Course Title Subdivision for Modeling and Animation

sample images

Course Materials SIGGRAPH 2000 Course Notes (30.2 MB PDF file).
Course materials and slides for the 1999 course.

Proposed length Full day

Summary Statement 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.

Course Presenters
Denis Zorin Peter Schröder
Media Research Laboratory
715 Broadway, Rm 1201
New York University
New York, NY 10003
fax: 212.995.4122
Caltech Multi-Res Modeling Group
Computer Science Dept. 256-80
California Institute of Technology
Pasadena, CA 91125
tel: 626.395.4269
fax: 626.792.4257


Tony DeRose

Leif Kobbelt
Studio Tools Group
Pixar Animation Studios
1001 West Cutting Blvd.
Richmond, CA 94804
tel: 510.620.6019
fax: 510.236.0388
Computer Graphics Group
Max-Planck-Institute for Computer Sciences
Im Stadtwald
66123 Saarbr ücken, Germany
tel: +49.681.9325.408
fax: +49.681.9325.499

Adi Levin Wim Sweldens
School of Mathematics
Tel-Aviv University
69978 Tel-Aviv Israel
tel: 972.3.5343496
Bell Laboratories, Lucent Technologies
600 Moutain Avenue
Murray Hill, NJ 07974
tel: 908.582.3288
fax: 908.582.3288
Expanded Statement 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.

Prerequisites 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 are required.

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 (without proofs) 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.

Course Syllabus 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 subdivision 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)
  • 60 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.
  • Foundations II: Subdivision Schemes for Surfaces (Zorin), 90 min.
    • 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.
      1. Catmull-Clark
      2. Doo-Sabin
      3. Loop
      4. Butterfly
      5. Midedge
      6. Kobbelt
    • Explicit Evaluation of subdivision surfaces. 15 min.
    • Questions and answers; 10 min.
    Afternoon: 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.
    • 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 conditions. 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 algorithm. (Sweldens)
      • 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 developed. (DeRose)
    • 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.

    Course History 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 None.

    Special Notes Requirements None.

    Speaker Biographies
    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.


    Peter Schröder

    Peter 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 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.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.


    Leif Kobbelt Leif Kobbelt is a senior researcher at the Max-Planck-Institute for Computer Sciences in 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.


    Adi Levin Adi Levin 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.


    Wim Sweldens Wim Sweldens is a researcher at Bell Laboratories, Lucent Technologies. 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 Wavelet Digest.


    Copyright © 1997-99 Peter Schröder and Denis Zorin