Grassmann Mechanics, Multivector Derivatives and Geometric Algebra

Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we explore here the benefits of performing the relevant computations in a Geometric Algebra framework, which is particularly suited to describing reflections. Starting from the Coxeter generators of the reflections, we describe how the relevant chiral (rotational), full (Coxeter) and binary polyhedral groups can be easily generated and treated in a unified way in a versor formalism. In particular, this yields a simple construction of the binary polyhedral groups as discrete spinor groups. These in turn are known to generate Lie and Coxeter groups in dimension four, notably the exceptional groups D_4, F_4 and H_4. A Clifford algebra approach thus reveals an unexpected connection between Coxeter groups of ranks 3 and 4. We discuss how to extend these considerations and computations to the Conformal Geometric Algebra setup, in particular for the non-crystallographic groups, and construct root systems and quasicrystalline point arrays. We finally show how a Clifford versor framework sheds light on the geometry of the Coxeter element and the Coxeter plane for the examples of the two-dimensional non-crystallographic Coxeter groups I_2(n) and the three-dimensional groups A_3, B_3, as well as the icosahedral group H_3.

A position and attitude control law is developed using geometric algebra (GA). GA is a powerful representational and computational system for geometry. The rigid body motion can be represented by the versor product in GA framework. Using the kinematics of the motor (the versor which represents the rigid body motion in GA), the control law of the rigid body motion can be developed. This paper provides a GA-based position and attitude control law by using the negative feedback of the motor. The stability of the control law is validated by the Lyapunov theorem and the numerical simulation.

On computer vision field Structure from Motion (SfM) algorithms offer good advantages for numerous applications (augmented reality, autonomous navigation, motion capture, remote sensing, object recognition, image-base 3D modeling, among others), nevertheless, these algorithms show some weakness; in the present paper we propose the use of Bio-inspired Aging Model-PSO (BAM-PSO) to improve the accuracy of SfM algorithms. The BAM-PSO algorithm is used over a Geometric Algebra (GA) framework in order to compute the rigid movement on images and this allows us to obtain a numerically stable algorithm.

In signal processing, the approach of the analytic signal is a capable and often used method. For signals of finite length, quadrature filters yield a bandpass filtered approximation of the analytic signal. In the case of multidimensional signals, the quadrature filters can only be applied with respect to a preference direction. Therefore, the orientation has to be sampled, steered or orientation adaptive filters have to be used. Up to now, there has been no linear approach to obtain an isotropic analytic signal which means that the amplitude is independent of the local orientation. In this paper, we present such an approach using the framework of geometric algebra. Our result is closely related to the Riesz transform and the structure tensor. It is seamless embedded in the framework of Clifford analysis. In a suitable coordinate system, the filter response contains information about local amplitude, local phase and local orientation of intrinsically one-dimensional signals. We have tested our filters on two- and three-dimensional signals.Keywordsquadrature filteranalytic signalRiesz transform

A simple and general design procedure is presented for the polarisation diversity of arbitrary conformal arrays; this procedure is based on the mathematical framework of geometric algebra and can be solved optimally using convex optimisation. Aside from being simpler and more direct than other derivations in the literature, this derivation is also entirely general in that it expresses the transformations in terms of rotors in geometric algebra which can easily be formulated for any arbitrary conformal array geometry. Convex optimisation has a number of advantages; solvers are widespread and freely available, the process generally requires a small number of iterations and a wide variety of constraints can be readily incorporated. The study outlines a two-step approach for addressing polarisation diversity in arbitrary conformal arrays: first, the authors obtain the array polarisation patterns using geometric algebra and secondly use a convex optimisation approach to find the optimal weights for the polarisation diversity problem. The versatility of this approach is illustrated via simulations of a 7×10 cylindrical conformal array.

