Geometric Algebra Computing: in Engineering and Computer Science

ISBN-13: 9781849961073 / Angielski / Twarda / 2010 / 526 str.

Eduardo Bayro-Corrochano;Gerik Scheuermann

Opis

This book presents new results on applications of geometric algebra. The time when researchers and engineers were starting to realize the potential of quaternions for - plications in electrical, mechanic, and control engineering passed a long time ago. Since the publication of Space-Time Algebra by David Hestenes (1966) and Clifford Algebra to Geometric Calculus: A Uni?ed Language for Mathematics and Physics by David Hestenes and Garret Sobczyk (1984), consistent progress in the app- cations of geometric algebra has taken place. Particularly due to the great dev- opments in computer technology and the Internet, researchers have proposed new ideas and algorithms to tackle a variety of problems in the areas of computer science and engineering using the powerful language of geometric algebra. In this process, pioneer groups started the conference series entitled "Applications of Geometric Algebra in Computer Science and Engineering" (AGACSE) in order to promote the research activity in the domain of the application of geometric algebra. The ?rst conference, AGACSE'1999, organized by Eduardo Bayro-Corrochano and Garret Sobczyk, took place in Ixtapa-Zihuatanejo, Mexico, in July 1999. The contri- tions were published in Geometric Algebra with Applications in Science and En- neering, Birkhauser, 2001. The second conference, ACACSE'2001, was held in the Engineering Department of the Cambridge University on 9-13 July 2001 and was organizedbyLeoDorst, ChrisDoran, andJoanLasenby. Thebestconferencecont- butions appeared as a book entitled Applications of Geometric Algebra in Computer Science and Engineering, Birkhauser, 2002. The third conference, AGACSE'2008, took place in August 2008 in Grimma, Leipzig, Germany.

Recenzje

From the reviews:

"This book is a result of the edited proceedings of the 2008 conference. It contains many advanced ideas from mathematics, physics, and computer science, and ... serve as a reference book on geometric algebra and its applications. ... includes numerous color illustrations, and the chapters end with references to the literature. ... This book should be treasured for presenting various geometric algebra applications in several areas ... . It will be useful to physicists, computer scientists, and engineers. ... this is a very useful book." (S. V. Nagaraj, ACM Computing Reviews, February, 2012)

Spis treści

Part I: Geometric Algebra New Tools for Computational Geometry and Rejuvenation of Screw Theory David Hestenes Tutorial: Structure Preserving Representation of Euclidean Motions through Conformal Geometric Algebra Leo Dorst Engineering Graphics in Geometric Algebra Alyn Rockwood and Dietmar Hildenbrand Parametrization of 3D Conformal Transformations in Conformal Geometric Algebra Hongbo Li Part II: Clifford Fourier Transform Two-Dimensional Clifford Windowed Fourier Transform Mawardi Bahri, Eckhard M. S. Hitzer and Sriwulan Adji The Cylindrical Fourier Transform Fred Brackx, Nele De Schepper, and Frank Sommen Analyzing Real Vector Fields with Clifford Convolution and Clifford Fourier Transform Wieland Reich and Gerik Scheuermann Clifford Fourier Transform for Color Image Processing Thomas Batard, Michel Berthier and Christophe Saint-Jean Hilbert Transforms in Clifford Analysis Fred Brackx, Bram De Knock and Hennie De Schepper Part III: Image Processing, Wavelets and Neurocomputing Geometric Neural Computing for 2D Contour and 3D Surface Reconstruction Jorge Rivera-Rovelo, Eduardo Bayro-Corrochano and Ruediger Dillmann Geometric Associative Memories and their Applications to Pattern Classification Benjamin Cruz, Ricardo Barron, Humberto Sossa Classification and Clustering of Spatial Patterns with Geometric Algebra Minh Tuan Pham, Kanta Tachibana, Eckhard M. S. Hitzer, Tomohiro Yoshikawa, and Takeshi Furuhashi QWT: Retrospective and New Applications Yi Xu, Xiaokang Yang, Li Song, Leonardo Traversoni and Wei Lu Part IV: Computer Vision Image Sensor Model using Geometric Algebra: from Calibration to Motion Estimation Thibaud Debaecker, Ryad Benosman and Sio H. Ieng Model-Based Visual Self-Localization Using Gaussian Spheres D. Gonzalez-Aguirre, T. Asfour, E. Bayro-Corrochano and R. Dillmann Part V: Conformal Mapping and Fluid Analysis Geometric Characterization of M-conformal Mappings K. Gürlebeck and J. Morais Fluid Flow Problems with Quaternionic Analysis: An Alternative Conception K. Gürlebeck and W. Sprössig Part VI: Cristalography, Holography and Complexity Interactive 3D Space Group Visualization with CLUCalc and Crystallographic Subperiodic Groups in Geometric Algebra Eckhard Hitzer, Christian Perwass and Daisuke Ichikawa Geometric Algebra Model of Distributed Representations Agnieszka Patyk Computational Complexity Reductions using Clifford Algebras René Schott and G. Stacey Staples Part VII: Efficient Computing with Clifford (Geometric) Algebra Efficient Algorithms for Factorization and Join of Blades Daniel Fontijne, Leo Dorst Gaalop - High Performance Parallel Computing based on Conformal Geometric Algebra Dietmar Hildenbrand, Joachim Pitt, Andreas Koch Some Applications of Gröbner Bases in Robotics and Engineering Rafal Ablamowicz

Blurb

Geometric algebra provides a rich and general mathematical framework for the development of solutions, concepts and computer algorithms without losing geometric insight into the problem in question. Many current mathematical subjects can be treated in an unified manner without abandoning the mathematical system of geometric algebra, such as multilinear algebra, projective and affine geometry, calculus on manifolds, Riemann geometry, the representation of Lie algebras and Lie groups using bivector algebras, and conformal geometry.

Geometric Algebra Computing in Engineering and Computer Science presents contributions from an international selection of experts in the field. This useful text/reference offers new insights and solutions for the development of theorems, algorithms and advanced methods for real-time applications across a range of disciplines. The book also provides an introduction to advanced screw theory and conformal geometry. Written in an accessible style, the discussion of all applications is enhanced by the inclusion of numerous examples, figures and experimental analysis.

Topics and features:

• Provides a thorough discussion of several tasks for image processing, pattern recognition, computer vision, robotics and computer graphics using the geometric algebra framework
• Introduces nonspecialists to screw theory in the geometric algebra framework, offering a tutorial on conformal geometric algebra and an overview of recent applications of geometric algebra
• Explores new developments in the domain of Clifford Fourier Transforms and Clifford Wavelet Transform, including novel applications of Clifford Fourier transforms for 3D visualization and colour image spectral analysis
• Presents a detailed study of fluid flow problems with quaternionic analysis
• Examines new algorithms for geometric neural computing and cognitive systems
• Analyzes computer software packages for extensive calculations in geometric algebra, investigating the algorithmic complexity of key geometric operations and how the program code can be optimized for real-time computations

The book is an essential resource for computer scientists, applied physicists, AI researchers and mechanical and electrical engineers. It will also be of value to graduate students and researchers interested in a modern language for geometric computing.

Prof. Dr. Eng. Eduardo Bayro-Corrochano is a Full Professor of Geometric Computing at Cinvestav, Mexico. He is the author of the Springer titles Geometric Computing for Perception Action Systems, Handbook of Geometric Computing, and Geometric Computing for Wavelet Transforms, Robot Vision, Learning, Control and Action.

Prof. Dr. Gerik Scheuermann is a Full Professor at the University of Leipzig, Germany. He is the author of the Springer title Topology-Based Methods in Visualization II.