AHCI RESEARCH GROUP
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Papers published in international journals,
proceedings of conferences, workshops and books.
OUR RESEARCH
Scientific Publications
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2022
Vitabile, Salvatore; Franchini, Silvia; Vassallo, Giorgio
An Optimized Architecture for CGA Operations and Its Application to a Simulated Robotic Arm Journal Article
In: Electronics (Switzerland), vol. 11, no. 21, 2022, ISSN: 2079-9292.
Abstract | Links | BibTeX | Tags: Application-specific processors, Clifford algebra, Computational geometry, Conformal geometric algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra, grasping, human-like robotic arms, inverse kinematics, Robotics
@article{vitabileOptimizedArchitectureCGA2022,
title = {An Optimized Architecture for CGA Operations and Its Application to a Simulated Robotic Arm},
author = { Salvatore Vitabile and Silvia Franchini and Giorgio Vassallo},
doi = {10.3390/electronics11213508},
issn = {2079-9292},
year = {2022},
date = {2022-01-01},
journal = {Electronics (Switzerland)},
volume = {11},
number = {21},
abstract = {Conformal geometric algebra (CGA) is a new geometric computation tool that is attracting growing attention in many research fields, such as computer graphics, robotics, and computer vision. Regarding the robotic applications, new approaches based on CGA have been proposed to efficiently solve problems as the inverse kinematics and grasping of a robotic arm. The hardware acceleration of CGA operations is required to meet real-time performance requirements in embedded robotic platforms. In this paper, we present a novel embedded coprocessor for accelerating CGA operations in robotic tasks. Two robotic algorithms, namely, inverse kinematics and grasping of a human-arm-like kinematics chain, are used to prove the effectiveness of the proposed approach. The coprocessor natively supports the entire set of CGA operations including both basic operations (products, sums/differences, and unary operations) and complex operations as rigid body motion operations (reflections, rotations, translations, and dilations). The coprocessor prototype is implemented on the Xilinx ML510 development platform as a complete system-on-chip (SoC), integrating both a PowerPC processing core and a CGA coprocessing core on the same Xilinx Virtex-5 FPGA chip. Experimental results show speedups of 78texttimes and 246texttimes for inverse kinematics and grasping algorithms, respectively, with respect to the execution on the PowerPC processor. textcopyright 2022 by the authors.},
keywords = {Application-specific processors, Clifford algebra, Computational geometry, Conformal geometric algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra, grasping, human-like robotic arms, inverse kinematics, Robotics},
pubstate = {published},
tppubtype = {article}
}
Conformal geometric algebra (CGA) is a new geometric computation tool that is attracting growing attention in many research fields, such as computer graphics, robotics, and computer vision. Regarding the robotic applications, new approaches based on CGA have been proposed to efficiently solve problems as the inverse kinematics and grasping of a robotic arm. The hardware acceleration of CGA operations is required to meet real-time performance requirements in embedded robotic platforms. In this paper, we present a novel embedded coprocessor for accelerating CGA operations in robotic tasks. Two robotic algorithms, namely, inverse kinematics and grasping of a human-arm-like kinematics chain, are used to prove the effectiveness of the proposed approach. The coprocessor natively supports the entire set of CGA operations including both basic operations (products, sums/differences, and unary operations) and complex operations as rigid body motion operations (reflections, rotations, translations, and dilations). The coprocessor prototype is implemented on the Xilinx ML510 development platform as a complete system-on-chip (SoC), integrating both a PowerPC processing core and a CGA coprocessing core on the same Xilinx Virtex-5 FPGA chip. Experimental results show speedups of 78texttimes and 246texttimes for inverse kinematics and grasping algorithms, respectively, with respect to the execution on the PowerPC processor. textcopyright 2022 by the authors.
2021
Franchini, Silvia; Vitabile, Salvatore
Geometric Calculus Applications to Medical Imaging: Status and Perspectives Proceedings Article
In: Xambó-Descamps, Sebasti`a (Ed.): Systems, Patterns and Data Engineering with Geometric Calculi, pp. 31–46, Springer International Publishing, Cham, 2021, ISBN: 978-3-030-74486-1.
