AT&T Bell Labs., Murray Hill, NJ.
IEEE Trans Image Process. 1996;5(6):976-86. doi: 10.1109/83.503913.
Many important problems in image processing and computer vision can be formulated as the solution of a system of simultaneous polynomial equations. Crucial issues include the uniqueness of solution and the number of solutions (if not unique), and how to find numerically all the solutions. The goal of this paper is to introduce to engineers and scientists some mathematical tools from algebraic geometry which are very useful in resolving these issues. Three-dimensional motion/structure estimation is used as the context. However, these tools should also be helpful in other areas including surface intersection in computer-aided design, and inverse position problems in kinematics/robotics. The tools to be described are Bezout numbers, Grobner bases, homotopy methods, and a powerful theorem which states that under rather general conditions one can draw general conclusions on the number of solutions of a polynomial system from a single numerical example.
图像处理和计算机视觉中的许多重要问题都可以表述为一个联立多项式方程组的解。关键问题包括解的唯一性和（如果不唯一）解的数量，以及如何数值地找到所有解。本文的目的是向工程师和科学家介绍代数几何中的一些数学工具，这些工具在解决这些问题时非常有用。三维运动/结构估计作为背景。然而，这些工具在其他领域也应该有帮助，包括计算机辅助设计中的曲面相交，以及运动学/机器人学中的逆位置问题。将要描述的工具是 Bezout 数、Grobner 基、同伦方法，以及一个强大的定理，该定理指出，在相当一般的条件下，可以从单个数值实例得出关于多项式系统解的数量的一般结论。
