which outperform all previous approaches.Dado que la función que aplica a cada cuaternión su matrix de rotación correspondiente es 2 a 1, la inversa de esta función no es diferenciable en todo su dominio. Por consiguiente, a veces se asume erróneamente que todas las inversiones deben contener necesariamente singularidades que surgen en forma de cocientes donde el divisor puede ser arbitrariamente pequeño. Esta idea errónea se aclaró cuando encontramos un nuevo método de conversión sin división. Este resultado desencadenó el trabajo de investigación presentado en esta tesis. A primera vista, la conversión de matriz a cuaternión no parece un problema relevante. En realidad, la mayoría de los investigadores lo consideran un problema bien resuelto cuya revisión no es probable que proporcione nuevos resultados en ningún área de interés práctico. Sin embargo, mostramos en esta tesis cómo la resolución del problema de la matriz de rotación más cercana según la norma de Frobenius se puede reducir a una conversión de matriz a cuaternión. Muchos problemas, como el de la calibración mano-cámara, el de la estimación de la pose de una cámara, el de la identificación de una ubicación, el del solapamiento de imágenes, etc. requieren encontrar la matriz de rotación más cercana a una matriz dada. Por lo tanto, la conversión de matriz a cuaternión se vuelve de suma importancia. Mientras que una rotación en 3D se puede representar mediante un cuaternión, una rotación en 4D se puede representar mediante un cuaternión doble. Como consecuencia, el cálculo de la matriz de rotación más cercana en 4D, utilizando nuestro enfoque, sigue esencialmente los mismos pasos que en el caso 3D. Aunque el caso 4D pueda parecer de interés teórico únicamente, mostramos en esta tesis su relevancia práctica gracias a una función poco conocida que relaciona desplazamientos en 3D con rotaciones en 4D. En esta tesis nos centramos en la obtención de soluciones de forma cerrada, en particular aquellas que solo requieren las cuatro operaciones aritméticas básicas porque se pueden implementar fácilmente en microcomputadores con recursos computacionales limitados. Además, los métodos de forma cerrada son preferibles por al menos dos razones: proporcionan la respuesta más significativa porque permiten analizar la influencia de cada variable en el resultado; y su costo computacional, en términos de operaciones aritméticas, es fijo y evaluable de antemano. De hecho, hemos derivado nuevos métodos de forma cerrada diseñados específicamente para resolver el problema de la calibración mano-cámara y el del registro de nubes de puntos cuya eficiencia supera la de todos los métodos anteriores.Postprint (published version
Intelligent OFDM telecommunication system. Part 1. Model of complex and quaternion systems
In this paper, we aim to investigate the superiority and practicability of many-parameter transforms (MPTs) from the physical layer security (PHY-LS) perspective. We propose novel Intelligent OFDM-telecommunication systems based on complex and quaternion MPTs. The new systems use inverse MPT (IMPT) for modulation at the transmitter and MPT for demodulation at the receiver. The purpose of employing the MPT is to improve: 1) the PHY-LS of wireless transmissions against to the wide-band anti-jamming and anti-eavesdropping communication; 2) the bit error rate (BER) performance with respect to the conventional OFDM-TCS; 3) the peak to average power ratio (PAPR). Each MPT depends on finite set of independent parameters (angles). When parameters are changed, many-parametric transform is also changed taking form of a set known (and unknown) orthogonal (or unitary) transforms. For this reason, the concrete values of parameters are specific "key" for entry into OFDM-TCS. Vector of parameters belong to multi-dimension torus space. Scanning of this space for find out the "key" (the concrete values of parameters) is hard problem. MPT has the form of the product of the Jacobi rotation matrixes and it describes a fast algorithm for MPT. The main advantage of using MPT in OFDM TCS is that it is a very flexible anti-eavesdropping and anti-jamming Intelligent OFDM TCS. To the best of our knowledge, this is the first work that utilizes the MPT theory to facilitate the PHY-LS through parameterization of unitary transforms. © 2019 IOP Publishing Ltd. All rights reserved
Geometric information in eight dimensions vs. quantum information
Complementary idempotent paravectors and their ordered compositions, are used
to represent multivector basis elements of geometric Clifford algebra for 3D
Euclidean space as the states of a geometric byte in a given frame of
reference. Two layers of information, available in real numbers, are
distinguished. The first layer is a continuous one. It is used to identify
spatial orientations of similar geometric objects in the same computational
basis. The second layer is a binary one. It is used to manipulate with 8D
structure elements inside the computational basis itself. An oriented unit cube
representation, rather than a matrix one, is used to visualize an inner
structure of basis multivectors. Both layers of information are used to
describe unitary operations -- reflections and rotations -- in Euclidian and
Hilbert spaces. The results are compared with ones for quantum gates. Some
consequences for quantum and classical information technologies are discussed.Comment: 14 pages, presented at International Symposium "Quantum Informatics
2007", October 3rd - 5th, 2007, Moscow Zvenigorod, Russi
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