Abstract | Links | BibTeX | Tags: 3D modeling, Clifford algebra, Deep learning, Geometric algebra, Geometric Calculus, Medical image classification, Medical image registration, Medical image segmentation, Medical Imaging, radiomics
@inproceedings{franchiniGeometricCalculusApplications2021,
title = {Geometric Calculus Applications to Medical Imaging: Status and Perspectives},
author = { Silvia Franchini and Salvatore Vitabile},
editor = { Sebasti{`a} {Xambó-Descamps}},
doi = {10.1007/978-3-030-74486-1_3},
isbn = {978-3-030-74486-1},
year = {2021},
date = {2021-01-01},
booktitle = {Systems, Patterns and Data Engineering with Geometric Calculi},
pages = {31--46},
publisher = {Springer International Publishing},
address = {Cham},
series = {SEMA SIMAI Springer Series},
abstract = {Medical imaging data coming from different acquisition modalities requires automatic tools to extract useful information and support clinicians in the formulation of accurate diagnoses. Geometric Calculus (GC) offers a powerful mathematical and computational model for the development of effective medical imaging algorithms. The practical use of GC-based methods in medical imaging requires fast and efficient implementations to meet real-time processing constraints as well as accuracy and robustness requirements. The purpose of this article is to present the state of the art of the GC-based techniques for medical image analysis and processing. The use of GC-based paradigms in Radiomics and Deep Learning, i.e. a comprehensive quantification of tumor phenotypes by applying a large number of quantitative image features and its classification, is also outlined.},
keywords = {3D modeling, Clifford algebra, Deep learning, Geometric algebra, Geometric Calculus, Medical image classification, Medical image registration, Medical image segmentation, Medical Imaging, radiomics},
pubstate = {published},
tppubtype = {inproceedings}
}
Medical imaging data coming from different acquisition modalities requires automatic tools to extract useful information and support clinicians in the formulation of accurate diagnoses. Geometric Calculus (GC) offers a powerful mathematical and computational model for the development of effective medical imaging algorithms. The practical use of GC-based methods in medical imaging requires fast and efficient implementations to meet real-time processing constraints as well as accuracy and robustness requirements. The purpose of this article is to present the state of the art of the GC-based techniques for medical image analysis and processing. The use of GC-based paradigms in Radiomics and Deep Learning, i.e. a comprehensive quantification of tumor phenotypes by applying a large number of quantitative image features and its classification, is also outlined.
2020
Franchini, Silvia; Gentile, Antonio; Vassallo, Giorgio; Vitabile, Salvatore
Implementation and Evaluation of Medical Imaging Techniques Based on Conformal Geometric Algebra Journal Article
In: International Journal of Applied Mathematics and Computer Science, vol. 30, no. 3, pp. 415–433, 2020, ISSN: 1641-876X.
Abstract | Links | BibTeX | Tags: 3D modeling, Clifford algebra, Computational geometry, Conformal geometric algebra, Geometric algebra, Medical image registration, Medical image segmentation, Medical Imaging
@article{franchiniImplementationEvaluationMedical2020,
title = {Implementation and Evaluation of Medical Imaging Techniques Based on Conformal Geometric Algebra},
author = { Silvia Franchini and Antonio Gentile and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.34768/amcs-2020-0031},
issn = {1641-876X},
year = {2020},
date = {2020-01-01},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {30},
number = {3},
pages = {415--433},
abstract = {Medical imaging tasks, such as segmentation, 3D modeling, and registration of medical images, involve complex geometric problems, usually solved by standard linear algebra and matrix calculations. In the last few decades, conformal geometric algebra (CGA) has emerged as a new approach to geometric computing that offers a simple and efficient representation of geometric objects and transformations. However, the practical use of CGA-based methods for big data image processing in medical imaging requires fast and efficient implementations of CGA operations to meet both real-time processing constraints and accuracy requirements. The purpose of this study is to present a novel implementation of CGA-based medical imaging techniques that makes them effective and practically usable. The paper exploits a new simplified formulation of CGA operators that allows significantly reduced execution times while maintaining the needed result precision. We have exploited this novel CGA formulation to re-design a suite of medical imaging automatic methods, including image segmentation, 3D reconstruction and registration. Experimental tests show that the re-formulated CGA-based methods lead to both higher precision results and reduced computation times, which makes them suitable for big data image processing applications. The segmentation algorithm provides the Dice index, sensitivity and specificity values of 98.14%, 98.05% and 97.73%, respectively, while the order of magnitude of the errors measured for the registration methods is 10-5. textcopyright 2020 Sciendo. All rights reserved.},
keywords = {3D modeling, Clifford algebra, Computational geometry, Conformal geometric algebra, Geometric algebra, Medical image registration, Medical image segmentation, Medical Imaging},
pubstate = {published},
tppubtype = {article}
}
Medical imaging tasks, such as segmentation, 3D modeling, and registration of medical images, involve complex geometric problems, usually solved by standard linear algebra and matrix calculations. In the last few decades, conformal geometric algebra (CGA) has emerged as a new approach to geometric computing that offers a simple and efficient representation of geometric objects and transformations. However, the practical use of CGA-based methods for big data image processing in medical imaging requires fast and efficient implementations of CGA operations to meet both real-time processing constraints and accuracy requirements. The purpose of this study is to present a novel implementation of CGA-based medical imaging techniques that makes them effective and practically usable. The paper exploits a new simplified formulation of CGA operators that allows significantly reduced execution times while maintaining the needed result precision. We have exploited this novel CGA formulation to re-design a suite of medical imaging automatic methods, including image segmentation, 3D reconstruction and registration. Experimental tests show that the re-formulated CGA-based methods lead to both higher precision results and reduced computation times, which makes them suitable for big data image processing applications. The segmentation algorithm provides the Dice index, sensitivity and specificity values of 98.14%, 98.05% and 97.73%, respectively, while the order of magnitude of the errors measured for the registration methods is 10-5. textcopyright 2020 Sciendo. All rights reserved.
