Geometric Algebra: A Powerful Tool for Solving Geometric Problems in Visual Computing

Leandro A. F. Fernandes, M. M. O. Neto
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引用次数: 5

Abstract

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 Plücker 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 Plücker 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.
几何代数:在视觉计算中解决几何问题的强大工具
视觉计算中的几何问题(计算机图形学、计算机视觉和图像处理)通常使用线性代数(LA)建模和解决。因此,向量被用来表示空间中的方向和点,而矩阵被用来模拟变换。然而,对于执行几何计算,LA提出了一些众所周知的限制。因此,人们经常需要聚合不同的形式(例如,四元数和plicker坐标)来获得完整的解。不幸的是,这些扩展本身并不是完全兼容的,人们必须习惯于在各种形式之间来回跳跃,填补它们之间的空白。另一方面,几何代数(GA)是一种数学框架,它自然地将有用的形式化(如复数、四元数和plicker坐标)推广并集成到用于几何运算的高级规范语言中。遗传算法由于其结构的一致性,往往具有通用性和普遍适用性。它们将相同的解扩展到更高的维度和各种几何元素,而不必像传统技术那样处理特殊情况。本教程旨在介绍遗传算法作为一种强大的数学工具来描述和解决视觉计算中的几何问题的基本概念。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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