Embedded Coprocessors for Native Execution of Geometric Algebra Operations

Clifford Algebra (CA) is a powerful mathematical language that allows for a simple and intuitive representation of geometric objects and their transformations. It has important applications in many research fields, such as computer graphics, robotics, and machine vision. Direct hardware support of Clifford data types and operators is needed to accelerate applications based on Clifford Algebra. This paper proposes a mixed software-hardware system that exploits the computational power of Graphics Processing Units (GPUs) to accelerate Clifford operations. A code generator, namely OpenCLifford, is presented that automatically generates Java and C libraries for the direct support of Clifford elements and operations as well as OpenCL kernels to be executed on the GPU. Experimental tests have been performed to evaluate the speedup of the OpenCL parallel code executed on the GPU against the baseline C code executed on the CPU. Average speedups of 47x and 27x have been measured for 3D and 5D Clifford Algebra, respectively. The paper also presents an execution analysis of an application for fractal generation showing a 35x speedup with respect to the baseline CPU execution.

Computer graphics applications require efficient tools to model geometric objects. The traditional approach based on compute-intensive matrix calculations is error-prone due to a lack of integration between geometric reasoning and matrix-based algorithms. Clifford algebra offers a solution to these issues since it permits specification of geometry at a coordinate-free level. The best way to exploit the symbolic computing power of geometric (Clifford) algebra is supporting its data types and operators directly in hardware. This paper outlines the architecture of S-CliffoSor (Sliced Clifford coprocessor), a parallelizable embedded coprocessor that executes native Clifford algebra operations. S-CliffoSor is a sliced coprocessor that can be replicated for parallel execution of concurrent Clifford operations. A single slice has been designed, implemented and tested on the Celoxica Inc. RC1000 board. The experimental results show the potential to achieve a 3times speedup for Clifford sums and 4times speedup for Clifford products compared to against the analogous operations in the software library generator GAIGEN.

Geometric or Clifford Algebra (CA) is a powerful mathematical tool that is attracting a growing attention in many research fields such as computer graphics, computer vision, robotics and medical imaging for its natural and intuitive way to represent geometric objects and their transformations. This paper introduces the architecture of CliffordCoreDuo, an embedded dual-core coprocessor that offers direct hardware support to four-dimensional (4D) Clifford algebra operations. A prototype implementation on an FPGA board is detailed. Experimental results show a 1.6x average speedup of CliffordCoreDuo in comparison with the baseline mono-core architecture. A potential cycle speedup of about 40x over Gaigen 2, a geometric algebra software library generator for general-purpose processors, is also demonstrated.