2017
Franchini, Silvia; Gentile, Antonio; Sorbello, Filippo; Vassallo, Giorgio; Vitabile, Salvatore
Embedded Coprocessors for Native Execution of Geometric Algebra Operations Journal Article
In: Advances in Applied Clifford Algebras, vol. 27, no. 1, pp. 559–580, 2017, ISSN: 0188-7009.
Abstract | Links | BibTeX | Tags: Application-specific processors, Clifford algebra, Conformal geometric algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra
@article{franchiniEmbeddedCoprocessorsNative2017,
title = {Embedded Coprocessors for Native Execution of Geometric Algebra Operations},
author = { Silvia Franchini and Antonio Gentile and Filippo Sorbello and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1007/s00006-016-0662-1},
issn = {0188-7009},
year = {2017},
date = {2017-01-01},
journal = {Advances in Applied Clifford Algebras},
volume = {27},
number = {1},
pages = {559--580},
abstract = {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. textcopyright 2016, Springer International Publishing.},
keywords = {Application-specific processors, Clifford algebra, Conformal geometric algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra},
pubstate = {published},
tppubtype = {article}
}
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. textcopyright 2016, Springer International Publishing.
Hildenbrand, Dietmar; Franchini, Silvia; Gentile, Antonio; Vassallo, Giorgio; Vitabile, Salvatore
GAPPCO: An Easy to Configure Geometric Algebra Coprocessor Based on GAPP Programs Journal Article
In: Advances in Applied Clifford Algebras, vol. 27, no. 3, pp. 2115–2132, 2017, ISSN: 0188-7009.
Abstract | Links | BibTeX | Tags: Clifford algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Gaalop, GAPP, GAPPCO, Geometric algebra, Geometric Algebra computing, Pre-compilers, Software hardware co-design
@article{hildenbrandGAPPCOEasyConfigure2017,
title = {GAPPCO: An Easy to Configure Geometric Algebra Coprocessor Based on GAPP Programs},
author = { Dietmar Hildenbrand and Silvia Franchini and Antonio Gentile and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1007/s00006-016-0755-x},
issn = {0188-7009},
year = {2017},
date = {2017-01-01},
journal = {Advances in Applied Clifford Algebras},
volume = {27},
number = {3},
pages = {2115--2132},
abstract = {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. textcopyright 2017, Springer International Publishing.},
keywords = {Clifford algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Gaalop, GAPP, GAPPCO, Geometric algebra, Geometric Algebra computing, Pre-compilers, Software hardware co-design},
pubstate = {published},
tppubtype = {article}
}
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. textcopyright 2017, Springer International Publishing.
2015
Franchini, Silvia; Gentile, Antonio; Vassallo, Giorgio; Vitabile, Salvatore
Accelerating Clifford Algebra Operations Using GPUs and an OpenCL Code Generator Proceedings Article
In: pp. 57–64, 2015, ISBN: 978-1-4673-8035-5.
Abstract | Links | BibTeX | Tags: Clifford algebra, Geometric algebra, GPU, Graphics Processing Units, Hardware-software co-design, Metaprogramming, OpenCL
@inproceedings{franchiniAcceleratingCliffordAlgebra2015,
title = {Accelerating Clifford Algebra Operations Using GPUs and an OpenCL Code Generator},
author = { Silvia Franchini and Antonio Gentile and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1109/DSD.2015.44},
isbn = {978-1-4673-8035-5},
year = {2015},
date = {2015-01-01},
pages = {57--64},
abstract = {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. textcopyright 2015 IEEE.},
keywords = {Clifford algebra, Geometric algebra, GPU, Graphics Processing Units, Hardware-software co-design, Metaprogramming, OpenCL},
pubstate = {published},
tppubtype = {inproceedings}
}
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. textcopyright 2015 IEEE.
Franchini, Silvia; Gentile, Antonio; Sorbello, Filippo; Vassallo, Giorgio; Vitabile, Salvatore
ConformalALU: A Conformal Geometric Algebra Coprocessor for Medical Image Processing Journal Article
In: IEEE Transactions on Computers, vol. 64, no. 4, pp. 955–970, 2015, ISSN: 0018-9340.
