In this tutorial a gentle introduction to geometric algebra will be given. In order
that the attendee ... interesting problems in engineering and computer science.
Tutorial Proposal for PSIVT 2013 Guanajuato “Geometric computing for robot vision, graphics and learning”
Lecturer: Prof. Eduardo Bayro Corrochano (CINVESTAV Campus Guadalajara, México) Main areas of expertise: computer vision, neural computing, robotics, geometric cybernetics, applications of geometric algebra to perception action systems. e-mail:
[email protected], Tel: +52 (33) 3777-3600 Ext. 1027, Fax: +52 (33) 3777-3609 Personal WebSite: http://www.gdl.cinvestav.mx/edb
Type of Tutorial: Lecture. Duration: 4 hours (2 sessions, 2 hours each), with handouts. Abstract In this tutorial a gentle introduction to geometric algebra will be given. In order that the attendee makes progress to understand the subject, the lecture will start with the modelling of basic geometric primitives and the spinors in Euclidean 2D and 3D Euclidean geometric algebras. Screw theory and the kinematic modelling of points, lines and planes will be studied using 4D motor algebra. Having explained these two subjects, then it is easier for the student to move to the 5D conformal geometric algebra: to treat conformal transformations and algebra of incidence of points, lines, circles, planes, spheres, hyper-planes and hyper-spheres. An important aim of the tutorial is to show the advantages of the conformal geometric algebra which permits to deal with projective geometry, algebra of incidence, and spinor computations, for example to compute problems of computer vision and robotics without the need to abandon this mathematical framework. We will discuss the modelling and development of algorithms for a variety of applications and how to speed up algorithms for real time. The tutorial will be enriched with interesting examples in the areas of image processing, computer vision, robot vision, mobile robots and manipulators as well as with techniques for learning, neural computing and control.
Outline Part 1 Geometric Algebra In this part, we will introduce the basic ideas behind the mathematics of Geometric Algebra (GA). We will briefly discuss the advantages of the use of the geometric algebra framework for geometric computing in the domains of neural computing, graphic engineering, computer vision, robotics, and perception action systems. Part 2 2D and 3D Geometric Algebras In this part, we will introduce the geometric algebras of the 2D and 3D space, and study some of their relevant features. This will enable us to build up a picture of how geometric algebra can be employed to solve interesting problems in engineering and computer science.
1. 2. 3. 4.
The geometric algebra of 2D space. The geometric algebra of 3D space Planes, volumes, and the vector cross product rediscovered. Rotations in 3D. Geometric Algebra provides a very clear and compact method for encoding rotations, which is considerably more powerful than working with matrices. 5. Angular momentum as a bivector. 6. Rigid body spatial velocity. Part 3 Quaternion, Bivector and Motor algebras In this part, we will study the quaternion, bivector and motor algebras for the 3D and 4D spaces, respectively. We will clarify concepts and study their relevant features. This will help us to use rotors and screw theory to tackle different problems in graphics engineering, kinematics and dynamics in Robotics. 1. Motivation. 2. 4D geometric algebra of the 3D kinematics. 2. Motor algebra. + 3. Motors, rotors, and translators in G 3,0,1. 4. Properties of motors. Part 4 Kinematics of the 3D and 4D Spaces In this part, we will study the modeling of the 3D kinematics of the geometric primitives points, lines and planes using vector calculus, 3D Euclidean geometric algebra and motor algebra. We will compare the three representations and draw conclusions. The modeling of the kinematics of these geometric primitives helps us to tackle various problems in graphic engineering, computer vision and robotics. 1. Motivation. 2. Representation using vector calculus. 3. Modeling using the 3D Euclidean geometric algebra. 4. Modeling using the 4D motor algebra. 5. Comparison. Part 5 This lecture part offers an introduction to Conformal Geometric Algebra (CGA). The computational entity of this framework is the sphere which helps for modeling the other geometric primitives as points, lines and planes and ruled surfaces as well. By a robust camera calibration, one can get read off the involved affine transformation, thus CGA can be used in its full extend in robotics for closing the gap between the visual and mechanical world representations. 1. Motivation. 2. Null cone, Hyperplane and Horosphere. 3. Stereographic projection. 4. Duality. 5. Meet and Join. 6. Sphere, line, plane, pair of points. Part 6 This last lecture part presents interesting cases of study and real applications in image processing, neuralcomputing, computer vision, robotics an geometric control. Most of these applications use the conformal geometric algebra framework. We discuss the issues related to the programming for real time applications and current hardware for fast computing. We conclude discussing related problems and new trends for geometric computing using the geometric algebra framework.
Justification Nowadays, the importance of geometric algebra in the areas of computer science and engineering has been finally recognized and we are eye witnessing an increasing interest in different communities of researchers and practitioners. It is common to hear certain complain of beginners that the geometric algebra is an unusual mathematical framework which unfortunately it is not though in the regular university teaching programs. The advantages of the use of geometric algebra are immense particularly to represent the physics of the problem without losing the geometric insight of the problem through the algebraic manipulation. Also the use of fewer redundant coefficients for the representation of geometric primitives and for the spinors for Lie groups is important. All of these advantageously characteristics makes of geometric algebra a promising framework for real time applications. This tutorial uses material tested in various courses given in a postgraduate courses in different universities of Europe and Latin America, where we have learned what and how should be thought to facilitate the student to understand and use fast the language of geometric algebra. Since we have been applying geometric algebra in the last fifteen years in diverse domains like in image processing, computer vision, robot vision, mobile robots and manipulators as well as for learning, neural computing and control, we can illustrate the tutorial for the benefit of the attendees with very interesting applications.
Requirements Tutorial room for 10 to 30 attendees, projector and internet connection.
