Research On Curve And Surface Fitting From Cloud Data

The problem of curve surface fitting meets frequently in the fundamental research and the practical application. In the pattern recognition and the computer vision, the model fitting of the graph (or image) data is a basic work. It has a wide range of applications in the field of engineering, statistics and computer graphics, etc. Many problems of CAD and CAGD are related with fitting problems. The curve or the surface fitting to the given data set of points is an important stage in the imagery processing, the pattern recognition and the computer vision, for example side examination, object restructuring and so on.Quadratic curves and surfaces become one of the basic models, and play an important role in computer graphics and computer-aided geometric design and other fields, because of its good geometric characteristics, lower degree and flexible control parameters. The fitting with the conic or quadratic surface also obtains the widespread application in the daily life and the industrial production. The question request the error of the fitting with the conic or quadratic surface to the plane or spatial data points is smallest under some kind of significance. The method to resolve the fitting problem basically divided into two categories: the objective function based on the algebra distance and the objective function based on the vertical distance. The greatest benefit of algebra distance is the rapid calculation, but the result of fitting is ineffective under normal circumstances. But, the vertical distance is the most accurate miss distance, and fitting based on vertical distance is also called the best fitting. But because of nonlinear of the question, there did not have the very good fitting method so far, therefore it has the necessity to do the further research.First, the paper describes the research background and significance of this paper, and the relevant work done on the issue of quadratic curve and surface fitting, as well as the current status of the study.Second, it introduces the curve and surface fitting on the reverse engineering and some solutions of this problem. There carried on the summary of the curve fitting methods of order points and the curve fitting methods of disorderly scattered points. The disorderly scattered points' fitting is a important problem in the fitting field. The present work of this problem has four kinds: the least-squares fitting method, the model fitting method, the backbone method and the separate algorithm. According to the representation of surface form, the surface restructuring algorithm may approximately divide into three kinds: The method based on grid, the method based on parameter expression and the method based on implicit expression, the article has made the detailed summary to each kind of method. And it also refers to a more detailed analysis solution for some typical algorithms. In this paper, the basic idea of solving quadratic curve and surface fitting problem is least-squares method, therefore, there is detailed description focused on the basic idea of least-squares.Finally, this paper proposes a new method for the conicoid fitting. It chooses the objective function based on the vertical distance - the minimum distance in all the distances of the point to all the points on the surface. Then solves the weight by defining objective function on the basis of least-squares theory, and gains parameter of implicit surface equation through minimizing the square sum of vertical distances, the fitting result is theoretically best. It uses the fitting result of objective function based on the algebraic distance as the starting value in advance, with linearization using the Newton-Raphson method, so transforms the non-linear problem into the linearity to reduce the computation complexity.This paper provides the examples for comparing errors of the surfaces produced by the new method and prevenient methods based on algebraic distance, It is proved this new method usually produces a good fit, and the results show that the algorithm is reliable and effective.