Subdivision for Modeling and Animation

Subdivision for Modeling and Animation
SIGGRAPH 99 Course Notes
Subdivision for Modeling and Animation
Organizers: Denis Zorin, New York University
Peter Schröder, California Institute of Technology
Lecturers
Denis Zorin
Media Research Laboratory
719 Broadway,rm. 1201
New York University
New York, NY 10012
net: [email protected]
Peter Schröder
Caltech Multi-Res Modeling Group
Computer Science Department 256-80
California Institute of Technology
Pasadena, CA 91125
net: [email protected]
Tony DeRose
Studio Tools Group
Pixar Animation Studios
1001 West Cutting Blvd.
Richmond, CA 94804
net: [email protected]
Jos Stam
Member Technical Staff
Aliasjwavefront
1218 First Avenue, 8th Floor
Seattle, WA 98104
net: [email protected]
Leif Kobbelt
Computer Graphics Group
Max-Planck-Institute for Computer Sciences
Im Stadtwald
66123 Saarbrücken, Germany
net: [email protected]
Joe Warren
Computer Graphics and Geometric Design Group
Computer Science Department
Rice University
Houston, Tx 77584
net: [email protected]
Schedule
Morning Session: Introductory Material 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.
Foundations I: Basic Ideas
Peter Schröder and Denis Zorin
Foundations II: Evaluation and survey of subdivision schemes
Denis Zorin and Jos Stam
Afternoon Session: Applications and Algorithms 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.
Implementing Subdivision and Multiresolution Meshes
Denis Zorin
A Variational Approach to Subdivision
Leif Kobbelt
Variational Subdivision Cookbook
Joe Warren
Subdivision Surfaces in the Making of Geri’s Game
Tony DeRose
5
6
Lecturers’ Biographies
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 is an associate professor of computer science at the California Institute of Technology,
Pasadena, where he directs the Caltech Multi-Res Modeling Group. He received a Master’s degree from
the MIT Media Lab and a PhD from Princeton University. For the past 6 years his work has concentrated
on exploiting wavelets and multiresolution techniques to build efficient representations and algorithms
for many fundamental computer graphics problems. 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.
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.
7
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 the
University of Karlsruhe, Germany. During the last 7 years he did research in various fields of computer
graphics and CAGD.
Jos Stam is currently a member of technical staff at Aliasjwavefront. 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 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.
8
Contents
1 Introduction
11
2 Foundations I: Basic Ideas
15
2.1
The Idea of Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2
Review of Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2.1
Piecewise Polynomial Curves . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2.2
Definition of B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.3
Refinability of B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.2.4
Refinement for Spline Curves . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2.5
Subdivision for Spline Curves . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Subdivision as Repeated Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.3.1
Discrete Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.3.2
Convergence of Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.3.3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
Analysis of Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.4.1
Invariant Neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.4.2
Eigen Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.4.3
Convergence of Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.4.4
Invariance under Affine Transformations
. . . . . . . . . . . . . . . . . . . . .
40
2.4.5
Geometric Behavior of Repeated Subdivision . . . . . . . . . . . . . . . . . . .
42
2.4.6
Size of the Invariant Neighborhood . . . . . . . . . . . . . . . . . . . . . . . .
42
2.4.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2.3
2.4
9
3 Subdivision Surfaces
3.1 Subdivision Surfaces: an Example . . . . . . . . . . . . . . . . . . . .
3.2 Natural Parameterization of Subdivision Surfaces . . . . . . . . . . . .
3.3 Subdivision Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Smoothness of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 C1 -continuity and Tangent Plane Continuity . . . . . . . . . . .
3.5 Analysis of Subdivision Surfaces . . . . . . . . . . . . . . . . . . . . .
3.5.1 C1 -continuity of Subdivision away from Extraordinary Vertices
3.5.2 Smoothness Near Extraordinary Vertices . . . . . . . . . . . .
3.5.3 Characteristic Map . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Piecewise-smooth surfaces and subdivision . . . . . . . . . . . . . . .
4 Subdivision Zoo
4.1 Overview of Subdivision Schemes . .
4.1.1 Notation and Terminology . .
4.2 Loop Scheme . . . . . . . . . . . . .
4.3 Modified Butterfly Scheme . . . . . .
4.4 Catmull-Clark Scheme . . . . . . . .
4.5 Kobbelt Scheme . . . . . . . . . . . .
4.6 Doo-Sabin and Midedge Schemes . .
4.7 Limitations of Stationary Subdivision
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5 Evaluation of Subdivision Surfaces
6 Implementing Subdivision and Multiresolution Meshes
7 Interpolatory Subdivision for Quad Meshes
8 A Variational Approach to Subdivision
9 Subdivision Cookbook
10 Subdivision Surfaces in the Making of Geri’s Game
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Chapter 1
Introduction
Twenty years ago the publication of the papers by Catmull and Clark [3] and Doo and Sabin [4] marked
the beginning of subdivision for surface modeling. This year, another milestone occurred when subdivision hit the big screen in Pixar’s short “Geri’s Game,” for which Pixar received an Academy award for
“Best Animated Short Film.” The basic ideas behind subdivision are very old indeed and can be traced
as far back as the late 40s and early 50s when G. de Rham used “corner cutting” to describe smooth
curves. It was only recently though that subdivision surfaces have found their way into wide application
in computer graphics and computer assisted geometric design (CAGD). One reason for this development is the importance of multiresolution techniques to address the challenges of ever larger and more
complex geometry: subdivision is intricately linked to multiresolution and traditional mathematical tools
such as wavelets.
Constructing surfaces through subdivision elegantly addresses many issues that computer graphics
practitioners are confronted with
Arbitrary Topology: Subdivision generalizes classical spline patch approaches to arbitrary topology. This implies that there is no need for trim curves or awkward constraint management between
patches.
Scalability: Because of its recursive structure, subdivision naturally accommodates level-of-detail
rendering and adaptive approximation with error bounds. The result are algorithms which can
make the best of limited hardware resources, such as those found on low end PCs.
Uniformity of Representation: Much of traditional modeling uses either polygonal meshes or
spline patches. Subdivision spans the spectrum between these two extremes. Surfaces can behave
11
as if they are made of patches, or they can be treated as if consisting of many small polygons.
Numerical Stability: The meshes produced by subdivision have many of the nice properties finite element solvers require. As a result subdivision representations are also highly suitable for
many numerical simulation tasks which are of importance in engineering and computer animation
settings.
Code Simplicity: Last but not least the basic ideas behind subdivision are simple to implement and
execute very efficiently. While some of the deeper mathematical analyses can get quite involved
this is of little concern for the final implementation and runtime performance.
In this course and its accompanying notes we hope to convince you, the reader, that in fact the above
claims are true!
The main focus or our notes will be on covering the basic principles behind subdivision; how subdivision rules are constructed; to indicate how their analysis is approached; and, most importantly, to address
some of the practical issues in turning these ideas and techniques into real applications.
The following 2 chapters will be devoted to understanding the basic principles. We begin with some
examples in the curve, i.e., 1D setting. This simplifies the exposition considerably, but still allows us to
introduce all the basic ideas which are equally applicable in the surface setting. Proceeding to the surface
setting we cover a variety of different subdivision schemes and their properties.
With these basics in place we proceed to the second, applications oriented part, covering algorithms
and implementations addressing
Interactive Multiresolution Mesh Editing: This section discusses many of the data structure
and algorithmic issues which need to be addressed to realize high performance. The result is a
system which allows for interactive, multiresolution editing of fairly complex geometry on PC
class machines with little hardware graphics support.
Subdivision Surfaces and Wavelets: This section shows how subdivision is the key element in
generalizing the traditional wavelet machinery to arbitrary topology surfaces. The result are a
class of algorithms which open up applications such as compression for subdivision surfaces, for
example.
A Variational Approach to Subdivision: Most subdivision methods are stationary, i.e., they use a
fixed set of rules. They are generally designed to exhibit some order of differentiability. In practice
it is often much more important to consider the fairness of the resulting surfaces. Variational
subdivision incorporates fairness measures into the subdivision process.
12
Exploiting Subdivision in Modeling and Animation: One reason subdivision is becoming very
popular is that it supports hierarchical editing and animation semantics. This was discovered
originally in the traditional spline setup and lead to the development of hierarchical splines. From
that technique it is only a small step to multiresolution modeling using subdivision. This section
discusses some of the issues in controlling animation hierarchically.
Subdivision Surfaces in the Making of Geri’s Game: This section discusses how subdivision
surfaces successfully address the needs of very high end production environments. in the process
new techniques had to be developed which are detailed in this part of the notes.
Beyond these Notes
One of the reasons that subdivision is enjoying so much interest right now is that it is very easy to
implement and very efficient. In fact it is used in many computer graphics courses at universities as a
homework exercise. The mathematical theory behind it is very beautiful, but also very subtle and at times
technical. We are not treating the mathematical details in these notes, which are primarily intended for
the computer graphics practitioners. However, for those interested in the theory there are many pointers
to the literature.
These notes as well as other materials such as presentation slides, applets and snippets of code are
available on the web at http://www.multires.caltech.edu/teaching/courses/subdivision/
and all readers are encouraged to explore the online resources. A repository of additional information
beyond this course is maintained at http://www.mrl.nyu.edu/dzorin/subdivision.
13
14
Chapter 2
Foundations I: Basic Ideas
Peter Schröder, Caltech
In this chapter we focus on the 1D case to introduce all the basic ideas and concepts before going
on to the 2D setting. Examples will be used throughout to motivate these ideas and concepts. We
begin initially with an example from interpolating subdivision, before talking about splines and their
subdivision generalizations.
Figure 2.1: Example of subdivision for curves in the plane. On the left 4 points connected with straight
line segments. To the right of it a refined version: 3 new points have been inserted “inbetween” the old
points and again a piecewise linear curve connecting them is drawn. After two more steps of subdivision
the curve starts to become rather smooth.
15
2.1 The Idea of Subdivision
We can summarize the basic idea of subdivision as follows:
Subdivision defines a smooth curve or surface as the limit of a sequence of successive refinements.
Of course this is a rather loose description with many details as yet undetermined, but it captures the
essence.
Figure 2.1 shows an example in the case of a curve connecting some number of initial points in the
plane. On the left we begin with 4 points connected through straight line segments. Next to it is a refined
version. This time we have the original 4 points and additionally 3 more points “inbetween” the old
points. Repeating the process we get a smoother looking piecewise linear curve. Repeating once more
the curve starts to look quite nice already. It is easy to see that after a few more steps of this procedure
the resulting curve would be as well resolved as one could hope when using finite resolution such as that
offered by a computer monitor or a laser printer.
Figure 2.2: Example of subdivision for a surface, showing 3 successive levels of refinement. On the left
an initial triangular mesh approximating the surface. Each triangle is split into 4 according to a particular
subdivision rule (middle). On the right the mesh is subdivided in this fashion once again.
16
An example of subdivision for surfaces is shown in Figure 2.2. In this case each triangle in the original
mesh on the left is split into 4 new triangles quadrupling the number of triangles in the mesh. Applying
the same subdivision rule once again gives the mesh on the right.
Both of these examples show what is known as interpolating subdivision. The original points remain
undisturbed while new points are inserted. We will see below that splines, which are generally not
interpolating, can also be generated through subdivision. Albeit in that case new points are inserted and
old points are moved in each step of subdivision.
How were the new points determined? One could imagine many ways to decide where the new points
should go. Clearly, the shape and smoothness of the resulting curve or surface depends on the chosen
rule. Here we list a number of properties that we might look for in such rules:
Efficiency: the location of new points should be computed with a small number of floating point
operations;
Compact support: the region over which a point influences the shape of the final curve or surface
should be small and finite;
Local definition: the rules used to determine where new points go should not depend on “far
away” places;
Affine invariance: if the original set of points is transformed, e.g., translated, scaled, or rotated,
the resulting shape should undergo the same transformation;
Simplicity: determining the rules themselves should preferably be an offline process and there
should only be a small number of rules;
Continuity: what kind of properties can we prove about the resulting curves and surfaces, for
example, are they differentiable?
For example, the rule used to construct the curve in Figure 2.1 computed new points by taking a weighted
average of nearby old points: two to the left and two to the right with weights 1=16 (,1; 9; 9; ,1) respectively (we are ignoring the boundaries for the moment). It is very efficient since it only involves 4
multiplies and 3 adds (per coordinate); has compact support since only 2 neighbors on either side are
involved; its definition is local since the weights do not depend on anything in the arrangement of the
points; the rule is affinely invariant since the weights used sum to 1; it is very simple since only 1 rule is
used (there is one more rule if one wants to account for the boundaries); finally the limit curves one gets
by repeating this process ad infinitum are C1 .
17
Before delving into the details of how these rules are derived we quickly compare subdivision to other
possible modeling approaches for smooth surfaces: traditional splines, implicit surfaces, and variational
surfaces.
1. Efficiency: Computational cost is an important aspect of a modeling method. Subdivision is
easy to implement and is computationally efficient. Only a small number of neighboring old
points are used in the computation of the new points. This is similar to knot insertion methods
found in spline modeling, and in fact many subdivision methods are simply generalization of knot
insertion. On the other hand implicit surfaces, for example, are much more costly. An algorithm
such as marching cubes is required to generate the polygonal approximation needed for rendering.
Variational surfaces can be even worse: a global optimization problem has to be solved each time
the surface is changed.
2. Arbitrary topology: It is desirable to build surfaces of arbitrary topology. This is a great strength
of implicit modeling methods. They can even deal with changing topology during a modeling
session. Classic spline approaches on the other hand have great difficulty with control meshes of
arbitrary topology. Here, “arbitrary topology” captures two properties. First, the topological genus
of the mesh and associated surface can be arbitrary. Second, the structure of the graph formed by
the edges and vertices of the mesh can be arbitrary; specifically, each vertex may be of arbitrary
degree.
These last two aspects are related: if we insist on all vertices having degree 4 (for quadrilateral)
control meshes, or having degree 6 (for triangular) control meshes, the Euler characteristic for a
planar graph tells us that such meshes can only be constructed if the overall topology of the shape
is that of the infinite plane, the infinite cylinder, or the torus. Any other shape, for example a
sphere, cannot be built from a quadrilateral (triangular) control mesh having vertices of degree 4
(6).
When rectangular spline patches are used in arbitrary control meshes, enforcing higher order continuity at extraordinary vertices becomes difficult and considerably increases the complexity of the
representation (see Figure 2.3 for an example of points not having valence 4). Implicit surfaces
can be of arbitrary topological genus, but the genus, precise location, and connectivity of a surface
are typically difficult to control. Variational surfaces can handle arbitrary topology better than
any other representation, but the computational cost can be high. Subdivision can handle arbitrary
topology quite well without losing efficiency; this is one of its key advantages. Historically subdivision arose when researchers were looking for ways to address the arbitrary topology modeling
18
Figure 2.3: A mesh with two extraordinary vertices, one with valence 6 the other with valence 3. In
the case of quadrilateral patches the standard valence is 4. Special efforts are required to guarantee high
order of continuity between spline patches meeting at the extraordinary points; subdivision handles such
situations in a natural way.
challenge for splines.
3. Surface features: Often it is desirable to control the shape and size of features, such as creases,
grooves, or sharp edges. Variational surfaces provide the most flexibility and exact control for creating features. Implicit surfaces, on the other hand, are very difficult to control, since all modeling
is performed indirectly and there is much potential for undesirable interactions between different
parts of the surface. Spline surfaces allow very precise control, but it is computationally expensive and awkward to incorporate features, in particular if one wants to do so in arbitrary locations.
Subdivision allows more flexible controls than is possible with splines. In addition to choosing
locations of control points, one can manipulate the coefficients of subdivision to achieve effects
such as sharp creases or control the behavior of the boundary curves.
4. Complex geometry: For interactive applications, efficiency is of paramount importance. Because
subdivision is based on repeated refinement it is very straightforward to incorporate ideas such
as level-of-detail rendering and compression for the internet. During interactive editing locally
adaptive subdivision can generate just enough refinement based on geometric criteria, for example.
19
For applications that only require the visualization of fixed geometry, other representations, such
as progressive meshes, are likely to be more suitable.
Since most subdivision techniques used today are based upon and generalize splines we begin with
a quick review of some basic facts of splines which we will need to understand the connection between
splines and subdivision.
2.2 Review of Splines
2.2.1 Piecewise Polynomial Curves
Splines are piecewise polynomial curves of some chosen degree. In the case of cubic splines, for example, each polynomial segment of the curve can be written as
x(t )
=
ai3t 3 + ai2t 2 + ai1t + ai0
y(t )
=
bi3t 3 + bi2t 2 + bi1t + bi0 ;
where (a; b) are constant coefficients which control the shape of the curve over the associated segment.
This representation uses monomials (t 3 ; t 2 ; t 1 ; t 0 ), which are restricted to the given segment, as basis
functions.
1.0
0.5
0.0
-0.5
-4
-3
-2
-1
0
1
2
3
4
Figure 2.4: Graph of the cubic B-spline. It is zero for the independent parameter outside the interval
[,2; 2].
Typically one wants the curve to have some order of continuity along its entire length. In the case of
cubic splines one would typically want C2 continuity. This places constraints on the coefficients (a; b)
of neighboring curve segments. Manipulating the shape of the desired curves through these coefficients,
20
while maintaining the constraints, is very awkward and difficult. Instead of using monomials as the basic
building blocks, we can write the spline curve as a linear combination of shifted B-splines, each with a
coefficient known as a control point
x(t )
=
y(t )
=
∑ xi B(t , i)
∑ yi B(t , i)
:
The new basis function B(t ) is chosen in such a way that the resulting curves are always continuous and
that the influence of a control point is local. One way to ensure higher order continuity is to use basis
functions which are differentiable of the appropriate order. Since polynomials themselves are inifinitely
smooth, we only have to make sure that derivatives match at the points where two polynomial segments
meet. The higher the degree of the polynomial, the more derivatives we are able to match. We also
want the influence of a control point to be maximal over a region of the curve which is close to the
control point. Its influence should decrease as we move away along the curve and disappear entirely at
some distance. Finally, we want the basis functions to be piecewise polynomial so that we can represent
any piecewise polynomial curve of a given degree with the associated basis functions. B-splines are
constructed to exactly satisfy these requirements (for a cubic B-spline see Figure 2.4) and in a moment
we will show how they are constructed.
The advantage of using this representation rather than the earlier one of monomials, is that the continuity conditions at the segment boundaries are already “hardwired” into the basis functions. No matter
how we move the control points, the spline curve will always maintain its continuity, for example, C2 in
the case of cubic B-splines.1 Furthermore, moving a control point has the greatest effect on the part of
the curve near that control point, and no effect whatsoever beyond a certain range. These features make
B-splines a much more appropriate tool for modeling piecewise polynomial curves.
Note: When we talk about curves, it is important to distinguish the curve itself and the graphs of the
coordinate functions of the curve, which can also be thought of as curves. For example, a curve can
be described by equations x(t ) = sin(t ), y(t ) = cos(t ). The curve itself is a circle, but the coordinate
functions are sinusoids. For the moment, we are going to concentrate on representing the coordinate
functions.
1 The
differentiability of the basis functions guarantees the differentiability of the coordinate functions of the curve. However, it does not guarantee the geometric smoothness of the curve. We will return to this distinction in our discussion of
subdivision surfaces.
21
2.2.2 Definition of B-Splines
There are many ways to derive B-splines. Here we choose repeated convolution, since we can see from
it directly how splines can be generated through subdivision.
We start with the simplest case: piecewise constant coordinate functions. Any piecewise constant
function can be written as
x(t ) = ∑ xi Bi0 (t );
where B0 (t ) is the box function defined as
B0 (t )
0t <1
=
1 if
=
0 otherwise;
and the functions Bi0(t ) = B0 (t , i) are translates of B0 (t ). Furthermore, let us represent the continuous
convolution of two functions f (t )and g(t ) with
(f
g)(t ) =
Z
f (s)g(t , s)ds:
A B-spline basis function of degree n can be obtained by convolving the basis function of degree n , 1
with the box B0 (t ).2 For example, the B-spline of degree 1 is defined as the convolution of B0 (t ) with
itself
B1 (t ) =
Z
B0 (s)B0 (t , s)ds:
Graphically (see Figure 2.5), this convolution can be evaluated by sliding one box function along the
coordinate axis from minus to plus infinity while keeping the second box fixed. The value of the convolution for a given position of the moving box is the area under the product of the boxes, which is just
the length of the interval where both boxes are non-zero. At first the two boxes do not have common
support. Once the moving box reaches 0, there is a growing overlap between the supports of the graphs.
The value of the convolution grows with t until t = 1. Then the overlap starts decreasing, and the value
of the convolution decreases down to zero at t = 2. The function B1 (t ) is the linear hat function as shown
in Figure 2.5.
We can compute the B-spline of degree 2 convolving B1 (t ) with the box B0 (t ) again
B2 (t ) =
Z
B1 (s)B0 (t , s)ds:
2 The
degree of a polynomial is the highest order exponent which occurs, while the order counts the number of coefficients
and is 1 larger. For example, a cubic curve is of degree 3 and order 4.
22
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Figure 2.5: The definition of degree 1 B-Spline B1(t ) (right side) through convolution of B0 (t ) with itself
(left side).
In this case, the resulting curve consists of three quadratic segments defined on intervals (0; 1), (1; 2) and
(2; 3). In general, by convolving l times, we can get a B-spline of degree l
Bl (t ) =
Z
Bl,1 (s)B0 (t , s)ds:
Defining B-splines in this way a number of important properties immediately follow. The first concerns
the continuity of splines
Theorem 1 If f (t ) is Ck -continuous, then (B0 f )(t ) is Ck+1 -continuous.
23
This is a direct consequence of convolution with a box function. From this it follows that the B-spline of
degree n is Cn,1 continuous because the B-spline of degree 1 is C0 -continuous.
2.2.3 Refinability of B-splines
Another remarkable property of B-splines is that they obey a refinement equation. This is the key
observation to connect splines and subdivision. The refinement equation for B-splines of degree l is
given by
Bl (t ) =
1
2l
l +1
∑
l + 1
k
k=0
Bl (2t , k):
(2.1)
In other words, the B-spline of degree l can be written as a linear combination of translated (k) and
dilated (2t) copies of itself. For a function to be refineable in this way is a rather special property. As an
example of the above equation at work consider the hat function shown in Figure 2.5. It is easy to see that
it can be written as a linear combination of dilated hat functions with weights (1=2; 1; 1=2) respectively.
The property of refinability is the key to subdivision and so we will take a moment to prove it. We
start by observing that the box function, i.e., the B-spline of degree 0 can be written in terms of dilates
and translates of itself
B0 (t ) = B0 (2t ) + B0 (2t , 1);
(2.2)
which is easily checked by direct inspection. Recall that we defined the B-spline of degree l as
O
l
Bl (t ) =
O
l
B0 (t ) =
i=0
(B0 (2t ) + B0 (2t
, 1))
(2.3)
i=0
This expression can be “multiplied” out by using the following properties of convolution for functions
f (t ), g(t ), and h(t )
f (t ) (g(t ) + h(t ))
f (t , i) g(t , k)
f (2t ) g(2t )
=
=
=
f (t ) g(t ) + f (t ) h(t )
linearity
m(t , i , k)
time shift
1
time scaling
2 m(2t )
where m(t ) = f (t ) g(t ). These properties are easy to check by substituting the definition of convolution
and amount to simple change of variables in the integration.
24
For example, in the case of B1 we get
B1 (t )
=
=
=
=
=
=
B0 (t ) B0(t )
, 1)) (B0(2t ) + B0(2t , 1))
B0 (2t ) B0 (2t ) + B0 (2t ) B0 (2t , 1) + B0(2t , 1) B0(2t ) + B0 (2t , 1) B0(2t , 1)
1
1
1
1
B1 (2t ) + B1 (2t , 1) + B1 (2t , 1) + B1 (2t , 1 , 1)
2
2
2
2
(B0 (2t ) + B0 (2t
1
(B1 (2t ) + 2B1 (2t , 1) + B1 (2t , 2))
2
1 2 2
B1(2t , k):
k
21 k∑
=0
The general statement for B-splines of degree l now follows from the binomial theorem
l +1
(x + y)
l +1
=
∑
l + 1
xl+1,k yk ;
k
k=0
with B0 (2t ) in place of x and B0 (2t , 1) in place of y.
2.2.4 Refinement for Spline Curves
"
#
With this machinery in hand let’s revisit spline curves. Let
γ(t ) =
x(t )
y(t )
=
∑ pi Bil (t )
i
be such a spline curve of degree l with control points (xi ; yi )T = pi 2 R2 . Since we don’t want to worry
about boundaries for now we leave the index set i unspecified. We will also drop the subscript l since the
degree, whatever it might be, is fixed for all our examples. Due to the definition of Bi (t ) = B(t , i) each
control point exerts influence over a small part of the curve with parameter values t 2 [i; i + l ].
Now consider p, the vector of control points of a given curve:
2 .. 3
66 . 77
66 p,2 77
66 p,1 77
p=6
66 p0 777
66 p1 77
64 p2 75
.
..
25
and the vector B(t ), which has as its elements the translates of the function B as defined above
B(t ) =
h
:::
B(t + 2) B(t + 1) B(t ) B(t , 1) B(t , 2)
i
:::
:
In this notation we can denote our curve as B(t )p.
Using the refinement relation derived earlier, we can rewrite each of the elements of B in terms of its
dilates
B(2t ) =
h
:::
B(2t + 2) B(2t + 1) B(2t ) B(2t , 1) B(2t , 2)
i
:::
;
using a matrix S to encode the refinement equations
B(t ) = B(2t )S:
The entries of S are given by Equation 2.1
S2i+k i = sk =
;
1 l+1
k
2l
:
The only non-zero entries in each column are the weights of the refinement equation, while successive
columns are copies of one another save for a shift down by two rows.
We can use this relation to rewrite γ(t )
γ(t ) = B(t )p = B(2t )Sp:
It is still the same curve, but desribed with respect to dilated B-splines, i.e., B-splines whose support is
half as wide and which are spaced twice as dense. We performed a change from the old basis B(t ) to the
new basis B(2t ) and concurrently changed the old control points p to the appropriate new control points
Sp. This process can be repeated
γ(t )
=
B(t )p0
B(2t )p1
..
.
=
B(2t )Sp0
=
B(2 j t )p j
=
B(2 j t )S j p0 ;
=
from which we can define the relationship between control points at different levels of subdivision
p j+1 = Sp j ;
where S is our infinite subdivision matrix.
26
Looking more closely at one component, i, of our control points we see that
j+1
pi
=
∑ Si l plj
;
:
l
To find out exactly which sk is affecting which term, we can divide the above into odd and even entries.
For the odd entries we have
p2i+1 = ∑ S2i+1 l pl
j+1
j
;
l
=
∑ s2 i,l
(
j
)+1
pl
l
and for the even entries we have
j+1
p2i
=
∑ S2i l plj = ∑ s2 i,l
(
;
l
j
)
pl :
l
From which we essentially get two different subdivision rules one for the new even control points of the
curve and one for the new odd control points. As examples of the above, let us consider two concrete
cases. For piecewise linear subdivision, the basis functions are hat functions. The odd coefficients are 12
and 12 , and a lone 1 for the even point. For cubic splines the odd coefficients turn out to be 12 and 12 , while
the even coefficients are 18 , 68 , and 18 .
Another way to look at the distinction between even and odd is to notice that odd points at level j + 1
are newly inserted, while even points at level j + 1 correspond directly to the old points from level j.
In the case of linear splines the even points are in fact the same at level j + 1 as they were at level j.
Subdivision schemes that have this property will later be called interpolating, since points, once they
have been computed, will never move again. In contrast to this consider cubic splines. In that case even
j+1
j
points at level j + 1 are local averages of points at level j so that p2i 6= pi . Schemes of this type will
later be called approximating.