Abstract | Links | BibTeX | Tags: 3D modeling, Clifford algebra, Computational geometry, Conformal geometric algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra, Growing Neural Gas, iterative closest point (ICP), marching spheres, Medical image registration, Medical Imaging, Segmentation, systems-on-programmable-chip, thin-plate spline robust point matching (TPS-RPM), Volume registration
@article{franchiniConformalALUConformalGeometric2015,
title = {ConformalALU: A Conformal Geometric Algebra Coprocessor for Medical Image Processing},
author = { Silvia Franchini and Antonio Gentile and Filippo Sorbello and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1109/TC.2014.2315652},
issn = {0018-9340},
year = {2015},
date = {2015-01-01},
journal = {IEEE Transactions on Computers},
volume = {64},
number = {4},
pages = {955--970},
abstract = {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. textcopyright 2015 IEEE.},
keywords = {3D modeling, Clifford algebra, Computational geometry, Conformal geometric algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra, Growing Neural Gas, iterative closest point (ICP), marching spheres, Medical image registration, Medical Imaging, Segmentation, systems-on-programmable-chip, thin-plate spline robust point matching (TPS-RPM), Volume registration},
pubstate = {published},
tppubtype = {article}
}
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. textcopyright 2015 IEEE.
2013
Augello, Agnese; Gaglio, Salvatore; Pilato, Giovanni; Vassallo, Giorgio
Clifford Rotors for Conceptual Representation in Chatbots Journal Article
In: Advances in Intelligent Systems and Computing, vol. 196 AISC, pp. 369–370, 2013, ISSN: 21945357.
Abstract | Links | BibTeX | Tags: Chatbots, Clifford algebra, Conceptual Spaces, Geometric algebra, Knowledge Representation, Latent Semantic Analysis, Natural Language Processing, Semantic Computing
@article{augelloCliffordRotorsConceptual2013,
title = {Clifford Rotors for Conceptual Representation in Chatbots},
author = { Agnese Augello and Salvatore Gaglio and Giovanni Pilato and Giorgio Vassallo},
doi = {10.1007/978-3-642-34274-5_64},
issn = {21945357},
year = {2013},
date = {2013-01-01},
journal = {Advances in Intelligent Systems and Computing},
volume = {196 AISC},
pages = {369--370},
abstract = {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 schema [3], that we call TW-ICW (term weight-inverse conceptual coordinate weight), to weigh the relevance of each term on each conceptual coordinate. textcopyright 2013 Springer-Verlag.},
keywords = {Chatbots, Clifford algebra, Conceptual Spaces, Geometric algebra, Knowledge Representation, Latent Semantic Analysis, Natural Language Processing, Semantic Computing},
pubstate = {published},
tppubtype = {article}
}
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 schema [3], that we call TW-ICW (term weight-inverse conceptual coordinate weight), to weigh the relevance of each term on each conceptual coordinate. textcopyright 2013 Springer-Verlag.
Franchini, Silvia; Gentile, Antonio; Sorbello, Filippo; Vassallo, Giorgio; Vitabile, Salvatore
Design and Implementation of an Embedded Coprocessor with Native Support for 5D, Quadruple-Based Clifford Algebra Journal Article
In: IEEE Transactions on Computers, vol. 62, no. 12, pp. 2366–2381, 2013, ISSN: 0018-9340.
Abstract | Links | BibTeX | Tags: Application-specific processors, Clifford algebra, Computational geometry, Computer graphics, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra, inverse kinematics, Motion capture, Raytracing, robotic arm, Robotics
@article{franchiniDesignImplementationEmbedded2013,
title = {Design and Implementation of an Embedded Coprocessor with Native Support for 5D, Quadruple-Based Clifford Algebra},
author = { Silvia Franchini and Antonio Gentile and Filippo Sorbello and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1109/TC.2012.225},
issn = {0018-9340},
year = {2013},
date = {2013-01-01},
journal = {IEEE Transactions on Computers},
volume = {62},
number = {12},
pages = {2366--2381},
abstract = {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 5texttimes average speedup for 4D Clifford products and a 4texttimes 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 3texttimes and 4texttimes with respect to the baseline Gaigen 2 implementation. Finally, a multicore approach to higher dimensional CA based on CliffordALU5 is discussed. textcopyright 1968-2012 IEEE.},
keywords = {Application-specific processors, Clifford algebra, Computational geometry, Computer graphics, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra, inverse kinematics, Motion capture, Raytracing, robotic arm, Robotics},
pubstate = {published},
tppubtype = {article}
}
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 5texttimes average speedup for 4D Clifford products and a 4texttimes 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 3texttimes and 4texttimes with respect to the baseline Gaigen 2 implementation. Finally, a multicore approach to higher dimensional CA based on CliffordALU5 is discussed. textcopyright 1968-2012 IEEE.
Franchini, Silvia; Gentile, Antonio; Vassallo, Giorgio; Sorbello, Filippo; Vitabile, Salvatore
A Specialized Architecture for Color Image Edge Detection Based on Clifford Algebra Proceedings Article
In: pp. 128–135, 2013, ISBN: 978-0-7695-4992-7.
