Reconstruction of 3D shapes with B-spline surface using diagonal approximation BFGS methods

The problem of surface reconstruction is a challenging problem in the fields of data visualization, virtual reality and engineering design. In this study, we investigate the topic of fitting B-spline surface to a set of 3D measured data points. Surface reconstruction in the proposed method consists of two main parts: (1) rewrite the problem as a nonlinear least squares optimization problem and compute the Jacobin matrix and (2) employ the diagonal approximation BFGS method to find the control points and the location parameters simultaneously. The space complexity and the time complexity of proposed method are O(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal {O}(n) $\end{document}. We perform numerical experiments with five test problems, including complex shapes, self intersections, large number of data points and high genus to evaluate the performance of the suggested approach. The experimental results demonstrate that the introduced approach is easy to implement, fast convergence rate, extremely small fitting errors, flexibility, very general and applicable to real time simulations.