A global approach to the refinement of manifold data

A refinement of manifold data is a computational process, which produces a denser set of discrete data from a given one. Such refinements are closely related to multiresolution representations of manifold data by pyramid transforms, and approximation of manifold-valued functions by repeated refinements schemes. Most refinement methods compute each refined element separately, independently of the computations of the other elements. Here we propose a global method which computes all the refined elements simultaneously, using geodesic averages. We analyse repeated refinements schemes based on this global approach, and derive conditions guaranteeing strong convergence.Comment: arXiv admin note: text overlap with arXiv:1407.836

Spline Subdivision Schemes for Compact Sets. A Survey

Dedicated to the memory of our colleague Vasil Popov January 14, 1942 – May 31, 1990 * Partially supported by ISF-Center of Excellence, and by The Hermann Minkowski Center for Geometry at Tel Aviv University, IsraelAttempts at extending spline subdivision schemes to operate on compact sets are reviewed. The aim is to develop a procedure for approximating a set-valued function with compact images from a finite set of its samples. This is motivated by the problem of reconstructing a 3D object from a finite set of its parallel cross sections. The first attempt is limited to the case of convex sets, where the Minkowski sum of sets is successfully applied to replace addition of scalars. Since for nonconvex sets the Minkowski sum is too big and there is no approximation result as in the case of convex sets, a binary operation, called metric average, is used instead. With the metric average, spline subdivision schemes constitute approximating operators for set-valued functions which are Lipschitz continuous in the Hausdorff metric. Yet this result is not completely satisfactory, since 3D objects are not continuous in the Hausdorff metric near points of change of topology, and a special treatment near such points has yet to be designed