Abstract | Links | BibTeX | Tags: Application-specific processors, Clifford algebra, Color image edge detection, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra, Image processing, Medical Imaging, Multispectral Magnetic Resonance images
@inproceedings{franchiniSpecializedArchitectureColor2013,
title = {A Specialized Architecture for Color Image Edge Detection Based on Clifford Algebra},
author = { Silvia Franchini and Antonio Gentile and Giorgio Vassallo and Filippo Sorbello and Salvatore Vitabile},
doi = {10.1109/CISIS.2013.29},
isbn = {978-0-7695-4992-7},
year = {2013},
date = {2013-01-01},
pages = {128--135},
abstract = {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. textcopyright 2013 IEEE.},
keywords = {Application-specific processors, Clifford algebra, Color image edge detection, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra, Image processing, Medical Imaging, Multispectral Magnetic Resonance images},
pubstate = {published},
tppubtype = {inproceedings}
}
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. textcopyright 2013 IEEE.
2012
Augello, Agnese; Gentile, Manuel; Pilato, Giovanni; Vassallo, Giorgio
Geometric Encoding of Sentences Based on Clifford Algebra Proceedings Article
In: KDIR 2012 - Proceedings of the International Conference on Knowledge Discovery and Information Retrieval, pp. 457–462, 2012, ISBN: 978-989-8565-29-7.
Abstract | BibTeX | Tags: Geometric algebra, Information Retrieval, Knowledge Representation, Natural Language Processing, Semantic Computing, Semantic Spaces
@inproceedings{augelloGeometricEncodingSentences2012,
title = {Geometric Encoding of Sentences Based on Clifford Algebra},
author = { Agnese Augello and Manuel Gentile and Giovanni Pilato and Giorgio Vassallo},
isbn = {978-989-8565-29-7},
year = {2012},
date = {2012-01-01},
booktitle = {KDIR 2012 - Proceedings of the International Conference on Knowledge Discovery and Information Retrieval},
pages = {457--462},
abstract = {Natural language sentences can be represented as vectors in a high dimensional vector space. Generally, these models are based on bag of words approaches, and therefore they do not fully capture the semantics of sentences which depends both by the semantics of the words, and their order in in the phrase. In this work we propose a sub-symbolic methodology to encode natural language sentences considering both these two aspects. The proposed approach exploits the properties of Geometric Algebra rotation operators, called rotors, to code sentences through the rotation of an orthogonal basis of a semantic space. The methodology is based on three main steps: the construction of a semantic space, the association of ad-hoc rotors to sentence bigrams, and finally the coding of the sentence through the application of the obtained rotors to a standard basis in the semantic space. Copyright textcopyright 2012 SciTePress - Science and Technology Publications.},
keywords = {Geometric algebra, Information Retrieval, Knowledge Representation, Natural Language Processing, Semantic Computing, Semantic Spaces},
pubstate = {published},
tppubtype = {inproceedings}
}
Natural language sentences can be represented as vectors in a high dimensional vector space. Generally, these models are based on bag of words approaches, and therefore they do not fully capture the semantics of sentences which depends both by the semantics of the words, and their order in in the phrase. In this work we propose a sub-symbolic methodology to encode natural language sentences considering both these two aspects. The proposed approach exploits the properties of Geometric Algebra rotation operators, called rotors, to code sentences through the rotation of an orthogonal basis of a semantic space. The methodology is based on three main steps: the construction of a semantic space, the association of ad-hoc rotors to sentence bigrams, and finally the coding of the sentence through the application of the obtained rotors to a standard basis in the semantic space. Copyright textcopyright 2012 SciTePress - Science and Technology Publications.
Franchini, Silvia; Gentile, Antonio; Sorbello, Filippo; Vassallo, Giorgio; Vitabile, Salvatore
Clifford Algebra Based Edge Detector for Color Images Proceedings Article
In: pp. 84–91, 2012, ISBN: 978-0-7695-4687-2.
Abstract | Links | BibTeX | Tags: Clifford algebra, Clifford convolution, Clifford Fourier transform, Color image edge detection, Edge detection, Geometric algebra, Image processing, Segmentation
@inproceedings{franchiniCliffordAlgebraBased2012,
title = {Clifford Algebra Based Edge Detector for Color Images},
author = { Silvia Franchini and Antonio Gentile and Filippo Sorbello and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1109/CISIS.2012.128},
isbn = {978-0-7695-4687-2},
year = {2012},
date = {2012-01-01},
pages = {84--91},
abstract = {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. textcopyright 2012 Crown Copyright.},
keywords = {Clifford algebra, Clifford convolution, Clifford Fourier transform, Color image edge detection, Edge detection, Geometric algebra, Image processing, Segmentation},
pubstate = {published},
tppubtype = {inproceedings}
}
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. textcopyright 2012 Crown Copyright.
Franchini, Silvia; Gentile, Antonio; Sorbello, Filippo; Vassallo, Giorgio; Vitabile, Salvatore
Design Space Exploration of Parallel Embedded Architectures for Native Clifford Algebra Operations Journal Article
In: IEEE Design and Test of Computers, vol. 29, no. 3, pp. 60–69, 2012, ISSN: 0740-7475.
