A Dual-Core Coprocessor with Native 4D Clifford Algebra Support

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.

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)

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.

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.

Clifford algebra or geometric algebra (GA) is a simple and intuitive way to model geometric objects and their transformations. Operating in high-dimensional vector spaces with significant computational costs, the practical use of GA requires dedicated software and/or hardware architectures to directly support Clifford data types and operators. In this paper, a family of embedded coprocessors for the native execution of GA operations is presented. The paper shows the evolution of the coprocessor family focusing on the latest two architectures that offer direct hardware support to up to five-dimensional Clifford operations. The proposed coprocessors exploit hardware-oriented representations of GA elements and operators properly conceived to obtain fast performing implementations. The coprocessor prototypes, implemented on field programmable gate arrays development boards, show significant speedups of about one order of magnitude with respect to the baseline software library Gaigen running on a general-purpose processor. The paper also presents an execution analysis of different GA-based applications, namely inverse kinematics of a robot, optical motion capture, raytracing, and medical image processing, showing good speedups with respect to the baseline general-purpose implementation.

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

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In this paper, we discuss a generalization of Vieta theorem (Vieta’s formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta’s formulas with the ordinary Vieta’s formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand – Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering.

In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras $$\mathcal {G}_{p,q}$$ of vector space of dimension $$n=p+q$$ . We present basis-free formulas for all characteristic polynomial coefficients in the cases $$n\le 6$$ , alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas are verified using computer calculations. We present an analytical proof of all formulas in the case $$n=4$$ , and one of the formulas in the case $$n=5$$ . We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade 1) and basis elements are presented in the case of arbitrary n, the formulas for rotors (elements of spin groups) are presented in the cases $$n\le 5$$ . The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation.

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.

The problem of measuring geometrical distortions is not trivial for volumetric domains. There exist intrinsic restrictions and constrains on higher dimensional mappings. Moreover, according to Liouville theorem, most existing techniques for 2D data can not be directly applied to volumetric objects. In this work we approximate continuous deformations by piecewise affine functions defined on tetrahedral meshes. Our aim is to study a few types of geometrical distortions that can be expressed as functions of singular values of a Jacobian. We employ the proposed methods of estimating conformal and isometric distortions to analyze volumetric data. In particular, we examine parametrization of tetrahedral models to a ball. Distortions produced by the resulting spatial mappings depict intrinsic structure of domains, and therefore can be employed in such tasks as detection of abnormalities and comparison (i.e. similarity assessment) between 3D objects. This geometric approach and results are highly relevant to various applications in Computer Vision, Computer Graphics, 3D Printing and Medical Imaging.

This thesis work presents the implementation and validation of image processing problems in hardware to estimate the performance and precision gain. It compares the implementation for the addressed problem on a Field Programmable Gate Array (FPGA) with a software implementation for a General Purpose Processor (GPP) architecture. For both solutions the implementation costs for their development is an important aspect in the validation. The analysis of the flexibility and extendability that can be achieved by a modular implementation for the FPGA design was another major aspect. This work is based upon approaches from previous work, which included the detection of Binary Large OBjects (BLOBs) in static images and continuous video streams [13, 15]. One addressed problem of this work is the tracking of the detected BLOBs in continuous image material. This has been implemented for the FPGA platform and the GPP architecture. Both approaches have been compared with respect to performance and precision. This research project is motivated by the MI6 project of the Computer Vision research group, which is located at the Bonn-Rhein-Sieg University of Applied Sciences. The intent of the MI6 project is the tracking of a user in an immersive environment. The proposed solution is to attach a light emitting device to the user for tracking the created light dots on the projection surface of the immersive environment. Having the center points of those light dots would allow the estimation of the user’s position and orientation. One major issue that makes Computer Vision problems computationally expensive is the high amount of data that has to be processed in real-time. Therefore, one major target for the implementation was to get a processing speed of more than 30 frames per second. This would allow the system to realize feedback to the user in a response time which is faster than the human visual perception. One problem that comes with the idea of using a light emitting device to represent the user, is the precision error. Dependent on the resolution of the tracked projection surface of the immersive environment, a pixel might have a size in cm2. Having a precision error of only a few pixels, might lead to an offset in the estimated user’s position of several cm. In this research work the development and validation of a detection and tracking system for BLOBs on a Cyclone II FPGA from Altera has been realized. The system supports different input devices for the image acquisition and can perform detection and tracking for five to eight BLOBs. A further extension of the design has been evaluated and is possible with some constraints. Additional modules for compressing the image data based on run-length encoding and sub-pixel precision for the computed BLOB center-points have been designed. For the comparison of the FPGA approach for BLOB tracking a similar implementation in software using a multi-threaded approach has been realized. The system can transmit the detection or tracking results on two available communication interfaces, USB and RS232. The analysis of the hardware solution showed a similar precision for the BLOB detection and tracking as the software approach. One problem is the strong increase of the allocated resources when extending the system to process more BLOBs. With one of the applied target platforms, the DE2-70 board from Altera, the BLOB detection could be extended to process up to thirty BLOBs. The implementation of the tracking approach in hardware required much more effort than the software solution. The design of high level problems in hardware for this case are more expensive than the software implementation. The search and match steps in the tracking approach could be realized more efficiently and reliably in software. The additional pre-processing modules for sub-pixel precision and run-length-encoding helped to increase the system’s performance and precision.

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.