Quasi-interpolation And Subdivision Based On Cubic B-splines With Tension Parameters

As an approximation method, quasi-interpolation is widely used in computer-aided geometric design (CAGD), data processing, especially in the field of scattered point cloud reconstruction. And quasi-interpolation could fit data points directly, without all the interpolation points on the curve or surface, especially effective in dealing with dead pixels and sharp point. The advantages of Quasi-interpolation have received a considerable attention by many authors, such as low computational cost, efficient approximants to a given set of data. What's more, many scholars have been beginning to focus on the shape control of splines with parameters in recent years. They have constructed kinds of expansion of B-splines models with shape parameters, however, these models generally lacked of the essence of splines'refinability, and couldn't express curves and surfaces'discrete generation by subdivision schemes. This paper discusses cubic B-splines with tension parameters constructed by Manni and the feature of cubic B-splines' refinement, further more, the paper focuses on quasi-interpolation based on cubic B-splines with tension parameters. The main research results in this article are as follows:First, the paper discusses the construction and properties of cubic B-splines curve with tension parameters, transformation theorem between cubic B-splines curve with tension parameters and Beta-spline curves or Gamma-spline curves. What's more, it gives a detailed discussion about the influences of tension parameters on the curve.Second, the paper makes research on the subdivision schemes of cubic B-splines curve with tension parameters, and then summarizes a unified rules of M-band subdivision in the condition of C 1 continuous, where 2≤M≤5,M∈Ν, especially obtains ternary subdivision mask. Beta-spline curves and Gamma-spline curves subdivision conditions are discussed, the binary and ternary subdivision numerical examples are given to illustrate the advantage of approximating to the limit curve. In particular, it studies subdivision conditions and binary subdivision rules of cubic B-splines curve with tension parameters in the condition of C 2 continuous.Third, the paper studies quasi-interpolation based on cubic B-splines with tension parameters, moreover, summarizes its M-band subdivision rules as well as numerical examples based on binary and ternary subdivision. Finally, it further extends the curve results to surface (including cubic B-splines surface with tension parameters and quasi-interpolation surface based on cubic B-splines with tension parameters).