Spline Subdivision Schemes for Compact Sets. A Survey

Dyn, Nira, and Farkhi, Elza. "Spline Subdivision Schemes for Compact Sets. A Survey." Serdica Mathematical Journal 28.4 (2002): 349-360. <http://eudml.org/doc/11568>.

@article{Dyn2002,
abstract = {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.},
author = {Dyn, Nira, Farkhi, Elza},
journal = {Serdica Mathematical Journal},
keywords = {Compact Sets; Spline Subdivision Schemes; Metric Average; Minkowski Sum; subdivision; splines; Minkowski sum; metric average; set-valued function; image; reconstruction},
language = {eng},
number = {4},
pages = {349-360},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Spline Subdivision Schemes for Compact Sets. A Survey},
url = {http://eudml.org/doc/11568},
volume = {28},
year = {2002},
}

TY - JOUR
AU - Dyn, Nira
AU - Farkhi, Elza
TI - Spline Subdivision Schemes for Compact Sets. A Survey
JO - Serdica Mathematical Journal
PY - 2002
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 28
IS - 4
SP - 349
EP - 360
AB - 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.
LA - eng
KW - Compact Sets; Spline Subdivision Schemes; Metric Average; Minkowski Sum; subdivision; splines; Minkowski sum; metric average; set-valued function; image; reconstruction
UR - http://eudml.org/doc/11568
ER -