Geometric Algebra Applications Vol. I

This chapter gives a detailed outline of geometric algebra and explains the related traditional algebras in common use by mathematicians, physicists, computer scientists, and engineers.

This chapter gives a detailed outline of differentiation, linear, and multilinear functions, eigenblades, and tensors formulated in geometric algebra and explains the related operators and transformations in common use by mathematicians, physicists, computer scientists, and engineers.

We have learned that readers of the work of D. Hestenes and G. Sobzyk (Hestenes and Sobczyk (1984). Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics.) [138] Chap. 8 and a late article of Ch. Doran, D. Hestenes and F. Sommen (Doran, Hestenes, Sommen and Van Acker (1993). Journal of Mathematical Physics, 34(8), pp. 3642–3669.) [72] Sect. IV may have difficulties to understand the subject and practitioners have difficulties to try the equations in certain applications. For this reason, this chapter reviews concepts and equations most of them introduced by D.Hestenes and G. Sobzyk (Hestenes and Sobczyk (1984). Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics.) [138] Chap. 8 and the article of Ch. Doran, D. Hestenes and F. Sommen (Doran, Hestenes, Sommen and Van Acker (1993). Journal of Mathematical Physics, 34(8), pp. 3642–3669.) [72] Sect. IV. This chapter is written in a clear manner for readers interested in applications in computer science and engineering. The explained equations will be required to understand advanced applications in next chapters.

It is the believe that imaginary numbers appeared for the first time around 1540 when the mathematicians Tartaglia and Cardano represented real roots of a cubic equation in terms of conjugated complex numbers. A Norwegian surveyor, Caspar Wessel, was in 1798 the first one to represent complex numbers by points on a plane with its vertical axis imaginary and horizontal axis real. This diagram was later known as the Argand diagram, although the true Argand’s achievement was an interpretation of i \( = \sqrt{(}-1)\) as a rotation by a right angle in the plane. Complex numbers received their name by Gauss and their formal definition as pair of real numbers was introduced by Hamilton in 1835.

This chapter presents the geometric algebra framework for dealing with 3D kinematics. The reader will see the usefulness of this mathematical approach for applications in computer vision and kinematics. We start with an introduction to 4D geometric algebra for 3D kinematics. Then we reformulate, using 3D and 4D geometric algebras, the classic model for the 3D motion of vectors. Finally, we compare both models, that is, the one using 3D Euclidean geometric algebra and our model, which uses 4D motor algebra.

The geometric algebra of a 3D Euclidean space \(G_{3,0,0}\) has a point basis and the motor algebra \(G_{3,0,1}\) a line basis. In the latter geometric algebra, the lines expressed in terms of Plücker coordinates can be used to represent points and planes as well. The reader can find a comparison of representations of points, lines, and planes using \(G_{3,0,0}\) and \(G_{3,0,1}\) in Chap. 7.

The geometric algebra of a 3D Euclidean space \(G_{3,0,0}\) has a point basis and the motor algebra \(G_{3,0,1}^+\) a line basis. In the latter, the lines are expressed in terms of Plücker coordinates and the points and planes in terms of bivectors. The reader can find a comparison of representations of points, lines, and planes using vector calculus, \(G_{3,0,0}\) and \(G_{3,0,1}^+\) in Chap. 7. Extending the degrees of freedom of the mathematical system, in the conformal geometric algebra \(G_{4,1}\), using the horosphere framework points, one can model lines, planes, circles, and spheres and also certain Lie groups as versors.

In this chapter, we will discuss the programming issues to compute in the geometric algebra framework. We will explain the technicalities for the programming which you have to take into account to generate a sound source code. At the end, we will discuss the use of specialized hardware as FPGA and NVidia CUDA to improve the efficiency of the code processing for applications in real time.

This chapter presents the theory and use of the Clifford Fourier transforms and Clifford wavelet transforms. We will show that using the mathematical system of the geometric algebra, it is possible to develop different kinds of Clifford Fourier and wavelet transforms which are very useful for image filtering, pattern recognition, feature detection, image segmentation, texture analysis, and image analysis in frequency and wavelet domains. These techniques are fundamental for automated visual inspection, robot guidance, medical image processing, analysis of image sequences, as well as for satellite and aerial photogrammetry.

