Talk
Abstracts:
Nira
Dyn
(School of Mathematical Sciences, Tel Aviv University, Israel)
niradyn@post.tau.ac.il
Spline
Subdivision Schemes for Compact Sets
Motivated by the problem of the reconstruction of 3D objects
from their 2D cross sections, we consider spline subdivision
schemes operating on data consisting of compact sets. A spline
subdivision scheme generates from such initial data a sequence
of set-valued functions, with compact sets as images, which
converges to a limit set-valued function. In the case of 2D
sets, the limit set valued function, with 2D sets as images,
describes a 3D object.
For the case of data consisting of convex sets, we replace addition
by Minkowski sums of sets. Then the spline subdivision schemes
generate limit set-valued functions which can be expressed as
linear combinations of integer shifts of a B-spline, with the
initial sets as coefficients. The subdivision techniques are
used to conclude that these limit "set-valued spline functions"
have shape preserving properties similar to those of scalar
spline functions. We obtain O(h2) rate of approximation
by the limit function, under mild smoothness assumptions on
the set-valued function, from which the initial data is sampled.
For the case of non-convex sets we show that the limit of the
spline subdivision schemes, using the Minkowski sums, is too
large to be a good approximation.
To define spline subdivision schemes for general compact sets,
we use the representation of spline subdivision schemes in terms
of repeated averages, and replace the usual average by a binary
operation between two compact sets, termed the "metric average".
These schemes are shown to converge in the Hausdorff metric,
and provide O(h) rate of approximation.
The results presented here, were obtained in collaboration with
E. Farkhi.
Bernd Hamann
(Co-Director and Professor, Center for Image Processing and
Integrated Computing and Department of Computer Science,
University of California, Davis)
hamann@cs.ucdavis.edu
Hierarchical Approaches for the
Visualization of Massive Scientific Data
Jorg
Peters
(Computer & Information Sciences & Enginering, University
of Florida) jorg@cise.ufl.edu
Curvature
Continuous Free-Form Surfaces
A
new technique for creating curvature continuous free-form surfaces
of unrestricted patch layout employs patches of degree 3 by
3 (bicubics) and some patches of degree 4 by (d+2), d>0.
The surfaces have the flexibility at extraordinary points of
C2 splines of degree d. The construction is expalined, in particular,
of curvature cntinuous free-formsplines of degree at most 3
by 5.
Jorg
Peters
(Computer & Information Sciences & Enginering, University
of Florida) jorg@cise.ufl.edu
Surface
Envelopes
Surface
envelopes are tight, two-sided enclosures of composite spline
surfaces. This talk hows how to construct the two hulls of the
enclosure so that matched triangle pairs sandwich a given nonlinear,
curved surface consisting of tensor-product B\'ezier patches.
The envelope of a surface may be viewed as a low cost approximate
piecewise linear implicitization with a precise and easily computed
error bound.
Hans-Peter
Seidel
(Max-Planck-Institut for Computer Science, Saarbrücken, Germany)
hpseidel@mpi-sb.mpg.de
Efficient
Processing of Large 3D Meshes
Due to their simplicity triangle meshes are often used to represent
geometric surfaces. Their main drawback is the large number
of triangles that are required to represent a smooth surface.
This problem has been addressed by a large number of mesh simplification
algorithms which reduce the number of triangles and approximate
the initial mesh. Hierarchical triangle mesh representations
provide access to a triangle mesh at a desired resolution, without
omitting any information.
In this talk we present an infrastructure for discrete geometry
processing, including algorithms for 3D reconstruction, curvature
computation, mesh reduction, geometric mesh smoothing, and multiresolution
editing of arbitrary unstructured tringle meshes.
In particular, we will demonstrate how mesh reduction and geometric
mesh smoothing can be combined to provide a powerful and numerically
efficient multiresolution smoothing and editing paradigm.
Joe
Warren (Department of Computer Science, Rice University)
jwarren@cs.rice.edu
A
Subdivision Scheme for Surfaces of Revolution
This
talk will describe a new non-stationary variant of Catmull-Clark
subdivision that is capable of reproducing surfaces of revolution.
Geometric
Design
2000-2001
Program: Mathematics in Multimedia
