Progressive Iterative Approximation for Extended B-Spline Interpolation Surfaces

Yeqing Yi,Chengzhi Liu,Zixuan Tang,Giovanni Garcea

doi:10.1155/2021/5556771

Abstract

In order to improve the computational efficiency of data interpolation, we study the progressive iterative approximation (PIA) for tensor product extended cubic uniform B-spline surfaces. By solving the optimal shape parameters, we can minimize the spectral radius of PIA’s iteration matrix, and hence the convergence rate of PIA is accelerated. Stated numerical examples show that the optimal shape parameters make the PIA have the fastest convergence rate.

Highlights

Data interpolation plays important roles in scientific research and engineering applications
An iterative method, namely, progressive iterative approximation (PIA), has attracted a lot of attention and has become a very hot research area. e PIA stands out because it has the advantages of clear geometric meaning, stable convergence, simple iterative format, local modification, and so on
We further study the PIA format for tensor product extended cubic uniform B-spline surfaces, which is an extension of the PIA for the classic bicubic uniform B-spline curves

Summary

Introduction

Data interpolation plays important roles in scientific research and engineering applications. Despite the fact that the PIA offers many advantages, there is a disadvantage, that is, slow rate of convergence To overcome this limitation and further improve the computational efficiency, a great deal of acceleration techniques have been conducted. The eigenvalues of the collocation matrix were expressed explicitly, and the optimal shape parameters were solved to make the PIA have the fastest convergence rate Based on this conclusion, we further study the PIA format for tensor product extended cubic uniform B-spline surfaces, which is an extension of the PIA for the classic bicubic uniform B-spline curves. By solving the optimal shape parameters, the convergence rate of PIA is accelerated, and the computational efficiency of data interpolation can be improved.

PIA for Extended Cubic Uniform B-Spline Surface

Optimal Shape Parameters

Numerical Examples

Conclusion