1999 SIGGRAPH Course: Subdivision for Modeling and Animation

Course Title

Subdivision for Modeling and Animation

sample images

Organizers	Peter Schröder, Denis Zorin
Speakers	Tony DeRose, Leif Kobbelt, Jos Stam, Joe Warren

Course Summary

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 covers the basic ideas of subdivision and a variety of different subdivision schemes detailing their properties, suitability for particular applications, and compare their relative merits.

Course Notes

PDF version of course nodes , as appeared on the CD-ROM (1.9 MB).

Course Syllabus and Slides

Morning: 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 applications.

Introduction and overview (Schröder); 15 min.

Course outline and schedule
High-level introduction to the basic ideas of subdivision
Quick overview of application examples

Foundations I: Basic Ideas (Schröder)

Constructing smooth curves through subdivision.
examples: b-spline knot insertion and interpolating subdivision
Subdivision for surfaces.
an example of a subdivision scheme: Loop
Properties of subdivision schemes: smoothness, locality, hierarchical structure.
How splines are related to subdivision.
Advantages of subdivision: arbitrary topology, efficiency, controllable surface features such as creases and cusps.

Foundations II: Subdivision for Surfaces (Zorin).

Overview of subdivision for surfaces.
Subdivision matrices for surface schemes; computing tangents and limit positions.
Subdivision rules for special surface features; boundaries and creases.

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 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. (Stam)

Subdivision Zoo (Zorin).

Classic schemes, their definition, and basic properties.
1. Catmull-Clark
2. Doo-Sabin
3. Loop
4. Butterfly
5. Midedge
6. Kobbelt

Afternoon: 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 subdivision surfaces.

Applications and Algorithms:
- Implementing Subdivision and Multiresolution Surfaces. 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)
- Subdivision Schemes for Fluid Flows. (no slides). 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. 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. 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)

Course Presenters'
Information

Organizers
Peter Schröder	Denis Zorin
Caltech Multi-Res Modeling Group Computer Science Dept. 256-80 California Institute of Technology Pasadena, CA 91125 vox: 626.395.4269 fax: 626.792.4257 net: ps@cs.caltech.edu	Media Research Laboratory 715 Broadway, Rm 1201 New York University New York, NY 10003 vox: 212.398.3405 fax: 212.995.4122 net: dzorin@mrl.nyu.edu

Speakers
Tony DeRose	Joe Warren
Studio Tools Group Pixar Animation Studios 1001 West Cutting Blvd. Richmond, CA 94804 vox: 510.620.6019 fax: 510.236.0388 net: derose@pixar.com	Department of Computer Science Rice University 6100 South Main Houston, Tx 77251 vox: 713.737.5728 fax: 713.285.5930 net: jwarren@rice.edu

Leif Kobbelt	Jos Stam
Computer Graphics Group University of Erlangen Am Weichselgarten 9 91058 Erlangen, GERMANY vox: +49.9131.859927 fax: +49.9131.859931 net: kobbelt@informatik.uni-erlangen.de	Member Technical Staff Alias \| wavefront 1218 First Avenue, 8th Floor Seattle, WA 98104 vox: 206.287.5624 fax: 206.667.9049 net: jstam@aw.sgi.com

Speaker Biographies
Peter Schröder	Peter 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 industry.
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 Stanford University. 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	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 Science 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 finalist 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	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.
Joe Warren	Joe Warren 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.
Leif Kobbelt	Leif 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.