Subdivision for Modeling and Animation
Course notes (30MB PDF).
|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
| Tony DeRose
Studio Tools Group
Pixar Animation Studios
1001 West Cutting Blvd.
Richmond, CA 94804
| Leif Kobbelt|
Computer Graphics Group
Max-Planck-Institute for Computer Sciences
66123 Saarbr ücken, Germany
| Adi Levin
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
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
|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
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.
- 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.
- Explicit Evaluation of subdivision surfaces. 15 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.
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)
- 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)