Weighted Progressive Iterative Approximation Method And Its Application

In computer aided geometric design and reverse engineering,it is an important task to construct a sequence of curves(surfaces)which meet the accuracy requirements to interpolate or approximate a given set of ordered points.A large-scale linear system of equations has to be solved for the computation of control points in reverse engineering.It is hard to popularize for its big computation amount.Many scholars have put forward different forms of interpolating and fitting methods.The progressive iterative approximation method(PIA,also known as geometric iterative method)is favored by most scholars for its good adaptability and convergent stability.One can continuously adjust and iterate the control points,getting a sequence of curves(surfaces)with increasing accuracy.The method not only greatly reduces the calculation,but also has obvious geometric meaning.In recent years,PIA method has been widely used in many fields.Although the classic PIA can guarantee the final curve and surface interpolating a set of data points,but the premise is to take all of the data points as the control points in each iteration.When the size of original data is large,the classic PIA method will appear weak flexibility,slow speed and other issues.PIA method has been improved recently,such as,local PIA,weighted PIA,Extended PIA,and least square PIA method,which expand the field of application,speed up the convergence rate,improve the flexibility,and also enrich the contents of the PIA method.In consideration of the types and advantages of the PIA method,the main work of this paper is as follows:1.Review the development of PIA,and improve weighted progressive iterative1.Review the development of PIA,and improve weighted progressive iterative approximation method(WPIA)by adjusting each weight for different vectors,and analyze its convergence and iteration effect;2.Modify the local PIA method,and realize the acceleration of data points which should be adjusted.Also,the relationship of local PIA between local algebraic interpolation has been studied.