Abstract | Links | BibTeX | Tags: Application-specific processors, Clifford algebra, Computational geometry, Design space exploration, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra
@article{franchiniDesignSpaceExploration2012,
title = {Design Space Exploration of Parallel Embedded Architectures for Native Clifford Algebra Operations},
author = { Silvia Franchini and Antonio Gentile and Filippo Sorbello and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1109/MDT.2012.2206150},
issn = {0740-7475},
year = {2012},
date = {2012-01-01},
journal = {IEEE Design and Test of Computers},
volume = {29},
number = {3},
pages = {60--69},
abstract = {The design space exploration of parallel embedded architectures that natively support Clifford algebra with different costs, performance and precision is studied. First, the most common applications of geometric algebra have been profiled in order to find the most frequent Clifford algebra operations to be natively supported on the coprocessors. The coprocessor design space has been explored using different design parameters. The parallel execution flow, as well as pipelining techniques, have been chosen for quadruple-based architectures to permit the fastest execution of the most frequent operation. Experimental tests concerned design space exploration, in terms of area cost, relative error, latencies and speedup, of the various implemented architectures based on different sets of architectural parameters, such as the number of multipliers and the coefficient precision. The coprocessor CliffordALU shows an effective 5x average speedup for Clifford products against the same operations in Gaigen-2, a geometric algebra software library generator for general-purpose processors.},
keywords = {Application-specific processors, Clifford algebra, Computational geometry, Design space exploration, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra},
pubstate = {published},
tppubtype = {article}
}
The design space exploration of parallel embedded architectures that natively support Clifford algebra with different costs, performance and precision is studied. First, the most common applications of geometric algebra have been profiled in order to find the most frequent Clifford algebra operations to be natively supported on the coprocessors. The coprocessor design space has been explored using different design parameters. The parallel execution flow, as well as pipelining techniques, have been chosen for quadruple-based architectures to permit the fastest execution of the most frequent operation. Experimental tests concerned design space exploration, in terms of area cost, relative error, latencies and speedup, of the various implemented architectures based on different sets of architectural parameters, such as the number of multipliers and the coefficient precision. The coprocessor CliffordALU shows an effective 5x average speedup for Clifford products against the same operations in Gaigen-2, a geometric algebra software library generator for general-purpose processors.
Franchini, Silvia; Gentile, Antonio; Sorbello, Filippo; Vassallo, Giorgio; Vitabile, Salvatore
A Dual-Core Coprocessor with Native 4D Clifford Algebra Support Proceedings Article
In: pp. 419–422, 2012, ISBN: 978-0-7695-4798-5.
Abstract | Links | BibTeX | Tags: Clifford algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra, Medical Imaging, multi-core architectures
@inproceedings{franchiniDualcoreCoprocessorNative2012,
title = {A Dual-Core Coprocessor with Native 4D Clifford Algebra Support},
author = { Silvia Franchini and Antonio Gentile and Filippo Sorbello and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1109/DSD.2012.2},
isbn = {978-0-7695-4798-5},
year = {2012},
date = {2012-01-01},
pages = {419--422},
abstract = {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. textcopyright 2012 IEEE.},
keywords = {Clifford algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra, Medical Imaging, multi-core architectures},
pubstate = {published},
tppubtype = {inproceedings}
}
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. textcopyright 2012 IEEE.
2011
Franchini, Silvia; Gentile, Antonio; Sorbello, Filippo; Vassallo, Giorgio; Vitabile, Salvatore
Fixed-Size Quadruples for a New, Hardware-Oriented Representation of the 4D Clifford Algebra Journal Article
In: Advances in Applied Clifford Algebras, vol. 21, no. 2, pp. 315–340, 2011, ISSN: 1661-4909.
Abstract | Links | BibTeX | Tags: Application-specific processors, Clifford algebra, Computational geometry, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra
@article{franchiniFixedSizeQuadruplesNew2011,
title = {Fixed-Size Quadruples for a New, Hardware-Oriented Representation of the 4D Clifford Algebra},
author = { Silvia Franchini and Antonio Gentile and Filippo Sorbello and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1007/s00006-010-0258-0},
issn = {1661-4909},
year = {2011},
date = {2011-01-01},
journal = {Advances in Applied Clifford Algebras},
volume = {21},
number = {2},
pages = {315--340},
abstract = {Clifford algebra (geometric algebra) offers a natural and intuitive way to model geometry in fields as robotics, machine vision and computer graphics. This paper proposes a new representation based on fixed-size elements (quadruples) of 4D Clifford algebra and demonstrates that this choice leads to an algorithmic simplification which in turn leads to a simpler and more compact hardware implementation of the algebraic operations. In order to prove the advantages of the new, quadruple-based representation over the classical representation based on homogeneous elements, a coprocessing core supporting the new fixed-size Clifford operands, namely Quad-CliffoSor (Quadruple-based Clifford coprocesSor) was designed and prototyped on an FPGA board. Test results show the potential to achieve a 23texttimes speedup for Clifford products and a 33texttimes speedup for Clifford sums and differences compared to the same operations executed by a software library running on a general-purpose processor. textcopyright 2010 Springer Basel AG.},
keywords = {Application-specific processors, Clifford algebra, Computational geometry, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra},
pubstate = {published},
tppubtype = {article}
}
Clifford algebra (geometric algebra) offers a natural and intuitive way to model geometry in fields as robotics, machine vision and computer graphics. This paper proposes a new representation based on fixed-size elements (quadruples) of 4D Clifford algebra and demonstrates that this choice leads to an algorithmic simplification which in turn leads to a simpler and more compact hardware implementation of the algebraic operations. In order to prove the advantages of the new, quadruple-based representation over the classical representation based on homogeneous elements, a coprocessing core supporting the new fixed-size Clifford operands, namely Quad-CliffoSor (Quadruple-based Clifford coprocesSor) was designed and prototyped on an FPGA board. Test results show the potential to achieve a 23texttimes speedup for Clifford products and a 33texttimes speedup for Clifford sums and differences compared to the same operations executed by a software library running on a general-purpose processor. textcopyright 2010 Springer Basel AG.
