Four-point Interpolation Subdivision Method For Fitting Ordered Point Sequence And Its Optimization Method

The fitting of discrete data points has always been an important research topic in computer aided geometric design and related fields.Subdivision modeling method constructs smooth curves and surfaces in discrete form,and is applicable to any topological structure,so it is widely used.Among them,the four-point interpolation subdivision method is easy to operate,and the resulting subdivision curve can be continuous,which has the advantages of geometric intuition and high subdivision efficiency.Based on the four-point interpolation subdivision method,the four-point interpolation subdivision curve is used to fit the ordered discrete point sequence.At the same time,the optimization algorithm of the four-point interpolation subdivision curve is proposed.In the algorithm of four-point interpolation subdivision curve fitting ordered discrete point sequence,the discrete data point sequence is parameterized first;Then,the limit point calculation formula of the four-point interpolation subdivision method is used,and the initial control point sequence of the fitting curve is solved by combining the parameterization of the discrete point sequence and the least square method;Finally,according to the initial control point sequence,the four-point interpolation subdivision curve approximating the discrete point sequence is obtained.The algorithm example shows that compared with the fitting algorithm of cubic uniform B-spline curve,the algorithm can use fewer initial control points to achieve the same fitting error.In practical applications such as data simplification and multi-resolution analysis,for large-scale discrete data point columns,the algorithm subdivides a small number of initial control point columns step by step to obtain a fitting curve composed of any number of point columns.When constructing curves with four-point interpolation subdivision method,if the relative distance between the initial control points is too close,the limit curves of the initial control points will appear self-intersecting,sharp corners and other unreasonable phenomena.Therefore,in view of these unreasonable phenomena,an optimization algorithm is proposed.First,parameterize the initial control points;Secondly,the limit point calculation formula of the four-point interpolation subdivision method and the parameterization of the initial control point column are used to optimize the original initial control point column,and the new initial control point column is calculated;Finally,a new four-point interpolation subdivision curve is constructed with this initial control vertex column.The new curve not only interpolates the original initial control points,but also avoids the phenomenon of self-intersection.Numerical experiments show that the algorithm can solve the self-intersection phenomenon of the four-point interpolation subdivision method in the process of interpolation subdivision,and obtain more reasonable interpolation curve,which meets the practical application requirements.