2.2.5 Subdivision for Spline Curves
In the previous section we saw that we can refine the control point sequence for a given spline by multiplying the control point vector p by the matrix S, which encodes the refinement equation for the B-spline
used in the definition of the curve. What happens if we keep repeating this process over and over, generating ever denser sets of control points? It turns out the control point sequence converges to the actual
spline curve. The speed of convergence is geometric, which is to say that the difference between the
curve and its control points decreases by a constant factor on every subdivision step. Loosely speaking
this means that the actual curve is hard to distinguish from the sequence of control points after only a
few subdivision steps.
27
We can turn this last observation into an algorithm and the core of the subdivision paradigm. Instead
of drawing the curve itself on the screen we draw the control polygon, i.e., the piecewise linear curve
through the control points. Applying the subdivision matrix to the control points defines a sequence of
piecewise linear curves which quickly converge to the spline curve itself.
In order to make these observations more precise we need to introduce a little more machinery in the
next section.
2.3 Subdivision as Repeated Refinement
2.3.1 Discrete Convolution
The coefficients sk of the B-spline refinement equation can also be derived from another perspective,
namely discrete convolution. This approach mimics closely the definition of B-splines through continuous convolution. Using this machinery we can derive and check many useful properties of subdivision
by looking at simple polynomials.
Recall that the generating function of a sequence ak is defined as
A(z) = ∑ ak zk ;
k
where A(z) is the z-transform of the sequence ak . This representation is closely related to the discrete
Fourier transform of a sequence by restricting the argument z to the unit circle, z = exp(iθ). For the case
of two coefficient sequences ak and bk their convolution is defined as
ck = (a b)k = ∑ ak,n bn :
n
In terms of generating functions this can be stated succinctly as
C(z) = A(z)B(z);
which comes as no surprise since convolution in the time domain is multiplication in the Fourier domain.
The main advantage of generating functions, and the reason why we use them here, is that manipulations of sequences can be turned into simple operations on the generating functions. A very useful
example of this is the next observation. Suppose we have two functions that each satisfy a refinement
equation
f (t )
=
∑ ak f (2t , k)
k
g(t )
=
∑ bk g(2t , k)
k
28
:
In that case the convolution h = f g of f and g also satisfies a refinement equation
h(t ) = ∑ ck h(2t , k);
k
whose coefficients ck are given by the convolution of the coefficients of the individual refinement equations
ck =
1
ak,i bi :
2∑
i
With this little observation we can quickly find the refinement equation, and thus the coefficients of the
subdivision matrix S, by repeated multiplication of generating functions. Recall that the box function
B0 (t ) satisfies the refinement equation B0 (t ) = B0 (2t ) + B0 (2t , 1). The generating function of this
refinement equation is A(z) = (1 + z) since the only non-zero terms of the refinement equation are those
belonging to indices 0 and 1. Now recall the definition of B-splines of degree l
O
l
Bl (t ) =
B0 (t );
k=0
from which we immediately get the associated generating function
S(z) =
1
l +1
(1 + z)
:
2l
The values sk used for the definition of the subdivision matrix are simply the coefficients of the various
powers of z in the polynomial S(z)
S(z) =
1
2l
l +1
∑
k=0
l + 1
k
zk ;
where we used the binomial theorem to expand S(z). Note how this matches the definition of sk in
Equation 2.1.
Recall Theorem 1, which we used to argue that B-splines of degree n are Cn,1 continuous. That same
theorem can now be expressed in terms of generating functions as follows
Theorem 2 If S(z) defines a convergent subdivision scheme yielding a C k -continuous limit function then
1
k+1 -continuous limit functions.
2 (1 + z)S(z) defines a convergent subdivision scheme with C
We will put this theorem to work in analyzing a given subdivision scheme by peeling off as many factors of 12 (1 + z) as possible, while still being able to prove that the remainder converges to a continuous
29
limit function. With this trick in hand all we have left to do is establish criteria for the convergence of
a subdivision scheme to a continuous function. Once we can verify such a condition for the subdivision scheme associated with B-spline control points we will be justified in drawing the piecewise linear
approximations of control polygons as approximations for the spline curve itself. We now turn to this
task.
2.3.2 Convergence of Subdivision
There are many ways to talk about the convergence of a sequence of functions to a limit. One can use
different norms and different notions of convergence. For our purposes the simplest form will suffice,
uniform convergence.
We say that a sequence of functions fi defined on some interval [a; b] R converges uniformly to a
limit function f if for all ε > 0 there exists an n0 > 0 such that for all n > n0
max j f (t ) , fn (t )j < ε:
t 2[a;b]
Or in words, as of a certain index (n0 ) all functions in the sequence “live” within an ε sized tube around
the limit function f . This form of convergence is sufficient for our purposes and it has the nice property that if a sequence of continuous functions converges uniformly to some limit function f , that limit
function is itself continuous.
For later use we introduce some norm symbols
k f (t )k
kpk
kSk
=
sup j f (t )j
=
sup j pi j
=
sup ∑ jSik j;
t
i
i
k
which are compatible in the sense that, for example, kSpk kSkkpk.
The sequence of functions we want to analyze now are the control polygons as we refine them with
the subdivision rule S. Recall that the control polygon is the piecewise linear curve through the control
points p j at level j. Independent of the subdivision rule S we can use the linear B-splines to define the
piecewise linear curve through the control points as P j (t ) = B1 (2 j t )p j .
One way to show that a given subdivision scheme S converges to a continuous limit function is to
prove that (1) the limit
P∞ (t ) = lim P j (t )
j!∞
30
exists for all t and (2) that the sequence P j (t ) converges uniformly. In order to show this property we
need to make the assumption that all rows of the matrix S sum to 1, i.e., the odd and even coefficients of
the refinement relation separately sum to 1. This is a reasonable requirement since it is needed to ensure
the affine invariance of the subdivision process, as we will later see. In matrix notation this means S1 = 1,
or in other words, the vector of all 1’s is an eigenvector of the subdivision matrix with eigenvalue 1. In
terms of generating functions this means S(,1) = 0, which is easily verified for the generating functions
we have seen so far.
Recall that the definition of continuity in the function setting is based on differences. We say f (t )
is continuous at t0 if for any ε > 0 there exists a δ > 0 so that j f (t0 ) , f (t )j < ε as long as jt0 , t j < δ.
The corresponding tool in the subdivision setting is the difference between two adjacent control points
j
j
pi+1 , pi = (∆p j )i . We will show that if the differences between neighboring control points shrink fast
enough, the limit curve will exist and be continuous:
Lemma 3 If k∆p j k < cγ j for some constant c > 0 and a shrinkage factor 0 < γ < 1 for all j > j0 0
then P j (t ) converges to a continuous limit function P∞ (t ).
Proof: Let S be the subdivision rule at hand, p1 = Sp0 and S1 be the subdivision rule for B-splines of
degree 1. Notice that the rows of S , S1 sum to 0
(S
, S1 )1 = S1 , S1 1 = 1 , 1 = 0
:
This implies that there exists a matrix D such that S , S1 = D∆, where ∆ computes the difference of
adjacent elements (∆)ii = ,1, (∆)i i+1 = 1, and zero otherwise. The entries of D are given as Di j =
, ∑kj=i (S , S1 )ik . Now consider the difference between two successive piecewise linear approximations
of the control points
;
kP j
+1
(t )
, P j (t )k
kB1(2 j 1t )p j 1 , B1(2 j t )p j k
j 1
= kB1 (2
t )Sp j , B1(2 j 1t )S1 p j k
j 1
= kB1 (2
t )(S , S1 )p j k
kB1(2 j 1t )kkD∆p j k
kDkk∆p j k
kDkcγ j
+
=
+
+
+
+
+
:
This implies that the telescoping sum P0 (t ) + ∑k=0 (Pk+1 , Pk )(t ) converges to a well defined limit function since the norms of each summand are bounded by a constant times a geometric term γ j . Let P∞ (t )
j
31
as j ! ∞, then
kP∞ (t ) , P j (t )k k1D,kγc γ j
<
;
since the latter is the tail of a geometric series. This implies uniform convergence and thus continuity of
P∞ (t ) as claimed.
How do we check such a condition for a given subdivision scheme? Suppose we had a derived
subdivision scheme D for the differences themselves
∆p j+1 = D∆p j ;
defined as the scheme that satisfies
∆S = D∆:
Or in words, we are looking for a difference scheme D such that taking differences after subdivision is
the same as applying the difference scheme to the differences. Does D always exist? The answer is yes
if S is affinely invariant, i.e., S(,1) = 0. This follows from the following argument. Multiplying S by ∆
computes a matrix whose rows are differences of adjacent rows in S. Since odd and even numbered rows
of S each sum to one, the rows of ∆S must each sum to zero. Now the existence of a matrix D such that
∆S = D∆ follows as in the argument above.
Given this difference scheme D all we would have to show is that some power m > 0 of D has norm
less than 1, kDm k = γ < 1. In that case k∆p j k < c(γ1 m ) j . (We will see in a moment that the extra degree
of freedom provided by the parameter m is needed in some cases.)
As an example, let us check this condition for cubic B-splines. Recall that B3(z) = 18 (1 + z)4 , i.e.,
=
p2ij++11
=
j +1
=
p2i
1
j
j
(4pi + 4pi+1 )
8
1 j
j
j
(p
+ 6pi + pi+1 ):
8 i,1
Taking differences we have
(∆p
j+1
)2i
=
=
p2i+1 , p2i
1
j
j
j
(, pi,1 , 2pi + 3pi+1 )
8
1
1
j
j
j
j
j
j
(3( pi+1 , pi ) + 1( pi , pi,1 )) = (3(∆p )i + 1(∆p )i,1 );
8
8
j+1
j+1
=
and similarly for the odd entries so that D(z) = 18 (1 + z)3 , from which we conclude that kDk = 12 , and that
the subdivision scheme for cubic B-splines converges uniformly to a continuous limit function, namely
the B-spline itself.
32
Another example, which is not a spline, is the so called 4 point scheme [5]. It was used to create
the curve in Figure 2.1, which is interpolating rather than approximating as is the case with splines. The
generating function for the 4 point scheme is
S(z) =
1
,3
,2 ,1
4
(,z + 4z , z )(1 + z)
16
Recall that each additional factor of 12 (1 + z) in the generating function increases the order of continuity of
the subdivision scheme. If we want to show that the limit function of the 4 point scheme is differentiable
we need to show that 18 (,z,3 + 4z,2 , z,1 )(1 + z)3 converges to a continuous limit function. This in
turn requires that D(z) = 18 (,z,3 + 4z,2 , z,1 )(1 + z)2 satisfy a norm estimate as before. The rows of D
have non-zero entries of ( 14 ; 14 ), and ( ,81 ; 68 ; ,81 ) respectively. Thus kDk = 1, which is not strong enough.
However, with a little bit more work one can show that kD2 k = 34 , so that indeed the 4 point scheme is
C1 .
In general, the difficult part is to find a set of coefficients for which subdivision converges. There
is no general method to achieve this. Once a convergent subdivision scheme is found, one can always
obtain a desired order of continuity by convolving with the box function.
2.3.3 Summary
So far we have considered subdivision only in the context of splines where the subdivision rule, i.e., the
coefficients used to compute a refined set of control points, was fixed and everywhere the same. There
is no pressing reason for this to be so. We can create a variety of different curves by manipulating the
coefficients of the subdivision matrix. This could be done globally or locally. I.e., we could change the
coefficients within a subdivision level and/or between subdivision levels. In this regard, splines are just
a special case of the more general class of curves, subdivision curves. For example, at the beginning of
this chapter we briefly outlined an interpolating subdivision method, while spline based subdivision is
approximating rather than interpolating.
Why would one want to draw a spline curve by means of subdivision? In fact there is no sufficiently
strong reason for using subdivision in one dimension and none of the commercial line drawing packages
do so, but the argument becomes much more compelling in higher dimensions as we will see in later
chapters.
In the next section we use the subdivision matrix to study the behavior of the resulting curve at a point
or in the neighborhood of a point. We will see that it is quite easy, for example, to evaluate the curve
exactly at a point, or to compute a tangent vector, simply from a deeper understanding of the subdivision
matrix.
33
2.4 Analysis of Subdivision
In the previous section we have shown that uniform spline curves can be thought of as a special case of
subdivision curves. So far, we have seen only examples for which we use a fixed set of coefficients to
compute the control points everywhere. The coefficients define the appearance of the curve, for example,
whether it is differentiable or has sharp corners. Consequently it is possible to control the appearance of
the curve by modifying the subdivision coefficients locally. So far we have not seen a compelling reason
to do so in the 1D setting. However, in the surface setting it will be essential to change the subdivision
rule locally around extraordinary vertices to ensure maximal order of continuity. But before studying this
question we once again look at the curve setting first since the treatment is considerably easier to follow
in that setting.
To study properties such as differentiability of the curve (or surface) we need to understand which
of the control points influences the neighborhood of the point of interest. This notion is captured by the
concept of invariant neighborhoods to which we turn now.
2.4.1 Invariant Neighborhoods
Suppose we want to study the limit curve of a given subdivision scheme in the vicinity of a particular
control point.3 To determine local properties of a subdivision curve, we do not need the whole infinite
vector of control points or the infinite matrix describing subdivision of the entire curve. Differentiability,
for example, is a local property of a curve. To study it we need consider only an arbitrarily small piece
of the curve around the origin. This leads to the question of which control points influence the curve in
the neighborhood of the origin?
As a first example consider cubic B-spline subdivision. There is one cubic segment to the left of the
origin with parameter values t 2 [,1; 0] and one segment to the right with parameter range t 2 [0; 1].
Figure 2.6 illustrates that we need 5 control points at the coarsest level to reach any point of the limit
curve which is associated with a parameter value between ,1 and 1, no matter how close it is to the
origin. We say that the invariant neighborhood has size 5. This size depends on the number of non-zero
entries in each row of the subdivision matrix, which is 2 for odd points and 3 for even points. The latter
implies that we need one extra control point to the left of ,1 and one to the right of 1.
Another way to see this argument is to consider the basis functions associated with a given subdivision
scheme. Once those are found we can find all basis functions overlapping a region of interest and
3 Here
and in the following we assume that the point of interest is the origin. This can always be achieved through renumbering of the control points.
34
-2
-1
0
1
4
-1
6
1
2
1
4
0
1
Figure 2.6: In the case of cubic B-spline subdivision the invariant neighborhood is of size 5. It takes
5 control points at the coarsest level to determine the behavior of the subdivision limit curve over the
two segments adjacent to the origin. At each level we need one more control point on the outside of
the interval t 2 [,1; 1] in order to continue on to the next subdivision level. 3 initial control points for
example would not be enough.
their control points will give us the control set for that region. How do we find these basis functions
in the setting when we don’t neccessarily produce B-splines through subdivision? The argument is
35
straightforward and also applies to surfaces. Recall that the subdivision operator is linear, i.e.,
P j (t )
=
=
=
!
B1 (2 j t )S j p0
j
B1 (2 t )S
∑
j
∑
p0i (ei )0
i
0
j
pi B1 (2 t )S j (ei )0
i
=
∑ p0i ϕij (t )
i
In this expression ei 0 stands for the vector consisting of all 0s except a single 1 in position i. In other
words the final curve is always a linear combination with weights p0i of fundamental solutions
lim ϕi (t ) = ϕi (t ):
j!∞
j
If we used the same subdivision weights throughout the domain it is easy to see that ϕi (t ) = ϕ(t ,
i), i.e., there is a single function ϕ(t ) such that all curves produced through subdivision from some
initial sequence of points p0 are linear combinations of translates of ϕ(t ). This function is called the
fundamental solution of the subdivision scheme. Questions such as differentiability of the limit curve
can now be studied by examining this one function
ϕ(t ) = lim S j (e0 )0 :
j!∞
For example, we can read off from the support of this function how far the influence of a control point
will be felt. Similarly, the shape of this function tells us something about how the curve (or surface) will
change when we pull on a control point. Note that in the surface case the rules we apply will depend on
the valence of the vertex in question. In that case we won’t get only a single fundamental solution, but a
different one for each valence. More on this later.
With this we can revisit the argument for the size of the invariant neighborhood. The basis functions
of cubic B-spline subdivision have support width of 4 intervals. If we are interested in a small open
neighborhood of the origin we notice that 5 basis functions will overlap that small neighborhood. The
fact that the central 5 control points control the behavior of the limit curve at the origin holds independent
of the level. With the central 5 control points at level j we can compute the central 5 control points at
level j + 1. This implies that in order to study the behavior of the curve at the origin all we have to
36
analyze is a small 5 5 subblock of the subdivision matrix
0
BB
BB
BB
@
-3
-2
j+1
p,2
j+1
p,1
j+1
p0
j+1
p1
j+1
p2
1 01
CC B
0
CC = 1 B
B
0
CC 8 B
B
B
A @0
6
4
1
0
0 0
-1
1
4
6
4
1
0
0
1
4
6
0
0
0
0
0
1
10 pj 1
,2
C
B
C
j
C
B
p,1 C
C
B
C
j
C
B
p0 C
C
B
C
C
B
j C
p
[email protected] 1 A
:
j
p2
1
2
3
16
-1
9
-1
9
-1
0
1
Figure 2.7: In the case of the 4 point subdivision rule the invariant neighborhood is of size 7. It takes 7
control points at the coarsest level to determine the behavior of the subdivision limit curve over the two
segments adjacent to the origin. One extra point at p2j is needed to compute p1j+1 . The other is needed
j+1
j
to compute p3 , which requires p3 . Two extra points on the left and right result in a total of 7 in the
invariant neighborhood.
The 4 point subdivision scheme provides another example. This time we do not have recourse to
37
splines to argue the properties of the limit curve. In this case each basis function has a support ranging
over 6 intervals. An easy way to see this is to start with the sequence e0 0 , i.e., a single 1 at the origin
surrounded by zeros. Repeatedly applying subdivision we can see that no points outside the original
[,3; 3] interval will become non-zero. Consequently for the invariant neighborhood of the origin we
need to consider 3 basis functions to the left, the center function, and 3 basis functions to the right. The
4 point scheme has an invariant neighborhood of 7 (see Figure 2.7). In this case the local subdivision
matrix is given by
0
BB
BB
BB
BB
BB
[email protected]
j+1
p,3
j+1
p,2
j+1
p,1
j+1
p0
j+1
p1
j+1
p2
j+1
p3
1 0 ,1 9 9 ,1 0 0
CC BB 0 0 16 0 0 0
CC BB
CC 1 BB 0 ,1 9 9 ,1 0
CC = 16 BB 0 0 0 16 0 0
CC BB 0 0 ,1 9 9 ,1
CA [email protected] 0 0 0 0 16 0
0
0
,1
0
9
9
0
0
0
0
0
0
,1
10
C
B
C
B
C
B
C
B
C
B
C
B
C
B
C
B
C
B
C
B
C
AB
@
j+1
p,3
j+1
p,2
j+1
p,1
j+1
p0
j+1
p1
j+1
p2
j+1
p3
1
C
C
C
C
C
C
C
C
C
C
C
A
Since the local subdivision matrix controls the behavior of the curve in a neighborhood of the origin,
it comes as no surprise that many properties of curves generated by subdivision can be inferred from
the properties of the local subdivision matrix. In particular, differentiability properties of the curve are
related to the eigenstructure of the local subdivision matrix to which we now turn. From now on the
symbol S will denote the local subdivision matrix.
2.4.2 Eigen Analysis
Recall from linear algebra that an eigenvector x of the matrix M is a non-zero vector such that Mx = λx,
where λ is a scalar. We say that λ is the eigenvalue corresponding to the right eigenvector x.
Assume the local subdivision matrix S has size n n and has real eigenvectors x0 ; x1 ; : : : ; xn,1 , which
form a basis, with corresponding real eigenvalues λ0 λ1 : : : λn,1 . For example, in the case of
38
cubic splines n = 5 and
(λ0 ; λ1 ; λ2 ; λ3 ; λ4 )
=
(x0 ; x1 ; x2 ; x3 ; x4 )
Given these eigenvectors we have
S(x0 ; x1 ; x2 ; x3 ; x4 )
=
(1;
0
B
B
B
B
B
B
@
=
1 1 1 1
;
;
; )
2 4 8 8
1 ,1
1
1
2
1 ,2
11
1
1 0 , 11
2
1 12
11
1 1
1
0λ
0
B
B
0
B
(x0 x1 x2 x3 x4 ) B
0
B
B
@0
;
;
;
;
0
SX
X ,1SX
=
XD
=
D:
1
0
0
0
0
0
0
0
0
1
1
C
C
C
C
C
C
A
:
0 0 0 0
λ1 0 0 0
0 λ2 0 0
0 0 λ3 0
0 0 0 λ4
1
C
C
C
C
C
C
A
The rows x̃i of X ,1 are called left eigenvectors since they satisfy x̃i S = λi x̃i , which can be seen by
multiplying the last equality with X ,1 on the right.
Note: not all subdivision schemes have only real eigenvalues or a complete set of eigenvectors. For
example, the 4 point scheme has eigenvalues
(λ0 ; λ1 ; λ2 ; λ3 ; λ4 ; λ5 ; λ6 ) = (1;
1 1 1 1
1
1
; ;
;
;,
;,
);
2 4 4 8 16 16
but it does not have a complete set of eigenvectors. These degeneracies are the cause of much technical
difficulty in the theory of subdivision. To keep our exposition simple and communicate the essential
ideas we will ignore these cases and assume from now on that we have a complete set of eigenvectors.
In this setting we can write any vector p of length n as a linear combination of eigenvectors:
p=
n,1
∑ ai xi
;
i=0
where the ai are given by the inner products ai = x̃i p. This decomposition works also when the entries
of p are n 2-D points (or 3-D points in the case of surfaces) rather than single numbers. In this case each
“coefficient” ai is a 2-D (3-D) point. The eigenvectors x0 ; : : : ; xn,1 are simply vectors of n real numbers.
39
In the basis of eigenvectors we can easily compute the result of application of the subdivision matrix
to a vector of control points, that is, the control points on the next level
Sp0
=
=
=
n,1
S ∑ ai xi
i=0
n,1
∑ ai Sxi
by linearity of S
i=0
n,1
∑ ai λixi
i=0
Applying S j times, we obtain
n,1
∑ ai λij xi
p j = S j p0 =
:
i=0
2.4.3 Convergence of Subdivision
If λ0 > 1, then S j x0 would grow without bound as j increased and subdivision would not be convergent.
Hence, we can see that in order for the sequence S j p0 to converge at all, it is necessary that all eigenvalues
are at most 1. It is also possible to show that only a single eigenvalue may have magnitude 1 [25].
A simple consequence of this analysis is that we can compute the limit position directly in the eigenbasis
P∞ (0) = lim S j p0 = lim
j!∞
j!∞
n,1
∑ ai λij xi = a0
;
i=0
since all eigen components jλi j < 1 decay to zero. For example, in the case of cubic B-spline subdivision
j
we can compute the limit position of pi as a0 = x̃0 p j , which amounts to
p∞
i = a0 =
1 j
j
j
(p
+ 4pi + pi+1 ):
6 i,1
Note that this expression is completely independent of the level j at which it is computed.
2.4.4 Invariance under Affine Transformations
If we moved all the control points simultaneously by the same amount, we would expect the curve defined
by these control points to move in the same way as a rigid object. In other words, the curve should be
invariant under distance-preserving transformations, such as translation and rotation. It follows from
40
linearity of subdivision that if subdivision is invariant with respect to distance-preserving transformations, it also should be invariant under any affine transformations. The family of affine transformations
in addition to distance-preserving transformations, contains shears.
a
Figure 2.8: Invariance under translation.
Let 1 be an n-vector of 1’s and a 2 R2 a displacement in the plane (see Figure 2.8) Then 1 a represents
a displacement of our seven points by a vector a. Applying subdivision to the transformed points, we get
S(p j + 1 a)
=
=
Sp j + S(1 a) by linearity of S
p j+1 + S(1 a):
From this we see that for translational invariance we need
S(1 a) = 1 a
Therefore, 1 should be the eigenvector of S with eigenvalue λ0 = 1.
Recall that when proving convergence of subdivision we assumed that 1 is an eigenvector with eigenvalue 1. We now see that this assumption is satisfied by any reasonable subdivision scheme. It would be
rather unnatural if the shape of the curve changed as we translate control points.
41
2.4.5 Geometric Behavior of Repeated Subdivision
If we assume that λ0 is 1, and all other eigenvalues are less than 1, we can choose our coordinate system
in such a way that a0 is the origin in R2 . In that case we have
pj =
n,1
∑ ai λij xi
i=1
Dividing both sides by λ1 , we obtain
j
1
n,1
λi j
λ1
i=2
λ1
j
j p = a1 x1 + ∑ ai
xi :
If we assume that jλ2 j; : : : ; jλn,1 j < jλ1 j, the sum on the right approaches zero as j ! ∞. In other words
the term corresponding to λ1 will “dominate” the behavior of the vector of control points. In the limit,
we get a set of n points arranged along the vector a1 . Geometrically, this is a vector tangent to our curve
at the center point (see Figure 2.9).
Just as in the case of computing the limit point of cubic B-spline subdivision by computing a0 we can
j
compute the tangent vector at pi by computing a1 = x̃1 p j
ti∞ = a1 = pij+1 , pij,1 :
If there were two equal eigenvalues, say λ1 = λ2 , as j increases, the points in the limit configuration
will be linear combinations of two vectors a1 and a2 , and in general would not be on the same line. This
indicates that there will be no tangent vector at the central point. This leads us to the following condition,
that, under some additional assumptions, is necessary for the existence of a tangent
All eigenvalues of S except λ0 = 1 should be less than λ1 .
2.4.6 Size of the Invariant Neighborhood
We have argued above that the size of the invariant neighborhood for cubic splines is 5 (7 for the 4pt
scheme). This was motivated by the question of which basis functions overlap a finite sized, however
small, neighborhood of the origin. Yet, when we computed the limit position as well as the tangent
vector for the cubic spline subdivision we used left eigenvectors, whose non-zero entries did not extend
beyond the immediate neighbors of the vertex at the origin. This turns out to be a general observation.
While the larger invariant neighborhood is needed for analysis, we can actually get away with a smaller
neighborhood if we are only interested in computation of point positions and tangents at those points
42
successive levels of
subdivision
displaced relative
to each other
enlarged versions of
left hand side curves
"zooming in" on the
center vertex
zoom factor 1
zoom factor 2
zoom factor 4
zoom factor 8
Figure 2.9: Repeatedly applying the subdivision matrix to our set of n control points results in the control
points converging to a configuration aligned with the tangent vector. The various subdivision levels have
been offset vertically for clarity.
corresponding to one of the original vertices. The value of the subdivision curve at the center point only
depends on those basis functions which are non-zero at that point. In the case of cubic spline subdivision
there are only 3 basis functions with this property. Similarly the first derivatives at the origin of the basis
functions centered at -2 and +2 are zero as well. Hence the derivative only depends on the immediate
neighbors as well. This must be so since the subdivision scheme is C1 . The basis functions have zero
derivative at the edge of their support by C1 -continuity assumption, because outside of the support the
derivative is identically zero.
For curves this distinction does not make too much of a difference in terms of computations, but
in the case of surfaces life will be much easier if we can use a smaller invariant neighborhood for the
computation of limit positions and tangents. For example, for Loop’s scheme we will be able to use
a 1-ring (only immediate neighbors) rather than a 2-ring. For the Butterfly scheme we will find that a
2-ring, rather than a 3-ring is sufficient to compute tangents.
43
2.4.7 Summary
For our subdivision matrix S we desire the following characteristics
the eigenvectors should form a basis;
the first eigenvalue λ0 should be 1;
the second eigenvalue λ1 should be less than 1;
all other eigenvalues should be less than λ1 .