Franchini, Silvia; Gentile, Antonio; Sorbello, Filippo; Vassallo, Giorgio; Vitabile, Salvatore
A New Embedded Coprocessor for Clifford Algebra Based Software Intensive Systems Proceedings Article
In: pp. 335–342, 2011, ISBN: 978-0-7695-4373-4.
Abstract | Links | BibTeX | Tags: Clifford algebra, Computational geometry, Compute-intensive algorithms, Computer graphics, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra, Software intensive systems
@inproceedings{franchiniNewEmbeddedCoprocessor2011,
title = {A New Embedded Coprocessor for Clifford Algebra Based Software Intensive Systems},
author = { Silvia Franchini and Antonio Gentile and Filippo Sorbello and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1109/CISIS.2011.55},
isbn = {978-0-7695-4373-4},
year = {2011},
date = {2011-01-01},
pages = {335--342},
abstract = {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 architecture of CliffoSorII (Clifford coprocessor II), an embedded coprocessor that offers direct hardware support to Clifford algebra operations, is introduced. The coprocessor has been designed, implemented and tested on a Field Programmable Gate Array (FPGA) board. The experimental results show the potential to achieve a 20x speedup for Clifford sums and differences and a 5x speedup for Clifford products against the analogous operations in Gaigen, a standard geometric algebra software library generator for general purpose processors. An execution analysis of a ray tracing application is also presented. textcopyright 2011 IEEE.},
keywords = {Clifford algebra, Computational geometry, Compute-intensive algorithms, Computer graphics, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra, Software intensive systems},
pubstate = {published},
tppubtype = {inproceedings}
}
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 architecture of CliffoSorII (Clifford coprocessor II), an embedded coprocessor that offers direct hardware support to Clifford algebra operations, is introduced. The coprocessor has been designed, implemented and tested on a Field Programmable Gate Array (FPGA) board. The experimental results show the potential to achieve a 20x speedup for Clifford sums and differences and a 5x speedup for Clifford products against the analogous operations in Gaigen, a standard geometric algebra software library generator for general purpose processors. An execution analysis of a ray tracing application is also presented. textcopyright 2011 IEEE.
2009
Franchini, Silvia; Gentile, Antonio; Sorbello, Filippo; Vassallo, Giorgio; Vitabile, Salvatore
An Embedded, FPGA-based Computer Graphics Coprocessor with Native Geometric Algebra Support Journal Article
In: Integration, the VLSI Journal, vol. 42, no. 3, pp. 346–355, 2009, ISSN: 0167-9260.
Abstract | Links | BibTeX | Tags: Application-specific processors, Clifford algebra, Computational geometry, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra
@article{franchiniEmbeddedFPGAbasedComputer2009,
title = {An Embedded, FPGA-based Computer Graphics Coprocessor with Native Geometric Algebra Support},
author = { Silvia Franchini and Antonio Gentile and Filippo Sorbello and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1016/j.vlsi.2008.09.010},
issn = {0167-9260},
year = {2009},
date = {2009-01-01},
journal = {Integration, the VLSI Journal},
volume = {42},
number = {3},
pages = {346--355},
abstract = {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 20texttimes speedup for 3D vector rotations, a 12texttimes speedup for Clifford sums and differences, and more than a 4texttimes 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. textcopyright 2008 Elsevier B.V. All rights reserved.},
keywords = {Application-specific processors, Clifford algebra, Computational geometry, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra},
pubstate = {published},
tppubtype = {article}
}
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 20texttimes speedup for 3D vector rotations, a 12texttimes speedup for Clifford sums and differences, and more than a 4texttimes 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. textcopyright 2008 Elsevier B.V. All rights reserved.
2008
Augello, Agnese; Vassallo, Giorgio; Gaglio, Salvatore; Pilato, Giovanni
Sentence Induced Transformations in "Conceptual" Spaces Proceedings Article
In: Proceedings - IEEE International Conference on Semantic Computing 2008, ICSC 2008, pp. 34–41, 2008, ISBN: 978-0-7695-3279-0.
