Fairing Method Of Curves/surfaces Based On Progressive-iteration Approximation And Sparse Optimization

Data fitting is an important tool to solve practical problems in scientific research and engineering practice.Since the geometric iteration method was raised,it has been paid more and more attention due to its excellent performance in data fitting.Taking curve fitting as an example,the geometric iteration method starts from an initial curve and constantly adjusts the control points to finally achieve the purpose of fitting a given data point.It is easy to implement and greatly saves computational resources,because the geometric iteration method only needs to calculate the difference vector of the data points and the adjustment vector of the control points in each iteration.In the traditional geometric iterative method,the adjustment direction is fixed no matter how the control points are adjusted each time,and only the adjustment step length can be controlled,which will bring inconvenience to the user,especially when the data set to be fitted is enormous.Based on the existing Least Squares Progressive-Iteration Approximation(LSPIA),a new least-squares geometric iterative method with multi-directional in-band parameters is proposed,which adjust the moving direction of control points by changing the weight of data points during iteration,thus increasing the flexibility of the geometric iteration method.We also analyze the convergence of the iterative format so as to add constraints according to the demands.In recent years,increasingly importance has been attached to sparse optimization as the development of mathematics tools and the rise of big data science.In this paper,sparse optimization and geometric iteration method are combined.Under the newly proposed iterative format,a sparse model of cubic B-spline curve fairing is established and solved by The Alternating Direction Method of Multipliers(ADMM).Experiments show that our method is better than other fairing methods when the fitting error is at the same level.