Preface Contributors Part I. Algebra and Geometry Point Groups and Space Groups in Geometric Algebra (D. Hestenes) The Inner Products of Geometric Algebra (L. Dorst) Unification of Grassmann's Progressive and Regressive Products using the Principle of Duality (S. Blake) From Unoriented Subspaces to Blade Operators (T.A. Bouma) Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra (H. Li) Rotations in n Dimensions as Spherical Vectors (W.E. Baylis/S. Hadi) Geometric and Algebraic Canonical Forms (N. Gordon) Functions of Clifford Numbers or Square Matrices (J. Snygg) Compound Matrices and Pfaffians: A Representation of Geometric Algebra (U. Prells/M.I. Friswell/S.D. Garvey) Analysis Using Abstract Vector Variables (F. Sommen) A Multivector Data Structure for Differential Forms and Equations (J.A. Chard/V. Shapiro) Jet Bundles and the Formal Theory of Partial Differential Equations (R. Baker/C. Doran) Imaginary Eigenvalues and Complex Eigenvectors Explained by Real Geometry (E.M.S. Hitzer) Symbolic Processing of Clifford Numbers in C++ (J.P. Fletcher) Clifford Numbers and their Inverses Calculated using the Matrix Representation (J.P. Fletcher) A Toy Vector Field Based on Geometric Algebra (A. Rockwood/S. Binderwala) Quadratic Transformations in the Projective Plane (G. Georgiev) Annihilators of Principal Ideals in the Grassmann Algebra (C. Koc/S. Esin) Part II. Applications to Physics Homogeneous Rigid Body Mechanics with Elastic Coupling (D. Hestenes/E.D. Fasse) Analysis of One and Two Particle Quantum Systems using Geometric Algebra (R. Parker/C. Doran) Interaction and Entanglement in the Multiparticle Spacetime Algebra (T.F. Havel/C.J.L. Doran) Laws of Reflection from Two or More Plane Mirrors in Succession (M. Derome) Exact Kinetic Energy Operators for Polyatomic Molecules (J. Pesonen) Geometry of Quantum Computing by Hamiltonian Dynamics of Spin Ensembles (T. Schulte-Herbruggen/K. Huper/U. Helmke/S.J. Glaser) Is the Brain a 'Clifford Algebra Quantum Computer'? (V. Labunets/E. Rundblad/J. Astola) A Hestenes Spacetime Algebra Approach to Light Polarization (Q.M. Sugon/D. McNamara) Quaternions, Clifford Algebra and Symmetry Groups (P.R. Girard) Part III. Computer Vision and Robotics A Generic Framework for Image Geometry (J.J. Koenderink) Color Edge Detection Using Rotors (E. Bayro-Corrochano/S. Flores) Numerical Evaluation of Versors with Clifford Algebra (C.B.U. Perwass/G. Sommer) The Role of Clifford Algebra in Structure-Preserving Transformations for Second-Order Systems (S.D. Garvey/M.I. Friswell/U. Prells) Applications of Algebra of Incidence in Visually Guided Robotics (E. Bayro-Corrochano/P. Lounesto/L.R. Lozano) Monocular Pose Estimation of Kinematic Chains (B. Rosenhahn/O. Granert/G. Sommer) Stabilization of 3D Pose Estimation (W. Neddermeyer/M. Schnell/W. Winkler/A. Lilienthal) Inferring Dynamical Information from 3D Position Data using Geometric Algebra (H. Udugama/G.S. Sajeewa/J. Lasenby) Clifford Algebra Space Singularities of Inline Planar Platforms (M.A. Baswell/R. Ablamowicz/J.N. Anderson) Part IV. Signal Processing and Other Applications Fast Quantum Fourier--Heisenberg--Weyl Transforms (V. Labunets/E. Rundblad/J. Astola) The Structure Multivector (M. Felsberg/G. Sommer) The Application of Clifford Algebra to Calculations of Multicomponent Chemical Composition (J.P. Fletcher) An Algorithm to Solve the Inverse IFS-Problem (E. Hocevar) Fast Quantum n-D Fourier and Radon Transforms (V. Labunets/E. Rundblad/J. Astola)

Representations of geometric (Clifford) algebras with real square matrices are reviewed by providing the general theorem as well as examples of lowest dimensions. New definitions for isometry and norm are proposed. Direct and indirect isometries are identified respectively with automorphisms and antiautomorphisms of the geometric algebra, while the norm of every element is defined as the $n^\textit{th}$-root of the absolute value of the determinant of its matrix representation of order $n$. It is deduced in which geometric algebras direct isometries are inner automorphisms (similarity transformations of matrices). Indirect isometries need reversion too. Finally, the most common isometries are reviewed in order to write them in this way.

Geometric algebra was introduced in the nineteenth century by the English mathematician William Kingdon Clifford (figure 2.1). Clifford appears to have been one of the small number of mathematicians at the time to be significantly influenced by Grassmann's work. Clifford introduced his geometric algebra by uniting the inner and outer products into a single geometric product. This is associative, like Grassmann's product, but has the crucial extra feature of being invertible , like Hamilton's quaternion algebra. Indeed, Clifford's original motivation was to unite Grassmann's and Hamilton's work into a single structure. In the mathematical literature one often sees this subject referred to as Clifford algebra . We have chosen to follow the example of David Hestenes, and many other modern researchers, by returning to Clifford's original choice of name – geometric algebra . One reason for this is that the first published definition of the geometric product was due to Grassmann, who introduced it in the second Ausdehnungslehre . It was Clifford, however, who realised the great potential of this product and who was responsible for advancing the subject. In this chapter we introduce the basics of geometric algebra in two and three dimensions in a way that is intended to appear natural and geometric, if somewhat informal. A more formal, axiomatic approach is delayed until chapter 4, where geometric algebra is defined in arbitrary dimensions.