In this study, a general and systematic method for three-dimensional pattern analysis of arbitrary conformal arrays is presented; this analysis is based on the mathematical framework of geometric algebra and considers both directional and polarised element patterns. A compact representation of the conformal array pattern is presented. Aside from being simpler and more direct than other derivations in the literature, this derivation is also entirely general in that it expresses the transformations in terms of rotors that can easily be formulated for any arbitrary conformal array geometry. Analysing conformal arrays using geometric algebra is simpler than the traditional Euler rotation angles and matrix representations. As well as presenting the new derivation and comparing it with conventional methods, the authors also present simulation results on cylindrical and conical arrays to illustrate the practicality and conciseness of the proposed method.

<abstract> <p>The identification of harmonic generating loads and the assignation of responsibility for harmonic pollution is an important first step for harmonic control in modern power systems. In this paper, a previously introduced power multivector is examined as a possible tool for the identification of such loads. This representation of power is based on the mathematical framework of Geometric Algebra (GA). Components of the power multivector derived at the point of connection of a load are grouped into a single quantity, which is a bivector in GA and is characterized by a magnitude, direction and sense. The magnitude of this bivector can serve as an indicator of the distortion at the terminals of the load. Furthermore, in contrast to indices based solely on magnitude, such as components derived from any apparent power equation, the proposed bivectorial representation can differentiate between loads that enhance distortion and those with a mitigating effect. Its conservative nature permits an association between the distortion at specific load terminals and the common point of connection. When several loads connected along a distribution line are considered, then an evaluation of the impact of each one of these loads on the distortion at a specific point is possible. Simulation results confirm that information included in the proposed bivector can provide helpful guidance when quantities derived from apparent power equations deliver ambiguous results.</p> </abstract>

Body sensor calibration and construction of 3D maps for robot navigation using the framework of conformal geometric algebra

Under geometric algebra framework, generalized definition of instantaneous power quantity in the multiphase system without any co-ordinate transformation is possible. In the area of active power filtering, with an objective to reduce inverter capacity, the Series Hybrid Active Power Filter (SHAPF) has been taken into account increasingly. This paper proposes a novel concept of decomposing the multiphase voltage vector into different quantities that correspond to different components of the power quantity. The separated components of voltage vector are directly used to derive reference for SHAPF. The mathematical formulation of proposed control scheme with its applications to SHAPF is presented. The validity of the proposed control scheme is verified by the simulation study.

The paper focus on the analysis and computing of the projective structure and motion using geometric invariants. This work relates current approaches in the geometric algebra framework; as a result the approach gains geometric transparency and elegance. The paper presents experiments regarding projective reconstruction of shape and motion using both simulated and real images.

This paper demonstrates that numerous calculations involving polarization transformations can be condensed by employing suitable geometric algebra formalism. For example, to describe polarization mode dispersion and polarization-dependent loss, both the material birefringence and differential loss enter as bivectors and can be combined into a single symmetric quantity. Their frequency and distance evolution, as well as that of the Stokes vector through an optical system, can then each be expressed as a single compact expression, in contrast to the corresponding Mueller matrix formulations. The intrinsic advantage of the geometric algebra framework is further demonstrated by presenting a simplified derivation of generalized Stokes parameters that include the electric field phase. This procedure simultaneously establishes the tensor transformation properties of these parameters.

Given a quadratic form on a vector space, the geometric algebra of the corresponding pseudo-euclidean space is defined in terms of a simple set of rules which characterizes the geometric product of vectors. We develop geometric algebra in such a way that it augments, but remains fully compatible with, the more traditional tools of matrix algebra. Indeed, matrix multiplication arises naturally from the geometric multiplication of vectors by introducing a spectral basis of mutually annihiliating idempotents in the geometric algebra. With the help of a few more algebraic identities, and given the proper geometric interpretation, the geometric algebra can be applied to the study of affine, projective, conformal and other geometries. The advantage of geometric algebra is that it provides a single algebraic framework with a comprehensive, but flexible, geometric interpretation. For example, the affine plane of rays is obtained from the euclidean plane of points by adding a single anti-commuting vector to the underlying vector space. The key to the study of noneuclidean geometries is the definition of the operations of meet and join, in terms of which incidence relationships are expressed. The horosphere provides a homogeneous model of euclidean space, and is obtained by adding a second anti-commuting vector to the underlying vector space of the affine plane. Linear orthogonal transformations on the higher dimensional vector space correspond to conformal or Mobius transformations on the horosphere. The horosphere was first constructed by F.A. Wachter (1792–1817), but has only recently attracted attention byo ering a host of new computational tools in projective and hyperbolic geometries when formulated in terms of geometric algebra.