It appears that for biological creatures, the external world may be internalized in terms of intrinsic geometric representations. We can formalize the relationships between the physical signals of external objects and the internal signals of a biological creature by using extrinsic vectors to represent those signals coming from the world and intrinsic vectors to represent those signals originating in the internal world. We can also assume that external and internal worlds employ different reference coordinate systems.

This chapter first presents Lie operators for key points detection working in the affine plane. This approach is stimulated by certain evidence of the human visual system; therefore, these Lie filters appear to be very useful for implementing in near future of an humanoid vision system.

This chapter will demonstrate that geometric algebra provides a simple mechanism for unifying current approaches in the computation and application of projective invariants using n-uncalibrated cameras. First, we describe Pascal’s theorem as a type of projective invariant, and then the theorem is applied for computing camera-intrinsic parameters. The fundamental projective invariant cross-ratio is studied in one, two, and three dimensions, using a single view and then n views. Next, by using the observations of two and three cameras, we apply projective invariants to the tasks of computing the view-center of a moving camera and to simplified visually guided grasping. The chapter also presents a geometric approach for the computation of shape and motion using projective invariants within a purely geometric algebra framework (Hestenes and Sobczyk (1984). Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, Hestenes and Ziegler (1991). Acta Applicandae Mathematicae 23: 25–63.) [138, 139]. Different approaches for projective reconstruction have utilized projective depth (Shashua (1994). IEEE Transations on Pattern Analysis and Machine Intelligence 16 (8): 778–790 and Sparr (1994). Proceedings of the European Conference on Computer Vision II.) [269, 284], projective invariants (Csurka and Faugeras (1998). Journal of Image and Vision Computing 16: 3–12.) [61], and factorization methods (Poelman and Kanade (1994). European Conference on Computer Vision, Tomasi and Kanade (1992). International Journal of Computer Vision 9 (2): 137–154 and Triggs (1995). IEEE Proceedings of the International Conference on Computer Vision (ICCV’95).) [237, 296, 300] (factorization methods incorporate projective depth calculations). We compute projective depth using projective invariants, which depend on the use of the fundamental matrix or trifocal tensor. Using these projective depths, we are then able to initiate a projective reconstruction procedure to compute shape and motion. We also apply the algebra of incidence in the development of geometric inference rules to extend 3D reconstruction.

The first section presents a non-iterative algorithm that combines the power of expression of geometric algebra with the robustness of Tensor Voting to find the correspondences between two sets of 3D points with an underlying rigid transformation. In addition, we present experiments of the conformal geometric algebra voting scheme using synthetic and real images.

In this chapter, we present a series of experiments in order to demonstrate the capabilities of geometric neural networks. We show cases of learning of a high nonlinear mapping and prediction. In the second part experiments of multiclass classification, object recognition, and robot trajectories interpolation using CSVM are included.

In geometric algebra, there exist specific operators named versors to model rotations, translations, and dilations, and are called rotors, translators and dilators respectively. In general, a versor \({\varvec{G}}\) is a multivector which can be expressed as the geometric product of non-singular vectors

Clifford algebras were created and classified by William K. Clifford (1878–1882) Clifford (Proc. London Math Soc, 4:381–395, 1873, [59]), Clifford (Am J Math, 1:350–358, 1878, [57], Clifford (In Mathematical Papers, Macmillan, London, [58], when he presented a new multiplication rule for vectors in Grassmann’s exterior algebra [112]. In the special case of the Clifford algebra \(CL_3\) for , the mathematical system embodied Hamilton’s quaternions Hamilton (Lectures on Quaternions, Hodges and Smith, Dublin, 1853, [119]). In Chap. 2 Sect. 2.​2 we explain a further geometric interpretation of the Clifford algebras developed by David Hestenes and called geometric algebra. Throughout this book we will utilize geometric algebra.