Algebraic Nonplanar Curve and Surface Estimation in 3-Space with Applications to Position Estimation

摘要

The paper addresses the problem of minimal parameter representation and estimation for complex planar and nonplanar curves, and surfaces in 3-D. The representation is based on concepts from Algebraic Geometry; a planar curve is the set of roots of a polynomial of two variables; a surface in 3-space is the set of roots of a polynomial of three variables; and a nonplanar curve is the intersection of two different surfaces. The authors show that the surface of interesting complex objects in 3-space can be represented by single high degree polynomials, and a similar statement applies to the representation of complex two-dimensional (planar) curves singly or jointly. An approximate expression for the mean square distance from a set of points to a curve or surface is developed, not only for quadratic surfaces, but for surfaces and curves defined by smooth functions, particularly for polynomials of higher degree. The authors show that the minimization of the approximate mean square distance is equivalent to the solution of an associated eigenproblem, and present a computationally efficient algorithm to solve it using known matrix computation techniques. The approximate mean square distance is not restricted to be used in dimension 3. It has the same meaning for any other space dimension. These modeling and estimation techniques are directly applicable to curve modeling and estimation for 2-D images. The application of the techniques is not restricted to vision systems because the data set could be provided by robotic tactile sensors or other sources of three dimensional data.