Orthogonal distance fitting of implicit curves and surfaces

Dimensional model fitting finds its applications in various fields of science and engineering and is a relevant subject in computer/machine vision and coordinate metrology. In this paper, we present two new fitting algorithms, distance-based and coordinate-based algorithm, for implicit surfaces and plane curves, which minimize the square sum of the orthogonal error distances between the model feature and the given data points. Each of the two algorithms has its own advantages and is to be purposefully applied to a specific fitting task, considering the implementation and memory space cost, and possibilities of observation weighting. By the new algorithms, the model feature parameters are grouped and simultaneously estimated in terms of form, position, and rotation parameters. The form parameters determine the shape of the model feature and the position/rotation parameters describe the rigid body motion of the model feature. The proposed algorithms are applicable to any kind of implicit surface and plane curve. In this paper, we also describe algorithm implementation and show various examples of orthogonal distance fit.

1 Introduction

With image processing, pattern recognition, and computer/machine vision, dimensional model (curve and surface) fitting to a set of given data points is a very common task carried out during a working project, e.g., edge detection, information extraction from 2D-image or 3D-range image, and object reconstruction. For the purpose of dimensional model fitting, we can consider three methods, namely, moment method [15], [32], [35], [42], Hough transform [8], [19], [28], and least-squares method (LSM) [23]. The moment method and Hough transform are efficient for fitting of relatively simple models, while their application to a complex object model or to an object model with a large number of model parameters is not encouraged. In this paper, we consider the LS-fitting algorithms for implicit model features. By data modeling and analysis in various disciplines of science and engineering, implicit features are very often used because of their compact description in form of and because of the possibility of a simple on-off and inside-outside decision.