Subdivision for Modeling and
This course provides an introduction
to Subdivision, a technique to generate smooth curves and surfaces, which
extends classical spline modeling approaches. The course will cover the
basic ideas of subdivision as well as the particulars of a number of different
subdivision algorithms; we will present the most recent contributions to
the area in a form accessible to a wide audience. The emphasis will be on
practical issues in using subdivision for geometric modeling and animation.
| Denis Zorin
Media Research Laboratory
715 Broadway, Rm 1201
New York University
New York, NY 10003
Caltech Multi-Res Modeling Group
Computer Science Dept. 256-80
California Institute of Technology
Pasadena, CA 91125
Studio Tools Group
Pixar Animation Studios
1001 West Cutting Blvd.
Richmond, CA 94804
Computer Graphics Group
Max-Planck-Institute for Computer Sciences
66123 Saarbr ücken, Germany
School of Mathematics
69978 Tel-Aviv Israel
Bell Laboratories, Lucent Technologies
600 Moutain Avenue
Murray Hill, NJ 07974
Subdivision is an algorithmic technique to generate smooth surfaces as
a sequence of successively refined polyhedral meshes. Its origins go
back to 1978 when Catmull and Clark, and Doo and Sabin first proposed
to generalize spline-patch methods to meshes of arbitrary
topology. Subdivision algorithms are exceptionally simple, work for
arbitrary control meshes and produce globally smooth surfaces. Special
choices of subdivision rules allow for the introduction of features
into a surface in a simple way. Subdivision-based representations of
complex geometry can be manipulated and rendered very efficiently,
which makes subdivision a highly suitable tool for interactive
animation and modeling systems.
This course will cover the basic ideas of subdivision and a variety of
different subdivision schemes detailing their properties, suitability
for particular applications, and compare their relative merits. Strong
emphasis will be placed on practical issues. At the end of the course
participants will be well prepared to implement the basic techniques
as well as delve into the research literature on the subject.
The course will be self contained
and does not assume prior knowledge of subdivision. Prior exposure
to parametric representations of curves and surfaces and associated algorithms
is required. Beyond that only basic linear algebra and introductory calculus
Topics Beyond the Prerequisites
The course will teach the basic ideas of subdivision for curves
and surfaces. We will explain a number of different subdivision
methods for surfaces and briefly discuss the theoretical results
useful in practice. The course will cover a variety of
subdivision-based algorithms for modeling and animation, including
adaptive evaluation, level-of-detail rendering and manipulation. A
number of specific applications will be presented.
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 subdivision 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: Subdivision Schemes for
Surfaces (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.
- Questions and answers; 10 min.
The afternoon session will focus on applications of subdivision and
the algorithmic issues practitioners need to address to build
efficient, well behaving systems for modeling and animation with
subdivision surfaces. Each presentation will be 30 min. long, with
10 min. allocated for questions and discussion.
- Overview of subdivision for surfaces, 15 min.
- Subdivision matrices for surface schemes; computing tangents and
limit positions 15 min.
- Subdivision rules for special surface features; boundaries
and creases; 10 min.
- Classic schemes, their definition, basic properties and comparison, 25 min.
- Explicit Evaluation of subdivision surfaces. 15 min.
- Questions and answers; 10 min.
- Applications and Algorithms:
- Implementing Subdivision and Multiresolution Surfaces, 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. In this part, we will focus on implementation
concerns common for subdivision and multiresolution
surfaces based on subdivision. (Zorin)
- Combined Subdivision Schemes, 40 min.
The speaker will present a class of subdivision schemes
called "Combined Subdivision Schemes". These are subdivision schemes
whose limit surfaces can satisfy prescribed boundary conditions.
Every combined subdivision
scheme consists of an ordinary subdivision scheme that operates in
the interior of the mesh, and special rules that operate near tagged
edges of the mesh and take into consideration the given boundary
The limit surfaces are smooth and they satisfy the boundary conditions.
The speaker will present examples of combined subdivision schemes, and
discuss their applications. (Levin)
- Parameterization, remeshing, and compression using subdivision, 40 min.
Subdivision methods typically use a simple mesh refinement
procedure such as triangle or square quadrisection. Iterating this
refinement step starting starting from a coarse arbitrary connectivity
control mesh generates so-called semi-regular meshes.
However meshes coming from scanning devices are fully
irregular and do not have semi-regular connectivity. In order to use
multiresolution and subdivision based algorithms for such meshes they
first need to be remeshed onto semi-regular connectivity. In this talk
we show how to use mesh simplification to build a smooth
parameterization of dense irregular connectivity meshes and to convert
them to semi-regular connectivity. Our method supports both
fully automatic operation as well as and user defined point and edge
constraints. We also show how semi-regular meshes
can be compressed using a wavelet and zero-tree based
- 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, A
Bug's Life, and Toy Story 2, 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 major
role the feature films 'A Bug's Life' and 'Toy Story 2' 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
- Summary and Wrapup: (all speakers)
|Innovative Methods of Presentation
Aside from the usual slide and video projection used during the
presentations we will have live demonstrations running on a PC
illustrating some of the concepts as well as showing full blown
applications using subdivision. While this is not new as such, we
believe it will significantly enhance the learning experience of
the participants. We plan to make some of the applications
available for experimentation at CAL.
