Geometric Algebra: A powerful tool for solving geometric problems in visual computing

摘要

Geometric problems in visual computing (computer graphics, computer vision, and image processing) are typically modeled and solved using linear algebra (LA). Thus, vectors are used to represent directions and points in space, while matrices are used to model transformations. LA, however, presents some well-known limitations for performing geometric computations. As a result, one often needs to aggregate different formalisms (e.g., quaternions and Pliicker coordinates) to obtain complete solutions. Unfortunately, such extensions are not fully compatible among themselves, and one has to get used to jumping back and forth between formalisms, filling in the gaps between them. Geometric algebra (GA), on the other hand, is a mathematical framework that naturally generalizes and integrates useful formalisms such as complex numbers, quaternions and Pliicker coordinates into a high-level specification language for geometric operations. Due to its consistent structure, GA equations are often universal and generally applicable. They extend the same solution to higher dimensions and to all kinds of geometric elements, without having to handle special cases, as it happens in conventional techniques. This tutorial aims at introducing the fundamental concepts of GA as a powerful mathematical tool to describe and solve geometric problems in visual computing.