Application Of Geometric Iteration Method In Fitting Discrete Data Points

Data point fitting is an important tool in many fields of science and technology such as geometric modeling and image analysis.The methods used can be divided into interpolation and fitting.Because the measured data are often discrete and have errors,it is difficult to use a polynomial to approximate its shape.In this case,B-spline is one of the most suitable approximation functions.Among them,the factors that affect the fitting effect of B-spline curve include control vertex,parameterization method of data points and selection of knot vector.Geometric iterative method,also known as progressive iterative approximation,is an iterative method with obvious geometric significance.The progressive iterative approximation method adjusts the control vertices iteratively,and gradually realizes interpolation or approximation of a given data point.In the iterative process,constraints can be flexibly added to make the final limit curve and surface meet these conditions.Parameterization method of data points and selection of knot vectors have great influence on the shape and precision of fitting curves and surfaces.If the parameters and knots are fixed and only the influence of the control vertex on the fitting curve and surface is considered,the accuracy of the fitting results is often not high.If data point parameterization and knots are considered,the problem under consideration becomes a multivariable and multidimensional nonlinear problem,which brings difficulties to solve the problem.On the basis of previous research,this thesis considers the influence of control vertex and data point parameters on the fitting effect,combines the idea of progressive iterative approximation method,and does the following work with the help of unconstrained optimization algorithm:1.The weight of the least squares progressive iterative approximation method(LSPIA)is optimized.In each iteration,the distance between the data point and the corresponding point on the curve of each iteration is considered,and the remainder of the objective function is minimized to obtain the weight.Combined with the optimal weight of LSPIA,an adaptive weight method is obtained.Numerical examples show that the selection of new weights accelerates the convergence speed of LSPIA method.2.The data point fitting problem is transformed into solving unconstrained optimization problem,and the control vertices are optimized by BFGS method.By adding scaling parameters,an adaptive BFGS fitting method is constructed.On the surface of numerical examples,compared with the existing three iterative methods,the convergence speed is faster in the previous iterative process,and it has the same shape preservation as the least square progressive iterative approximation algorithm.3.On the basis of control vertex iteration,the data point parameters are optimized by step acceleration method,which ensures that the optimized parameters are increasing in sequence.The fitting curve and surface after optimizing the parameters on the surface of numerical examples are closer to the data points,and the fitting error is smaller.