B-spline surface fitting by iterative geometric interpolation/approximation algorithms

Recently, the use of B-spline curves/surfaces to fit point clouds by iteratively repositioning the B-spline's control points on the basis of geometrical rules has gained in popularity because of its simplicity, scalability, and generality. We distinguish between two types of fitting, interpolation and approximation. Interpolation generates a B-spline surface that passes through the data points, whereas approximation generates a B-spline surface that passes near the data points, minimizing the deviation of the surface from the data points. For surface interpolation, the data points are assumed to be in grids, whereas for surface approximation the data points are assumed to be randomly distributed. In this paper, an iterative geometric interpolation method, as well as an approximation method, which is based on the framework of the iterative geometric interpolation algorithm, is discussed. These two iterative methods are compared with standard fitting methods using some complex examples, and the advantages and shortcomings of our algorithms are discussed. Furthermore, we introduce two methods to accelerate the iterative geometric interpolation algorithm, as well as a method to impose geometric constraints, such as reflectional symmetry, on the iterative geometric interpolation process, and a novel fairing method for non-uniform complex data points. Complex examples are provided to demonstrate the effectiveness of the proposed algorithms.