Schröder is currently an assodiate professor
of computer science at the
California Institute of Technology, Pasadena, where he directs the
Caltech Multi-Res Modeling Group. For the past 7 years his work has
concentrated on exploiting wavelets and multiresolution techniques to
build efficient representations and algorithms for many fundamental
computer graphics problems. He has taught in a number of Siggraph
courses and most recently co-led the course on Wavelets in Computer
Graphics (1996) and the course on Subdivision for Modeling and
Animation (1998). His current research focuses on subdivision as a
fundamental paradigm for geometric modeling and rapid manipulation of
large, complex geometric models. The results of his work have been
published in venues ranging from Siggraph to special journal issues on
wavelets and WIRED magazine, and he is a frequent consultant to
|Denis Zorin || Denis Zorin is an
assistant professor at the Courant Institute of Mathematical
Sciences, New York University. He
received a BS degree from the Moscow Institute of Physics and Technology,
a MS degree in Mathematics from Ohio State University and a PhD in
Computer Science from the California
Institute of Technology. In 1997-98, he was a research associate at
the Computer Science Department of
His research interests include
multiresolution modeling, the theory of subdivision, and applications
of subdivision surfaces in Computer Graphics. He is also interested in
perceptually-based computer graphics algorithms. He has published
several papers in Siggraph proceedings.
Tony DeRose is currently a member of the Tools Group at Pixar
Animation Studios. He received a BS in Physics in 1981 from the University
of California, Davis; in 1985 he received a Ph.D. in Computer
from the University of California,
Berkeley. He received a Presidential Young Investigator award from
the National Science Foundation in 1989. In 1995 he was selected as a
in the software category of the Discover
Awards for Technical Innovation.
From September 1986 to December 1995 Dr. DeRose was a Professor of
Computer Science and Engineering at the University of
Washington. From September 1991 to August 1992 he was on
sabbatical leave at the Xerox
Palo Alto Research Center and at Apple Computer. He has served on
various technical program committees including SIGGRAPH, and from 1988
through 1994 was an associate editor of ACM Transactions on Graphics.
His research has focused on mathematical methods for surface
modeling, data fitting, and more recently, in the use of
multiresolution techniques. Recent projects include object
acquisition from laser range data and multiresolution/wavelet methods
for high-performance computer graphics.
Jos Stam is currently a member of technical staff at Alias|wavefront.
He received BS degrees in computer science and mathematics from the University
of Geneva, Switzerland in 1988 and 1989, and he received a MS and a PhD in
computer science both from the University of Toronto in 1991 and 1995,
respectively. His research interests cover most areas of computer graphics:
natural phenomena, rendering, animation and surface modeling. He has published
papers at SIGGRAPH and elsewhere in all of these areas.
Recently, his research has focused on the fundamentals of subdivision surfaces and
their practical use in a commercial product. Stam is a leading expert in both the
theory and application of subdivision surfaces. His work on evaluating subdivision
surfaces presented at last years SIGGRAPH conference has been widely acclaimed as
being a landmark paper in the area.
is currently an Associate Professor in the
Department of Computer Science
at Rice University. He received his master's and
Ph.D. degrees in 1986 from Cornell University.
His research interests focus on the relationship
between computers, mathematics and geometry. During the
course of his research career, he has made fundamental
contributions to topics such as algebraic surfaces, rational
surfaces, finite element mesh generation and subdivision.
Currently, he is investigating the relationship between
subdivision and systems of partial differential equations.
Kobbelt currently holds a position as a post-doctoral
research fellow at the
University of Erlangen , Germany. His major research interest is
sophisticated free-form modeling based on polygonal meshes. He
received his master's (1992) and Ph.D. (1994) degrees from the University of Karlsruhe
, Germany. He then spent one year at the
University of Wisconsin, Madison as a visiting researcher in Carl
de Boor's group. Since 1996 he has been working in the geometric modeling
unit of the Computer Graphics Group at Erlangen. During the last 5
years he made significant contributions to the construction and
analysis of subdivision schemes and pioneered the combination of the
subdivision paradigm with variational methods from CAGD.
The morning section will focus on the foundations of subdivision,
starting with subdivision curves and moving on to
surfaces. We will review and compare a number of different schemes
and discuss the relation between subdivision and splines. The
emphasis will be on properties of subdivsion most relevant for
Introduction and overview (Schröder); 15 min.
Foundations I: Basic Ideas (Schröder)
- Course outline and schedule
- High-level introduction to the basic ideas of subdivision
- Quick overview of application examples
Foundations II: Construction and Analysis
of Subdivision Schemes (Zorin), 90 min.
