Geodesic
Spline Subdivision Schemes for Compact Sets. A Survey
Serdica Mathematical Journal, Tome 28 (2002) no. 4, pp. 349-360.

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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.
Keywords: Compact Sets, Spline Subdivision Schemes, Metric Average, Minkowski Sum 
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Dyn, Nira; Farkhi, Elza. Spline Subdivision Schemes for Compact Sets. A Survey. Serdica Mathematical Journal, Tome 28 (2002) no. 4, pp. 349-360. http://geodesic.mathdoc.fr/item/SMJ2_2002_28_4_a5/