# Estimation of Planar Curves, Surfaces, and Nonplanar Space Curves Defined by Implicit Equations with Applications to Edge and Range Image Segmentation

## Abstract

This paper addresses the problem of parametric representation and estimation of complex planar curves in 2-D, surfaces in 3-D and nonplanar space curves in 3-D. Curves and surfaces can be defined either parametrically or implicitly, and we use the latter representation. A planar curve is the set of zeros of a smooth function of two variables x-y, a surface is the set of zeros of a smooth function of three variables x-y-z, and a space curve is the intersection of two surfaces, which are the set of zeros of two linearly independent smooth functions of three variables x-y-z, For example, the surface of a complex object in 3-D can be represented as a subset of a single implicit surface, with similar results for planar and space curves. We show how this unified representation can be used for object recognition, object position estimation, and segmentation of objects into meaningful subobjects, that is, the detection of “interest regions” that are more complex than high curvature regions and, hence, more useful as features for object recognition. Fitting implicit curves and surfaces to data would be ideally based on minimizing the mean square distance from the data points to the curve or surface. Since the distance from a point to a curve or surface cannot be computed exactly by direct methods, the approximate distance, which is a first-order approximation of the real distance, is introduced, generalizing and unifying previous results. We fit implicit curves and surfaces to data minimizing the approximate mean square distance, which is a nonlinear least squares problem. We show that in certain cases, this problem reduces to the generalized eigenvector fit, which is the minimization of the sum of squares of the values of the functions that define the curves or surfaces under a quadratic constraint function of the data. This fit is computationally reasonable to compute, is readily parallelizable, and, hence, is easily computed in real time. In general, the generalized eigenvector fit provides a very good initial estimate for the iterative minimization of the approximate mean square distance. Although we are primarily interested in the 2-D and 3-D cases, the methods developed herein are dimension independent. We show that in the case of algebraic curves and surfaces, i.e., those defined by sets of zeros of polynomials, the minimizers of the approximate mean square distance and the generalized eigenvector fit are invariant with respect to similarity transformations. Thus, the generalized eigenvector fit is independent of the choice of coordinate system, which is a very desirable property for object recognition, position estimation, and the stereo matching problem. Finally, as applications of the previous techniques, we illustrate the concept of “interest regions” This paper addresses the problem of parametric for object recognition and describe a variable-order segmentation algorithm that applies to the three cases of interest. © 1991, IEEE. All rights reserved.