Search results for "Ford"
showing 10 items of 785 documents
A brief introduction to Clifford algebra
2010
Geometric algebra (also known as Clifford algebra) is a powerful mathematical tool that offers a natural and direct way to model geometric objects and their transformations. It is gaining growing attention in different research fields as physics, robotics, CAD/CAM and computer graphics. Clifford algebra makes geometric objects (points, lines and planes) into basic elements of computation and defines few universal operators that are applicable to all types of geometric elements. This paper provides an introduction to Clifford algebra elements and operators.
Clifford Algebra based Edge Detector for Color Images
2012
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 …
ConformalALU: A Conformal Geometric Algebra Coprocessor for Medical Image Processing
2015
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 …
A Sliced Coprocessor for Native Clifford Algebra Operations
2007
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. …
Design and implementation of an embedded coprocessor with native support for 5D, quadruple-based Clifford algebra
2013
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 t…
A New Embedded Coprocessor for Clifford Algebra based Software Intensive Systems
2011
Computer graphics applications require efficient tools to model geometric objects and their transformations. Clifford algebra (also known as geometric algebra) is receiving a growing attention in many research fields, such as computer graphics, machine vision and robotics, as a new, interesting computational paradigm that offers a natural and intuitive way to perform geometric calculations. At the same time, compute-intensive graphics algorithms require the execution of million Clifford operations. Clifford algebra based software intensive systems need therefore the support of specialized hardware architectures capable of accelerating Clifford operations execution. In this paper the archite…
Clifford Rotors for Conceptual Representation in Chatbots
2013
In this abstract we introduce an unsupervised sub-symbolic natural language sentences encoding procedure aimed at catching and representing into a Chatbot Knowledge Base (KB) the concepts expressed by an user interacting with a robot. The chatbot KB is coded in a conceptual space induced from the application of the Latent Semantic Analysis (LSA) paradigm on a corpus of documents. LSA has the effect of decomposing the original relationships between elements into linearly-independent vectors. Each basis vector can be considered therefore as a "conceptual coordinate", which can be tagged by the words which better characterize it. This tagging is obtained by performing a (TF-IDF)-like weighting…
Maxwell’s Equations and Occam’s Razor
2017
In this paper a straightforward application of Occam’s razor principle to Maxwell’s equation shows that only one entity, the electro-magnetic four-potential, is at the origin of a plurality of concepts and entities in physics. The application of the so called “Lorenz gauge” in Maxwell’s equations denies the status of real physical entity to a scalar field that has a gradient in space-time with clear physical meaning: the four-current density field. The mathematical formalism of space-time Clifford algebra is introduced and then used to encode Maxwell’s equations starting only from the electromagnetic four-potential. This approach suggests a particular Zitterbewegung (ZBW) model for charged …
An Embedded, FPGA-based Computer Graphics Coprocessor with Native Geometric Algebra Support
2009
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…
A Dual-Core Coprocessor with Native 4D Clifford Algebra Support
2012
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 1.6× average speedup of CliffordCoreDuo in comparison with the baseline mono-core architecture. A potential cycle speedup of about 40× o…