B-spline scale-space of spline curves and surfaces

We present iterative algorithms for $B$-spline scale-space smoothing of geometric data and recovery of high frequency information in the smoothing process. The scale-space representation is based on a directional smoothing process using $B$-splines. If the geometric data are approximated or modelled by uniform $B$-splines or box-splines then the scale-space smoothing produces $B$-spline curves or box-spline surfaces. The method is applicable to geometric data processing and geometric modelling of free-form curves and surfaces from quadrilateral polyhedra with extraordinary vertices.     

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.     

Equations of parametric surfaces with base points via syzygies

Let $S$ be a tensor product parametrized surface in $P^3$; that is, $S$ is given as the image of $\phi :P^1\times P^1\to P^3$. This paper will show that the use of syzygies in the form of a combination of moving planes and moving quadrics provides a valid method for finding the implicit equation of $S$ when certain base points are present. This work extends the algorithm provided by Cox [Cox, D.A., 2001. Equations of parametric curves and surfaces via syzygies. In: Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering. Contemporary Mathematics vol. 286, pp. 1–20] for when $\phi$ has no base points, and it is analogous to some of the results of Busé et al. [Busé, L., Cox, D., D’Andrea, C., 2003. Implicitization of surfaces in $P^3$ in the presence of base points. J. Algebra Appl. 2 (2), 189–214] for the case of a triangular parametrization $\phi :P^2\to P^3$ with base points.