Although subdivision as a basic geometric modeling method
was first proposed in 1978 it is only in the last 9 years that it has received
greater attention in the CAGD and computer graphics community. Its attraction
lies in the simplicity of the fundamental algorithms and the ability of
subdivision to deal with large models of arbitrary topology in interactive
settings. It was only very recently that fundamental results were
proved and efficient algorithms developed to move this technique into the mainstream.
During the past two years, we have seen
subdivision integrated into a growing number of commercial tools
(Alias|wavefront Maya 2.0, Pixar's RenderMan, 3D MAX, Nichimen's
Murai) and used in a
number of productions (Pixar's Geri's Game, A Bug's Life, Toy Story
2). We could observe a fast growth of interest in subdivision both in
computer animation, CAD and gaming industries and among researchers.
At the same time, information on
subdivision is scattered among numerous research articles, not always
suitable for practitioners interested primarily in basic principles
and implementation. Our course aims to meet the demand for a
comprehensive but accessible introduction to subdivision.
This course was taught in 1998 and 1999 with a somewhat different team.
The attendance was considerable both in 98 and 99 (several hundred people)
and we expect it to increase the next year.
The main difference to last year's
course is a number of changes in the presentations and the composition
of the team. Based on the feedback received we are adding more
demos, code examples, and Q&A sessions. In addition,
we plan to update the course notes to reflect recent developments
and further streamline the presentation, and make sample code
available as a part of the electronic version of the notes.
We have added two new speakers, Adi Levin and Wim Sweldens, who will
present some of the interesting recent work, as well as a more
developed but extremely important topic of conversion of
arbitrary meshes to subdivision connectivity. At the same time we
are changing the designation from intermediate to advanced as the
material does require a solid familiarity with standard techniques
such as spline patches (though no prior knowledge of subdivision).
|Course Notes Description
The course notes will contain tutorial material with pointers to the
relevant literature for those who want to study individual issues in
greater depth. The overall outline will follow the 1998-99 materials,
with updates and greater integration of the presentation. In
essence the course notes will be a revised and expanded version of
the 1999 notes. The electronic version will include sample source code.
|Special Presentation Requirements
|Special Notes Requirements
||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.
Schröder is currently an associate professor
of computer science at the
California Institute of Technology, Pasadena, where he directs the
Caltech Multi-Res Modeling Group. For the past 8 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 (1998/99). His current research focuses on subdivision as a
fundamental paradigm for geometric modeling and rapid manipulation of
large, complex geometric models.He was recently named a Packard Fellow
in recognition of his research work.
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.In 1998, he was a major contributor to the
Oscar winning short film "Geri's game", and in 1999 he received
SIGGRAPH's Computer Graphics Achievement Award.
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 the use
of subdivision surfaces in production animation, including the instrumentation of subdivision surfaces with high level animation controls.
is a senior researcher at the
Max-Planck-Institute for Computer
Saarbrücken, Germany. His major research interests
include multiresolution and free-form modeling as well
as the efficient handling of polygonal mesh data. He
received his habilitation degree from the
University of Erlangen Germany where he worked from 1996 to 1999. In
1995/96 he spent one post-doc year at the
University of Wisconsin, Madison
He received his master's (1992) and
Ph.D.\ (1994) degrees from
University of Karlsruhe,
Germany. During the last 7 years he did research in
various fields of computer graphics and CAGD.
has recently completed a PhD in Applied Mathematics at
Tel-Aviv University . He received a BS degree in Applied Mathematics
from Tel-Aviv university. Currently he is a visiting
researcher at the
Caltech Department of Computer Science.
His research interests include surface representation for
Computer Aided Geometric Design, the theory and applications of
Subdivision methods and geometric algorithms for Computer Graphics and
CAGD. He has published a paper in Siggraph'99.
is a researcher at
His work concerns the generalization of signal
processing techniques to complex geometries. He is the inventor of
the "lifting scheme", a technique for building wavelets and
multiresolution transforms on irregularly sampled data and surfaces in
3D. More recently he worked on parameterization, remeshing, and
compression of subdivision surfaces. He has lectured widely on the use
of wavelets and subdivision in computer graphics and participated in
three previous SIGGRAPH courses. MIT's Technology Review recently
selected him as one of a 100 top young technological innovators.
He is the founder and editor-in-chief on the
Copyright © 1997-99 Peter Schröder and Denis Zorin