Serdica Math J SPLINE SUBDIVISION SCHEMES FOR COMPACT SETS A SURVEY
نویسندگان
چکیده
Attempts at extending spline subdivision schemes to operate on compact sets are reviewed The aim is to develop a procedure for ap proximating a set valued function with compact images from a nite set of its samples This is motivated by the problem of reconstructing a D object from a nite set of its parallel cross sections The rst attempt is limited to the case of convex sets where the Minkowski sum of sets is successfully ap plied 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 opera tors for set valued functions which are Lipschitz continuous in the Hausdor metric Yet this result is not completely satisfactory since D objects are not continuous in the Hausdor metric near points of change of topology and a special treatment near such points has yet to be designed Mathematics Subject Classi cation E B A A
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