44
Chapter 3
Subdivision Surfaces
Denis Zorin, New York University
In this chapter we review the basic principles of subdivision surfaces. These principles can be applied
to a variety of subdivision schemes described in Chapter 4: Doo-Sabin, Catmull-Clark, Loop, Modified
Butterfly, Kobbelt, Midedge.
Some of these schemes were around for a while: the 1978 papers of Doo and Sabin and Catmull and
Clark were the first papers describing subdivision algorithms for surfaces. Other schemes are relatively
new. Remarkably, during the period from 1978 until 1995 little progress was made in the area. In
fact, until Reif’s work [23] on C1 -continuity of subdivision most basic questions about the behavior
of subdivision surfaces near extraordinary vertices were not answered. Since then there was a steady
stream of new theoretical and practical results: classical subdivision schemes were analyzed [24, 16],
new schemes were proposed [30, 10, 8, 17], and general theory was developed for C1 and Ck -continuity
of subdivision [23, 18, 26, 28]. Smoothness analysis was performed in some form for allmost all known
schemes, for all of them, definitive results were obtained during the last 2 years only.
One of the goals of this chapter is to provide an accessible introduction to the mathematics of subdivision surfaces (Sections 3.4 and 3.5). Building on the material of the first chapter, we concentrate on
the few general concepts that we believe to be of primary importance: subdivision surfaces as parametric
surfaces, C1 -continuity, eigenstructure of subdivision matrices, characteristic maps.
The developments of recent years have convinced us of the importance of understanding the mathematical foundations of subdivision. A Computer Graphics professional who wishes to use subdivision,
probably is not interested in the subtle points of a theoretical argument. However, understanding the
45
general concepts that are used to construct and analyze subdivision schemes allows one to choose the
most appropriate subdivision algorithm or customize one for a specific application.
3.1 Subdivision Surfaces: an Example
One of the simplest subdivision schemes is the Loop scheme, invented by Charles Loop [14]. We will
use this scheme as an example to introduce some basic features of subdivision for surfaces.
The Loop scheme is defined for triangular meshes. The general pattern of refinement, which we call
vertex insertion, is shown in Figure 3.1.
Figure 3.1: Refinement of a triangular mesh. New vertices are shown as black dots. Each edge of the
control mesh is split into two, and new vertices are reconnected to form 4 new triangles, replacing each
triangle of the mesh.
Like most (but not all) other subdivision schemes, this scheme is based on a spline basis function,
called the three-directional quartic box spline. Unlike more conventional splines, such as the bicubic
spline, the three-directional box spline is defined on the regular triangular grid; the generating polynomial for this spline is
1
2
2
2
(1 + z1 ) (1 + z2 ) (1 + z1 z2 ) :
S(z1 ; z2 ) =
16
Note that the generating polynomial for surfaces has two variables, while the generating polynomials for
curves descibed in Chapter 2, had only one. This spline basis function is C2 -continuous. Subdivision
rules for it are shown in Figure 3.2.
46
1
8
3
8
3
8
1
8
1
16
1
16
10
16
1
16
1
16
1
16
1
16
Figure 3.2: Subdivision coefficients for a three directional box spline.
In one dimension, once a spline basis is chosen, all the coefficients of the subdivision rules that are
needed to generate a curve are completely determined. The situation is radically different and more
complex for surfaces. The structure of the control polygon for curves is always very simple: the vertices
are arranged into a chain, and any two pieces of the chain of the same length always have identical
structure. For two-dimensional meshes, the local structure of the mesh may vary: the number of edges
connected to a vertex may be different from vertex to vertex. As a result the rules derived from the spline
basis function may be applied only to parts of the mesh that are locally regular; that is, only to those
vertices that have a valence of 6 (in the case of triangular schemes). In other cases, we have to design
new rules for vertices with different valences. Such vertices are called extraordinary.
For the time being, we consider only meshes without a boundary. Note that the quartic box spline
rule used to compute the control point inserted at an edge (Figure 3.2,left) can be applied anywhere. The
only rule that needs modification is the rule used to compute new positions of control points inherited
from the previous level.
Loop proposed to use coefficients shown in Figure 3.3. It turns out that this choice of coefficients
guarantees that the limit surface of the scheme is “smooth.”
Note that these new rules only influence local behavior of the surface near extraordinary vertices. All
vertices inserted in the course of subdivision are always regular, i.e., have valence 6.
This example demonstrates the main challenge in the design of subdivision schemes for surfaces:
one has to define additional rules for irregular parts of the mesh in such a way that the limit surfaces
47
Figure 3.3: Loop scheme: coefficients for extraordinary vertices. The choice of β is not unique;
2
Loop [14] suggests 1k (5=8 , ( 38 + 14 cos 2π
k ) ).
have desired properties, in particular, are smooth. In this chapter one of our main goals is to describe
the conditions that guarantee that a subdivision scheme produces smooth surfaces. We start with defining subdivision surfaces more rigorously (Section 3.2), and defining subdivision matrices (Section 3.3).
Subdivision matrices have many applications, including computing limit positions of the points on the
surface, normals, and explicit evaluation of the surface (Chapter 4). Next, we define more precisely what
a smooth surface is (Section 3.4), introducing two concepts of geometric smoothness—tangent plane
continuity and C1 -continuity. Then we explain how it is possible to understand local behavior of subdivision near extraordinary vertices using characteristic maps (Section 3.5). In Chapter 4 we discuss a
variety of subdivision rules in a systematic way.
3.2 Natural Parameterization of Subdivision Surfaces
The subdivision process produces a sequence of polyhedra with increasing numbers of faces and vertices.
Intuitively, the subdivision surface is the limit of this sequence. The problem is that we have to define
what we mean by the limit more precisely. For this, and many other purposes, it is convenient to represent
subdivision surfaces as functions defined on some parametric domain with values in R3 . In the regular
case, the plane or a part of the plane is the domain. However, for arbitrary control meshes, it might be
impossible to parameterize the surface continuously over a planar domain.
48
Fortunately, there is a simple construction that allows one to use the initial control mesh, or more
precisely, the corresponding polygonal complex, as the domain for the surface.
Parameterization over the initial control mesh. We start with the simplest case: suppose the initial
control mesh is a simple polyhedron, i.e., it does not have self-intersections.
Suppose each time we apply the subdivision rules to compute the finer control mesh, we also apply
midpoint subdivision to a copy of the initial control polyhedron (see Figure 3.4). This means that we
leave the old vertices where they are, and insert new vertices splitting each edge in two. Note that
each control point that we insert in the mesh using subdivision corresponds to a point in the midpointsubdivided polyhedron. Another important fact is that midpoint subdivision does not alter the control
polyhedron regarded as a set of points; and no new vertices inserted by midpoint subdivision can possibly
coincide.
Figure 3.4: Natural parameterization of the subdivision surface
We will use the second copy of the control polyhedron as our domain. We denote it as K, when it is
49
regarded as a polyhedron with identified vertices, edges and faces, and jK j when it is regarded simply as
a subset of R3 .
Important remark on notation: we will refer to the points computed by subdivision as control
points; the word vertex is reserved for the vertices of the polyhedron that serves as the domain and
new vertices added to it by midpoint subdivision. We will use the letter v to denote vertices, and p j (v) to
denote the control point corresponding to v after j subdivision steps.
As we repeatedly subdivide, we get a mapping from a denser and denser subset of the domain to the
control points of a finer and finer control mesh. At each step, we linearly interpolate between control
vertices, and regard the mesh generated by subdivision as a piecewise linear function on the domain K.
Now we have the same situation that we had for curves: a sequence of piecewise linear functions defined
on a common domain. If this sequence of functions converges uniformly, the limit is a map f from jK j
into R3 . This is the limit surface of subdivision.
An important fact about the parameterization that we have just constructed is that for a regular mesh
the domain can be taken to be the plane with a regular triangular grid. If in the regular case the subdivision
scheme reduces to spline subdivision, our parameterization is precisely the standard (u; v) parameterization of the spline, which is guaranteed to be smooth.
To understand the general idea, this definition is sufficient, and a reader not interested in the subtle details can proceed to the next section and assume from now on that the initial mesh has no selfintersections.
General case. The crucial fact that we needed to parameterize the surface over its control polyhedron
was the absence of self-intersections. Otherwise, it could happen that a vertex on the control polyhedron
has more than one control point associated with it.
In general, we cannot rely on this assumption: quite often control meshes have self-intersections or
coinciding control points. We can observe though that the positions of vertices of the control polyhedron
are of no importance for our purposes: we can deform it in any way we want. In many cases, this
is sufficient to eliminate the problem with self intersections; however, there are cases when the selfintersection cannot be removed by any deformation (example: Klein bottle, Figure 3.5). It is always
possible to do that if we place our mesh in a higher-dimensional space; in fact, 4 dimensions are always
enough.
This leads us to the following general choice of the domain: a polyhedron with no self-intersections,
possibly in four-dimensional space. The polyhedron has to have the same structure as the initial control
50
Figure 3.5: The surface (Klein bottle) has an intersection that cannot be removed in 3D.
mesh of the surface, that is, there is a one-to-one correspondence between vertices, edges and faces of
the domain and the initial control mesh. Note that now we are completely free to chose the control points
of the initial mesh any way we like.
3.3 Subdivision Matrix
An important tool both for understanding and using subdivision is the subdivision matrix, similar to
the subdivision matrix for the curves introduced in Chapter 2. In this section we define the subdivision
matrix and discuss how it can be used to compute tangent vectors and limit positions of points. Another
application of subdivision matrices is explicit evaluation of subdivision surfaces described in Chapter 4.
Subdivision matrix. Similarly to the one-dimensional case, the subdivision matrix relates the control points in a fixed neighborhood of a vertex on two sequential subdivision levels. Unlike the onedimensional case, there is not a single subdivision matrix for a given surface subdivision scheme: a
separate matrix is defined for each valence.
For the Loop scheme control points for only two rings of vertices around an extraordinary vertex B
define f (U ) completely. We will call the set of vertices in these two rings the control set of U.
j
Let p0 be the value at level j of the control point corresponding to B. Assign numbers to the vertices
in the two rings (there are 3k vertices). Note that U j and U j+1 are similar: one can establish a one-to-one
correspondence between the vertices simply by shrinking U j by a factor of 2. Enumerate the vertices
51
5
2
7
5
2
1
0
8
4
9
1
0
8
3
3
7
4
9
6
6
Figure 3.6: The Loop subdivision scheme near a vertex of degree 3. Note that 3 3 + 1 = 10 points in
two rings are required.
in the rings; there are 3k vertices, plus the vertex in the center. Let pij , i = 1 : : : 3k be the corresponding
control points.
By definition of the control set, we can compute all values pij+1 from the values pij . Because we only
consider subdivision which computes finer levels by linear combination of points from the coarser level,
the relation between the vectors of points p j+1 and p j is given by a (3k + 1) (3k + 1) matrix:
0
BB
@
j+1
p0
..
.
j+1
p3k
1
C
C
A
0
B
SB
@
=
j
p0
..
.
j
p3k
1
C
C
A
:
It is important to remember that each component of p j is a point in the three-dimensional space. The
matrix S is the subdivision matrix, which, in general, can change from level to level. We consider only
schemes for which it is fixed. Such schemes are called stationary.
We can now rewrite each of the coordinate vectors in terms of the eigenvectors of the matrix S (compare to the use of eigen vectors in the 1D setting). Thus,
p0 = ∑ ai xi
i
and
p j = (S) j p0 = ∑(λi ) j ai xi
i
52
where the xi are the eigenvectors of S, and the λi are the corresponding eigenvalues, arranged in nonincreasing order. As discussed for the one-dimensional case, λ0 has to be 1 for all subdivision schemes, in
order to guarantee invariance with respect to translations and rotations. Furthermore, all stable, converging subdivision schemes will have all the remaining λi less than 1.
Subdominant eigenvalues and eigenvectors It is clear that as we subdivide, the behavior of p j , which
determines the behavior of the surface in the immediate vicinity of our point of interest, will depend only
on the eigenvectors corresponding to the largest eigenvalues of S.
To proceed with the derivation, we will assume for simplicity that λ = λ1 = λ2 > λ3 . We will call
λ1 and λ2 subdominant eigenvalues. Furthermore, we let a0 = 0; this corresponds to choosing the origin
of our coordinate system in the limit position of the vertex of interest (just as we did in the 1D setting).
Then we can write
pj
(λ) j
= a1 x1 + a2 x2 + a3
λ j
3
λ
x3 : : :
(3.1)
where the higher-order terms disappear in the limit.
This formula is very important, and deserves careful consideration. Recall that p j is a vector of 3k + 1
3D points, while xi are vectors of 3k + 1 numbers. Hence the coefficients ai in the decomposition above
have to be 3D points.
This means that, up to a scaling by (λ) j , the control set for f (U ) approaches a fixed configuration.
This configuration is determined by x1 and x2 , which depend only on the subdivision scheme, and on a1
and a2 which depend on the initial control mesh.
Each vertex in p j for sufficiently large j is a linear combination of a1 and a2 , up to a vanishing term.
This indicates that a1 and a2 span the tangent plane. Also note that if we apply an affine transform A,
taking a1 and a2 to coordinate vectors e1 and e2 in the plane, then, up to a vanishing term, the scaled
configuration will be independent of the initial control mesh. The transformed configuration consists of
2D points with coordinates (x1 i ; x2 i ), i = 0 : : : 3k, which depend on the subdivision matrix.
Informally, this indicates that up to a vanishing term, all subdivision surfaces generated by a scheme
differ near an extraordinary point only by an affine transform. In fact, this is not quite true: it may happen
that a particular configuration (x1 i ; x2 i ), i = 0 : : : 3k does not generate a surface patch, but, say, a curve.
In that case, the vanishing terms will have influence on the smoothness of the surface.
;
;
53
Tangents and limit positions. We have observed that similar to the one-dimensional case, the coefficients a0 a1 and a2 in the decomposition 3.1 are the limit position of the control point for the central
vertex v0 , and two tangents respectively. To compute these coefficients, we need corresponding left
eigenvectors:
a0 = (l0 ; p);
a1 = (l1 ; p);
a0 = (l2 ; p)
Similarly to the one-dimensional case, the left eigenvectors can be computed using only a smaller
submatrix of the full subdivision matrix. For example, for the Loop scheme we need to consider the
k + 1 k + 1 matrix acting on the control points of 1-neighborhood of the central vertex, not on the
points of the 2-neighborhood.
In the descriptions of subdivision schemes in the next section we describe these left eigenvectors
whenever information is available.
3.4 Smoothness of Surfaces
Intuitively, we call a surface smooth, if, at a close distance, it becomes indistinguishable from a plane.
Before discussing smoothness of subdivision surfaces in greater detail, we have to define more precisely
what we mean by a surface, in a way that is convenient for analysis of subdivision.
The discussion in the section is somewhat informal; for a more rigorous treatment, see [23, 22, 26],
3.4.1 C1 -continuity and Tangent Plane Continuity
Recall that we have defined the subdivision surface as a function f : jK j ! R3 on a polyhedron. Now
we can formalize our intuitive notion of smoothness, namely local similarity to a piece of the plane. A
surface is smooth at a point x of its domain jK j, if for a sufficiently small neighborhood Ux of that point
the image f (Ux ) can be smoothly deformed into a planar disk. More precisely,
Definition 1 A surface f : jK j ! R3 is C1 -continuous, if for every point x 2 jK j there exists a regular
parameterization π : D ! f (Ux ) of f (Ux ) over a unit disk D in the plane, where Ux is the neighborhood
in jK j of x. A regular parameterization π is one that is continuously differentiable, one-to-one, and has
a Jacobi matrix of maximum rank.
The condition that the Jacobi matrix of p has maximum rank is necessary to make sure that we have no
degeneracies, i.e., that we really do have a surface, not a curve or point. If p = ( p1 ; p2 ; p3 ) and the disc
54
is parameterized by x1 and x2 , the condition is that the matrix
0
B
B
B
@
∂p1
∂x1
∂p1
∂x2
∂p2
∂x1
∂p2
∂x2
∂p3
∂x1
∂p3
∂x2
1
C
C
C
A
have maximal rank (2).
There is another, weaker, definition of smoothness, which is often useful. This definition captures the
intuitive idea that the tangent plane to a surface changes continuously near a smooth point. Recall that a
tangent plane is unquely characterized by its normal. This leads us to the following definition:
Definition 2 A surface f : jK j ! R3 is tangent plane continuous at x 2 jK j if and only if surface normals
are defined in a neighborhood around x and there exists a limit of normals at x.
This is a useful definition, since it is easier to prove surfaces are tangent plane continuous. Tangent
plane continuity, however, is weaker than C1 -continuity.
As a simple example of a surface that is tangent plane continuous but not C1 -continuous, consider the
shape in Figure 3.7. Points in the vicinity of the central point are “wrapped around twice.” There exists a
tangent plane at that point, but the surface does not “locally look like a plane.” Formally speaking, there
is no regular parameterization of the neighborhood of the central point, even though it has a well-defined
tangent plane.
From the previous example, we see how the definition of tangent plane continuity must be strengthened to become C1 :
Lemma 4 If a surface is tangent plane continuous at a point and the projection of the surface onto the
tangent plane at that point is one-to-one, the surface is C1 .
The proof can be found in [26].
3.5 Analysis of Subdivision Surfaces
In this section we discuss how to determine if a subdivision scheme produces smooth surfaces. Typically,
it is known in advance that a scheme produces C1 -continuous (or better) surfaces in the regular setting.
For local schemes this means that the surfaces generated on arbitrary meshes are C1 -continuous away
from the extraordinary vertices. We start with a brief discussion of this fact, and then concentrate on
55
Figure 3.7: Example of a surface that is tangent plane continuous but not C1 -continous.
analysis of the behavior of the schemes near extraordinary vertices. Our goal is to formulate and provide
some motivation for Reif’s sufficient condition for C1 -continuity of subdivision.
We assume a subdivision scheme defined on a triangular mesh, with certain restrictions on the structure of the subdivision matrix, defined in Section 3.5.2. Similar derivations can be performed without
56
these assumptions, but they become significantly more complicated. We consider the simplest case so as
not to obscure the main ideas of the analysis.
3.5.1 C1 -continuity of Subdivision away from Extraordinary Vertices
Most subdivision schemes are constructed from regular schemes, which are known to produce at least
C1 -continuous surfaces in the regular setting for almost any initial configuration of control points. If our
subdivision rules are local, we can take advantage of this knowledge to show that the surfaces generated
by the scheme are C1 -continuous for almost any choice of control points anywhere away from extraordinary vertices. We call a subdivision scheme local, if only a finite number of control points is used to
compute any new control point, and does not exceed a fixed number for all subdivision levels and all
control points.
One can demonstrate, as we did for the curves, that for any triangle T of the domain the surface f (T )
is completely determined by only a finite number of control points corresponding to vertices around
T . For example, for the Loop scheme, we need only control points for vertices that are adjacent to the
triangle. (see Figure 3.8). This is true for triangles at any subdivision level.
Figure 3.8: Control set for a triangle for the three-directional box spline.
To show this, fix a point x of the domain jK j (not necessarily a vertex). For any level j, x is contained
in a face of the domain; if x is a vertex, it is shared by several faces. Let U j (x) be the collection of faces
on level j containing x, the 1-neighborhood of x. The 1-neighborhood of a vertex can be identified with a
k-gon in the plane, where k is the valence. We need j to be large enough so that all neighbors of triangles
in U j (x) are free of extraordinary vertices. Unless x is an extraordinary vertex, this is easily achieved.
f (U j (x)) will be regular (see Figure 3.9).
This means that f (U j (x)) is identical to a part of the surface corresponding to a regular mesh, and
is therefore C1 -continuous for almost any choice of control points, because we have assumed that our
57
C
A
B
Figure 3.9: 2-neighborhoods (1-neighborhood of 1-neighborhood) of vertices A, C contain only regular
vertices; this is not the case for B, which is an extraordinary vertex.
scheme generates C1 -continuous surfaces over regular meshes.1
3.5.2 Smoothness Near Extraordinary Vertices
Now that we know that surfaces generated by our scheme are (at least) C1 -continuous away from the
extraordinary vertices, all we have to do is find a a smooth parameterization near each extraordinary
vertex, or establish that no such parameterization exists.
Consider the extraordinary vertex B in Figure 3.9. After sufficient number of subdivision steps, we
will get a 1-neighborhood U j of B, such that all control points defining f (U j ) are regular, except B itself.
This demonstrates that it is sufficient to determine if the scheme generates C1 -continuous surfaces for
a very specific type of domains K: triangulations of the plane which have a single extraordinary vertex
in their center, surrounded by regular vertices. We can assume all triangles of these triangulations to be
identical (see Figure 3.10) and call such triangulations k-regular.
At first, the task still seems to be very difficult: for any configuration of control vertices, we have to
find a parameterization of f (U j ). However, it turns out that the problem can be further simplified.
We outline the idea behind a sufficient condition for C1 -continuity proposed by Reif [23]. This criterion tells us when the scheme is guaranteed to produce C1 -continuous surfaces, but if it fails, it is still
possible that the scheme might be C1 -continuous.
In addition to the subdivision matrix described in Section 3.3 , we need one more tool to formulate
the criterion: the characteristic map. It turns out that rather than trying to consider all possible surfaces
generated by subdivision, it is typically sufficient to look at a single map—the characteristic map.
1 Our
argument is informal, and there are certain unusual cases when it fails; see [26] for details.
58
Figure 3.10: k-regular triangulation for k = 9.
3.5.3 Characteristic Map
Our observations made in Section 3.3 motivate the definition of the characteristic map. Recall that the
control points near a vertex converge to a limit configuration independent, up to an affine transformation,
from the control points of the original mesh. This limit configuration defines a map. Informally speaking,
any subdivision surface generated by a scheme looks near an extraordinary vertex of valence k like the
characteristic map of that scheme for valence k.
Figure 3.11: Control set of the characteristic map for k = 9.
Note that when we described subdivision as a function from the plane to R3 , we may use control
vertices not from R3 , but from R2 ; clearly, subdivision rules can be applied in the plane rather then in
59
space. Then in the limit we obtain a map from the plane into the plane. The characteristic map is a map
of this type.
As we have seen, the configuration of control points near an extraordinary vertex approaches a1 x1 +
a2 x2 , up to a scaling transformation. This means that the part of the surface defined on the k-gon U j
as j ! ∞, and scaled by the factor 1=λ j , approaches the surface defined by the vector of control points
a1 x1 + a2 x2 . Let f [p] : U ! R3 be the limit surface generated by subdivision on U from the control set
p.
Definition 3 The characteristic map of a subdivision scheme for a valence k is the map Φ : U ! R2
generated by the vector of 2D control points e1 x1 + e2 x2 : Φ = f [e1 x1 + e2 x2 ], where e1 and e2 are unit
coordinate vectors, and x1 and x2 are subdominant eigenvectors.
Regularity of the characteristic map Inside each triangle of the k-gon U, the map is C1 : the argument of Section 3.5.1 can be used to show this. Moreover, the map has one-sided derivatives on the
boundaries of the triangles, except at the extraordinary vertex, so we can define one-sided Jacobians on
the boundaries of triangles too. We will say that the characteristic map is regular if its Jacobian is not
zero anywhere on U excluding the extraordinary vertex but including the boundaries between triangles.
The regularity of the characteristic map has a geometric meaning: any subdivision surface can be
written, up to a scale factor λ j , as
,
f [p j ](t ) = AΦ(t ) + a(t )O (λ3 =λ) j
;
t 2 U j , a(t ) a bounded function U j ! R3 , and A is a linear transform taking the unit coordinate vectors
in the plane to a1 and a2 . Differentiating along the two coordinate directions t1 and t2 in the parametric
domain U j , and taking a cross product, after some calculations, we get the expression for the normal to
the surface:
,
n(t ) = (a1 a2 )J [Φ(t )] + O (λ3 =λ)2 j ã(t )
where J [Φ] is the Jacobian, and ã(t ) some bounded vector function on U j .
The fact that the Jacobian does not vanish for Φ means that the normal is guaranteed to converge to
a1 a2 ; therefore, the surface is tangent plane continuous.
Now we need to take only one more step. If, in addition to regularity, we assume that Φ is injective,
we can invert it and parameterize any surface as f (Φ,1 (s)), where s 2 Φ(U ). Intuitively, it is clear that
up to a vanishing term this map is just an affine map, and is differentiable. We omit a rigorous proof
here. For a complete treatment see [23]; for more recent developments, see [26] and [28].
60
We arrive at the following condition, which is the basis of smoothness analysis of all subdivision
schemes considered in these notes.
Reif’s sufficient condition for smoothness. Suppose the eigenvectors of a subdivision matrix form a
basis, the largest three eigenvalues are real and satisfy
λ0 = 1 > λ1 = λ2 > jλ3 j
If the characteristic map is regular, then almost all surfaces generated by subdivision are tangent
plane continuous; if the characteristic map is also injective, then almost all surfaces generated by
subdivision are C1 -continuous.
Note: Reif’s original condition is somewhat different, because he defines the characteristic map on an
annular region, rather than on a k-gon. This is necessary for applications, but makes it somewhat more
difficult to understand.
In Chapter 4, we will discuss the most popular stationary subdivision schemes, all of which have
been proved to be C1 -continuous at extraordinary vertices. These proofs are far from trivial: checking
the conditions of Reif’s criterion is quite difficult, especially checking for injectivity. In most cases
calculations are done in symbolic form and use closed-form expressions for the limit surfaces of subdivision [24, 8, 16, 17]. In [27] an interval-based approach is described, which does not rely on closed-form
expressions for limit surfaces, and can be applied, for example, to interpolating schemes.
3.6 Piecewise-smooth surfaces and subdivision
Piecewise smooth surfaces. So far, we have assumed that we consider only closed smooth surfaces.
However, in reality we typically need to model more general classes of surfaces: surfaces with boundaries, which may have corners, creases, cusps and other features. One of the significant advantages of
subdivision is that it is possible to introduce features into surfaces using simple modifications of rules.
Here we briefly describe a class of surfaces (piecewise smooth surfaces) which appears to be adequate
for many applications. This is the class of surfaces that includes, for example, quadrilateral free-form
patches, and other common modeling primitives. At the same time, we have excluded from consideration surfaces with various other types of singularities. To generate surfaces from this class, in addition to
vertex and edge rules such as the Loop rules (Section 3.1), we need to define several other types of rules.
To define piecewise smooth surfaces, we start with smooth surfaces that have a piecewise-smooth
boundary. For simplicity, assume that our surfaces do not have self-intersections. Recall that for closed
61
D
H
Q1
Q3
Q0
Figure 3.12: The charts for a surface with piecewise smooth boundary.
C1 -continuous surface M in R3 each point has a neighborhood that can be smoothly deformed into an
open planar disk D.
A surface with a smooth boundary is defined in a similar way, but the neighborhoods of points on the
boundary can be smoothly deformed into a half-disk H, with closed boundary. To define a surface with
piecewise smooth boundaries, we introduce two additional types of local charts: concave and convex
corner charts, Q3 and Q1 (Figure 3.12). Thus, a C1 -continuous surface with piecewise smooth boundary
locally looks like one of the domains D, H, Q1 and Q3 .
Piecewise-smooth surfaces are the surfaces that can be constructed out of surfaces with piecewise
smooth boundaries joined together.
If the resulting surface is not C1 -continuous at the common boundary of two pieces, this common
boundary is a crease. We allow two adjacent smooth segments of a boundary to be joined, producing a
crease ending in a dart (cf. [9]). For dart vertices an additional chart Q0 is required; the surface near a
dart can be deformed into this chart smoothly everywhere except at an open edge starting at the center of
the disk.
Subdivision schemes for piecewise smooth surfaces. An important observation for constructing subdivision rules for the boundary is that the last two corner types are not equivalent, that is, there is no
smooth nondegenerate map from Q1 to Q3 . It follows from the theory of subdivision [26], that a single
subdivision rule cannot produce both types of corners. In general, any complete set of subdivision rules
should contain separate rules for all chart types. Most, if not all, known schemes provide rules for charts
of type D and H (smooth boundary and interior vertices); rules for charts of type Q1 and Q0 (convex
corners and darts) are typically easy to construct; however, Q3 (concave corner) is more of a challenge,
and no rules were known until recently.
