# Spline Subdivision Schemes for Compact Sets. A Survey

Serdica Mathematical Journal (2002)

- Volume: 28, Issue: 4, page 349-360
- ISSN: 1310-6600

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topDyn, 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 -

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