Fitting B-Spline Curves to Point Clouds in the Presence of Obstacles

We consider the computation of an approximating curve to fit a given set of points. Such a curve fitting is a common problem in CAGD and we review three different approaches based upon minimization of the squared distance function: the point distance, tangent distance and squared distance error term. We enhance the classic setup comprising a point cloud and an approximating B-Spline curve by obstacles a final solution must not penetrate. Two algorithms for solving the emerging constrained optimization problems are presented and used to enclose point clouds from inside as well as from outside and to avoid arbitrary smooth bounded obstacles. Moreover, approximations of guaranteed quality and shaking objects are examined within this context. In a next step, we extend our work and study the curve fitting problem on parametrized surfaces. The approximation is still performed in the two dimensional parameter space while geometric properties of the manifold enter the computation at the same time. Additionally, we use this new error term to approximate the borders of point clouds on manifolds. Finally, as a minimization of the squared distance function is sensitive to outliers, a curve fitting in the L norm based on the signed distance function is developed. We describe the necessary results of non-smooth optimization theory and apply this robust optimization to curve fitting in the presence of obstacles.