- Constructing smooth curves through subdivision; 10 min.
examples: b-spline knot insertion and interpolating subdivision
- Subdivision for surfaces; 10 min.
an example of a subdivision scheme: Loop
- Properties of subdivision schemes: smoothness, locality,
hierarchical structure; 10 min.
- How splines are related to subdivision; 10 min.
- Advantages of subdivision: arbitrary topology, efficiency,
controllable surface features such as creases and cusps; 10 min.
The afternoon session will focus on applications of subdivision and
the algorithmic issues practictioners need to address to build
efficient, well behaving systems for modeling and animation with
- Overview of subdivision for surfaces; geometric smoothness. 15 min.
- Subdivision matrices for surface schemes; computing tangents and
limit positions 15 min.
- Classic schemes, their definition, and basic properties; 25 min.
- Subdivision rules for special surface features; boundaries
and creases; 15 min.
- Methods for constructing subdivision schemes; improving
smoothness, curvature continuity, mesh quality; 10 min.
- Computation of moments; 10 min.
- Basic algorithms and data structures for implementing
subdivision; adaptive evaluation, level-of-detail rendering; 10 min.
- Applications and Algorithms:
- Interactive Multiresolution Mesh Editing, 40 min.
Subdivision can model smooth surfaces, but in many
applications one is interested in surfaces which carry
details at many levels of resolution. Multiresolution mesh
editing extends subdivision by including detail offsets at
every level of subdivision, unifying patch based editing
with the flexibility of high resolution polyhedral
meshes. The result is a hierarchical editing system built
around highly adaptive algorithms and datastructures to
deliver interactive performance on low end workstations
for complex geometric models. This section will detail
the underlying ideas and the algorithms necessary to build
a scalable multiresolution editing system. (Zorin)
- Exact Evaluation of Subdivision Surfaces
Until recently it was believed that subdivision surfaces
(Catmull-Clark, Loop, Doo-Sabin, ...) could not be
evaluated exactly everywhere. This talk disproves this
belief and presents the ideas and techniques that enable
exact evaluation. Evaluation is important as it allows
many standard algorithms developed for parametric
surfaces to be applied to subdivision surfaces. The
evaluation technique relies on a new set of eigen-basis
functions which depend directly on the eigenvectors of
the subdivision matrix. The cost of the resulting
evaluation scheme is comparable to that of a bi-cubic
spline for the case of Catmull-Clark subdivision. The
emphasis of this talk is on an intuitive understanding of
the mathematical techniques and on practical applications
of the evaluation schemes. (Stam)
- Subdivision Schemes for Fluid Flows, 40 min.
The motion of fluids has been a topic of study for
hundreds of years. In its most general setting, fluid
flow is governed by a system of non-linear partial
differential equations known as the Navier-Stokes
equations. However, in several important setting, these
equations degenerate into simpler systems of linear PDEs.
This section will show that flows corresponding to these
linear cases can be modeled using subdivision schemes for
vectors. These schemes expressed the flow as the limit of
an increasing dense set of vector fields. The beauty of
this approach is that realistic flows can now be modeled
and manipulated in real time using their associated
subdivision scheme. The section will conclude by
discussing a number of practical details that arose in the
implementation of such a scheme. (Warren)
- A Variational Approach to Subdivision, 40 min.
Surfaces generated using subdivision have certain orders
of continuity. However, it is well known from geometric
modeling that high quality surfaces often require
additional optimization (fairing). In the variational
approach to subdivision, refined meshes are not prescribed
by static rules, but are chosen so as to minimize some
energy functional. The approach combines the advantages of
subdivision (arbitrary topology) with those of variational
design (high quality surfaces). This section will describe
the theory of variational subdivision and highly efficient
algorithms to construct fair surfaces. (Kobbelt)
- Subdivision Surfaces in the Making of Geri's Game and A
Bug's Life, 40 min.
Geri's Game is a 3.5 minute computer animated film that
Pixar completed in 1997. The film marks the first
time that Pixar has used subdivision surfaces in a
production. In fact, subdivision surfaces were used to
model virtually everything that moves. Subdivision
surfaces went on to play a large role in the recently
released feature film 'A Bug's Life' from Disney/Pixar.
This section will describe what led Pixar to use
subdivision surfaces, discuss several issues that were
encountered along the way, and present several of the solutions
that were developed. (DeRose)
- Summary and Wrapup: (all speakers)