Abstract | Links | BibTeX | Tags: Clifford algebra, Geometric algebra, Latent Semantic Analysis, Natural Language Processing, Semantic Computing
@inproceedings{augelloSentenceInducedTransformations2008,
title = {Sentence Induced Transformations in "Conceptual" Spaces},
author = { Agnese Augello and Giorgio Vassallo and Salvatore Gaglio and Giovanni Pilato},
doi = {10.1109/ICSC.2008.74},
isbn = {978-0-7695-3279-0},
year = {2008},
date = {2008-01-01},
booktitle = {Proceedings - IEEE International Conference on Semantic Computing 2008, ICSC 2008},
pages = {34--41},
abstract = {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. textcopyright 2008 IEEE.},
keywords = {Clifford algebra, Geometric algebra, Latent Semantic Analysis, Natural Language Processing, Semantic Computing},
pubstate = {published},
tppubtype = {inproceedings}
}
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. textcopyright 2008 IEEE.
Franchini, Silvia; Gentile, Antonio; Sorbello, Filippo; Vassallo, Giorgio; Vitabile, Salvatore
An FPGA Implementation of a Quadruple-Based Multiplier for 4D Clifford Algebra Proceedings Article
In: pp. 743–751, 2008, ISBN: 978-0-7695-3277-6.
Abstract | Links | BibTeX | Tags: Clifford algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra
@inproceedings{franchiniFPGAImplementationQuadruplebased2008,
title = {An FPGA Implementation of a Quadruple-Based Multiplier for 4D Clifford Algebra},
author = { Silvia Franchini and Antonio Gentile and Filippo Sorbello and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1109/DSD.2008.91},
isbn = {978-0-7695-3277-6},
year = {2008},
date = {2008-01-01},
pages = {743--751},
abstract = {Geometric or Clifford algebra is an interesting paradigm for geometric modeling infields 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 40 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. textcopyright.2008 IEEE.},
keywords = {Clifford algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra},
pubstate = {published},
tppubtype = {inproceedings}
}
Geometric or Clifford algebra is an interesting paradigm for geometric modeling infields 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 40 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. textcopyright.2008 IEEE.
2007
Franchini, Silvia; Gentile, Antonio; Grimaudo, Marco; Hung, Carlo Alberto; Impastato, Sandro; Sorbello, Filippo; Vassallo, Giorgio; Vitabile, Salvatore
A Sliced Coprocessor for Native Clifford Algebra Operations Proceedings Article
In: pp. 436–439, 2007, ISBN: 978-0-7695-2978-3.
Abstract | Links | BibTeX | Tags: Clifford algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra
@inproceedings{franchiniSlicedCoprocessorNative2007,
title = {A Sliced Coprocessor for Native Clifford Algebra Operations},
author = { Silvia Franchini and Antonio Gentile and Marco Grimaudo and Carlo Alberto Hung and Sandro Impastato and Filippo Sorbello and Giorgio Vassallo and Salvatore Vitabile},
doi = {10.1109/DSD.2007.4341505},
isbn = {978-0-7695-2978-3},
year = {2007},
date = {2007-01-01},
pages = {436--439},
abstract = {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 3x speedup for Clifford sums and 4x speedup for Clifford products compared to against the analogous operations in the software library generator GAIGEN. textcopyright 2007 IEEE.},
keywords = {Clifford algebra, Embedded coprocessors, Field Programmable Gate Arrays, FPGA prototyping, Geometric algebra},
pubstate = {published},
tppubtype = {inproceedings}
}
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 3x speedup for Clifford sums and 4x speedup for Clifford products compared to against the analogous operations in the software library generator GAIGEN. textcopyright 2007 IEEE.
Pilato, Giovanni; Augello, Agnese; Vassallo, Giorgio; Gaglio, Salvatore
Geometrie Algebra Rotors for Sub-Symbolic Coding of Natural Language Sentences Journal Article
In: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4692 LNAI, no. PART 1, pp. 42–51, 2007, ISSN: 03029743.
Abstract | BibTeX | Tags: Clifford algebra, Geometric algebra, Latent Semantic Analysis, Natural Language Processing
@article{pilatoGeometrieAlgebraRotors2007,
title = {Geometrie Algebra Rotors for Sub-Symbolic Coding of Natural Language Sentences},
author = { Giovanni Pilato and Agnese Augello and Giorgio Vassallo and Salvatore Gaglio},
issn = {03029743},
year = {2007},
date = {2007-01-01},
journal = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)},
volume = {4692 LNAI},
number = {PART 1},
pages = {42--51},
abstract = {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. textcopyright Springer-Verlag Berlin Heidelberg 2007.},
keywords = {Clifford algebra, Geometric algebra, Latent Semantic Analysis, Natural Language Processing},
pubstate = {published},
tppubtype = {article}
}
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. textcopyright Springer-Verlag Berlin Heidelberg 2007.