In Chapter 4 we present descriptions of various rules for smooth (not piecewise smooth) surfaces with
boundary. For extensions of the Loop and Catmull-Clark schemes including concave corner rules, see
62
[2].
Interpolating boundaries. Quite often our goal is not just to generate a smooth surface of a given
topological type approximating or interpolating an initial mesh with boundary, but to interpolate a given
set of boundary or even an arbitrary set of curves. In this case, one can use a technique developed
by A. Levin [11, 12, 13]. The advantage of this approach is that the interpolated curves need not
be generated by subdivision; one can easily create blend subdivision surfaces with different types of
parametric surfaces (for a example, NURBS).
63
64
Chapter 4
Subdivision Zoo
Denis Zorin, New York University
4.1 Overview of Subdivision Schemes
In this section we describe most known stationary subdivision schemes generating C1 -continuous surfaces on arbitrary meshes. Without doubt, our discussion is not exhaustive even as far as stationary
schemes are concerned. There are even wholly different classes of subdivision schemes, most importantly variational schemes, that we do not discuss here. Different approaches to variational subdivision
are described in the parts of the notes written by Joe Warren and Leif Kobbelt.
At a first glance, the variety of existing schemes might appear chaotic. However, there is a straightforward way to classify most of the schemes based on three criteria:
the type of refinement rule (vertex insertion or corner-cutting);
the type of generated mesh (triangular or quadrilateral);
whether the scheme is approximating or interpolating.
The following table shows this classification:
Vertex insertion
Corner-cutting
Triangular meshes
Quadrilateral meshes
Approximating
Loop
Catmull-Clark
Interpolating
Modified Butterfly
Kobbelt
65
Doo-Sabin
Midedge
It can be seen from this table that there is little replication in functionality: most schemes produce
substantially different types of surfaces. Now we consider our classification criteria in greater detail.
First, we note that each subdivision scheme defined on meshes of arbitrary topology is based on a
regular subdivision scheme such as a subdivision schemes for splines, for example. Our classification is
primarily a classification of regular subdivision schemes—once such a scheme is fixed, additional rules
have to be specified only for extraordinary vertices or faces that cannot be part of a regular mesh.
Mesh type. Regular subdivision schemes act on regular control meshes, that is, vertices of the mesh
correspond to regularly spaced points in the plane. However, the faces of the mesh can be formed in
different ways. For a regular mesh, it is natural to use faces that are identical. If, in addition, we assume
that the faces are regular polygons, it turns out that there are only three ways to choose the face polygons:
we can use only squares, equilateral triangles and regular hexagons. Meshes consisting of hexagons are
not very common, and the first two types of tiling are the most convenient for practical purposes. These
leads to two types of regular subdivision schemes: those defined for quadrilateral tilings, and those
defined for triangular tilings.
Vertex insertion and corner-cutting. Once the tiling of the plane is fixed, we have to define how a
refined tiling generated by the scheme is related to the original tiling. There are two main approaches
that are used to generate a refined tiling: one is vertex insertion and the other is corner cutting (see
Figure 4.1). The schemes using the first method are often called primal, and the schemes using the
second method are called dual. In the first case, each edge of a triangular or a quadrilateral mesh is split
into two, old vertices of the mesh are retained, and new vertices inserted on edges are connected. For
quadrilaterals, an additional vertex is inserted for each face.
In the second case, for each old face, a new similar face is created inside of it and the newly created
faces are connected. As a result, we get two new vertices for each old edge, a new face for each edge
and each vertex. The old vertices are discarded. Geometrically, one can think about this process as first
cutting off the vertices, and then cutting off the edges of a polyhedron. For quadrilateral tilings, this can
be done in such a way that the refined tiling has only quadrilateral faces. For triangles, we can get only a
hexagonal tiling. Thus, a regular corner-cutting algorithm for triangles would have to alternate between
triangular and hexagonal tilings.
Approximation vs. Interpolation. Vertex insertion schemes can be interpolating or approximating: as
the vertices of the coarser tiling are also vertices of the refined tiling, for each vertex a sequence of control
66
Vertex insertion for quadrilaterals
Vertex insertion for triangles
Corner-cutting for quadrilaterals
Figure 4.1: Different refinement rules.
points, corresponding to different subdivision levels, is defined. If all points in the sequence are the same,
we say that the scheme is interpolating. Otherwise, we call it approximating. Interpolation is an attractive
feature in more than one way. First, the original control points defining the surface are also points of the
limit surface, which allows one to control it in a more intuitive manner. Second, many algorithms can be
considerably simplified, and many calculations can be performed “in place.” Unfortunately, the quality
of these surfaces is not as high as the quality of surfaces produced by approximating schemes, and the
schemes do not converge as fast to the limit surface as the approximating schemes.
We concentrate primarily on vertex insertion schemes; we briefly discuss two corner-cutting schemes,
67
Doo-Sabin and the Midedge subdivision scheme, proposed by Habib and Warren [7], and independently
discovered by Peters and Reif [17].
4.1.1 Notation and Terminology
Here we summarize the notation that we use in subsequent sections. Some of it was already introduced
earlier.
Regular and extraordinary vertices. We have already seen that subdivision schemes defined on triangular meshes create new vertices only of valence 6 in the interior. On the boundary, the newly
created vertices have valence 4. Similarly, on quadrilateral meshes both vertex-insertion and
corner-cutting schemes create only vertices of valence 4 in the interior, and 3 on the boundary.
Hence, after several subdivision steps, most vertices in a mesh will have one of these valences (6
in the interior, 4 on the boundary for triangular meshes, 4 in the interior, 3 on the boundary for
quadrilateral). The vertices with these valences are called regular, and vertices of other valences
extraordinary.
Notation for vertices near a fixed vertex. In Figure 4.2 we show the notation that we use for vertices
of quadrilateral and triangular subdivision schemes near a fixed vertex. Typically, we need it for
extraordinary vertices; we also use it for regular vertices, to describe calculations of limit positions
and tangent vectors. Note that this notation is for a fixed level; the names of vertices changes from
j
j
one level to the next. For brevity, we denote the value p j (vi l ) by pi l .
;
;
Odd and even vertices. For vertex insertion (primal) schemes, the vertices of the coarser mesh are also
vertices of the refined mesh. For any subdivision level, we call all new vertices that are created
at that level, odd vertices. This term comes from the one-dimensional case, when vertices of the
control polygons can be enumerated sequentially and on any level the newly inserted vertices are
assigned odd numbers. The vertices inherited from the previous level are called even. (See also
Chapter 2).
Face and edge vertices. For triangular schemes (Loop and Modified Butterfly), there is only one type
of odd vertex. For quadrilateral schemes, some vertices are inserted when edges of the coarser
mesh are split, other vertices are inserted for a face. These two types of odd vertices are called
edge and face vertices respectively.
Boundaries and creases. Typically, special rules have to be specified on the boundary of a mesh. These
rules are commonly chosen in such a way that the boundary curve of the limit surface does not
68
j
p i+1,5
j
p i+2,3
j
p i+1,6
j
p i+1,6
j
p i+1,5
j
p i+1,4
j
p i+2,3
j
p i+2,1
j
p i+1,3
j
j
p i+1,1
j
p i,3
j
p i,1
p 0j
j
p i-1,1
j
j
p i+2,3
p i+1,2
j
p i+1,2 p i,6
j
p i,5
j
p i,2
j
j
p i,4
j
p i,6
j
p i,11
j
p i,5
j
p i,4
j
p i,3
j
p i,10
j
p i,9
j
p i,8
j
p i,7
j
p i-1,6
j
p i-1,2
j
p i-1,3
j
p i-1,5
j
p i,12
j
p i,2
j
p i,1
j
p i-1,1
p i-1,6
j
p i+1,3
j
p i+1,1
p 0j
j
p i-1,3
j
p i-1,2
j
p i+2,1
j
p i+1,7
j
p i+1,4
j
p i-1,4
j
p i-1,5
j
p i-1,4
Figure 4.2: Enumeration of vertices of a mesh near an extraordinary vertex; for a boundary vertex, the
0 , th sector is adjacent to the boundary.
depend on any interior control vertices, and is smooth or piecewise smooth (C1 or C2 -continuous).
The same rules can be used to introduce sharp features into C1 -surfaces: some interior edges can
be tagged as crease edges, and boundary rules are applied for all vertices that are inserted on such
edges.
Masks. We often specify a subdivision rule by providing its mask. The mask is a picture showing which
control points are used to compute a new control point, which we denote with a black dot. The
numbers are the coefficients of the subdivision rule. For example, if p1 , p2 are vertices of an
edge, and v3 and v4 are the other two vertices of the triangles that share this edge, then the Loop
subdivision rule for an interior odd vertex v depicted in Figure 4.3, can be written as
p j+1 (v) =
3 j
3
1
1
p (v1 ) + p j (v2 ) + p j (v3 ) + p j (v4 )
8
8
8
8
69
4.2 Loop Scheme
The Loop scheme is a simple approximating vertex insertion scheme for triangular meshes proposed by
Charles Loop [14]. C1 -continuity of this scheme for valences up to 100, including the boundary case,
was proved by Schweitzer [24]. The proof for all valences can be found in [26].
The scheme is based on the three-directional box spline, which produces C2 -continuous surfaces on
the regular meshes. The Loop scheme produces surfaces that are C2 -continuous everywhere except at
extraordinary vertices, where they are C1 -continuous. Hoppe, DeRose, Duchamp et al. [9] proposed a
piecewise C1 -continuous extension of the Loop scheme, with special rules defined for edges; in [2], the
boundary rules are further improved, and new rules for concave corners and normal modification are
proposed.
The scheme can be applied to arbitrary polygonal meshes, after the mesh is converted to a triangular
mesh, for example, by triangulating each polygonal face.
Subdivision rules The masks for the Loop scheme are shown in Figure 4.3. For boundaries and edges
tagged as crease edges, special rules are used. These rules produce a cubic spline curve along the
boundary/crease. The curve only depends on control points on the boundary/crease.
In [9], the rules for extraordinary crease vertices and their neighbors on the crease were modified to
produce tangent plane continuous surfaces on either side of the crease (or on one side of the boundary).
In practice, for reasons discussed in [29], this modification does not lead to a significant difference in
the appearance of the surface. At the same time, as a result of this modification, the boundary curve
becomes dependent on the valences of vertices on the curve. This is a disadvantage in situations when
two surfaces have to be joined together along a boundary. It appears that in practically all cases it is
safe to use the rules shown in Figure 4.3. Although the surface will not be formally C1 -continuous near
vertices of valence greater than 7, the result will be visually indistinguishable from a C1 -surface obtained
with modified rules, with the additional advantage of independence of the boundary from the interior.
If it is necessary to ensure C1 -continuity, we propose a different modification. Rather than modifying
the rules for the boundary curve, and making it dependent on the valence of vertices, we modify rules
for interior odd vertices adjacent to an extraordinary vertex. For n < 7, no modification is necessary.
For n > 7, it is sufficient to use the mask shown in Figure 4.4. Then the limit surface can be shown to
be C1 -continuous on the boundary. A better, although slightly more complex modification can be found
1
1
2π
in [2]: instead of 12 and 14 we can use 14 + 14 cos k2π
,1 and 2 , 4 cos k,1 respectively, where k is the valence
of the boundary vertex.
70
2
Figure 4.3: Loop subdivision: in the picture above, β can be chosen to be either 1n (5=8 , ( 38 + 14 cos 2π
n ) )
3
(original choice of Loop [14]), or, for n > 3, β = 8n
as proposed by Warren [25]. For n = 3, β = 3=16
can be used.
Tangent vectors. The rules for computing tangent vectors for the Loop scheme are especially simple.
To compute a pair of tangent vectors at an interior vertex, use
t1 =
k,1
∑ cos
i=0
k,1
2πi
p(vi 1 )
k
;
(4.1)
2πi
p(vi 1 ):
t2 = ∑ sin
k
i=0
;
These formulas can be applied to the control points at any subdivision level.
Quite often, the tangent vectors are used to compute a normal. The normal obtained as the cross
product t1 t2 can be interpreted geometrically. This cross product can be written as a weighted sum of
normals to all possible triangles with vertices p(v), p(vi 1 ), p(vl 1 ), i; l = 0 : : : k , 1, i 6= l. The standard
way of obtaining vertex normals for a mesh by averaging the normals of triangles adjacent to a vertex,
;
71
;
1
8
1
2
1
4
extraordinary
vertex
1
8
Figure 4.4: Modified rule for odd vertices adjacent to a boundary extraordinary vertex (Loop scheme).
can be regarded as a first approximation to the normals given by the formulas above. At the same time,
it is worth observing that computing normals as t1 t2 is less expensive than averaging the normals of
triangles. The geometric nature of the normals obtained in this way suggests that they can be used to
compute approximate normals for other schemes, even if the precise normals require more complicated
expressions.
At a boundary vertex, the tangent along the curve is computed using talong = p(v0 1 ) , p(vk,1 1 ). The
tangent across the boundary/crease is computed as follows [9]:
;
tacross = p(v0 1 ) + p(v1 1 ) , 2p(v0 )
;
;
;
for k = 2
tacross = p(v2 1 ) , p(v0 ) for k = 3
;
(4.2)
k,2
tacross = sin θ ( p(v0 1 ) + p(vk,1 1 )) + (2 cos θ , 2) ∑ sin iθ p(vi 1 ) for k 4
;
;
;
i=1
where θ = π=(k , 1). These formulas apply whenever the scheme is tangent plane continuous at the
boundary; it does not matter which method was used to ensure tangent plane continuity.
Limit positions Another set of simple formulas allows one to compute limit positions of control points
for a fixed vertex, that is, the limit lim j!∞ p j (v) for a fixed v. For interior vertices, the mask for computing
the limit value at an interior vertex is the same as the mask for computing the value on the next level,
with β replaced by χ = 3 8β1 +n .
=
72
For boundary vertices, the formula is always
p∞ (v0 ) =
1
3
1
p(v0 1 ) + p(v0 ) + p(v1 k,1 )
5
5
5
;
;
This expression is similar to the rule for even boundary vertices, but with different coefficients. However,
different formulas have to be used if the rules on the boundary are modified as in [9].
4.3 Modified Butterfly Scheme
The Butterfly scheme was proposed by Dyn, Gregory and Levin in [6]. However, although the original
Butterfly scheme is defined on arbitrary triangular meshes, the limit surface is not C1 -continuous at
extraordinary points of valence k = 3 and k > 7 [26]. It is C1 on regular meshes.
Unlike approximating schemes based on splines, this scheme does not produce piecewise polynomial
surfaces in the limit. In [30] a modification of the Butterfly scheme was proposed, which guarantees that
the scheme produces C1 -continuous surfaces for arbitrary meshes (for a proof see [26]). The scheme is
known to be C1 but not C2 on regular meshes. The masks for the the scheme are shown in Figure 4.5.
The tangent vectors at extraordinary interior vertices can be computed using the same rules as for
the Loop scheme. For regular vertices, the formulas are more complex: in this case, we have to use
control points in a 2-neighborhood of a vertex. If the control points are arranged into a vector p =
[ p0 ; p0 1 ; p1 1 ; : : : ; p5 1 ; p0 2 ; p1 2 ; p2 2 ; : : : p5 3 ] of length 19, then the tangents are given by scalar products
(l1 p) and (l2 p), where the vectors l1 and l2 are
;
;
;
;
;
;
;
1 1
1 1
l1 = 0; 16; 8; ,8; ,16; ,8; 8; ,4; 0; 4; 4; 0; ,4; 1; ; , ; ,1; , ;
2 2
2 2
p
4 8 4 4 8 4 1 1
1 1
l2 = 3 0; 0; 8; 8; 0; ,8; ,8; , ; , ; , ; ; ; ; 0; ; ; 0; , ; ,
3 3 3 3 3 3 2 2
2 2
(4.3)
Because the scheme is interpolating, no formulas are needed to compute the limit positions: all control
points are on the surface. On the boundary, the four point subdivision scheme is used [5]. To achieve
C1 -continuity on the boundary, special coefficients have to be used (see [29] for details).
4.4 Catmull-Clark Scheme
The Catmull-Clark scheme was described in [3]. It is based on the tensor product bicubic spline. The
masks are shown in Figure 4.6. The scheme produces surfaces that are C2 everywhere except at extraordinary vertices, where they are C1 . The tangent plane continuity of the scheme was analyzed by Ball and
73
1
8
1
16
-
1
16
s2
1
2
1
2
s1
s3
s0
-
1
16
-
1
8
1
16
Mask for interior odd vertices with
regular neighbors
sk-1
sk-2
-
1
16
9
16
9
16
-
1
16
Mask for crease
and boundary vertices
a. Masks for odd vertices
b. Mask for odd vertices adjacent to
an extraordinary vertex
Figure 4.5: Modified Butterfly subdivision. The coefficients si are 1k
5
1
, s1 2 = , 12
; for k = 4, s0 = 38 , s2 = , 18 , s1 3 = 0.
For k = 3, s0 = 12
;
, 1 + cos 2iπ + 1 cos 4iπ for k
4
k
2
k
>
5.
;
Storry [1], and C1 -continuity by Peters and Reif [16]. The values of α and β can be chosen from a wide
range (see Figure 4.8). On the boundary, using the coefficients for the cubic spline produces acceptable
results, however, the resulting surface formally is not C1 -continuous. A modification similar to the one
performed in the case of Loop subdivision makes the scheme C1 -continuous (Figure 4.7). Again, a bet5
3
1
2π
ter, although a bit more complicated choice of coefficients is 38 + 14 cos k2π
,1 instead of 8 and 8 , 4 cos k,1
instead of 18 . See [29] for further details about the behavior on the boundary.
The rules of Catmull-Clark scheme are defined for meshes with quadrilateral faces. Arbitrary polygonal meshes can be reduced to a quadrilateral mesh using a more general form of Catmull-Clark rules [3]:
74
Figure 4.6: Catmull-Clark subdivision. Catmull and Clark [3] suggest the following coefficients for rules
3
1
and γ = 4k
at extraordinary vertices: β = 2k
a face control point for an n-gon is computed as the average of the corners of the polygon;
an edge control point as the average of the endpoints of the edge and newly computed face control
points of adjacent faces;
the formula for even control points can be chosen in different ways; the original formula is
p j+1 (v) =
k,2 j
1 k,1
1 k,1
f
p (v) + 2 ∑ p j (vi ) + 2 ∑ p j+1 (vi )
k
k i=0
k i=0
75
Figure 4.7: Modified rule for odd vertices adjacent to a boundary extraordinary vertex (Catmull-Clark
scheme).
Figure 4.8: Ranges for coefficients α and β of the Catmull-Clark scheme; α = 1 , γ , β is the coefficient
of the central vertex.
f
where vi are the vertices adjacent to v on level j, and vi are face vertices on level j + 1 corresponding to faces adjacent to v.
4.5 Kobbelt Scheme
This interpolating scheme was described by Kobbelt in [10]. For regular meshes, it reduces to the tensor
product of four point schemes. C1 -continuity of this scheme for interior vertices for all valences is proven
76
in [27].
1
256
-
9
256
-
9
256
1
256
-
9
256
-
81
256
81
256
81
256
81
256
9
256
-
1
256
9
256
-
9
256
-
9
256
1
256
9
256
Mask for a face vertex
-
1
16
9
16
9
16
-
1
16
Mask for edge, crease
and boundary vertices
b. Computing a face vertex adjacent to an extraordinary
vertex
a. Regular masks
Figure 4.9: Kobbelt subdivision.
Crucial for the construction of this scheme is the observation (valid for any tensor-product scheme)
that the face control points can be computed in two steps: first, all edge control points are computed.
Next, face vertices are computed using the edge rule applied to a sequence of edge control points on the
same level. As shown in Figure 4.9, there are two ways to compute a face vertex in this way. In the
regular case, the result is the same. Assuming this method of computing all face control points, only one
rule of the regular scheme is modified: the edge odd control points adjacent to an extraordinary vertex
77
are computed differently. Specifically,
1
1
,
w) p0j + ( , w) pij 1 + wpij + wpij 3
2
2
k,1
4 k,1 j
w
4w
j
j
j
j
j
j
j
p
,
(
p
+
p
+
p
)
,
(
p
+
p
+
p
+
p
)
+
∑ pij 2
i1
i,1 1
i1
i+1 1
i,2 2
i,1 2
i2
i+1 2
k i∑
1
=2 , w
(
1
=2 , w)k
=0
i=0
1
pij+
1 =(
;
j
vi
=
;
;
;
;
;
;
;
;
;
;
;
(4.4)
where w = ,1=16 (also, see Figure 4.2 for notation). On the boundaries and creases, the four point
subdivision rule is used.
Unlike other schemes, eigenvectors of the subdivision matrix cannot be computed explicitly; hence,
there are no precise expressions for tangents. In any case, the effective support of this scheme is too large
for such formulas to be of practical use: typically, it is sufficient to subdivide several times and then use,
for example, the formulas for the Loop scheme (see discussion in the section on the Loop scheme).
For more details on this scheme, see the part of the notes written by Leif Kobbelt.
4.6 Doo-Sabin and Midedge Schemes
The Doo-Sabin subdivision is quite simple conceptually: there is no distinction between odd and even
vertices, and a single mask is sufficient to define the scheme. A special rule is required only for the
boundaries, where the limit curve is a quadratic spline. It was observed by Doo that this can also be
achieved by replicating the boundary edge, i.e., creating a quadrilateral with two coinciding pairs of
vertices. Nasri [15] describes other ways of defining rules for boundaries. The rules for the Doo-Sabin
scheme are shown in Figure 4.10. C1 -continuity for schemes similar to the Doo-Sabin schemes was
analyzed by Peters and Reif [16].
An even simpler scheme was proposed by Habib and Warren [8] and by Peters and Reif [17]: this
scheme uses even smaller stencils than the Doo-Sabin scheme; for regular vertices, only three control
points are used (Figure 4.11).
The disadvantage of all dual schemes is that the mesh hierarchy associated with this schemes is
somewhat less natural: the vertices of coarser meshes cannot be identified with vertices of finer meshes.
4.7 Limitations of Stationary Subdivision
Stationary subdivision, while overcoming certain problems inherent in spline representations, still has
a number of limitations. Most problems are much more apparent for interpolating schemes than for
78
Figure 4.10: The Doo-Sabin subdivision. The coefficients are defined by the formulas α0 = 1=4 + 5=4k
and αi = (3 + 2 cos(2iπ=k))=4k, for i = 1 : : : k , 1
approximating schemes. In this section we briefly discuss a number of these problems.
Problems with curvature continuity While it is possible to obtain subdivision schemes which are
C2 -continuous, there are indications that such schemes either have very large support [21, 19], or necessarily have zero curvature at extraordinary vertices. A compromise solution was recently proposed by
Umlauf [20]. Nevertheless, this limitation is quite fundamental: degeneracy or discontinuity of curvature
typically leads to visible defects of the surface.
Decrease of smoothness with valence For some schemes, as the valence increases, the magnitude of
the third largest eigenvalue approaches the magnitude of the subdominant eigenvalues. As an example
we consider surfaces generated by the Loop scheme near vertices of high valence.
In Figure 4.12 (right side), one can see a typical problem that occurs because of “eigenvalue clustering:” a crease might appear, abruptly terminating at the vertex. In some cases this behavior may be
79
Figure 4.11: The Midedge subdivision. The coefficients are defined by the formulas αi
2 ∑n̄j=0 2, ji cos 2πik j , n̄ = n,2 1 for i = 0 : : : k , 1
=
desirable, but our goal is to make it controllable rather than let the artifacts appear by chance.
Ripples Another problem, presence of ripples in the surface close to an extraordinary point, is also
shown in Figure 4.12. It is not clear whether this artifact can be eliminated. It is closely related to the
curvature problem.
Uneven structure of the mesh On regular meshes, subdivision matrices of C1 -continuous schemes
always have subdominant eigenvalue 1=2. When the eigenvalues of subdivision matrices near extraordinary vertices significantly differ from 1=2, the structure of the mesh becomes uneven: the ratio of the size
of triangles on finer and coarser levels adjacent to a given vertex is roughly proportional to the magnitude
of the subdominant eigenvalue. This effect can be seen clearly in Figure 4.14.
80
Figure 4.12: Left: ripples on a surface generated by the Loop scheme near a vertex of large valence;
Right: mesh structure for the Loop scheme near an extraordinary vertex with a significant “highfrequency” component; a crease starting at the extraordinary vertex appears.
Optimization of subdivision rules It is possible to eliminate eigenvalue clustering, as well as the
difference in eigenvalues of the regular and extraordinary case by prescribing the eigenvalues of the
subdivision matrix and deriving suitable subdivision coefficients. This approach was used to derive
coefficients of the Butterfly scheme.
As expected, the meshes generated by the modified scheme have better structure near extraordinary
points (Figure 4.13). However, the ripples become larger, so one kind of artifact is traded for another. It
is, however, possible to seek an optimal solution or one close to optimal; alternatively, one may resort to
a family of schemes that would provide for a controlled tradeoff between the two artifacts.
81
Figure 4.13: Left: mesh structure for the Loop scheme and the modified Loop scheme near an extraordinary vertex; a crease does not appear for the modified Loop. Right: shaded images of the surfaces for
Loop and modified Loop; ripples are more apparent for modified Loop.
82
Loop
3
4
5
7
9
16
Modified
Loop
Loop
Modified
Loop
Figure 4.14: Comparison of control nets for Loop scheme and modified Loop scheme. Note that for
Loop scheme the size of the hole in the ring (1-neighborhood removed) is very small relatively to the
surrounding triangles for valence 3 and becomes larger as k grows. For modified Loop scheme this size
remains constant.
83
84
Bibliography
[1] BALL , A. A., AND S TORRY, D. J. T. Conditions for tangent plane continuity over recursively
generated B-spline surfaces. ACM Transactions on Graphics 7, 2 (1988), 83–102.
[2] B IERMANN , H., L EVIN , A., AND Z ORIN , D. Piecewise smooth subdivision surfaces with normal
control. Tech. Rep. TR1999-781, NYU, 1999.
[3] C ATMULL , E., AND C LARK , J. Recursively generated B-spline surfaces on arbitrary topological
meshes. Computer Aided Design 10, 6 (1978), 350–355.
[4] D OO , D., AND S ABIN , M. Analysis of the behaviour of recursive division surfaces near extraordinary points. Computer Aided Design 10, 6 (1978), 356–360.
[5] DYN , N., G REGORY, J. A., AND L EVIN , D. A four-point interpolatory subdivision scheme for
curve design. Computer Aided Geometric Design 4 (1987), 257–268.
[6] DYN , N., L EVIN , D., AND G REGORY, J. A. A butterfly subdivision scheme for surface interpolation with tension control. ACM Transactions on Graphics 9, 2 (April 1990), 160–169.
[7] H ABIB , A., AND WARREN , J. Edge and vertex insertion for a class of c1 subdivision surfaces.
presented at 4th SIAM COnference on Geometric Design, November 1995.
[8] H ABIB , A., AND WARREN , J. Edge and vertex insertion for a class of subdivision surfaces.
Preprint. Computer Science, Rice University, 1996.
[9] H OPPE , H., D E ROSE , T., D UCHAMP, T., H ALSTEAD , M., J IN , H., M C D ONALD , J.,
S CHWEITZER , J., AND S TUETZLE , W. Piecewise smooth surface reconsruction. In Computer
Graphics Proceedings (1994), Annual Conference Series, ACM Siggraph, pp. 295–302.
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[10] KOBBELT, L. Interpolatory subdivision on open quadrilateral nets with arbitrary topology. In
Proceedings of Eurographics 96 (1996), Computer Graphics Forum, pp. 409–420.
