This book presents the theory and applications of an advanced mathematical language called geometric algebra that greatly helps to express the ideas and concepts and to develop algorithms in the broad domain of robot physics.
In the history of science, without essential mathematical concepts, theories would have not been developed at all. We can observe that in various periods of the history of mathematics and physics, certain stagnation occurred; from time to time, thanks to new mathematical developments, astonishing progress took place. In addition, we see that the knowledge became unavoidably fragmented as researchers attempted to combine different mathematical systems. Each mathematical system brings about some parts of geometry; however, together, these systems constitute a highly redundant system due to an unnecessary multiplicity of representations for geometric concepts. The author expects that due to his persistent efforts to bring to the community geometric algebra for applications as a metalanguage for geometric reasoning, in the near future tremendous progress in computer vision, machine learning, and robotics should take place.
What is geometric algebra? Why is its application so promising? Why should researchers, practitioners, and students make the effort to understand geometric algebra and use it? We want to answer all these questions and convince the reader that becoming acquainted with geometric algebra for applications is a worthy undertaking.
The history of geometric algebra is unusual and quite surprising. In the 1870s, William Kingdon Clifford introduced his geometric algebra, building on the earlier works of Sir William Rowan Hamilton and Hermann Gunther Grassmann. In Clifford’s work, we perceive that he intended to describe the geometric properties of vectors, planes, and higher-dimensional objects. Most physicists encounter the algebra in the guise of Pauli and Dirac matrix algebras of quantum theory. Many roboticists or computer graphic engineers use quaternions for 3D rotation estimation and interpolation, as a pointwise approach is too difficult for them to formulate homogeneous transformations of high-order geometric entities.
1. Geometric Algebra for the Twenty-First Century Cybernetics
Part I. Fundamentals of Geometric Algebra
2. Introduction to Geometric Algebra
3. Differentiation, Linear, and Multilinear Functions in Geometric Algebra
4. Geometric Calculus
5. Lie Algebras, Lie Groups, and Algebra of Incidence
Part II. Euclidean, Pseudo-Euclidean Geometric Algebras, Incidence Algebra, Conformal and Projective Geometric Algebras
6. 2D, 3D, and 4D Geometric Algebras
7. Kinematics of the 2D and 3D Spaces
8. Conformal Geometric Algebra
9. The Geometric Algebras
10. Programming Issues
Part III. Image Processing and Computer Vision
11. Quaternion-Clifford Fourier and Wavelet Transforms
12. Geometric Algebra of Computer Vision
Part IV. Machine Learning
13. Geometric Neurocomputing
Part V. Applications of GA in Image Processing, Graphics and Computer Vision
14. Applications of Lie Filters, Quaternion Fourier, and Wavelet Transforms
15. Invariants Theory in Computer Vision and Omnidirectional Vision
16. Geometric Algebra Tensor Voting, Hough Transform, Voting and Perception Using Conformal Geometric Algebra
17. Modeling and Registration of Medical Data
Part VI. Applications of GA in Machine Learning
18. Applications in Neurocomputing
19. Neurocomputing for 2D Contour and 3D Surface Reconstruction
20. Clifford Algebras and Related Algebras
Appendix A: Glossary
Appendix B: Useful Formulas for Geometric Algebra
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