Optical Mueller matrices in terms of geometric algebra

Geometric or Clifford algebra is an interesting paradigm for geometric modeling in fields as computer graphics, machine vision and robotics. In these areas the research effort is actually aimed at finding an efficient implementation of geometric algebra. The best way to exploit the symbolic computing power of geometric algebra is to support its data types and operators directly in hardware. However the natural representation of the algebra elements as variable-length objects causes some problems in the case of a hardware implementation. This paper proposes a 4D Clifford algebra in which the variable-length elements are mapped into fixed-length elements (quadruples). This choice leads to a simpler and more compact hardware implementation of 4D geometric algebra. The paper also presents the architecture of CliffArchy, a coprocessing core supporting the new fixed-length Clifford operands. A prototype implementation on a FPGA board is described.

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.

With the widespread use of color images, the copyright protection of those images using watermarks is one of the latest research topics. The use of color images as watermarks has advantages over binary and irreplaceable grayscale images. Color images are intuitive, rich, and lively; they have large amounts of copyright protection information and more easily recognized by human vision. To improve the security of watermark information and embedding positions and improve the algorithm's robustness against various attacks, a Quaternion Fourier transform (QFT) based algorithm, based on Arnold transform and chaotic encryption, is proposed in this paper. Geometric algebra (GA) can deal with color images in vector form with each component of RGB handled individually. We used Quaternion, which is a sub-algebra of GA, and effectively handled color image processing by using Fourier transformation. After deriving the calculation process of the QFT with strong security by Arnold scrambling and chaotic encryption, this paper proposes a digital watermarking algorithm that resists geometric attacks by using color images as carriers. The robustness and quality of the proposed watermarking algorithm is tested with different with many statistical measures. Experimental outcomes show that the proposed approach is the best to solve conflict problems between quality and robustness. Also, the proposed approach exhibits worthy robustness against many attacks, such as, conventional attacks, and geometrical attacks.

The hidden universal Clifford algebra structure of M4(R) is ambiguous in the sense that this matrix algebra is the universal geometric algebra belonging to each of the real four-dimensional quadratic vector spaces R1,3 and R2,2. As a non-universal Clifford algebra, however, M4(R) is a geometric algebra of the unique five-dimensional quadratic vector space R2,3.

The main objective of this lecture is to discuss the proper place for Clifford’s geometric algebra in the mathematical curriculum at all levels.

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.

Recently, models of neural networks in the real domain have been extended into the high dimensional domain such as the complex number and quaternion domain, and several high-dimensional models have been proposed. These extensions are generalized by introducing Clifford algebra (geometric algebra). In this paper we extend conventional real-valued Hopfield-type neural networks into the octonion domain and discuss their dynamics. The octonions represent a particular extension of the quaternions which also represent a particular extension of the complex numbers and have 7 imaginary parts. They are non-commutative and non-associative on multiplication and do not belong to Clifford algebra due to the latter fact. With this in mind we propose four models of octonion Hopfield-type neural networks. We derive existence conditions of an energy function and construct energy function for each model.

The use of Clifford or geometric algebra for dealing with three dimensional geometry is discussed. One issue is the representation of the rigid body motions of rotations and translations as elements within the algebra. The approach used is to work with projective geometry and choose the square of an additional basis element to be large (infinite). This allows Euclidean points to be represented as vectors in the algebra and transforms on these to be handled using bivectors. This paper looks at the use of Clifford algebra for handling the types of transforms required in robotic applications. A number of example applications are given.