Geometric Computing with Clifford Algebras

1. Front Matter

   Pages I-XVIII

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2. A Unified Algebraic Approach for Classical Geometries

   1. Front Matter

      Pages 1-1

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   2. New Algebraic Tools for Classical Geometry

      • David Hestenes, Hongbo Li, Alyn Rockwood

      Pages 3-26

   3. Generalized Homogeneous Coordinates for Computational Geometry

      • Hongbo Li, David Hestenes, Alyn Rockwood

      Pages 27-59

   4. Spherical Conformal Geometry with Geometric Algebra

      • Hongbo Li, David Hestenes, Alyn Rockwood

      Pages 61-75

   5. A Universal Model for Conformal Geometries of Euclidean, Spherical and Double-Hyperbolic Spaces

      • Hongbo Li, David Hestenes, Alyn Rockwood

      Pages 77-104

   6. Geo-MAP Unification

      • Ambjörn Naeve, Lars Svensson

      Pages 105-126

   7. Honing Geometric Algebra for Its Use in the Computer Sciences

      • Leo Dorst

      Pages 127-152

3. Algebraic Embedding of Signal Theory and Neural Computation

   1. Front Matter

      Pages 153-153

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   2. Spatial-Color Clifford Algebras for Invariant Image Recognition

      • Ekaterina Rundblad-Labunets, Valeri Labunets

      Pages 155-184

   3. Non-commutative Hypercomplex Fourier Transforms of Multidimensional Signals

      • Thomas Bülow, Michael Felsberg, Gerald Sommer

      Pages 187-207

   4. Commutative Hypercomplex Fourier Transforms of Multidimensional Signals

      • Michael Felsberg, Thomas Bülow, Gerald Sommer

      Pages 209-229

   5. Fast Algorithms of Hypercomplex Fourier Transforms

      • Michael Felsberg, Thomas Bülow, Gerald Sommer, Vladimir M. Chernov

      Pages 231-254

   6. Local Hypercomplex Signal Representations and Applications

      • Thomas Bülow, Gerald Sommer

      Pages 255-289

   7. Introduction to Neural Computation in Clifford Algebra

      • Sven Buchholz, Gerald Sommer

      Pages 291-314

   8. Clifford Algebra Multilayer Perceptrons

      • Sven Buchholz, Gerald Sommer

      Pages 315-334

4. Geometric Algebra for Computer Vision and Robotics

   1. Front Matter

      Pages 335-335

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   2. A Unified Description of Multiple View Geometry

      • Christian B. U. Perwass, Joan Lasenby

      Pages 337-369

   3. 3D-Reconstruction from Vanishing Points

      • Christian B. U. Perwass, Joan Lasenby

      Pages 371-392

   4. Analysis and Computation of the Intrinsic Camera Parameters

      • Eduardo Bayro-Corrochano, Bodo Rosenhahn

      Pages 393-414

Clifford algebra, then called geometric algebra, was introduced more than a cenetury ago by William K. Clifford, building on work by Grassmann and Hamilton. Clifford or geometric algebra shows strong unifying aspects and turned out in the 1960s to be a most adequate formalism for describing different geometry-related algebraic systems as specializations of one "mother algebra" in various subfields of physics and engineering. Recent work outlines that Clifford algebra provides a universal and powerfull algebraic framework for an elegant and coherent representation of various problems occuring in computer science, signal processing, neural computing, image processing, pattern recognition, computer vision, and robotics. This monograph-like anthology introduces the concepts and framework of Clifford algebra and provides computer scientists, engineers, physicists, and mathematicians with a rich source of examples of how to work with this formalism.

• Algebra
• Algebraic Expressions
• Algebraic Geometry
• Clifford Algebras
• Computational Geometry
• Computer
• Computer Vision
• Geometric Computing
• Geometric Languages
• Neural Computation
• algorithms
• image processing
• robot
• robotics
• signal processing

From the reviews:

"This monograph-like anthology presents a collection of contributions concerning the problem of solving geometry related problems with suitable algebraic embeddings. It is not only directed at scientists who have already discovered the power of Clifford algebras ... but also at those scientists who are interested in Clifford algebras ... . Therefore, an effort is made to keep this book accessible to newcomers ... while still presenting up to date research and new developments. ... The 21 coherently written chapters cover all relevant issues ... ." (Vasily A. Chernecky, Mathematical Reviews, Issue 2003 m)

"This is a collection of contributions which describe the solution of geometry-related problems by suitable algebraic embeddings, especially into Clifford algebras. ... this book can serve as a reference to the state of the art concerning the use of Clifford algebras as a frame for geometric computing." (H. G. Feichtinger, Monatshefte für Mathematik, Vol. 140 (4), 2003)

"Clifford Algebras were introduced by W. K. Clifford in 1878. … The book is a collection of 21 chapters/papers written by experts in the field. These 21 papers are coherently written and the book can be read almost like a monograph. … The book is clearly written and well structured. It is recommended to mathematicians, physicists, computer scientists, engineers, and ... to graduate students." (K. Gürlebeck, Zeitschrift für Analysis und ihre Anwendungen, Vol. 21 (4), 2002)

• Institut für Informatik, Lehrstuhl für Kognitive Systeme, Christian-Albrechts-Universität zu Kiel, Kiel, Germany

  Gerald Sommer

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