[11] L EVIN , A. Boundary algorithms for subdivision surfaces. In Israel-Korea Bi-National Conference
on New Themes in Computerized Geometrical Modeling (1998), pp. 117–121.
[12] L EVIN , A. Combined subdivision schemes for the design of surfaces satisfying boundary conditions. To appear in CAGD, 1999.
[13] L EVIN , A. Interpolating nets of curves by smooth subdivision surfaces. to appear in SIGGRAPH’99 proceedings, 1999.
[14] L OOP, C. Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah,
Department of Mathematics, 1987.
[15] NASRI , A. H. Polyhedral subdivision methods for free-form surfaces. ACM Transactions on
Graphics 6, 1 (January 1987), 29–73.
[16] P ETERS , J., AND R EIF, U. Analysis of generalized B-spline subdivision algorithms. SIAM Jornal
of Numerical Analysis (1997).
[17] P ETERS , J., AND R EIF, U. The simplest subdivision scheme for smoothing polyhedra. ACM
Transactions on Graphics 16(4) (October 1997).
[18] P RAUTZSCH , H. Analysis of Ck -subdivision surfaces at extraordianry points. Preprint. Presented
at Oberwolfach, June, 1995, 1995.
[19] P RAUTZSCH , H., AND R EIF, U. Necessary conditions for subdivision surfaces. 1996.
[20] P RAUTZSCH , H., AND U MLAUF, G. G2 -continuous subdivision algorithm. Preprint, 1997.
[21] R EIF, U. A degree estimate for polynomial subdivision surface of higher regularity. Tech. rep.,
Universität Stuttgart, Mathematisches Institut A, 1995. preprint.
[22] R EIF, U. Some new results on subdivision algorithms for meshes of arbitrary topology. In Approximation Theory VIII, C. K. Chui and L. Schumaker, Eds., vol. 2. World Scientific, Singapore, 1995,
pp. 367–374.
[23] R EIF, U. A unified approach to subdivision algorithms near extraordinary points. Computer Aided
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[24] S CHWEITZER , J. E. Analysis and Application of Subdivision Surfaces. PhD thesis, University of
Washington, Seattle, 1996.
[25] WARREN , J. Subdivision methods for geometric design. Unpublished manuscript, November 1995.
[26] Z ORIN , D. Subdivision and Multiresolution Surface Representations.
Pasadena, 1997.
PhD thesis, Caltech,
[27] Z ORIN , D. A method for analysis of c1 -continuity of subdivision surfaces. submitted to SIAM
Journal of Numerical Analysis, 1998.
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[29] Z ORIN , D. Smoothness of subdivision surfaces on the boundary. preprint, Computer Science
Department, Stanford University, 1998.
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arbitrary topology. Computer Graphics Proceedings (SIGGRAPH 96) (1996), 189–192.
87
88
Exact Evaluation Of Catmull-Clark Subdivision
Surfaces
At Arbitrary Parameter Values
Jos Stam
Alias wavefront, Inc.
1218 Third Ave, 8th Floor,
Seattle, WA 98101, U.S.A.
[email protected]
Abstract
In this paper we disprove the belief widespread within the computer graphics community that
Catmull-Clark subdivision surfaces cannot be evaluated directly without explicitly subdividing.
We show that the surface and all its derivatives can be evaluated in terms of a set of eigenbasis
functions which depend only on the subdivision scheme and we derive analytical expressions for
these basis functions. In particular, on the regular part of the control mesh where Catmull-Clark
surfaces are bi-cubic B-splines, the eigenbasis is equal to the power basis. Also, our technique is
both efficient and easy to implement. We have used our implementation to compute high quality
curvature plots of subdivision surfaces. The cost of our evaluation scheme is comparable to that
of a bi-cubic spline. Therefore, our method allows many algorithms developed for parametric
surfaces to be applied to Catmull-Clark subdivision surfaces. This makes subdivision surfaces an
even more attractive tool for free-form surface modeling.
1 Introduction
Subdivision surfaces have emerged recently as a powerful and useful technique in modeling freeform surfaces. However, although in theory subdivision surfaces admit local parametrizations,
there is a strong belief within the computer graphics community that these parametrizations cannot be evaluated exactly for arbitrary parameter values. In this paper we disprove this belief and
provide a non-iterative technique that efficiently evaluates Catmull-Clark subdivision surfaces and
their derivatives up to any order. The cost of our technique is comparable to the evaluation of a
bi-cubic surface spline. The rapid and precise evaluation of surface parametrizations is crucial for
many standard operations on surfaces such as picking, rendering and texture mapping. Our evaluation technique allows a large body of useful techniques from parametric surfaces to be transferred
to subdivision surfaces, making them even more attractive as a free-form surface modeling tool.
1
Our evaluation is based on techniques first developed to prove smoothness theorems for subdivision schemes [3, 5, 1, 4, 8, 6]. These proofs are constructed by transforming the subdivision into
its eigenspace1 . In its eigenspace, the subdivision is equivalent to a simple scaling of each of its
eigenvectors by their eigenvalue. These techniques allow us to compute limit points and limit normals at the vertices of the mesh, for example. Most of the proofs, however, consider only a subset
of the entire eigenspace and do not address the problem of evaluating the surface everywhere. We,
on the other hand, use the entire eigenspace to derive an efficiently evaluated analytical form of the
subdivision surface everywhere, even in the neighborhood of extraordinary vertices. In this way,
we have extended a theoretical tool into a very practical one.
In this paper we present an evaluation scheme for Catmull-Clark subdivision surfaces [2]. However, our methodology is not limited to these surfaces. Whenever subdivision on the regular part
of the mesh coincides with a known parametric representation [8], our approach should be applicable. We have decided to present the technique for the special case of Catmull-Clark subdivision
surfaces in order to show a particular example fully worked out. In fact, we have implemented
a similar technique for Loop’s triangular subdivision scheme [5]. The details of that scheme are
given in another paper in these course notes. We believe that Catmull-Clark surfaces have many
properties which make them attractive as a free-form surface design tool. For example, after one
subdivision step each face of the initial mesh is a quadrilateral, and on the regular part of the mesh
the surface is equivalent to a piecewise uniform B-spline. Also, algorithms have been written to
fair these surfaces [4] and to dynamically animate them [7].
In order to define a parametrization, we introduce a new set of eigenbasis functions. These
functions were first introduced by Warren in a theoretical setting for curves [10] and used in a
more general setting by Zorin [11]. In this paper, we show that the eigenbasis of the CatmullClark subdivision scheme can be computed analytically. Also, we show that in the regular case
the eigenbasis is equal to the power basis and that the eigenvectors then correspond to the “change
of basis matrix” from the power basis to the bi-cubic B-spline basis. The eigenbasis introduced in
this paper can thus be thought of as a generalization of the power basis at extraordinary vertices.
Since our eigenbasis functions are analytical, the evaluation of Catmull-Clark subdivision surfaces
can be expressed analytically. As shown in the results section of this paper, we have implemented
our evaluation scheme and used it in many practical applications. In particular, we show for the
first time high resolution curvature plots of Catmull-Clark surfaces precisely computed around the
irregular parts of the mesh.
The paper is organized as follows. Section 2 is a brief review of the Catmull-Clark subdivision
scheme. In Section 3 we cast this subdivision scheme into a mathematical setting suitable for analysis. In Section 4 we compute the eigenstructure to derive our evaluation. Section 5 is a discussion
of implementation issues. In Section 6 we exhibit results created using our technique, comparing it
to straightforward subdivision. Finally in Section 7 we conclude, mentioning promising directions
for future research.
This paper is almost equivalent to our SIGGRAPH’98 paper [9]. We have corrected two errors
in the Appendices and added a small section devoted to stability issues in Section 5. Also we have
included on the CDROM course notes a data file which contains the eigenstructures up to valence
50.
1 To
be defined precisely below.
2
v
CCCCCCCCCC
CCCCCCCCCC
(1,1)
CCCCCCCCCC
CCCCCCCCCC
CCCCCCCCCC
CCCCCCCCCC
CCCCCCCCCC
CCCCCCCCCC
CCCCCCCCCC
(0,0)
CCCCCCCCCC
13
14
9
10
16
15
11
BBBBBBBB
BBBBBBBB
(1,1)
BBBBBBBB
BBBBBBBB
BBBBBBBB
BBBBBBBB
(0,0) 7
6 BBBBBBBB
BBBBBBBB
5
12
8
u
1
2
3
4
Figure 1: A bi-cubic B-spline is defined by 16 control vertices. The numbers on the right show the
ordering of the corresponding B-spline basis functions in the vector (u; v ).
b
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBB
Figure 2: Initial mesh and two levels of subdivision. The shaded faces correspond to regular
bi-cubic B-spline patches. The dots are extraordinary vertices.
1.1 Notations
In order to make the derivations below as clear and compact as possible we adopt the following
notational conventions. All vectors are assumed to be columns and are denoted by boldface lower
case roman characters, e.g., . The components of the vector are denoted by the corresponding
italicized character: the i-th component of a vector is thus denoted vi . The component of a vector
should not be confused with an indexed vector such as k . Matrices are denoted by uppercase
boldface characters, e.g., . The transpose of a vector (resp. matrix ) is denoted by T (resp.
T
). The transpose of a vector is simply the same vector written row-wise. Therefore the dot
product between two vectors and is written “ T ”. The vector or matrix having only zero
elements is denoted by . The size of this vector (matrix) should be obvious from the context.
v
v
M
M
0
u
v
uv
v
v
M
v
2 Catmull-Clark Subdivision Surfaces
The Catmull-Clark subdivision scheme was designed to generalize uniform B-spline knot insertion
to meshes of arbitrary topology [2]. An arbitrary mesh such as the one shown on the upper left
hand side of Figure 2 is used to define a smooth surface. The surface is defined as the limit of a
sequence of subdivision steps. At each step the vertices of the mesh are updated and new vertices
are introduced. Figure 2 illustrates this process. On each vertex of the initial mesh, the valence is
3
2N+1
2
9
3
1
2N+8
8
BBBBB
BBBBB
BBBBB
BBBBB
BBBBB
BBBBB
4
6
7
2N+7
5
2N+6
2N+5
2N+4
2N+2
2N+3
Figure 3: Surface patch near an extraordinary vertex with its control vertices. The ordering of the
control vertices is shown on the bottom. Vertex 1 is an extraordinary vertex of valence N = 5.
the number of edges that meet at the vertex. A vertex having a valence not equal to four is called
an extraordinary vertex. The mesh on the upper left hand side of Figure 2 has two extraordinary
vertices of valence three and one of valence five. Away from extraordinary vertices, the CatmullClark subdivision is equivalent to midpoint uniform B-spline knot insertion. Therefore, the 16
vertices surrounding a face that contains no extraordinary vertices are the control vertices of a
uniform bi-cubic B-spline patch (shown schematically in Figure 1). The faces which correspond to
a regular patch are shaded in Figure 2. This figure shows how the portion of the surface comprised
of regular patches grows with each subdivision step. In principle, the surface can thus be evaluated
whenever the holes surrounding the extraordinary vertices are sufficiently small. Unfortunately,
this iterative approach is too expensive near extraordinary vertices and does not provide exact
higher derivatives.
Because the control vertex structure near an extraordinary vertex is not a simple rectangular
grid, all faces that contain extraordinary vertices cannot be evaluated as uniform B-splines. We
assume that the initial mesh has been subdivided at least twice, isolating the extraordinary vertices
so that each face is a quadrilateral and contains at most one extraordinary vertex. In the rest of
the paper, we need to demonstrate only how to evaluate a patch corresponding to a face with just
one extraordinary vertex, such as the region near vertex 1 in Figure 3. Let us denote the valence
of that extraordinary vertex by N . Our task is then to find a surface patch (u; v ) defined over the
unit square = [0; 1] [0; 1] that can be evaluated directly in terms of the K = 2N + 8 vertices
that influence the shape of the patch corresponding to the face. We assume in the following that
the surface point corresponding to the extraordinary vertex is (0; 0) and that the orientation of is chosen such that u v points outside of the surface.
A simple argument shows that the influence on the limit surface of the seven “outer control
vertices” numbered 2N + 2 through 2N + 8 in Figure 3 can be accounted for directly. Indeed,
consider the situation depicted in Figure 4 where we show a mesh containing a vertex of valence 5
and a regular mesh side by side. Let us assume that all the control vertices are set to zero except for
the seven control vertices highlighted in Figure 4. If we repeat the Catmull-Clark subdivision rules
for both meshes we actually obtain the same limit surface, since the exceptional control vertex
s
s
s s
4
0
BBBBB
BBBBB
BBBBB
0
0
BBBBB
0
0
0
0
0
0
BBBB
BBBB
0BBBB
0
BBBB
0
0
0
0
0
0
0
0
0
BBBBB
BBBBB
BBBBB
BBBBB
BBBB
BBBB
BBBB
BBBB
BBBB
0 0 0
0 0 0
0
0 0 0
0 0 0
0 0 0
0 0
0
0
Figure 4: The effect of the seven outer control vertices does not depend on the valence of the
extraordinary vertex. When the 2N + 1 control vertices in the center are set to zero the same limit
surface is obtained.
at the center of the patch remains equal to zero after each subdivision step. Therefore, the effect
of the seven outer control vertices is simply each control vertex multiplied by its corresponding
bi-cubic B-spline tensor product basis function. In the derivation of our evaluation technique we
do not need to make use of this fact. However, it explains the simplifications which occur at the
end of the derivation.
2N+1
2
1
9
BBBBBBBB
BBBBBBBB
BBBBBBBB
BBBBBBBB
BBBBBBBB
BBBBBBBB
BBBBBBBB
BBBBBBBB
BBBBBBBB
3
2N+8
4
8
6
2N+17
2N+7
5
7
2N+16
2N+6
2N+5
2N+4
2N+15
2N+3
2N+13
2N+12
2N+11
2N+2
2N+14
2N+10
2N+9
Figure 5: Addition of new vertices by applying the Catmull-Clark subdivision rule to the vertices
in Figure 3.
5
3 Mathematical Setting
In this section we cast the informal description of the previous section into a rigorous mathematical
setting. We denote by
T
0 = ( 0;1 ; ; 0;K ) ;
C
c
c
the initial control vertices defining the surface patch shown in Figure 3. The ordering of these
vertices is defined on the bottom of Figure 3. This peculiar ordering is chosen so that later computations become more tractable. Note that the vertices do not result in the 16 control vertices of a
uniform bi-cubic B-spline patch, except when N = 4.
Through subdivision we can generate a new set of M = K + 9 vertices shown as circles superimposed on the initial vertices in Figure 5. Subsets of these new vertices are the control vertices
of three uniform B-spline patches. Therefore, three-quarters of our surface patch is parametrized,
and could be evaluated as simple bi-cubic B-splines (see top left of Figure 6). We denote this new
set of vertices by
CT1 = (c1;1; ; c1;K ) and C T1 = CT1 ; c1;K+1; ; c1;M
With these matrices, the subdivision step is a multiplication by an K
matrix :
1=
0:
A
C
AC
:
K (extended) subdivision
(1)
Due to the peculiar ordering that we have chosen for the vertices, the extended subdivision matrix
has the following block structure:
A = SS11 S012
S
!
;
(2)
where is the 2N +1 2N +1 subdivision matrix usually found in the literature [4]. The remaining
two matrices correspond to the regular midpoint knot insertion rules for B-splines. Their exact
definition can be found in Appendix A. The additional points needed to evaluate the three B-spline
patches are defined using a bigger matrix of size M K :
A
0;
C 1 = AC
0
1
S
0
A = [email protected] S11 S12 CA :
where
S
S
(3)
S21 S22
The matrices 21 and 22 are defined in Appendix A. The subdivision step of Equation 1 can be
repeated to create an infinite sequence of control vertices:
Cn = ACn,1 = AnC0 and
n,1 = AA
n,1C0;
C n = AC
n 1:
C
As noted above, for each level n 1, a subset of the vertices of n becomes the control vertices
of three B-spline patches. These control vertices can be defined by selecting 16 control vertices
from n and storing them in 16 3 matrices:
C
Bk;n = PkC n;
6
v
u
2N+17 2N+16 2N+15 2N+14
!!
!!!
!!
3
!!!
!!
1!!!
2
!!!
!!!
!!
!!!
!!!
P3
P2
P1
2N+7 2N+6
2N+8
2N+2
3
4
5
2N+3
2
1
6
2N+4
2N+16 2N+15 2N+14 2N+9
2N+2 2N+10
!!!!!
!!!!!
2N+3
!!!!!
!!!!!
!!!!
!!!!
!!!!
!!!!
!!!!
2N+7 2N+6
!!!!!
!!!!!
2N+2
!!!!!
!!!!!
2N+7 2N+6
2N+10
4
5
2N+11
1
6
2N+4 2N+12
4
5
2N+3 2N+11
8
7
2N+5 2N+13
1
6
2N+4 2N+12
Figure 6: Indices of the control vertices of the three bi-cubic B-spline patches obtained from
P
C n.
b
where k is a 16 M “picking” matrix and k = 1; 2; 3. Let (u; v ) be the vector containing the
16 cubic B-spline basis functions (see Appendix B). If the control vertices are ordered as shown
on the left of Figure 1, then the surface patch corresponding to each matrix of control vertices is
defined as
T
Tn Tk (u; v);
(4)
k;n (u; v ) = k;n (u; v ) =
s
B b
CPb
where (u; v ) 2 , n 1 and k = 1; 2; 3. Using the ordering convention for the B-spline control
vertices of Figure 1, the definition of the picking matrices is shown in Figure 6. Each row of k
is filled with zeros except for a one in the column corresponding to the index shown in Figure 6
(see Appendix B for more details). The infinite sequence of uniform B-spline patches defined by
Equation 4 form our surface (u; v ), when “stitched together”. More formally, let us partition the
unit square into an infinite set of tiles f
nk g, n 1; k = 1; 2; 3, as shown in Figure 7. Each tile
with index n is four times smaller than the tiles with index n , 1. More precisely:
P
s
1 1 1
1 = 2n ; 2n,1 0; 2n ;
n2 = 21n ; 2n1,1 21n ; 2n1,1 ;
n3 = 0; 21n 21n ; 2n1,1 :
n
s
(5)
A parametrization for (u; v ) is constructed by defining its restriction to each tile nk to be equal to
the B-spline patch defined by the control vertices k;n:
B
s(u; v) = sk;n(tk;n(u; v)):
n
k
The transformation
tk;n maps the tile nk onto the unit square :
t1;n(u; v) = (2nu , 1; 2nv);
t2;n(u; v) = (2nu , 1; 2nv , 1) and
t3;n(u; v) = (2nu; 2nv , 1):
7
(6)
(7)
(8)
(9)
v
1
Ω13
Ω2
2
Ω 3 Ω22
Ω33 Ω32
Ω31
Ω11
Ω21
u
Figure 7: Partition of the unit square into an infinite family of tiles.
Equation 6 gives an actual parametrization for the surface. However, it is very costly to evaluate,
since it involves n , 1 multiplications of the K K matrix . The evaluation can be simplified
considerably by computing the eigenstructure of . This is the key idea behind our new evaluation
technique and is the topic of the next section.
A
A
4 Eigenstructure, Eigenbases and Evaluation
A
The eigenstructure of the subdivision matrix is defined as the set of its eigenvalues and eigenvectors. In our case the matrix is non-defective for any valence. Consequently, there always exists K linearly independent eigenvectors [4]. Therefore we denote this eigenstructure by ( ; ),
where is the diagonal matrix containing the eigenvalues of , and
is an invertible matrix
whose columns are the corresponding eigenvectors. The computation of the eigenstructure is then
equivalent to the solution of the following matrix equation:
A
A
AV = V;
V
V
(10)
where the i-th diagonal element of is an eigenvalue with a corresponding eigenvector equal to
the i-th column of the matrix (i = 1; ; K ). There are many numerical algorithms which can
compute solutions for such equations. Unfortunately for our purposes, these numerical routines do
not always return the correct eigenstructure. For example, in some cases the solver returns complex
eigenvalues. For this reason, we must explicitly compute the eigenstructure. Since the subdivision
matrix has a definite block structure, our computation can be done in several steps. In Appendix
A we analytically compute the eigenstructure ( ; 0 ) (resp. ( ; 1 )) of the diagonal block
(resp. 12 ) of the subdivision matrix defined in Equation 2. The eigenvalues of the subdivision
matrix are the union of the eigenvalues of its diagonal blocks:
V
S
U
= 0 0
8
W
!
:
S
S
S
Using the eigenvectors of and 12 , it can be proven that the eigenvectors for the subdivision
matrix must have the following form:
!
U
0
0
V = U1 W1 :
U
The matrix 1 is unknown and is determined from Equation 10. If we replace the matrices
and by their block representations, we obtain the following matrix equation:
A
U
S11U0 + S12U1 = U1:
U
, V
(11)
Since 0 is known, 1 is computed by solving the 2N + 1 linear systems of Equation 11. In
principle, this equation could be solved symbolically. In practice, however, because of the small
sizes of the linear systems (7 7) we can compute the solution up to machine accuracy (see the
next section for details). The inverse of our eigenvector matrix is equal to
,1
U
0,1
0
,
1
,
1
,W1 U1 U0 W1
V ,1 =
U
where both 0 and
Equation 10:
!
;
(12)
W1 can be inverted exactly (see Appendix A). This fact allows us to rewrite
A = VV,1:
This decomposition is the crucial result that we use in constructing a fast evaluation scheme of the
surface patch. Indeed, the subdivided control vertices at level n are now equal to
C
V C
n,1C0 = AV
n,1V,1C0 = AV
n,1C^ 0;
C n = AA
where ^ 0 = ,1 0 is the projection of the K control vertices into the eigenspace of the subdivision matrix. Using this new expression for the control vertices at the n-th level of subdivision,
Equation 4 can be rewritten in the following form:
T b(u; v):
sk;n(u; v) = C^ T0 n,1 PkAV
We observe that the right most terms in this equation are independent of the control vertices and
the power n. Therefore, we can precompute this expression and define the following three vectors:
T b(u; v)
x(u; v; k) = PkAV
k
= 1; 2; 3:
(13)
The components of these three vectors correspond to a set of K bi-cubic splines. In Appendix
B we show how to compute these splines. Notice that the splines xi (u; v; k) depend only on the
valence of the extraordinary vertex. Consequently, we can rewrite the equation for each patch more
compactly as:
^ T0 n,1 (u; v; k) k = 1; 2; 3:
(14)
k;n (u; v ) =
To make the expression for the evaluation of the surface patch more concrete, let Ti denote the
rows of ^ 0 . Then the surface patch can be evaluated as:
s
C x
C
K
X
s(u; v) = (i)n,1 xi(tk;n(u; v); k)pi:
n
k
i=1
9
p
(15)
p
Therefore, in order to evaluate the surface patch, we must first compute the new vertices i (only
once for a given mesh). Next, for each evaluation we determine n and then scale the contribution
from each of the splines by the relevant eigenvalue to the power n , 1. Since all but the first of the
eigenvalues are smaller than one, their contribution decreases as n increases. Thus, for large n, i.e.,
for surface-points near the extraordinary vertex, only a few terms make a significant contribution.
In fact for (u; v ) = (0; 0) the surface point is 1 , which agrees with the definition of a limit point
in [4].
Alternatively, the bi-cubic spline functions (u; v; k ) can be used to define a set of eigenbasis
functions for the subdivision. For a given eigenvalue i we define the function 'i by its restrictions
on the domains nk as follows:
p
x
'i (u; v ) nk
= (i)n,1xi (tk;n(u; v); k);
with i = 1; ; K . By the above definition these functions satisfy the following scaling relation:
'i (u=2; v=2) = i 'i (u; v ):
The importance of these functions was first noted by Warren in the context of subdivision curves
[10]. More recently, Zorin has defined and used eigenbasis functions to prove smoothness conditions for very general classes of subdivision schemes [11]. However, explicit analytical expressions
for particular eigenbases have never appeared before. On the other hand, we can compute these
bases analytically. Figures 8 and 9 show the complete sets of eigenbasis functions for valences 3
and 5. In the figures we have normalized each function such that its range is bounded within ,1
and 1. In particular, the first eigenbasis corresponding to an eigenvalue of one is always a constant
function for any valence. A closer look at Figures 8 and 9 reveals that they share seven identical
functions. In fact as shown in Appendix B, the last seven eigenbasis functions for any valence are
always equal to
1
1
1
1
1
1
1
3
3
3
3
3
2
3
3
2
3
u v ; u ; u v; u v ; v ; uv ; u v :
36
6
6
2
6
6
2
Furthermore, by transforming these functions back from the eigenspace using
seven tensor B-spline basis functions
W1,1 we obtain the
b4 (u; v ); b8(u; v ); b12 (u; v ); ; b16 (u; v );
i.e., the basis functions corresponding to the “outer layer” of control vertices of Figure 3. This
should not come as a surprise since as we noted above, the influence of the outer layer does not
depend on the valence of the extraordinary vertex (see Figure 4).
In the regular bi-cubic B-spline case (N = 4), the remaining eigenbasis can be chosen to be
equal to the power basis
f1; u; v; u2; uv; v2; u2v; uv2; u2v2g:
The scaling property of the power basis is obvious. For example, the basis function u2 v corre-
sponds to the eigenvalue 1=8:
(u=2)2(v=2) = (1=2)2(1=2)u2v = 81 u2v:
This relationship between the Catmull-Clark subdivision and the power basis in the regular case
does not seem to have been noted before. Note also that the eigenvectors in this case correspond
10
to the “change of basis matrix” from the bi-cubic B-spline basis to the power basis. The eigenbasis functions at extraordinary vertices can thus be interpreted as a generalization of the power
basis. However, the eigenbases are in general not polynomials. In the case of the Catmull-Clark
subdivision they are piece-wise bi-cubic polynomials. The evaluation of the surface patch given
by Equation 15 can now be rewritten exactly as:
s(u; v) = X 'i(u; v)pi:
K
i=1
(16)
This is the key result of our paper, since this equation gives a parametrization for the surface
corresponding to any face of the control mesh, no matter what the valence is. There is no need to
subdivide. Equation 16 also allows us to compute derivatives of the surface up to any order. Only
the corresponding derivatives of the basis functions appearing in Equation 16 are required. For
example, the partial derivative of the i-th eigenbasis with respect to u is:
@
'i (u; v ) nk
@u
@
= 2n(i)n,1 @u
xi (tk;n(u; v ); k);
where the factor 2n is equal to the derivative of the affine transformation
2pn will be present when the order of differentiation is p.
tk;n. Generally a factor
5 Implementation
Although the derivation of our evaluation technique is mathematically involved, its implementation
is straightforward. The tedious task of computing the eigenstructure of the subdivision matrix only
has to be performed once and is provided in Appendix A. In practice, we have precomputed these
eigenstructures up to some maximum valence, say NMAX=50, and have stored them in a file. The
file and some C code that reads in the data can be found on the course CDROM. Any program using
our evaluation technique can read in these precomputed eigenstructures. In our implementation the
eigenstructure for each valence N is stored internally as
typedef
struct f
double L[K];
/* eigenvalues */
double iV[K][K];
/* inv of the eigenvectors */
double x[K][3][16]; /* coeffs of the splines */
g EIGENSTRUCT;
EIGENSTRUCT eigen[NMAX];,
where K=2*N+8. At the end of this section we describe how we computed these eigenstructures.
We emphasize that this step has to be performed only once and that its computational cost is
irrelevant to the efficiency of our evaluation scheme.
