Search results for "Geometric Algebra"

showing 10 items of 29 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.

Settore ING-INF/05 - Sistemi Di Elaborazione Delle InformazioniClifford algebrageometric algebra
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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 …

Settore ING-INF/05 - Sistemi Di Elaborazione Delle InformazioniCoprocessorComputer scienceClifford algebraConformal geometric algebraConformal mapImage processingParallel computingImage segmentationComputational geometryTheoretical Computer ScienceGeometric algebraOperator (computer programming)Computational Theory and MathematicsConformal geometric algebra five-dimensional clifford algebra computational geometry embedded coprocessors systems-on-programmable-chip FPGA-based prototyping medical imaging segmentation 3D modeling Volume registration Growing Neural Gas marching spheres iterative closest point (ICP) thin-plate spline robust point matching (TPS-RPM)Hardware and ArchitectureScalingSoftwareIEEE Transactions on Computers
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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…

Settore ING-INF/05 - Sistemi Di Elaborazione Delle InformazioniCoprocessorSpeedupComputer sciencebusiness.industryembedded coprocessorsClifford algebraParallel computingcomputer graphicComputer graphicsGeometric algebracompute-intensive algorithmSoftwaresoftware intensive systemComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATIONcomputational geometryGraphicsClifford algebraField-programmable gate arraybusiness
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GAPPCO: An Easy to Configure Geometric Algebra Coprocessor Based on GAPP Programs

2017

Because of the high numeric complexity of Geometric Algebra, its use in engineering applications relies heavily on tools and devices for efficient implementations. In this article, we present a novel hardware design for a Geometric Algebra coprocessor, called GAPPCO, which is based on Geometric Algebra Parallelism Programs (GAPP). GAPPCO is a design for a coprocessor combining the advantages of optimizing software with a configurable hardware able to implement arbitrary Geometric Algebra algorithms. The idea is to have a fixed hardware easily and fast to be configured for different algorithms. We describe the new hardware design together with the complete tool chain for its configuration.

Settore ING-INF/05 - Sistemi Di Elaborazione Delle InformazioniCoprocessorTheoretical computer sciencebusiness.industryApplied MathematicsConfigurable hardware02 engineering and technologyParallel computing01 natural sciences020202 computer hardware & architectureComputer Science::Hardware ArchitectureGeometric algebraSoftwareChain (algebraic topology)0103 physical sciencesGeometric Algebra Geometric Algebra computing Gaalop GAPP GAPPCO0202 electrical engineering electronic engineering information engineeringParallelism (grammar)GapP010307 mathematical physicsbusinessImplementationMathematicsAdvances in Applied Clifford Algebras
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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…

Settore ING-INF/05 - Sistemi Di Elaborazione Delle InformazioniDiscrete mathematicsComputer sciencebusiness.industryLatent semantic analysisInformationSystems_INFORMATIONSTORAGEANDRETRIEVALRepresentation (systemics)computer.software_genreChatbotGeometric algebraKnowledge baseArtificial IntelligenceEncoding (semiotics)chatbot clifford algebraArtificial intelligenceDialog systembusinesscomputerNatural language processingNatural language
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A Family of Embedded Coprocessors with Native Geometric Algebra Support

2015

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, however, 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 represe…

Settore ING-INF/05 - Sistemi Di Elaborazione Delle InformazioniGeometric Algebra Embedded Coprocessors Application- Specic Processors FPGA-based Prototyping
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Geometric Algebra Rotors for Sub-Symbolic Coding of Natural Language Sentences

2007

A sub-symbolic encoding methodology for natural language sentences is presented. The procedure is based on the creation of an LSA-inspired semantic space and associates rotation operators derived from Geometric Algebra to word bigrams of the sentence. The operators are subsequently applied to an orthonormal standard basis of the created semantic space according to the order in which words appear in the sentence. The final rotated basis is then coded as a vector and its orthogonal part constitutes the sub-symbolic coding of the sentence. Preliminary experimental results for a classification task, compared with the traditional LSA methodology, show the effectiveness of the approach.

Settore ING-INF/05 - Sistemi Di Elaborazione Delle InformazioniLatent Semantic AnalysisInformationSystems_INFORMATIONSTORAGEANDRETRIEVALGeometric Algebra RotorsGeometric Algebra Rotors Latent Semantic Analysis
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Sentence Induced Transformations in Conceptual Spaces

2008

The proposed work illustrates how "primitive concepts" can be automatically induced from a text corpus. The primitive concepts are identified by the orthonormal axis of a "conceptual" space induced by a methodology inspired to the latent semantic analysis approach. The methodology represents a natural language sentence by means of a set of rotations of an orthonormal basis in the "conceptual"space. The rotations, triggered by the sequence of words composing the sentence and realized by means of geometric algebra rotors, allow to highlight "conceptual" relations that can arise among the primitive concepts.

Settore ING-INF/05 - Sistemi Di Elaborazione Delle InformazioniSequenceComputer sciencebusiness.industryLatent semantic analysisChatbots conceptual spaces LSA semantic computingWord processingData_MISCELLANEOUScomputer.software_genreGeometric algebraSemantic role labelingOrthonormal basisArtificial intelligencebusinesscomputerNatural languageNatural language processingSentence
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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…

Settore ING-INF/05 - Sistemi Di Elaborazione Delle InformazioniSpeedupCoprocessorComputer scienceClifford algebraParallel computingRendering (computer graphics)Computer graphicsGeometric algebraHardware and ArchitectureComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATIONElectrical and Electronic EngineeringClifford algebra Computational geometry Embedded coprocessors Application-specific processor FPGA-based prototypingField-programmable gate arraySoftwareEuclidean vector
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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…

Settore ING-INF/05 - Sistemi Di Elaborazione Delle InformazioniSpeedupCoprocessorComputer sciencebusiness.industryClifford algebraParallel computingComputer graphicsGeometric algebraSoftwareClifford algebra embedded coprocessors multi-core architectures FPGA prototyping medical imagingField-programmable gate arraybusinessFPGA prototype2012 15th Euromicro Conference on Digital System Design
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