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Please use this identifier to cite or link to this item: http://hdl.handle.net/10525/510

Title: Spline Subdivision Schemes for Compact Sets. A Survey
Authors: Dyn, Nira
Farkhi, Elza
Keywords: Compact Sets
Spline Subdivision Schemes
Metric Average
Minkowski Sum
Issue Date: 2002
Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation: Serdica Mathematical Journal, Vol. 28, No 4, (2002), 349p-360p
Abstract: Attempts 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.
Description: 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, Israel
URI: http://hdl.handle.net/10525/510
ISSN: 1310-6600
Appears in Collections:Volume 28 Number 4

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