Given that the eigenstructures have been precomputed and read in from a file, we evaluate a
surface patch around an extraordinary vertex in two steps. First, we project the control vertices
surrounding the patch into the eigenspace of the subdivision matrix. Let the control vertices be
ordered as shown in Figure 3 and stored in an array C[K]. The projected vertices Cp[K] are then
easily computed by using the precomputed inverse of the eigenvectors:
11
ProjectPoints(point *Cp,point *C,int N)f
for ( i=0 ; i<2*N+8 ; i++ )f
Cp[i] = (0,0,0);
for ( j=0 ; j<2*N+8 ; j++ )f
Cp[i] += eigen[N].iV[i][j] * C[j];
g
g
g
This routine is called only whenever one of the patches is evaluated for the first time or after an
update of the mesh. This step is, therefore, called at most once per surface patch. The second
step of our evaluation, on the other hand, is called whenever the surface has to be evaluated at a
particular parameter value (u,v). The second step is a straightforward implementation of the
sum appearing in Equation 15. The following routine computes the surface patch at any parameter
value.
EvalSurf ( point P, double u, double v,
point *Cp, int N ) f
/* determine in which domain nk the parameter lies */
n = floor(min(-log2(u),-log2(v)))+1;
pow2 = pow(2,n-1);
u *= pow2; v *= pow2;
if ( v < 0.5 ) f
k=0; u=2*u-1; v=2*v;
g
else if ( u < 0.5 ) f
k=2; u=2*u; v=2*v-1;
g
else f
k=1; u=2*u-1; v=2*v-1;
g
/* Now evaluate the surface */
P = (0,0,0);
for ( i=0 ; i<2*N+8 ; i++ ) f
P += pow(eigen[N].L[i],n-1) *
EvalSpline(eigen[N].x[i][k],u,v)*Cp[i];
g
g
The function EvalSpline computes the bi-cubic B-spline polynomial whose coefficients are
given by its first argument at the parameter value (u,v). When either one of the parameter values
u or v is zero, we set it to a sufficiently small value near the precision of the machine, to avoid
an overflow that would be caused by the log2 function. Because EvalSpline evaluates a bicubic polynomial, the cost of EvalSurf is comparable to that of a bi-cubic surface spline. The
extra cost due to the logarithm and the elevation to an integer power is minimal, because these
operations are efficiently implemented on most current hardware. Since the projection step is only
called when the mesh is updated, the cost of our evaluation depends predominantly on EvalSurf.
12
The computation of the p-th derivative is entirely analogous. Instead of using the routine
EvalSpline we employ a routine that returns the p-th derivative of the bi-cubic B-spline. In
addition, the final result is scaled by a factor pow(2,n*p). The evaluation of derivatives is
essential in applications that require precise surface normals and curvature. For example, Newton
iteration schemes used in ray surface computations require higher derivatives of the surface at
arbitrary parameter values.
We now describe how we compute the eigenstructure of the subdivision matrix. This step
only has to be performed once for a given set of valences. The efficiency of this step is not
crucial. Accuracy is what matters here. As shown in the appendix, the eigenstructure of the
two matrices and 12 can be computed analytically. The corresponding eigenstructure of the
extended subdivision matrix requires the solution of the 2N + 1 linear systems of Equation 11.
We did not solve these analytically because these systems are only of size 7 7. Consequently,
these systems can be solved up to machine accuracy using standard linear solvers. We used the
dgesv routine from LINPACK to perform the task. The inverse of the eigenvectors is computed
by carrying out the matrix products appearing in Equation 12. Using the eigenvectors, we also
precompute the coefficients of the bi-cubic splines (u; v; k) as explained in Appendix B. For each
valence N we stored the results in the data structure eigen[NMAX] and saved them in a file to be
read in at the start of any application which uses the routines ProjectPoints and EvalSurf
described above. This data is provided in the file ccdata50.dat on the CDROM for valences
upto NMAX=50. The C program cctest.c demonstrates how to read in that data.
S
S
A
x
5.1 Some Remarks on Stability
As noted previously there is a problem when evaluating the surface at the extraordinary point
using EvalSurf since the log is ill-defined at (0; 0). One option as mentioned above is to clamp
the (u; v ) values below a certain threshold. A better option is to return the limit point Cp[0]
directly. When computing derivatives other instabilities can occur, although in practice we have
not encountered them in our implementation using the clamping of the u , v values. However,
instabilities could be a nuisance in other applications of the evaluation method. The problem is
that we have to multiply the derivative by a factor pow(2,n*p) which diverges for large n. One
possible solution is to include this factor when taking powers of the eigenvalues, i.e., line
P += pow(eigen[N].L[i],n-1) *
should be replaced by
P += p2*pow(p2*eigen[N].L[i],n-1) * ,
where p2=pow(2,p). Although this simple modification reduces instabilities it is not completely
satisfactory. The problem is that the inverse of the eigenvalues are not always powers of two so
that the products (2p )n either converge to zero or diverge.
The cleanest solution is to reparametrize the surface using the characteristic map introduced in
[8]. The characteristic map is simply the mapping defined by the eigenbasis functions '2 (u; v ) and
'3 (u; v ). Let
(u; v ) = (x(u; v ); y (u; v )) = ('2 (u; v ); '3(u; v ));
13
be the characteristic map. A more stable implementation would be to evaluate:
s(x; y) = s
,1 (u; v ) :
This reparametrization requires the inversion of the two eigenbasis functions '2 and '3 . Note
also that the evaluation of the derivatives requires the computation of the derivatives of the inverse
of the characteristic map as well. Also we point out that the evaluation of the curvature near
the extraordinary point is inherently unstable since the the curvature at these points is known not
to exist (diverge) for Catmull-Clark surfaces. In particular, this implies that Catmull-Clark are in
general not C 2 surfaces. As a corollary, for example, a perfect sphere cannot be represented exactly
by a Catmull-Clark surface.
6 Results
In Figure 10 we depict several Catmull-Clark subdivision surfaces. The extraordinary vertex whose
valence N is given in the figure is located in the center of each surface. The position information
within the blue patches surrounding the extraordinary vertex is computed using our new evaluation
technique. The remaining patches are evaluated as bi-cubic B-splines. Next to each surface we also
depict the curvature of the surface. We map the value of the Gaussian curvature onto a hue angle.
Red corresponds to a flat surface, while green indicates high curvature. We have purposely made
the curvature plot discontinuous in order to emphasize the iso-contour lines. Both the shaded
surface and the curvature plot illustrate the accuracy of our method. Notice especially how the
curvature varies smoothly across the boundary between the patches evaluated using our technique
and the regular bi-cubic B-spline patches. The curvature plots also indicate that for N 6= 4 the
Gaussian curvature takes on arbitrarily large values near the extraordinary vertex. The curvature at
the extraordinary vertex is in fact infinite, which explains the diverging energy functionals in [4].
Figure 11 depicts more complex surfaces. The blue patches are evaluated using our technique.
7 Conclusion and Future Work
In this paper we have presented a technique to evaluate Catmull-Clark subdivision surfaces. This is
an important contribution since the lack of such an evaluation schemes has been sited as the chief
argument against the use of subdivision scheme in free-form surface modelers. Our evaluation
scheme permits many algorithms and analysis techniques developed for parametric surfaces to be
extended to Catmull-Clark surfaces. The cost of our algorithm is comparable to the evaluation
of a bi-cubic spline. The implementation of our evaluation is straightforward and we have used
it to plot the curvature near extraordinary vertices. We believe that the same methodology can
be applied to many other subdivision schemes sharing the features of Catmull-Clark subdivision:
regular parametrization away from extraordinary vertices. We have worked out the details for
Loop’s triangular scheme, and the derivation can be found in the accompanying paper in these
course notes. Catmull-Clark surfaces and Loop surfaces (when the valence 6= 3) share the property
that their extended subdivision matrices are non-defective. In general, this is not the case. For
example, the extended subdivision matrix of Doo-Sabin surfaces cannot generally be diagonalized.
In that case, however, we can use the Jordan normal form of the extended subdivision matrix and
employ Zorin’s general scaling relations [11].
14
Acknowledgments
I wish to thank the following individuals for their help: Eugene Lee for assisting me in fine tuning
the math, Michael Lounsbery and Gary Herron for many helpful discussions, Darrek Rosen for
creating the models, Pamela Jackson for proofreading the paper, Gregg Silagyi for his help during
the submission, and Milan Novacek for his support during all stages of this work. Thanks also to
Markus Meister from Brown University who kindly pointed out a mistake in Appendix A of our
SIGGRAPH paper.
A Subdivision Matrices and Their Eigenstructures
S
The matrix corresponds to the extraordinary rules around the extraordinary vertex. With our
choice of ordering of the control vertices the matrix is:
0
BB adN
BB f
BB
S = BBBB fd
BB ..
BB .
@ d
f
bN cN bN cN bN
d e e 0 0
f f f 0 0
e e d e e
0 0 f f f
..
.
e
f
0
0
0
0
0
0
...
1
CC
C
0 0 0 C
C
0 0 0 C
C
CC
0 0 0 C
CC
..
.
C
e d e C
A
0 f f
bN cN bN cN
0 0 e e
0
0
0
0 e
0 0
where
1 ; f = 1:
= 1 , 47N ; bN = 2N3 2 ; cN = 4N1 2 ; d = 83 ; e = 16
4
Since the lower right 2N 2N block of S has a cyclical structure, we can use the discrete Fourier
aN
S
transform to compute the eigenstructure of . This was first used in the context of subdivision
surfaces by Doo and Sabin [3]. The discrete Fourier transform can be written compactly by introducing the following 2N 2N “Fourier matrix”;
0
1
1
0
1
0
1
0
BB 0 1 0
C
1 0
1 C
BB
,1
,(N ,1) 0 C
CC
1
0
!
0
!
BB
CC
,
1
,
(
N
,
1)
F = BBB 0 1 0. ! . 0. !
CC ;
.
.
.
BB
C
.
.
.
[email protected] 1 0 !,(N ,1) 0 !,(N ,1)2 0 CCA
0 1 0 !,(N ,1) 0 !,(N ,1)2
where ! = exp(i2=N ). Using these notations we can write down the “Fourier transform” of the
matrix
S compactly as:
0^
BB S00
S^ = BBB ..
@ .
0
0
^S1
0
0
0
0
...
0
15
0
0
0
S^ N ,1
1
CC
CC = T S T,1;
CA
where
!
T = 0 10F ; T,1 = 01 F0
0 N
1
aN NbN NcN
S^ 0 = [email protected] d 2f 2e CA and
1
!
;
0 f ,l2f l f
1
,l
e
!
+
!
+
d
e
1
+
!
A;
S^ l = @ f 1 + !l
f
S
l = 1; ; N , 1. The eigenstructure of the Fourier transform ^ is computed from the eigenstructures of its diagonal blocks. The first block ^ 0 has eigenvalues
S
1
= 1;
2 ; 3
p
= 81N ,7 + 3N 49 , 30N + 5N 2
0
1
1 1622 , 122 + 1 1623 , 123 + 1
CA :
62 , 1
63 , 1
K^ 0 = [email protected] 1
1
42 + 1
43 + 1
^ l (l = 1; ; N , 1) are equal to:
Similarly, the two eigenvalues of each block S
0
!
!v
!1
u
u
1
2
l
2
l
l
t
@
A
l =
16 5+cos N cos N 18 + 2 cos N ;
and eigenvectors
where we have used some trigonometric relations to simplify the resulting expressions. The corresponding eigenvectors of each block are
,
+
K^ l = 41l+,!l1 41l+,!l1
We have to single out the special case when N is even and l
block is
!
K^ N=2 = 10 01
!
:
= N=2. In this case the corresponding
:
S^ are the union of the eigenvalues of its blocks and the eigenvectors
0 ^
1
K
0
0
0
0
BB 0 K^
CC
0
0
1
CC :
^K = B
BB ..
...
0
0 CA
@ .
0 0 0 K^ N ,1
^ are similar, they have the same eigenSince the subdivision matrix S and its Fourier transform S
The eigenvalues of the matrix
are
values. The eigenvectors are computed by inverse Fourier transforming these eigenvectors:
K = T,1 K^ :
16
S
Consequently, we have computed the eigenvalues and eigenvectors of . However, in this form
the eigenvectors are complex valued and most of the eigenvalues are actually of multiplicity two,
+
+
,
since ,
l = N ,l and l = N ,l . We relabel these eigenvalues as follows:
4
= ,1 ; 5 = +1; 6 = ,2 ; 7 = +2; Since we have rearranged the eigenvalues, we have to rearrange the eigenvectors. At the same time
we make these eigenvectors real. Let 1 ; ; 2N +1 be the columns of , then we can construct
the columns of a matrix 0 as follows:
k
U
k
K
u1 = k1; u2 = k2; u3 = k3;
u2l+2 = 21 (kl+3 + k2N ,l+2) and
u2l+3 = 21i (kl+3 , k2N ,l+2) :
More precisely u1 , u2 , u3 , u2l+2 and u2l+3 are equal to
011 0162 , 12 + 11 0162 , 12 + 11
BB1CC BB 262 , 12 CC BB 363 , 13 CC
BB1CC BB 4 + 1 CC BB 4 + 1 CC
2
3
BB.CC ; BB
CC ; BB
CC ;
.
..
..
BB..CC BB
CC BB
C
[email protected] [email protected] 62 , 1 CA [email protected] 63., 1 CCA
1
42 + 1
43 + 1
1
0
1
0
0
0
CC
BB
CC
BB
0
4l+3 , 1
C
B
CC
BB
CC
BB
S (l)
1 + C(l)
BB
BB (4l+3 , 1)S(l) CCC
BB (4l+3 , 1)C(l) CCC
BB C(l) + C2(l) CC and BBB S(l) + S2(l) CCC ;
CC
BB
CC
BB
..
..
.
.
CC
BB
C
BB
@(4l+3 , 1)S(N ,1)(l)CA
@(4l+3 , 1)C(N ,1)(l)A
C(N ,1) (l) + 1
S(N ,1) (l)
respectively, where l = 1; ; N2 , N2 = N , 1 when N is odd and N2 = N , 2 when N is even,
(l) = (l + 1)=2 when l is odd and (l) = l=2 when l is even, and
!
!
2
k
2
k
and S = sin
:
C = cos
k
k
N
N
When N is even the last two eigenvectors are
uT2N = (0; 1; 0; ,1; 0; 1; 0; ; ,1; 0) and
uT2N +1 = (0; 0; 1; 0; ,1; 0; 1; ; 0; ,1) :
Finally, the diagonal matrix of eigenvalues is
= diag (1; 2; 3; 4; 4; ; N +2; N +2) :
17
U
The inverse of the eigenvectors 0 can be computed likewise by first computing the inverses of
each block ^ l in the Fourier domain and then setting
,1 = ^ ,1 :
1
With the same reshuffling as above we can then compute ,
0 . The resulting expressions are,
however, rather ugly and are not reproduced in this paper.
The remaining blocks of the subdivision matrix
directly follow from the usual B-spline
knot-insertion rules.
K
K
K T
U
A
0
BB 0c
BB
BB 0
S12 = BB 0
BB 0
[email protected] 0
0
1
b c 0 b c 0
C
e e 0 0 0 0C
C
c b c 0 0 0C
C
0 e e 0 0 0C
C;
CC
0 0 0 e e 0C
0 0 0 c b cC
A
0 0 0 0 e e
where
0
0
0
0
0
0 b a
0 e d
0 c b
0 0 0
0 d d
b 0 0
d 0 0
a b c
d d e
e 0 0
c b a b c 0 0
e e d d 0 0 0 0
0 1C
0 CC
0 CCC
0 CC ;
0 CCC
0A
0
3 and c = 1 :
32
64
For the case N = 3, there is no control vertex c8 (c8 = c2 ) and the second column of the matrix
S11 is equal to (0; 0; c; e; 0; c; e)T .
a=
9;
16
0
BB ec
BB
BB b
S11 = BB e
BB e
[email protected] b
b=
S
The eigenstructure of the matrix 12 can be computed manually, since this matrix has a simple
form. Its eigenvalues are:
= diag 641 ; 18 ; 161 ; 321 ; 18 ; 161 ; 321 ;
with corresponding eigenvectors:
0
BB 10
BB
BB 0
W1 = BB 0
BB 0
[email protected] 0
0
W
1
1
1
1
0
0
0
2
1
0
,1
0
0
0
11
2
,1
2
0
0
0
1
0
0
0
1
1
1
2
0
0
0
1
0
,1
The inverse 1,1 of this matrix is easily computed manually.
The other two matrices appearing in are:
00
BB 0
BB 0
BB
B0
S21 = BBB 0
BB 0
BB 0
[email protected] 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
e
f
A
0 f 0 0
0 d e 0
0 f f 0
0 e d e
0 0 f f
e d 0 0
f f 0 0
d e 0 0
f 0 0 0
0f
0 1C
BB e
0 CC
BB 0
0 CC
BB
0 CCC
B0
0 CC ; S22 = BBB 0
BB e
0 CCC
BB 0
0 CC
[email protected] 0
0A
0
0
18
1
11 C
0 C
C
0 C
C
0 C
CC :
2 C
C
,1 CA
2
1
f 0 0 f 0 0
d e 0 e 0 0C
CC
f f 0 0 0 0C
C
e d e 0 0 0C
C
CC :
0 f f 0 0 0C
e 0 0 d e 0C
C
0 0 0 f f 0C
C
A
0 0 0 e d eC
0 0 0 0 f f
B Eigenbasis Functions
x
In this appendix we compute the bi-cubic spline pieces (u; v; k ) of the eigenbasis defined in
Equation 13. The vector (u; v ) contains the 16 tensor B-spline basis functions (i = 1; ; 16):
b
bi (u; v ) = N(i,1)%4 (u)N(i,1)=4 (v );
where “%” and “=” stand for the remainder and the division respectively. The functions Ni (t) are
the uniform B-spline basis functions:
6N0 (t)
6N1 (t)
6N2 (t)
6N3 (t)
The projection matrices
vectors (see Figure 6):
= 1 , 3t + 3t2 , t3 ;
= 4 , 6t2 + 3t3;
= 1 + 3t + 3t2 , 3t3 and
= t3 :
P1, P2 and P3 are defined by introducing the following three permutation
q1 = (8; 7; 2N + 5; 2N + 13; 1; 6; 2N + 4; 2N + 12;
4; 5; 2N + 3; 2N + 11; 2N + 7; 2N + 6; 2N + 2;
2N + 10);
2
q = (1; 6; 2N + 4; 2N + 12; 4; 5; 2N + 3; 2N + 11;
2N + 7; 2N + 6; 2N + 2; 2N + 10; 2N + 16;
2N + 15; 2N + 14; 2N + 9);
3
q = (2; 1; 6; 2N + 4; 3; 4; 5; 2N + 3; 2N + 8; 2N + 7;
2N + 6; 2N + 2; 2N + 17; 2N + 16; 2N + 15;
2N + 14):
Since for the case N = 3 the vertices c2 and c8 are the same vertex, q11 = 2 instead of 8 for N = 3.
Using these permutation vectors we can compute each bi-cubic spline as follows:
xi (u; v; k) =
where i = 1; ; K and
16
X
j =1
Vqjk ;i bj (u; v );
.
V = AV
References
[1] A. A. Ball and J. T. Storry. Conditions For Tangent Plane Continuity Over Recursively
Defined B-spline Surfaces. ACM Transactions on Graphics, 7(2):83–102, April 1988.
[2] E. Catmull and J. Clark. Recursively Generated B-Spline Surfaces On Arbitrary Topological
Meshes. Computer Aided Design, 10(6):350–355, 1978.
[3] D. Doo and M. A. Sabin. Behaviour Of Recursive Subdivision Surfaces Near Extraordinary
Points. Computer Aided Design, 10(6):356–360, 1978.
19
Figure 8: The complete set of 14 eigenbasis functions for extraordinary vertices of valence N
= 3.
[4] M. Halstead, M. Kass, and T. DeRose. Efficient, Fair Interpolation Using Catmull-Clark Surfaces. In Proceedings of SIGGRAPH ’93, pages 35–44. Addison-Wesley Publishing Company, August 1993.
[5] C. T. Loop. Smooth Subdivision Surfaces Based on Triangles. M.S. Thesis, Department of
Mathematics, University of Utah, August 1987.
[6] J. Peters and U. Reif. Analysis Of Generalized B-Splines Subdivision Algorithms. To appear
in SIAM Journal of Numerical Analysis.
[7] H. Qin, C. Mandal, and C. Vemuri. Dynamic Catmull-Clark Subdivision Surfaces. IEEE
Transactions on Visualization and Computer Graphics, 4(3):215–229, July-September 1998.
[8] U. Reif. A Unified Approach To Subdivision Algorithms Near Extraordinary Vertices. Computer Aided Geometric Design, 12:153–174, 1995.
[9] J. Stam. Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter
Values. In Computer Graphics Proceedings, Annual Conference Series, 1998, pages 395–
404, July 1998.
[10] J. Warren. Subdivision Methods For Geometric Design. Unpublished Manuscript. Preprint
available on the web at
http://www.cs.rice.edu/˜jwarren/papers/book.ps.gz, 1994.
[11] D. N. Zorin. Subdivision and Multiresolution Surface Representations. PhD thesis, Caltech,
Pasadena, California, 1997.
20
Figure 9: The complete set of 18 eigenbasis functions for extraordinary vertices of valence N
N=3
= 5.
N=5
N=8
N=30
Figure 10: Surfaces having an extraordinary vertex in the center. For each surface we depict
the patches evaluated using our technique in blue. Next to them is a curvature plot. Derivative
information for curvature is also computed near the center vertex using our technique.
21
Figure 11: More complex surfaces rendered using our evaluation technique (in blue).
22
Evaluation of Loop Subdivision Surfaces
Jos Stam
Alias wavefront, Inc.
1218 Third Ave, 8th Floor,
Seattle, WA 98101, U.S.A.
[email protected]
Abstract
This paper describes a technique to evaluate Loop subdivision surfaces at arbitrary parameter values. The method is a straightforward extension of our evaluation work for Catmull-Clark
surfaces. The same ideas are applied here, with the differences being in the details only.
1 Introduction
Triangular meshes arise in many applications, such as solid modelling and finite element simulations. The ability to define a smooth surface from a given triangular mesh is therefore an important
problem. For topologically regular meshes a smooth triangular surface can be defined using box
splines [1]. In 1987 Loop generalized the recurrence relations for box splines to irregular meshes
[3]. Using his subdivision rules any triangular mesh can be refined. In the limit of an infinite
number of subdivisions a smooth surface is obtained. Away from extraordinary vertices (whose
valence N 6= 6) the surface can be parametrized using triangular Bezier patches derived from the
box splines [2]. Until recently it was believed that no parametrizations that lend themselves to
efficient evaluation existed at the extraordinary points. This paper disproves this belief. We define
parametrizations near extraordinary points and show how to evaluate them efficiently. The techniques are identical to those used in our previous work on evaluating Catmull-Clark subdivision
surfaces [5]. The differences are in the details only: different parameter domain, different subdivision rules and consequently a different eigenanalysis. We assume that the reader is familiar with
the content of [5].
The remainder of this short paper is organized as follows. The next section briefly reviews triangular Loop subdivision surfaces. Section 3 summarizes how we define and evaluate a parametrization for such surfaces. Section 4 discusses implementation details while Section 5 depicts some
results obtained using our scheme. Finally, some conclusions and possible extensions of this work
are given in Section 6. Material which is of a rather technical nature is explained in the appendices.
1
2
5
9
u=1
1
3
4 CCCCCCC
8
12
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
11
7 CCCCCCC
CCCCCCC
w=1
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCC
v=1
6
10
Figure 1: A single regular triangular patch defined by 12 control vertices.
2 Loop Subdivision Surfaces
Loop triangular splines generalize the box spline subdivision rules to meshes of arbitrary topology.
On a regular part of the mesh each triangular patch can be defined by 12 control vertices as shown
in Fig. 1. The basis functions corresponding to each of the control vertices are given in Appendix
A. We obtained these basis functions by using a conversion from box splines to triangular Bezier
patches developed by Lai [2]. This (regular) triangular patch can be denoted compactly as:
s(v; w) = C b(v; w); (v; w) 2 ;
T
C is a 12 3 matrix containing the control vertices of the patch ordered as in Fig. 1 and
b(v; w) is the vector of basis functions (see Appendix A). The surface is defined over the “unit
where
triangle”:
= f (v; w) j v 2 [0; 1] and w 2 [0; 1 , v] g:
The parameter domain is a subset of the plane such that v = 1 corresponds to the point (1; 0) and
w = 1 corresponds to the point (0; 1). We introduce the third parameter u = 1 , v , w such
that (u; v; w) forms a barycentric system of coordinates for the unit triangle. The value u = 1
corresponds to the origin (0; 0). The degree of the basis function is at most 4 in each parameter
and our surface patch is therefore a quartic spline.
The situation around an extraordinary vertex of valence N is depicted in Fig. 2. The shaded
triangle in this figure is defined by the K = N + 6 control vertices surrounding the patch. The
extraordinary vertex corresponds to the parameter value u = 1. Since the valence of the extraordinary vertex in the middle of the figure is N = 7, there are K = 13 control vertices in this case.
The figure also provides the labelling of the control vertices. We store the initial K control vertices
in a K 3 matrix
0 = ( 0 1; ; 0 ) :
C
T
c
c
;
2
;K
5
N+6
N
6
N+1
CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
N+2
2CCCCCCCCCCCC
CCCCCCCCCCCC
4
N+5
1
3
N+3
N+4
Figure 2: An irregular triangular patch defined by K = N + 6 = 13 control vertices. The vertex
labelled “1” in the middle of the figure is extraordinary of valence 7.
5
4
3
N+4
N+9
6
1
N
N+6
N+12
N+1
N+5
N+11
CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
N+2 N+10
2CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
CCCCCCCCCCCC
N+3CCCCCCCCCCCC
N+7
CCCCCCCCCCCC
N+8
Figure 3: The mesh of Fig. 2 after one Loop subdivision step. Notice that three-quarters of the
triangular patch can be evaluated.
CCCCCCCC
3
CCCCCCCC
CCCCCCCC
2
CCCCCCCC
CCCCCCCC
1
CCCCCCCC
CCCCCCCC
CCCCCCCC
1
N+1
1
2
N+3
N+5
N
N+6
N+12
N+1
N+5
N+11
CCCCCCC
3
CCCCCCC
CCCCCCC
CCCCCCC
N+2
N+10
CCCCCCC
CCCCCCC
N+7
N+3
N+4
2
CCCCCC
CCCCCC
2
CCCCCC
CCCCCC
1
N+1CCCCCC
CCCCCC
3
N+7
3
N+4
N+9
2
N+2
CCCCCC
CCCCCC
1
CCCCCC
CCCCCC
CCCCCC
N+7
N+3
CCCCCC
N+10
N+10
N+5
N+2
N+8
N+6
N
Figure 4: Three regular meshes corresponding to the three shaded patches. The labelling of the
control vertices defines the picking matrices.
3
3 Method of Evaluation
3.1 Setup
Through subdivision we can generate a new set of M = K + 6 = N + 12 control vertices as
shown in Fig. 3. Notice that we now have enough control vertices to evaluate three-quarters of the
triangular patch. We denote the new set of control vertices by:
C1 = (c1 1; ; c1 ) and
C 1 = (c1 1; ; c1 ; c1 +1; ; c1 ) :
T
;
;K
;
;K
T
;K
;M
The subdivision step in terms of these matrices is entirely described by an K K extended subdivision matrix :
1=
0;
where
!
