Double-layer Progressive And Iterative Approximation For Least Squares B-spline Curve And Surface Fitting

The problem of data fitting has always been a key research topic in the fields of computer graphics,computer-aided design and computer-aided manufacturing.Numerous data fitting methods have already existed.For B-spline curve fitting 2D data sets and B-spline surface fitting 3D data sets,it is still an important research direction due to its wide application in industrial production.Data sets are generally obtained from real objects,and then geometric methods are used to build their digital models in the field of reverse engineering.Since the distribution of data points is often irregular,B-spline curves are usually selected to fit such data sets.When B-spline curve fits data points,it is necessary to solve linear equations to adjust control vertices.In order to make B-spline curve fit large and scattered data sets quickly,double-layer least squares progressive iterative approximation(DL-LSPIA)algorithm for uniform 3 times B-spline curve fitting method is proposed.Firstly,by fitting arc between adjacent points,we can obtain corresponding discrete curvature for each data point.Feature points are selected from data points as interpolation points.At the same time,in addition to interpolation points,some data points in the data set are used as data points to be fitted in the first layer.An initial fitting curve is constructed.Then,LSPIA algorithm are used to optimize and update control vertices until the error of data points to be fitted satisfies the given accuracy.Secondly,the control vertices at the end of the first layer iteration are recorded as the initial control vertices at the beginning of the second layer LSPIA algorithm.All data points except interpolation points are used as fitting points,and control vertices are updated by LSPIA algorithm.Finally,the uniform 3 times B-spline fitting curve satisfying error precision is obtained through multiple iterations.Based on DL-LSPIA algorithm,uniform 3 times B-spline curve fitting method makes full use of the local properties of uniform B-spline curve and the advantages of LSPIA algorithm,which reduces the running time.Lots of empirical examples show the algorithm can obtain the fitting curve efficiently.For the fitting problem of 3D data sets,DL-LSPIA algorithm for uniform 3 × 3times B-spline surface fitting method is proposed,which inherited four advantages of LSPIA algorithm for B-spline surface fitting method.Moreover,for the data sets with huge data points,introducing the idea of layered iteration,we divide the data points into two layers for fitting,make use of the local properties of uniform 3 × 3 times B-spline surface and combine with the reuse of intermediate results,thus reducing the number of iterations under the condition of meeting the error accuracy and shortening the iteration time.Firstly,some data points in the data set are used as data points to be fitted in the first layer.An initial fitting surface is constructed.Then,LSPIA algorithm is used to optimize and update control vertices until error of the data points to be fitted satisfies the given accuracy.Secondly,the control vertices at the end of the first layer iteration are recorded as the initial control vertices at the beginning of the second layer LSPIA algorithm.All data points are used as fitting points,and control vertices are updated by LSPIA algorithm.Finally,the uniform 3 × 3 times B-spline fitting surface satisfying error precision is obtained through multiple iterations.Numerical examples illustrated in this paper show the DL-LSPIA surface fitting method is robust and can quickly generate the fitting surface for given data sets.