We present an algorithm in the Conformal Geometric Algebra framework which computes the transformation (rotation and translation) relating the coordinate system of an endoscopic camera, with the coordinate system of the markers placed on it. Such markers are placed in order to track the camera’s position in real time by using an optical tracking system. The problem is an adaptation of the so-called hand-eye calibration, well known in robotics; however we name it the endoscope-tracking system calibration. By this way, we can relate the preoperative data (3D model and planing of the surgery), with the data acquired in real time during the surgery by means of the optical tracking system and the endoscope. Results obtained with our approach are compared with other approach which computes separately (in a two-step algorithm) the rotation and translation, and they are promising.

We study integrable spin chains and quantum and classical cellular automata with interaction range ℓ≥3. This is a family of integrable models for which there was no general theory so far. We develop an algebraic framework for such models, generalizing known methods from nearest-neighbor interacting chains. This leads to a new integrability condition for medium-range Hamiltonians, which can be used to classify such models. A partial classification is performed in specific cases, including U(1)-symmetric three-site interacting models, and Hamiltonians that are relevant for interaction-round-a-face models. We find a number of models which appear to be new. As an application we consider quantum brickwork circuits of various types, including those that can accommodate the classical elementary cellular automata on light cone lattices. In this family we find that the so-called Rule150 and Rule105 models are Yang-Baxter integrable with three-site interactions. We present integrable quantum deformations of these models, and derive a set of local conserved charges for them. For the famous Rule54 model we find that it does not belong to the family of integrable three-site models, but we cannot exclude Yang-Baxter integrability with longer interaction ranges.

This book is to be an accessible book on patterns, their representation, and inference. There are a small number of ideas and techniques that, when mastered, make the subject more accessible. This book has arisen from ten years of a research program which the authors have embarked upon, building on the more abstract developments of metric pattern theory developed by one of the authors during the 1970s and 1980s. The material has been taught over multiple semesters as part of a second year graduate-level course in pattern theory, essentially an introduction for students interested in the representation of patterns which are observed in the natural world. The course has attracted students studying biomedical engineering, computer science, electrical engineering, and applied mathematics interested in speech recognition and computational linguistics, as well as areas of image analysis, and computer vision. Now the concept of patterns pervades the history of intellectual endeavor; it is one of the eternal followers in human thought. It appears again and again in science, taking on different forms in the various disciplines, and made rigorous through mathematical formalization. But the concept also lives in a less stringent form in the humanities, in novels and plays, even in everyday language. We use it all the time without attributing a formal meaning to it and yet with little risk of misunderstanding. So, what do we really mean by a pattern? Can we define it in strictly logical terms? And if we can, what use can we make of such a definition? These questions were answered by General Pattern Theory, a discipline initiated by Ulf Grenander in the late 1960s [1–5]. It has been an ambitious effort with the only original sketchy program having few if any practical applications, growing in mathematical maturity with a multitude of applications having appeared in biology/medicine and in computer vision, in language theory and object recognition, to mention but a few. Pattern theory attempts to provide an algebraic framework for describing patterns as structures regulated by rules, essentially a finite number of both local and global combinatory operations. Pattern theory takes a compositional view of the world, building more and more complex structures starting from simple ones. The basic rules for combining and building complex patterns from simpler ones are encoded via graphs and rules on transformation of these graphs.