A
C
AC
A = SS11 S012
;
(1)
and the blocks are defined in Appendix B. The additional vertices needed to evaluate the surface
are obtained from a bigger subdivision matrix :
A
0;
C 1 = AC
0
1
S
0
A = [email protected] S11 S12 CA ;
where
S
S21 S22
S
C
and 21 and 22 are defined in Appendix B. Three subsets of 12 control vertices from 1 define
three regular triangular patches which can now be evaluated. If we repeat the subdivision step, we
generate an infinite sequence of control vertices:
= ,1 = ,1 0; n 1:
C
AC
AA C
form the control vertices of a regular triangular
For each n 1 subsets of 12 vertices from C
patch. Let us denote these three sets of control vertices by the following three 12 3 matrices B ,
with k = 1; 2; 3. To compute these control vertices we introduce the 12 M “picking matrices”
P:
B = P C ; k = 1; 2; 3:
Each row of the picking matrix P is filled with zeros except for a one in the column corresponding
n
n
n
n
n;k
k
n;k
k
n
k
to the index shown in Fig. 4. Each surface patch is then defined as follows:
s (v; w) = B b(v; w) = C P b(v; w):
We seek a parametrization s(v; w) for our triangular surface for all (v; w ) 2 . As shown in Fig.
n;k
T
T
T
n;k
n
k
5 we can partition the parameter domain into an infinite set of tiles , with n 1 and k
These subdomains are defined for n 1 more precisely as:
n
h
i
h
io
1 = (v; w) j v 2 2, ; 2, +1 and w 2 0; 2, +1 , v
n
k
n
n
n
n
h
i
o
2 = (v; w) j v 2 0; 2, and w 2 [0; v]
i
h
n
h
3 = (v; w) j v 2 0; 2, and w 2 2, ; 2,
n
n
n
n
n
n
4
+1 , v
n
io
:
= 1; 2; 3.
3
3
Ω2
Ω31
Ω3
2
Ω2
2
Ω3
Ω13
1
Ω2
2
Ω1
Ω1
1
Figure 5: The parameter domain is partitioned into an infinite set of triangular tiles.
The surface patch is then defined by its restriction to each of these triangles:
,1 b(t (v; w));
s(v; w) = s (t (v; w)) = C0 P AA
(2)
where the transformation t maps the tile onto the unit tile (with the correct orientation of
T
n
k
n;k
n;k
n;k
n
T
k
n;k
n
k
Fig. 1):
t 1(v; w) = (2 v , 1; 2 w);
t 2(v; w) = (1 , 2 v; 1 , 2 w) and
t 3(v; w) = (2 v; 2 w , 1):
n
n
n;
n
n
n;
n
n
n;
Eq. 2 actually defines a parametrization for the surface patch. However, it is expensive to evaluate
since it involves taking powers of a certain matrix to any number n 1. To make the parametrization more efficient, we eigenanalyze.
3.2 Eigenstructure
When the valence N
can be diagonalized:
> 3, the extended subdivision matrix A is non-defective1 . Consequently, A
A = VV,1;
where is the diagonal matrix which contains the eigenvalues and
These matrices have the following block structure:
(3)
V contains the eigenvectors.
!
!
0
U
0
0
= 0 and V = U1 W1 :
The diagonal blocks and correspond to the eigenvalues of S and S12 , respectively, and their
corresponding eigenvectors are stored in U0 and W1 , respectively. The matrix U1 is computed by
extending the eigenvectors of S, i.e., by solving the following linear systems:
U1 , S12U1 = S11U0:
(4)
1 The
case
N = 3 has a non-trivial Jordan block and is treated in Appendix C.
5
C
In Appendix B we compute the entire eigenstructure for Loop’s scheme precisely. Let ^ 0 =
,1 0 be the projection of the initial control vertices onto the eigenspace of and let (v; w) be
the K -dimensional vector of eigenbasis functions defined by:
A
V C
P AV b(t (v; w)) n 1 and k = 1; 2; 3:
(5)
The eigenbasis functions for valences N = 5 and N = 7 are depicted in Fig. 6. Each function of
(v; w)
nk
=
,1
T
n
k
n;k
A
the eigenbasis corresponds to one of the eigenvectors of the matrix . Each eigenbasis function
is entirely defined by its restriction on the unit triangles 11 , 12 and 13 . On each of these domains
the eigenbasis is a quartic spline. The basis functions can be evaluated elsewhere since they satisfy
the following scaling relation:
(v=2; w=2) = (v; w):
The triangular surface patch can now be written solely in terms of the eigenbasis:
s(v; w) = C^ 0 (v; w):
T
(6)
In the next section we show how to implement this equation.
4 Implementation
The eigenstructures of the subdivision matrices for a meaningful range of valences have to be computed once only. Let NMAX be the maximum valence, then each eigenstructure is stored internally
in the following data structure:
typedef
struct f
double L[K];
/* eigenvalues */
double iV[K][K];
/* inverse of the eigenvectors */
double Phi[K][3][12]; /* Coefficients of the eigenbasis */
g EIGENSTRUCT;
EIGENSTRUCT eigen[NMAX];,
where K=N+6. The coefficients of the eigenbasis functions are given in the basis of Appendix A.
There are three sets of control vertices, one for each of the fundamental domains of the eigenbasis.
. The eigenstructure was computed from the
These control vertices are simply equal to
results of Appendix B and by solving the linear system defined by Eq. 4 numerically. Also, we
numerically inverted the eigenvectors without encountering any numerical instabilities. We have
included a data file called lpdata50.dat on the CDROM which contains these eigenstructures
up to NMAX=50. Also included on the CDROM is a C program which reads in and prints out the
data.
Using this eigenstructure the surface for any patch can be evaluated by first projecting the K
control vertices defining the patch into the eigenspace of the subdivision matrix with the following
routine.
P AV
k
ProjectPoints ( point *Cp, point *C, int N )
for ( i=0 ; i<N+6 ; i++ )f
Cp[i]=(0,0,0);
6
f
for ( j=0 ; j<N+6 ; j++ )f
Cp[i] += eigen[N].iV[i][j] * C[j];
g
g
g
This routine has to be called only once for a particular set of control vertices.
The evaluation at a parameter value (v,w) is performed by computing the product given in
Eq. 6.
EvalSurf ( point P, double v, double w, point *Cp, int N )
/* determine in which domain the parameter lies */
n = floor(1-log2(v+w));
pow2 = pow(2,n-1);
v *= pow2; w *= pow2;
if ( v > 0.5 ) f
k=0; v=2*v-1; w=2*w;
n
f
k
g
else if ( w > 0.5 ) f
k=2; v=2*v; w=2*w-1;
g
else f
k=1; v=1-2*v; w=1-2*w;
g
/* Now evaluate the surface */
P = (0,0,0);
for ( i=0 ; i<N+6 ; i++ ) f
P += pow(eigen[N].L[i],n-1) *
EvalBasis(eigen[N].Phi[i][k],v,w) * Cp[i];
g
g
where the routine EvalBasis evaluates a regular triangular patch using the basis of Appendix
A. To evaluate higher order derivatives, we replace EvalBasis with a function that evaluates a
derivative of the basis. In this case, the end result also must be multiplied by two to the power
n*p, where p is the order of differentiation. Therefore, the following line should be added at the
end of EvalSurf:
P = k==1 ?
pow(-2,n*p)*P : pow(2,n*p)*P;
5 Results
We have implemented our evaluation technique and have used it to compute the eigenbases for
different valences. Fig. 6 depicts the entire set of eigenbasis for valences 5 and 7. Notice that the
last 6 eigenbasis functions are the same regardless of the valence, since they depend only on the
eigenvectors of 21 , which are the same for any valence. In fact, as for Catmull-Clark surfaces,
S
7
(a)
(b)
Figure 6: Complete set of eigenbasis function for a patch of valence (a) N
8
= 5 and (b) N = 7.
(a)
(b)
(c)
(d)
Figure 7: Results created using our evaluation scheme: (a) The base mesh which contains vertices
with valences ranging from 3 to 9, (b) Isoparameter lines, (c) shaded surface and (d) Gaussian
curvature plot.
9
these eigenbasis functions are equal to simple monomials (see [5]). The eigenbasis functions
contain all the information necessary to analyze Loop subdivision surfaces.
To test our code we created the mesh shown in Figure 7.(a) which contains vertices of valence
3 to 9. Figure 7.(b) shows a closeup of the isoparameter lines generated by Loop subdivision and
evaluated using our technique. In Figure 7.(c) we evaluated both the surface and the normal. Figure
7.(d) shows a Gaussian curvature plot, where red denotes positive curvature, green flat curvature
and blue negative curvature.
6 Conclusions and Future Work
In this paper we have shown that our evaluation technique first developed for Catmull-Clark surfaces can be extended to the class of Loop subdivision surfaces. Our next step will be to present
these results in a more general setting in which Catmull-Clark and Loop are regarded as special
cases. The class of polynomial surfaces defined by Reif would be a good candidate [4].
Acknowledgments
Thanks to Michael Lounsbery for his assistance in computing the matrix
M of Appendix A.
A Regular Triangular Spline Basis Functions
The triangular surface defined by the control vertices shown in Fig. 1 can be expressed in terms of
12 basis functions. Since Loop’s scheme on the regular part of the mesh is a box spline, we can
find the corresponding Bezier patch control vertices of the triangle. Lai has developed FORTRAN
code which provides the conversion to the control vertices for the quartic triangular Bezier patches
corresponding to the box spline [2]. We have used his code (with L=2, M=2 and N=2) to get a
12 15 matrix which converts from the Bezier control vertices of the patch to the 12 control
vertices shown in Fig. 1. We get the 12 basis functions for our triangular patch by multiplying the
15 multivariate Bernstein polynomials by the matrix . Carrying out this multiplication leads to
the following result (thanks to Maple’s built in feature which converts to LaTeX):
M
M
b (v; w) = 121
4
T
u4 + 2 u3v; u4 + 2 u3w;
u + 2 u3w + 6 u3v + 6 u2vw + 12 u2 v 2 + 6 uv 2w + 6 uv 3 + 2 v 3w + v 4 ;
6 u4 + 24 u3w + 24 u2w2 + 8 uw3 + w4 + 24 u3v + 60 u2vw + 36 uvw2 +
6 vw3 + 24 u2v2 + 36 uv2w + 12 v2w2 + 8 uv3 + 6 v3w + v4;
u4 + 6u3w + 12 u2w2 + 6 uw3 + w4 + 2 u3v + 6 u2 vw + 6 uvw2 + 2 vw3;
2 uv3 + v4; u4 + 6 u3w + 12 u2w2 + 6 uw3 + w4 + 8 u3v + 36 u2vw +
36 uvw2 + 8 vw3 + 24 u2v2 + 60 uv2w + 24 v2w2 + 24 uv3 + 24 v3w + 6 v4;
u4 + 8 u3w + 24 u2w2 + 24 uw3 + 6 w4 + 6 u3v + 36 u2vw + 60 uvw2 +
24 vw3 + 12 u2v2 + 36 uv2w + 24 v2w2 + 6 uv3 + 8 v3w + v4;
2 uw3 + w4; 2 v3w + v4 ;
10
2 uw3 + w4 + 6 uvw2 + 6 vw3 + 6 uv2w + 12 v2w2 + 2 uv3 + 6 v3w + v4 ;
w4 + 2 vw3 ;
where u = 1 , v , w .
B Eigenstructure of the Subdivision Matrix
A
The subdivision matrix is composed of three blocks. The upper left block contains the “extraordinary rules” of Loop’s scheme. It is equal to
0
BB ac bc bd
B
S = BBB c d c.
[email protected]
..
c d 0
N
where
N
N
1
b b b
CC
0 0 0 0 d C
d 0 0 0 0 C
CC ;
.
b
b
N
N
N
N
...
..
0 0
0
N
d
c
CA
a = 1 , (N ); b = (N )=N; c = 3=8 and d = 1=8:
N
N
We have used the shorthand notation
(N ) =
5 , (3 + 2 cos (2=N ))2 :
8
64
If we Fourier transform the matrix we get:
0 a Nb
0
0
BB c c + 2d 0 0
BB 0
0
S^ = BBB 0 00 f (1)
0 f (2)
BB
@
0
0
0
0
N
where
N
0
0
0
0
f (N , 1)
...
1
CC
CC
CC ;
CC
CA
3 2
8 8
f (k) = + cos (2k=N ) :
The eigenvalues and the eigenvectors of the transformed matrix are trivial to compute because of
the almost-diagonal structure. They are
1 = 1; 2 =
5 , (N ); = f (1); ; 3
8
N
+1
= f (N , 1):
Notice that we have 23 = 2 . This is not surprising since Loop constructed his scheme from this
relation [3]. The eigenvalues 3 to ,1 are of multiplicity two, since f (k) = f (N , k), except
of course for the case when N is even, then 2+ 2 is only of multiplicity one. The corresponding
eigenvectors are (when stored column wise):
0 1 ,8 0 0 0 1
3
C
B
N
N=
BB 1
B
U^ 0 = BBB 00
BB
@
0
1
0
0
N
0
11
0 0 0 C
CC
1 0 0 C
0 1 0 C
CC :
..
. . . .. C
.
. A
0 0 1
S
By Fourier transforming these vectors back, we can compute the eigenvectors of the matrix . The
result is
0 1 ,8
1
0
0
0
3
B
C
1
1
1
BB 1 1
CC
BB 1 1
CC
E (1)
E (2)
E (N , 1)
U0 = BB 1 1
E (2)
E (4)
E (2(N , 1)) CCC ;
BB
..
..
...
CA
.
.
@
1 1 E (N , 1) E (2(N , 1)) E ((N , 1)(N , 1))
where E (k) = exp (2ik=N ). These are complex valued vectors. To get real-valued vectors we
N
just combine the two columns of each eigenvalue to obtain two corresponding real eigenvalues.
For example, the two real eigenvectors for the eigenvalue 3+ , k = 0; ; N , 1 are:
k
v = (0; C (0); C (k); C (2k); ; C ((N , 1)k)) and
w = (0; S (0); S (k); S (2k); ; S ((N , 1)k)) ;
T
k
T
k
where
C (k) = cos (2k=N ) and S (k) = sin (2k=N ) :
The corresponding matrix of diagonal vectors is equal to
= diag 1; 2; 3; 3; ; (
N
when N is odd, and is equal to
,1) 2 ; ( ,1) 2 ;
=
N
=
1
= diag 1; 2; 3; 3; ; 2,1; 2,1; 8 ;
N=
N=
S
when N is even. This completes the eigenanalysis of the matrix . Let us now turn to the remainder
of the matrix .
The remaining blocks of the matrix are now given.
A
0
BB 21
S12 = 161 BBBB 0
@1
0
S
0
1
0
0
0
0
1
2
0
0
0
0
0
1
0
A
1
0
0C
2
B
0C
C
BB 1
1B
0C
and
S
=
11
CC
BB 2
16
1A
@1
2
2
6
10
6
1
0
The matrix 12 has the following eigenvalues:
1
0
1
6
0
0
1
0
0
0
0
0
1 = 2 = 3 = ; and 4 = 5 = ;
8
16
i.e.,
= diag 81 ; 18 ; 81 ; 161 ; 161 :
And the corresponding eigenvectors are:
0
BB 01
W1 = BBBB 1
@0
0
1
,1 1 0 0 C
,1 1 0 1 CC
0 0 0 0C
CC :
0 1 1 0A
1 0 0 0
12
0
0
0
0
0
0
0
0
1
6
1
6 C
1 C
C
0 C
CC :
10 A
6
We point out that the following problem might occur when trying to solve Eq. 4. When N
is even, the column corresponding to the last eigenvector of gives rise to a degenerate linear
system, since the eigenvalue is 1=8. Fortunately, the system can be solved manually, and in this
case the last column of 1 is given by:
S
U
u1
= (0; 8; 0; ,8; 0) :
are
The remaining two blocks of the matrix A
0 0 3 0 0 0 0 1 1
03
BB 0 3 0 0 0 0 0 CC
BB 1
B
C
BB 0
BB 0 3 1 0 0 0 0 CC
1
1
S21 = 8 BB 0 1 0 0 0 0 3 CC and S22 = 8 BBB 3
[email protected] 0 0 0 0 0 0 3 CA
[email protected] 1
0 0 0 0 0 1 3
0
T
;N
+1
1
3
1
0
0
0
0
1
3
0
0
0
0
0
0
1
3
1
01
0C
C
CC
0C
:
0C
CC
1A
3
C Valence N = 3
When the valence of the extraordinary point is equal to three, the analysis of Section 3 breaks
down since, in that case, the extended subdivision matrix has a non-trivial Jordan block. This
means that the eigenvectors do not form a basis and the subdivision matrix cannot be diagonalized.
Fortunately, this case can be dealt with quite easily since the matrices involved are only of size
9 9. Most of the computations reported in this appendix were computed using Maple’s jordan
command. In this case the Jordan decomposition of the subdivision matrix is
A = VJV,1;
where
01
BB
BB 0
BB 0
BB
BB 0
B
J = BBB 0
BB 0
BB
BB 0
BB
[email protected] 0
0
0
1=4
0
0
0
0
0
0
0
0
0
1=4
0
0
0
0
0
0
0
0
0
1=8
0
0
0
0
0
0
0
0
0
1=8
0
0
0
0
13
0
0
0
0
0
1=8
0
0
0
0
0
0 1
C
0
0
0 C
CC
CC
0
0
0 C
C
0
0
0 C
CC
0
0
0 C
CC ;
C
0
0
0 C
CC
1=16 0
0 C
CC
CA
0 1=16 1 C
0
0 1=16
and
01 0
BB
BB 1 0
BB 1 ,1
BB
BB 1 1
B
V = BBB 1 3
BB 1 0
BB
BB 1 ,3
BB
[email protected] 1 4
1 0
0 2=5 1=5
BB
BB 0 ,1=3
BB 0 2=3
BB
BB ,8 0
B
V,1 = BBB ,4 0
BB ,8 3
BB
BB 7
26
BB 1116 33
[email protected] , 165 0
0
1
,1
0
3
4
0
0
,3
0
0
0
0
1
1
0
0
,1
0
0
0
0
,1
0
1
0
1
1=5 1=5
,1=3 2=3
,1=3 ,1=3
3
3
0
3
3
0
, 337 , 4033
0
0
0
0
0
0
0
1
1
0
0
0
1
0
1
0
0
0
0
0
0
0
0
1
0
33 1
C
,22 CCC
,22 CCC
C
,22 CCC
198 C
CC
165 473 C
CC
16
C
0 198 C
CC
165 438 C
CA
16
0 198
0
0
0
0
0
0
0
0
0
0
0
,1
0
0
0
1
1
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
,1
0
0
1
CC
CC
CC
CC
CC
CC
CC :
CC
CC
CC
CC
CA
16
16
16 16
16
165
165 , 165 165 , 165
1
, 1651 , 1651 , 1651 0 0 0
55
Using these matrices the surface can now be evaluated as in the other cases. Only the evaluation
routine has to be modified to account for the additional Jordan block. The modification relies on
the fact that powers of the Jordan block have a simple analytical expression:
1=16 1
0 1=16
!
n
,1 !
1
=
16
n=
16
=
:
0
1=16
n
n
n
With this in mind the last lines of routine EvalSurf should read:
/* Now evaluate the surface */
P = (0,0,0);
for ( i=0 ; i<N+6 ; i++ ) f
P += pow(eigen[N].L[i],n-1) *
EvalBasis(eigen[N].Phi[i][k],v,w) * Cp[i];
if ( i==N+4 && N==3 )
P += (n-1) * pow(eigen[N].L[i],n-2) *
EvalBasis(eigen[N].Phi[i][k],v,w) * Cp[i+1];
g
g
14
References
[1] C. de Boor, K. Hollig, and S. D. Riemenschneider. Box Splines. Springer Verlag, Berlin, 1993.
[2] M. J. Lai. Fortran Subroutines For B-Nets Of Box Splines On Three- and Four-Directional
Meshes. Numerical Algorithms, 2:33–38, 1992.
[3] C. T. Loop. Smooth Subdivision Surfaces Based on Triangles. M.S. Thesis, Department of
Mathematics, University of Utah, August 1987.
[4] U. Reif. A Unified Approach To Subdivision Algorithms Near Extraordinary Vertices. Computer Aided Geometric Design, 12:153–174, 1995.
[5] J. Stam. Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values. In Computer Graphics Proceedings, Annual Conference Series, 1998, pages 395–404,
July 1998.
15
Chapter 6
Implementing Subdivision and
Multiresolution Meshes
Speaker: Denis Zorin
Data structures for subdivision
In this section we briefly describe some considerations that we found useful when choosing appropriate data structures for implementing subdivision surfaces. We will consider only primal subdivision
schemes, such as Loop, Catmull-Clark or Butterly.
Representing meshes. In all cases, we need to start with data structures representing the top-level
mesh. For subdivision schemes we typically assume that the top level mesh satisfies several requirements
that allow us to apply the subdivision rules everywhere. These requirements are
no more than two triangles share an edge;
all triangles sharing a vertex form an open or closed neighborhood of the vertex; in other words,
can be arranged in such order that two sequential triangles always share an edge.
A variety of representations were proposed in the past for general meshes of this type, sometimes
with some of the assumptions relaxed, sometimes with more assumptions added, such as orientability of
the surface represented by the mesh. These representations include winged edge, quad edge, half edge
end other data structures. The most common one is the winged edge. However, this data structure is
far from being the most space efficient and convenient for subdivision. First, most data that we need to
store in a mesh, is naturally associated with vertices and polygons, not edges. Edge-based data structures
are more appropriate in the context of edge-collapse-based simplification. For subdivision, it is more
natural to consider data structures with explicit representations for faces and vertices, not for edges. One
possible and relatively simple data structure for polygons is
struct Polygon {
vector<Vertex*> vertices;
vector<Polygon*> neighbors;
vector<short> neighborEdges;
...
};
For each polygon, we store an array of pointers to vertices, an array of adjacent polygons (neighbors)
across corresponding edge numbers; and an array of adjacent edge numbers in neighbors. In addition,
if we allow nonorientable surfaces, we need to keep track of the orientation of the neighbors, which
1
can be achieved by using signed edge numbers in the array neighorEdges. To complete the mesh
representation, we add a data structure for vertices to the polygon data structure.
Let us compare this data structure to the winged edge. Let P be the number of polygons in the
mesh, V the number of vertices and E the number of edges. The storage required for the polygon-based
data structure is approximately 2:5 P VP 32-bit words, where VP is the average number of vertices per
polygon. Here we assuming that all polygons have fewer than 216 edges, so only 2 bytes are required to
store the edge number. Note that we disregard the geometric and other information stored in vertices and
polygons, counting only the memory used to maintain the data structure.
To estimate the value of 2:5 P VP in terms of V , we use the Euler formula. Recall that any mesh
satisfies V , E + P = g, where g is the genus, the number of “holes” in the surface. Assuming genus
small compared to the number of vertices, we get an approximate equation V , E + P = 0; we also
assume that the boundary vertices are a negligible fraction of the total number of vertices. Each polygon
on the average has VP vertices and the same number of edges. Each edge is shared by two polygons
which results in E = VP P=2. Let PV be the number of polygons per vertex. Then P = PV V =VP , and
E = V PV =2. This leads to
1
PV
+
1
VP
=
1
2
(1)
In addition, we know that VP , the average number of vertices per polygon, is at least 3. It follows
from (1) that PV 6. Therefore, the total memory spent in the polygon data structure is 2:5PV V 15V .
For the winged edge data structure, for each edge we use 8 pointers (4 pointers to adjacent edges, 2
pointers to adjacent faces, and two pointers to vertices); given that the total number of edges E is greater
than 3V , the total memory consumption is greater than 24V , significantly worse than the polygon data
structure.
One of the commonly mentioned advantages of the winged edge data structure is its constant size. It
is unclear if this has any consequence in the context of C++: it is relatively easy to create structures with
variable size. However, having a variety of dynamically allocated data of different small sizes may have
a negative impact on performance. We observe that after the first subdivision step all polygons will be
either triangles or quadrilaterals for all schemes that we have considered, so most of the data items will
have fixed size and the memory allocation can be easily optimized.
Hierarchical meshes: arrays vs. trees. Once a mesh is subdivided, we need to represent all the
polygons generated by subdivision. The choice of representation depends on many factors. One of the
2
important decisions to make is whether adaptive subdivision is necessary for a particular application or
not. For simplicity, lets assume that we have performed one subdivision step on an arbitrary polygonal
mesh, so we always start with a mesh consisting only of quads or triangles. If we assume that we
need only uniform subdivision, all vertices and polygons of each subdivided top-level quad or triangle
can be represented as a two-dimensional array. Thus, the complete data structure would consist of a
representation of a top level mesh, with each top level face containing a 2D array of vertex pointers.
The pointers on the border between two top-level neighbors point pairwise to the same vertices. The
advantage of this data structure is that it has practically no pointer overhead. The disadvantage is that
even though it is possible to do adaptive subdivision on this structure, a lot of space will be wasted.
If we do want adaptive subdivision and maintain efficient storage, the alternative is to use a tree
structure; each non-leaf polygon becomes a node in the tree, containing a pointer to a block of N children,
with N being 4 for non-top-level polygons (both for triangular and quadrilateral schemes). Similarly, the
number of vertices M is 3 or 4:
struct PolygonNode {
Vertex* v[M];
PolygonNode* ptr;
...
};
To compare the two approaches to organizing the hierarchies (arrays and trees), we need to compare
the representation overhead in these two cases. In the first case (arrays) all adjacency relations are
implicit, and there is no overhead. In the second case, there is overhead in the form of pointers to children
and vertices. For a given number of subdivision steps n the total overhead can be easily estimated. We
will consider triangular meshes; the estimates will be similar for quadrilateral meshes. For the purposes
of the estimate we can assume that the genus is 0, so the number of triangles P, the number of edges
E and the number of vertices V in the initial mesh are related by P , E + V = 0. The total number of
triangles in a complete tree P(4n+1 , 1)=3. For the triangular mesh VP = 3, and (1) yields PV = 6; thus,
the total number of triangles is P = 2V , and the total number of edges is E = 3V .
For each leaf and non-leaf node we need 4 words (1 pointer to the block of children and three
pointers to vertices). The total cost of the structure is 4P(4n+1 , 1)=3 = 8V (4n+1 , 1)=3 words, which is
approximately 11 V 4n .
To estimate when a tree is spatially more efficient than an array, we determine how many nodes have
to be removed from the tree for the gain from the adaptivity to exceed the loss from the overhead. For
3
this, we need a reasonable estimate of the size of the useful data stored in the structures – otherwise, the
array will always win.
The number of vertices inserted on subdivision step i is approximately 3 4i,1V . Assuming that for
each vertex we store all control points on all subdivision levels, and each control point takes 3 words, we
get the following estimate for the control point storage
3V
,
(n + 1) + 3n + 3
42(n , 1) +
:::
4n
=V
,
4n+1 , 1
The total number of vertices is V 4n ; assuming that at each vertex we store the normal vector, the
limit position vector (3 words), color (3 words) and some extra information, such as subdivision tags (1
word), we get 7V 4n more words. The total useful storage is approximately 11 V 4n , the same as the
cost of the structure.
Thus for our example the tree introduces a 100% overhead, which implies that it has an advantage
over the array if at least half of the nodes are absent. Wether this will happen, depends on the criterion
for adaptation. If the criterion attempts to measure how well the surface approximates the geometry,
and if only 3 or 4 subdivision levels are used, we have observed that fewer than 50% of the nodes were
removed. However, if different criteria are used (e.g. distance to the camera) the situation is likely to
be radically different. If more subdivision levels are used it is likely that almost all nodes on the bottom
level are absent.
4
Interactive Multiresolution Mesh Editing
Denis Zorin
Caltech
Peter Schrödery
Caltech
Abstract
We describe a multiresolution representation for meshes based on
subdivision, which is a natural extension of the existing patch-based
surface representations. Combining subdivision and the smoothing algorithms of Taubin [26] allows us to construct a set of algorithms for interactive multiresolution editing of complex hierarchical meshes of arbitrary topology. The simplicity of the underlying algorithms for refinement and coarsification enables us to make
them local and adaptive, thereby considerably improving their efficiency. We have built a scalable interactive multiresolution editing
system based on such algorithms.
1
Wim Sweldensz
Bell Laboratories
arbitrary topology setting and across a continuous range of scales
and hardware resources.
