Silvia Franchini

An embedded, FPGA-based computer graphics coprocessor with native geometric algebra support

The representation of geometric objects and their transformation are the two key aspects in computer graphics applications. Traditionally, computer-intensive matrix calculations are involved in modeling and rendering three-dimensional (3D) scenery. Geometric algebra (aka Clifford algebra) is attracting attention as a natural way to model geometric facts and as a powerful analytical tool for symbolic calculations. In this paper, the architecture of Clifford coprocessor (CliffoSor) is introduced. CliffoSor is an embedded parallel coprocessing core that offers direct hardware support to Clifford algebra operators. A prototype implementation on a programmable gate array (FPGA) board is detailed. Initial test results show the potential to achieve a 20x speedup for 3D vector rotations, a 12x speedup for Clifford sums and differences, and more than a 4x speedup for Clifford products, compared to the analogous operations in GAIGEN, a standard geometric algebra library generator for general-purpose processors. An execution analysis of a raytracing application is also presented.

A Sliced Coprocessor for Native Clifford Algebra Operations

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.

Design and Implementation of an Embedded Coprocessor with Native Support for 5D, Quadruple-Based Clifford Algebra

Geometric or Clifford algebra (CA) is a powerful mathematical tool that offers a natural and intuitive way to model geometric facts in a number of research fields, such as robotics, machine vision, and computer graphics. Operating in higher dimensional spaces, its practical use is hindered, however, by a significant computational cost, only partially addressed by dedicated software libraries and hardware/software codesigns. For low-dimensional algebras, several dedicated hardware accelerators and coprocessing architectures have been already proposed in the literature. This paper introduces the architecture of CliffordALU5, an embedded coprocessing core conceived for native execution of up to 5D CA operations. CliffordALU5 exploits a novel, hardware-oriented representation of the algebra elements that allows for faster execution of Clifford operations. In this paper, a prototype implementation of a complete system-on-chip (SOC) based on CliffordALU5 is presented. This prototype integrates an embedded processing soft-core based on the PowerPC 405 and a CliffordALU5 coprocessor on a Xilinx XUPV2P Field Programmable Gate Array (FPGA) board. Test results show a 5× average speedup for 4D Clifford products and a 4× average speedup for 5D Clifford products against the same operations in Gaigen 2, a CA software library generator running on the general-purpose PowerPC processor. This paper also presents an execution analysis of three different applications in three diverse domains, namely, inverse kinematics of a robot, optical motion capture, and raytracing, showing an average speedup between 3× and 4× with respect to the baseline Gaigen 2 implementation. Finally, a multicore approach to higher dimensional CA based on CliffordALU5 is discussed.

ConformalALU: A Conformal Geometric Algebra Coprocessor for Medical Image Processing

Medical imaging involves important computational geometric problems, such as image segmentation and analysis, shape approximation, three-dimensional (3D) modeling, and registration of volumetric data. In the last few years, Conformal Geometric Algebra (CGA), based on five-dimensional (5D) Clifford Algebra, is emerging as a new paradigm that offers simple and universal operators for the representation and solution of complex geometric problems. However, the widespread use of CGA has been so far hindered by its high dimensionality and computational complexity. This paper proposes a simplified formulation of the conformal geometric operations (reflections, rotations, translations, and uniform scaling) aimed at a parallel hardware implementation. A specialized coprocessing architecture (ConformalALU) that offers direct hardware support to the new CGA operators, is also presented. The ConformalALU has been prototyped as a complete System-on-Programmable-Chip (SoPC) on the Xilinx ML507 FPGA board, containing a Virtex-5 FPGA device. Experimental results show average speedups of one order of magnitude for CGA rotations, translations, and dilations with respect to the geometric algebra software library Gaigen running on the general-purpose PowerPC processor embedded in the target FPGA device. A suite of medical imaging applications, including segmentation, 3D modeling and registration of medical data, has been used as testbench to evaluate the coprocessor effectiveness.

An FPGA Implementation of a Quadruple-Based Multiplier for 4D Clifford 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.

Design Space Exploration of Parallel Embedded Architectures for Native Clifford Algebra Operations

Clifford (geometric) algebra is a natural and intuitive way to model geometric objects and their transformations. It has important applications in a variety of fields, including robotics, machine vision and computer graphics, where it has gained a growing interest. This paper presents the design space exploration of parallel embedded architectures that natively support Clifford algebra with different costs, performance and precision. Results show an effective 5x average speedup for Clifford products compared with a software library developed specifically for Clifford algebra.

Clifford Algebra Based Edge Detector for Color Images

Edge detection is one of the most used methods for feature extraction in computer vision applications. Feature extraction is traditionally founded on pattern recognition methods exploiting the basic concepts of convolution and Fourier transform. For color image edge detection the traditional methods used for gray-scale images are usually extended and applied to the three color channels separately. This leads to increased computational requirements and long execution times. In this paper we propose a new, enhanced version of an edge detection algorithm that treats color value triples as vectors and exploits the geometric product of vectors defined in the Clifford algebra framework to extend the traditional concepts of convolution and Fourier transform to vector fields. Experimental results presented in the paper show that the proposed algorithm achieves detection performance comparable to the classical edge detection methods allowing at the same time for a significant reduction (about 33%) of computational times.

A Specialized Architecture for Color Image Edge Detection Based on Clifford Algebra

Edge detection of color images is usually performed by applying the traditional techniques for gray-scale images to the three color channels separately. However, human visual perception does not differentiate colors and processes the image as a whole. Recently, new methods have been proposed that treat RGB color triples as vectors and color images as vector fields. In these approaches, edge detection is obtained extending the classical pattern matching and convolution techniques to vector fields. This paper proposes a hardware implementation of an edge detection method for color images that exploits the definition of geometric product of vectors given in the Clifford algebra framework to extend the convolution operator and the Fourier transform to vector fields. The proposed architecture has been prototyped on the Celoxica RC203E Field Programmable Gate Array (FPGA) board. Experimental tests on the FPGA prototype show that the proposed hardware architecture allows for an average speedup ranging between 6x and 18x for different image sizes against the execution on a conventional general-purpose processor. Clifford algebra based edge detector can be exploited to process not only color images but also multispectral gray-scale images. The proposed hardware architecture has been successfully used for feature extraction of multispectral magnetic resonance (MR) images.

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

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.

Accelerating Clifford Algebra Operations Using GPUs and an OpenCL Code Generator

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.