Jet Subdivision
If you are reading this page,
then perhaps you had passed the age to
find Pixar's animated movies such as Toy Story II
and A Bug's
Life highly amusing. However, if you are into multiscale
mathematics then your adrenaline level may rise a bit if you realize
that the
type of geometric modelling techiques  known as subdivision
methods  used in the production of these films are intriguingly
linked to things such as fractals and wavelets. Despite the
box office
successes in
computer graphics, subdivision methods have not
made a similar impact in the engineering community, where the
requirements for surface quality are much more demanding than those of
the entertainment and marketing communities. In particular, both the
automobile industry (e.g. BMW) and the aircraft industry (e.g. Boeing) would like to have
surfaces with good
curvature properties, yet all existing
subdivision surface schemes generate surfaces that are either
flat or have curvature singularities at extraordinary vertices. (Of
course, the latter is by no means the only
reason why the engineering design community has not picked up on
subdivision methods.)
A
new methodology: jet subdivision. Ordinary subdivision schemes
start
with an initial polyhedral surfaces, called a base mesh, and construct
a sequence of refined meshes converging to a smooth surface. The
initial mesh is determined by the positions of the vertices of the base
mesh, while the later meshes are formed by a twostep process of
subdivision involving a step of "topological refinement" followed by
an averaging step. Jet subdivision schemes incorporate higher order
data at each vertex of the base mesh. This higher order data may be
viewed as an approximation of the rth order Taylor approximation of
the limiting surface  defined in an intrinsic
way and is called an "rjet". From this perspective, ordinary
subdivision schemes use 0th order Taylor approximations of the
limiting surface, and thus a special case of a jet subdivision scheme
with r=0.
Collaborators: Thomas Duchamp, Bin Han, Bruce Piper
Students: Matthew Ferrara, Kyle
McDonald, Dr. Yonggang Xue
Optimization consultant: Michael Overton
Gallery
A Jet Subdvision Surface
Characteristics maps in Bnet
representations. The followings are the
characteristic
maps
of
the twopoint interpolatory jet
subdivision scheme constructed in the article Jet
Subdivision
Schemes on the kregular complex. This subdivision scheme
happens to have an intimate connection with the
classical PowellSabin
spline, as shown in
our earlier article Multivariate
Refinable
Hermite interpolant. As such,
the characteristic map of this
particular jet subdivision
scheme has a piecewise quadratic polynomial representation and
(consequently) can be expressed in Bnet
representations. Not only are such
Bnet representations used to
prove the
fact that these characteristic maps are regular and injective for
arbitrary valence k, they are also used to give explicit
paramaterization of the surfaces produced by our subdivision schemes.
You may notice another, somewhat
similar, set of figures in the paper
(Figure 11, first row which shows how the characteristic map
distorts
the lines of the refined kregular complex. It is
over a 'characteristic plane' that one can smoothly
parametrize
a subdivision surface in the vicinity of an extraordinary vertex  an
idea pioneered by Ulrich Reif about a decade ago. This
is why characteristic maps have to be injective and
regular, otherwise the 'characteristic planes' would not look like
planes at all...
Another Jet Subdvision Surface:
comparion with three variants of Loop surface
Original
Loop Scheme 
unbounded Curvature at k=3

A
variant with bounded (but
discontinuous)
curvature

A
zero
curvature variant

Our
flexible C2 JSS
See [section
5.2]









The extrinsic and intrinsic views of
differential geometry:
The following picture is an outcome of my attempt to alleviate my
headache when learning about geometry. I share it with you anyway.
Related Publications:
Jet
Subdivision Schemes on the kRegular Complex, Y. Xue, T.
P.Y. Yu and
T.
Duchamp (pdf)(bibtex entry)
Multivariate
Refinable Hermite
Interpolants, by B. Han, T. P.Y. Yu and B. Piper
(pdf)
(ps) (bibtex
entry)
Design
of Hermite Subdivision Schemes aided by Spectral Radius
Optimization, by B. Han, M. Overton and T. P.Y. Yu
(pdf)(ps)(bibtex
entry)
NonInterpolatory
Hermite
Subdivision Schemes, by B. Han, T. P.Y.
Yu and Y. Xue (pdf)(ps)(bibtex
entry)
Facebased Hermite Subdivision Schemes, by B. Han and
T. P.Y. Yu (pdf)(ps)(bibtex
entry)
Honeycomb and kfold Hermite Subdivision Schemes, Y.
Xue
and T. P.Y. Yu (pdf)(ps)(bibtex
entry)The Research presented at this
Web Site has been sponsored
by a National
Science Foundation CAREER award (CCR 9984501).
The PI is also grateful to a software
donation by Alias
This page © Copyright 20042005 by Thomas Yu
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