Introduction
Applications such as special effects and animation require creation
and manipulation of complex geometric models of arbitrary topology. Like real world geometry, these models often carry detail at
many scales (cf. Fig. 1). The model might be constructed from
scratch (ab initio design) in an interactive modeling environment or
be scanned-in either by hand or with automatic digitizing methods.
The latter is a common source of data particularly in the entertainment industry. When using laser range scanners, for example, individual models are often composed of high resolution meshes with
hundreds of thousands to millions of triangles.
Manipulating such fine meshes can be difficult, especially when
they are to be edited or animated. Interactivity, which is crucial in
these cases, is challenging to achieve. Even without accounting for
any computation on the mesh itself, available rendering resources
alone, may not be able to cope with the sheer size of the data. Possible approaches include mesh optimization [15, 13] to reduce the
size of the meshes.
Aside from considerations of economy, the choice of representation is also guided by the need for multiresolution editing semantics. The representation of the mesh needs to provide control at a large scale, so that one can change the mesh in a broad,
smooth manner, for example. Additionally designers will typically also want control over the minute features of the model (cf.
Fig. 1). Smoother approximations can be built through the use of
patches [14], though at the cost of loosing the high frequency details. Such detail can be reintroduced by combining patches with
displacement maps [17]. However, this is difficult to manage in the
[email protected]
y [email protected]
z [email protected]
Figure 1: Before the Armadillo started working out he was flabby,
complete with a double chin. Now he exercises regularly. The original is on the right (courtesy Venkat Krischnamurthy). The edited
version on the left illustrates large scale edits, such as his belly, and
smaller scale edits such as his double chin; all edits were performed
at about 5 frames per second on an Indigo R10000 Solid Impact.
For reasons of efficiency the algorithms should be highly adaptive and dynamically adjust to available resources. Our goal is to
have a single, simple, uniform representation with scalable algorithms. The system should be capable of delivering multiple frames
per second update rates even on small workstations taking advantage of lower resolution representations.
In this paper we present a system which possesses these properties
Multiresolution control: Both broad and general handles, as
well as small knobs to tweak minute detail are available.
Speed/fidelity tradeoff: All algorithms dynamically adapt to
available resources to maintain interactivity.
Simplicity/uniformity: A single primitive, triangular mesh, is
used to represent the surface across all levels of resolution.
Our system is inspired by a number of earlier approaches. We
mention multiresolution editing [11, 9, 12], arbitrary topology subdivision [6, 2, 19, 7, 28, 16], wavelet representations [21, 24, 8, 3],
and mesh simplification [13, 17]. Independently an approach similar to ours was developed by Pulli and Lounsbery [23].
It should be noted that our methods rely on the finest level mesh
having subdivision connectivity. This requires a remeshing step before external high resolution geometry can be imported into the editor. Eck et al. [8] have described a possible approach to remeshing
arbitrary finest level input meshes fully automatically. A method
that relies on a user's expertise was developed by Krishnamurthy
and Levoy [17].
1.1 Earlier Editing Approaches
H-splines were presented in pioneering work on hierarchical
editing by Forsey and Bartels [11]. Briefly, H-splines are obtained
by adding finer resolution B-splines onto an existing coarser resolution B-spline patch relative to the coordinate frame induced by the
coarser patch. Repeating this process, one can build very complicated shapes which are entirely parameterized over the unit square.
Forsey and Bartels observed that the hierarchy induced coordinate
frame for the offsets is essential to achieve correct editing semantics.
H-splines provide a uniform framework for representing both the
coarse and fine level details. Note however, that as more detail
is added to such a model the internal control mesh data structures
more and more resemble a fine polyhedral mesh.
While their original implementation allowed only for regular
topologies their approach could be extended to the general setting
by using surface splines or one of the spline derived general topology subdivision schemes [18]. However, these schemes have not
yet been made to work adaptively.
Forsey and Bartels' original work focused on the ab initio design setting. There the user's help is enlisted in defining what is
meant by different levels of resolution. The user decides where to
add detail and manipulates the corresponding controls. This way
the levels of the hierarchy are hand built by a human user and the
representation of the final object is a function of its editing history.
To edit an a priori given model it is crucial to have a general procedure to define coarser levels and compute details between levels.
We refer to this as the analysis algorithm. An H-spline analysis algorithm based on weighted least squares was introduced [10], but
is too expensive to run interactively. Note that even in an ab initio
design setting online analysis is needed, since after a long sequence
of editing steps the H-spline is likely to be overly refined and needs
to be consolidated.
Wavelets provide a framework in which to rigorously define multiresolution approximations and fast analysis algorithms.
Finkelstein and Salesin [9], for example, used B-spline wavelets
to describe multiresolution editing of curves. As in H-splines, parameterization of details with respect to a coordinate frame induced
by the coarser level approximation is required to get correct editing semantics. Gortler and Cohen [12], pointed out that wavelet
representations of detail tend to behave in undesirable ways during
editing and returned to a pure B-spline representation as used in
H-splines.
Carrying these constructions over into the arbitrary topology surface framework is not straightforward. In the work by Lounsbery et
al. [21] the connection between wavelets and subdivision was used
to define the different levels of resolution. The original constructions were limited to piecewise linear subdivision, but smoother
constructions are possible [24, 28].
An approach to surface modeling based on variational methods
was proposed by Welch and Witkin [27]. An attractive characteristic of their method is flexibility in the choice of control points.
However, they use a global optimization procedure to compute the
surface which is not suitable for interactive manipulation of complex surfaces.
Before we proceed to a more detailed discussion of editing we
first discuss different surface representations to motivate our choice
of synthesis (refinement) algorithm.
1.2 Surface Representations
There are many possible choices for surface representations.
Among the most popular are polynomial patches and polygons.
Patches are a powerful primitive for the construction of coarse
grain, smooth models using a small number of control parameters.
Combined with hardware support relatively fast implementations
are possible. However, when building complex models with many
patches the preservation of smoothness across patch boundaries can
be quite cumbersome and expensive. These difficulties are compounded in the arbitrary topology setting when polynomial parameterizations cease to exist everywhere. Surface splines [4, 20, 22]
provide one way to address the arbitrary topology challenge.
As more fine level detail is needed the proliferation of control
points and patches can quickly overwhelm both the user and the
most powerful hardware. With detail at finer levels, patches become
less suited and polygonal meshes are more appropriate.
Polygonal Meshes can represent arbitrary topology and resolve fine detail as found in laser scanned models, for example.
Given that most hardware rendering ultimately resolves to triangle
scan-conversion even for patches, polygonal meshes are a very basic primitive. Because of sheer size, polygonal meshes are difficult
to manipulate interactively. Mesh simplification algorithms [13]
provide one possible answer. However, we need a mesh simplification approach, that is hierarchical and gives us shape handles
for smooth changes over larger regions while maintaining high frequency details.
Patches and fine polygonal meshes represent two ends of a spectrum. Patches efficiently describe large smooth sections of a surface
but cannot model fine detail very well. Polygonal meshes are good
at describing very fine detail accurately using dense meshes, but do
not provide coarser manipulation semantics.
Subdivision connects and unifies these two extremes.
Figure 2: Subdivision describes a smooth surface as the limit of a
sequence of refined polyhedra. The meshes show several levels of
an adaptive Loop surface generated by our system (dataset courtesy
Hugues Hoppe, University of Washington).
Subdivision defines a smooth surface as the limit of a sequence
of successively refined polyhedral meshes (cf. Fig. 2). In the regular patch based setting, for example, this sequence can be defined
through well known knot insertion algorithms [5]. Some subdivision methods generalize spline based knot insertion to irregular
topology control meshes [2, 6, 19] while other subdivision schemes
are independent of splines and include a number of interpolating
schemes [7, 28, 16].
Since subdivision provides a path from patches to meshes, it can
serve as a good foundation for the unified infrastructure that we
seek. A single representation (hierarchical polyhedral meshes) supports the patch-type semantics of manipulation and finest level detail polyhedral edits equally well. The main challenge is to make
the basic algorithms fast enough to escape the exponential time and
space growth of naive subdivision. This is the core of our contribution.
We summarize the main features of subdivision important in our
context
Topological Generality: Vertices in a triangular (resp. quadrilateral) mesh need not have valence 6 (resp. 4). Generated surfaces are smooth everywhere, and efficient algorithms exist for
computing normals and limit positions of points on the surface.
Multiresolution: because they are the limit of successive refinement, subdivision surfaces support multiresolution algorithms,
such as level-of-detail rendering, multiresolution editing, compression, wavelets, and numerical multigrid.
Simplicity: subdivision algorithms are simple: the finer mesh
is built through insertion of new vertices followed by local
smoothing.
Maps to
Graph with vertices
Uniformity of Representation: subdivision provides a single
representation of a surface at all resolution levels. Boundaries
and features such as creases can be resolved through modified
rules [14, 25], reducing the need for trim curves, for example.
Initial mesh
Adaptive analysis
Adaptive render
Adaptive synthesis
Render
Begin dragging
Select group of vertices
at level i
Create dependent
submesh
Release selection
Drag
Local analysis
Local synthesis
Render
Figure 3: The relationship between various procedures as the user
moves a set of vertices.
2
Subdivision
We begin by defining subdivision and fixing our notation. There are
2 points of view that we must distinguish. On the one hand we are
dealing with an abstract graph and perform topological operations
on it. On the other hand we have a mesh which is the geometric
object in 3-space. The mesh is the image of a map defined on the
graph: it associates a point in 3D with every vertex in the graph
(cf. Fig. 4). A triangle denotes a face in the graph or the associated
polygon in 3-space.
Initially we have a triangular graph T 0 with vertices V 0 . By
recursively refining each triangle into 4 subtriangles we can build
a sequence of finer triangulations T i with vertices V i , i > 0
(cf. Fig. 4). The superscript i indicates the level of triangles and
vertices respectively. A triangle t 2 T i is a triple of indices
t = fva ; vb; vc g V i .
The vertex sets are nested as V j V i if j < i. We define
odd vertices on level i as M i = V i+1 n V i . V i+1 consists of two
disjoint sets: even vertices (V i ) and odd vertices (M i ). We define
the level of a vertex v as the smallest i for which v 2 V i . The level
of v is i + 1 if and only if v 2 M i .
refinement
Aside from our perspective, which unifies the earlier approaches,
our major contribution—and the main challenge in this program—
is the design of highly adaptive and dynamic data structures and
algorithms, which allow the system to function across a range of
computational resources from PCs to workstations, delivering as
much interactive fidelity as possible with a given polygon rendering performance. Our algorithms work for the class of 1-ring subdivision schemes (definition see below) and we demonstrate their
performance for the concrete case of Loop's subdivision scheme.
The particulars of those algorithms will be given later, but Fig. 3
already gives a preview of how the different algorithms make up
the editing system. In the next sections we first talk in more detail
about subdivision, smoothing, and multiresolution transforms.
3
i
T
i
s (3)
i
i
1
V
2
3
i+1
T
6
s (1)
i
s (2)
s (6)
s (3)
5
i+1
s (5)
i+1
1
i+1
i+1
i+1
4
subdivision
1.3 Our Contribution
V
Mesh with points
2
i+1
s (1) s (4)
i+1
s (2)
Figure 4: Left: the abstract graph. Vertices and triangles are members of sets V i and T i respectively. Their index indicates the level
of refinement when they first appeared. Right: the mapping to the
mesh and its subdivision in 3-space.
With each set V i we associate a map, i.e., for each vertex v and
each level i we have a 3D point si (v) 2 3 . The set si contains
all points on level i, si = fsi (v) j v 2 V i g. Finally, a subdivision
scheme is a linear operator S which takes the points from level i to
points on the finer level i + 1: si+1 = S si
Assuming that the subdivision converges, we can define a limit
surface as
R
=
k 0
lim
!1 S s :
k
(v) 2 R3 denotes the point on the limit surface associated with
vertex v.
In order to define our offsets with respect to a local frame we also
need tangent vectors and a normal. For the subdivision schemes
that we use, such vectors can be defined through the application of
linear operators Q and R acting on si so that qi (v) = (Qsi )(v)
and ri (v) = (Rsi )(v) are linearly independent tangent vectors at
(v). Together with an orientation they define a local orthonormal
frame F i (v) = (ni (v); qi (v); ri (v)). It is important to note that
in general it is not necessary to use precise normals and tangents
during editing; as long as the frame vectors are affinely related to
the positions of vertices of the mesh, we can expect intuitive editing
behavior.
1-ring at level i
1-ring at level i+1
Figure 5: An even vertex has a 1-ring of neighbors at each level of
refinement (left/middle). Odd vertices—in the middle of edges—
have 1-rings around each of the vertices at either end of their edge
(right).
Next we discuss two common subdivision schemes, both of
which belong to the class of 1-ring schemes. In these schemes
points at level i + 1 depend only on 1-ring neighborhoods of points
at level i. Let v V i (v even) then the point si+1 (v) is a function
of only those si (vn ), vn
V i , which are immediate neighbors
of v (cf. Fig. 5 left/middle). If m M i (m odd), it is the vertex
inserted when splitting an edge of the graph; we call such vertices
middle vertices of edges. In this case the point si+1 (m) is a function of the 1-rings around the vertices at the ends of the edge (cf.
Fig. 5 right).
2
2
2
1
1
1
1
1
a(k)
3
3
1
1
1
Figure 6: Stencils for Loop subdivision with unnormalized weights
for even and odd vertices.
Loop is a non-interpolating subdivision scheme based on a generalization of quartic triangular box splines [19]. For a given even
vertex v
V i , let vk V i with
1 k K be its K 1ring neighbors. The new point si+1 (v) is defined as si+1 (v) =
(a(K ) + K ),1(a(K ) si (v) + Kk=1 si (vk )) (cf. Fig. 6), a(K ) =
K (1 (K ))=(K ), and (K ) = 5=8 (3+2 cos(2=K ))2 =64.
For odd v the weights shown in Fig. 6 are used. Two independent tangent vectors t1 (v) and t2 (v) are given by tp (v) =
K cos(2(k + p)=K ) si (vk ).
k=1
Features such as boundaries and cusps can be accommodated
through simple modifications of the stencil weights [14, 25, 29].
P
2
,
P
,
Butterfly is an interpolating scheme, first proposed by Dyn et
al. [7] in the topologically regular setting and recently generalized to arbitrary topologies [28]. Since it is interpolating we have
si (v) = (v) for v V i even. The exact expressions for odd
vertices depend on the valence K and the reader is referred to the
original paper for the exact values [28].
2
For our implementation we have chosen the Loop scheme, since
more performance optimizations are possible in it. However, the
algorithms we discuss later work for any 1-ring scheme.
3
Multiresolution Transforms
So far we only discussed subdivision, i.e., how to go from coarse to
fine meshes. In this section we describe analysis which goes from
fine to coarse.
We first need smoothing, i.e., a linear operation H to build a
smooth coarse mesh at level i 1 from a fine mesh at level i:
,
si,1 = H si :
Several options are available here:
Least squares: One could define analysis to be optimal in the
least squares sense,
min
ksi , S si,1 k2 :
si,1
2
L
The solution may have unwanted undulations and is too expensive to compute interactively [10].
Fairing: A coarse surface could be obtained as the solution to
a global variational problem. This is too expensive as well. An
alternative is presented by Taubin [26], who uses a local nonshrinking smoothing approach.
,
2P
H := (I + L) (I + L):
With subdivision and smoothing in place, we can describe the
transform needed to support multiresolution editing. Recall that
for multiresolution editing we want the difference between successive levels expressed with respect to a frame induced by the coarser
level, i.e., the offsets are relative to the smoother level.
With each vertex v and each level i > 0 we associate a detail
3 . The set di contains all detail vectors on level i,
vector, di (v)
di = di (v) v V i . As indicated in Fig. 7 the detail vectors
are defined as
f
1
2
Because of its computational simplicity we decided to use a version
of Taubin smoothing. As before let v
V i have K neighbors
K si (vk ), to define
i
i
,
1
vk V . Use the average, s (v) = K
k=1
i
i
the discrete Laplacian (v) = s (v) s (v). On this basis Taubin
gives a Gaussian-like smoother which does not exhibit shrinkage
2R
j 2 g
di = (F i )t (si , S si,1 ) = (F i )t (I , S H ) si ;
i.e., the detail vectors at level i record how much the points at level
i differ from the result of subdividing the points at level i , 1. This
difference is then represented with respect to the local frame F i to
obtain coordinate independence.
Since detail vectors are sampled on the fine level mesh V i , this
transformation yields an overrepresentation in the spirit of the BurtAdelson Laplacian pyramid [1]. The only difference is that the
smoothing filters (Taubin) are not the dual of the subdivision filter
(Loop). Theoretically it would be possible to subsample the detail
vectors and only record a detail per odd vertex of M i,1 . This is
what happens in the wavelet transform. However, subsampling the
details severely restricts the family of smoothing operators that can
be used.
s
Smoothing
s
i
i-1
Subdivision
i
s -Ss
i-1
i t
(F )
d
i
Figure 7: Wiring diagram of the multiresolution transform.
4 Algorithms and Implementation
Before we describe the algorithms in detail let us recall the overall
structure of the mesh editor (cf. Fig 3). The analysis stage builds
a succession of coarser approximations to the surface, each with
fewer control parameters. Details or offsets between successive
levels are also computed. In general, the coarser approximations
are not visible; only their control points are rendered. These control points give rise to a virtual surface with respect to which the
remaining details are given. Figure 8 shows wireframe representations of virtual surfaces corresponding to control points on levels 0,
1, and 2.
When an edit level is selected, the surface is represented internally as an approximation at this level, plus the set of all finer level
details. The user can freely manipulate degrees of freedom at the
edit level, while the finer level details remain unchanged relative
to the coarser level. Meanwhile, the system will use the synthesis
algorithm to render the modified edit level with all the finer details
added in. In between edits, analysis enforces consistency on the
internal representation of coarser levels and details (cf. Fig. 9).
The basic algorithms Analysis and Synthesis are very
simple and we begin with their description.
Let i = 0 be the coarsest and i = n the finest level with N
vertices. For each vertex v and all levels i finer than the first level
thresholds. Three thresholds control this pruning: A for adaptive
analysis, S for adaptive synthesis, and R for adaptive rendering.
To make lazy evaluation fast enough several caches are maintained
explicitly and the order of computations is carefully staged to avoid
recomputation.
4.1 Adaptive Analysis
Figure 8: Wireframe renderings of virtual surfaces representing the
first three levels of control points.
The generic version of analysis traverses entire levels of the hierarchy starting at some finest level. Recall that the purpose of analysis
is to compute coarser approximations and detail offsets. In many
regions of a mesh, for example, if it is flat, no significant details
will be found. Adaptive analysis avoids the storage cost associated
with detail vectors below some threshold A by observing that small
detail vectors imply that the finer level almost coincides with the
subdivided coarser level. The storage savings are realized through
tree pruning.
For this purpose we need an integer v:nest
:=
maxi v:d[i]
A . Initially v:nest = n and the following precondition holds before calling Analysis(i):
The surface is uniformly subdivided to level i,
v V i : v:s[i] = si (v),
v V i i < j v:nest : v:d[j ] = dj (v).
Now Analysis(i) becomes:
fk
Figure 9: Analysis propagates the changes on finer levels to coarser
levels, keeping the magnitude of details under control. Left: The
initial mesh. Center: A simple edit on level 3. Right: The effect of
the edit on level 2. A significant part of the change was absorbed
by higher level details.
where the vertex v appears, there are storage locations v:s[i] and
v:d[i], each with 3 floats. With this the total storage adds to 2 3
(4N=3) floats. In general, v:s[i] holds si (v) and v:d[i] holds di (v);
temporarily, these locations can be used to store other quantities.
The local frame is computed by calling v:F (i).
Global analysis and synthesis are performed level wise:
Analysis
Synthesis
for i = n downto 1
Analysis(i)
for i = 1 to n
Synthesis(i)
With the action at each level described by
Analysis(i)
8v 2 V ii,1 :
8v 2 V :
8 2
8 2
k j
g
Analysis(i)
8v 2 V ii,1 : v:s[i , 1] := smooth (v; i)
8v 2 V :
v:d[i] := v:s[i] , subd(v; i , 1)
if v:nest > i or kv:d[i]k A then
v:d[i] := v:F (i)t v:d[i]
else
v:nest := i , 1
Prune(i , 1)
Triangles that do not contain details above the threshold are unrefined:
Prune(i)
v:s[i , 1] := smooth (v; i)
v:d[i] := v:F (i)t (v:s[i] , subd(v; i , 1))
8t 2 T i : If all middle vertices m have m:nest = i , 1
and all children are leaves, delete children.
This results in an adaptive mesh structure for the surface with
v:d[i] = di (v) for all v V i , i v:nest. Note that the resulting mesh is not restricted, i.e., two triangles that share a vertex
can differ in more than one level. Initial analysis has to be followed
by a synthesis pass which enforces restriction.
2
and
Synthesis(i)
8v 2 V i : s:v[i] := v:F (i) v:d[i] + subd(v; i , 1)
,
Analysis computes points on the coarser level i 1 using smoothing (smooth), subdivides si,1 (subd), and computes the detail
vectors di (cf. Fig. 7). Synthesis reconstructs level i by subdividing
level i 1 and adding the details.
So far we have assumed that all levels are uniformly refined, i.e.,
all neighbors at all levels exist. Since time and storage costs grow
exponentially with the number of levels, this approach is unsuitable
for an interactive implementation. In the next sections we explain
how these basic algorithms can be made memory and time efficient.
Adaptive and local versions of these generic algorithms (cf.
Fig. 3 for an overview of their use) are the key to these savings.
The underlying idea is to use lazy evaluation and pruning based on
,
4.2 Adaptive Synthesis
The main purpose of the general synthesis algorithm is to rebuild
the finest level of a mesh from its hierarchical representation. Just
as in the case of analysis we can get savings from noticing that in
flat regions, for example, little is gained from synthesis and one
might as well save the time and storage associated with synthesis. This is the basic idea behind adaptive synthesis, which has two
main purposes. First, ensure the mesh is restricted on each level,
(cf. Fig. 10). Second, refine triangles and recompute points until
the mesh has reached a certain measure of local flatness compared
against the threshold S .
The algorithm recomputes the points si (v) starting from the
coarsest level. Not all neighbors needed in the subdivision stencil
of a given point necessarily exist. Consequently adaptive synthesis
Refine(t; i; dir )
V
V
i
V
i
8 2
V
i
lazily creates all triangles needed for subdivision by temporarily refining their parents, then computes subdivision, and finally deletes
the newly created triangles unless they are needed to satisfy the
restriction criterion. The following precondition holds before entering AdaptiveSynthesis:
8t 2 T j j 0 j i : t is restricted
8v 2 V j j 0 j v:depth : v:s[j ] = sj (v)
fsi (v)has been recomputedg.
Delete children of t
then
Update(v; i)
v:s[i + 1] := subd(v; i)
v:depth := i + 1
if v:nest i + 1 then
v:s[i + 1] += v:F (i + 1) v:d[i + 1]
The condition v:depth = i + 1 indicates whether an earlier call to
Refine already recomputed si+1 (v). If not, call GetRing (v; i)
and Update (v; i) to do so. In case a detail vector lives at v at level
i (v:nest i + 1) add it in. Next compute si+1 (m) for middle vertices on level i + 1 around v (m
N (v; i + 1; 1), where
N (v; i; l) is the l-ring neighborhood of vertex v at level i). If m
has to be calculated, compute subd(m; i) and add in the detail if it
exists and record this fact in the flag forced which will prevent unrefinement later. At this point, all si+1 have been recomputed for the
vertices and middle vertices of t. Unrefine t and delete its children
if Refine was called directly, the triangle is sufficiently flat, and
none of the middle vertices contain details (i.e., forced = false).
The list current functions as a cache holding triangles from level
i 1 which are temporarily refined to build a 1-ring around the
vertices of t. If after processing all vertices and middle vertices of
t it is decided that t will remain refined, none of the coarser-level
triangles from current can be unrefined without violating restriction. Thus t:restrict is set for all of them. The function Flat(t)
measures how close to planar the corners and edge middle vertices
of t are.
2
,
AdaptiveSynthesis
8v 2 V 0 : v:depth := 0
for i = 0 to n , 1
temptri := fg
8t 2 T i :
current := fg
Refine (t; i; true )
8t 2 temptri : if not t:restrict
forced
8 2
Figure 10: A restricted mesh: the center triangle is in T i and its
vertices in V i . To subdivide it we need the 1-rings indicated by the
circular arrows. If these are present the graph is restricted and we
can compute si+1 for all vertices and middle vertices of the center
triangle.
where v:depth := maxi
t:leaf
8v 2 t : if v:depth < i + 1
V i+1
V i+1
T
then Create children for t
then
GetRing (v; i)
Update (v; i)
m N (v; i + 1; 1) :
Update (m; i)
if m:nest i + 1 then
forced := true
if dir and Flat(t) < S and not
Delete children of t
else
t current : t:restrict := true
if
i
i+1
then
Finally, GetRing(v; i) ensures that a complete ring of triangles
on level i adjacent to the vertex v exists. Because triangles on level
i are restricted triangles all triangles on level i 1 that contain v
exist (precondition). At least one of them is refined, since otherwise there would be no reason to call GetRing(v; i). All other
triangles could be leaves or temporarily refined. Any triangle that
was already temporarily refined may become permanently refined
to enforce restriction. Record such candidates in the current cache
for fast access later.
,
The list temptri serves as a cache holding triangles from levels
j < i which are temporarily refined. A triangle is appended to the
list if it was refined to compute a value at a vertex. After processing
level i these triangles are unrefined unless their t:restrict flag is
set, indicating that a temporarily created triangle was later found
to be needed permanently to ensure restriction. Since triangles are
appended to temptri , parents precede children. Deallocating the
list tail first guarantees that all unnecessary triangles are erased.
The function Refine(t; i; dir ) (see below) creates children of
T i and computes the values Ssi (v) for the vertices and mid-
t2
dle vertices of t. The results are stored in v:s[i + 1]. The boolean
argument dir indicates whether the call was made directly or recursively.
GetRing(v; i)
8t 2 T i,1 with v 2 t :
if t:leaf then
Refine (t; i , 1; false ); temptri :append(t)
t:restrict := false; t:temp := true
t:temp then
current :append(t)
if
4.3 Local Synthesis
Even though the above algorithms are adaptive, they are still run everywhere. During an edit, however, not all of the surface changes.
The most significant economy can be gained from performing analysis and synthesis only over submeshes which require it.
Assume the user edits level l and modifies the points sl (v) for
v V l V l . This
invalidates coarser level values si and di for
i
certain subsets V
V i , i l, and finer level points si for subsets
V i V i for i > l. Finer level detail vectors di for i > l remain
correct by definition. Recomputing the coarser levels is done by
local incremental analysis described in Section 4.4, recomputing
the finer level is done by local synthesis described in this section.
The set of vertices V i which are affected depends on the support
of the subdivision scheme. If the support fits into an m-ring around
the computed vertex, then all modified vertices on level i + 1 can
be found recursively as
2
[ N v; i
V i+1 =
v2V i
(
+1
; m):
We assume that m = 2 (Loop-like schemes) or m = 3 (Butterfly
type schemes). We define the subtriangulation T i to be the subset
of triangles of T i with vertices in V i .
LocalSynthesis is only slightly modified from
AdaptiveSynthesis: iteration starts at level l and iterates only over the submesh T i .
4.4 Local Incremental Analysis
After an edit on level l local incremental analysis will recompute
si (v) and di (v) locally for coarser level vertices (i l) which are
affected by the edit. As in the previous section, we assume that
the user edited a set of vertices v on level l and call V i the set of
vertices affected on level i. For a given vertex v V i we define
2
v
3
v
v
2
f1
v
4
v
1
ve
v
v
2
ve
v
v
f2
7
v
6
Figure 11: Sets of even vertices affected through smoothing by either an even v or odd m vertex.
Ri,1 (v)
V i,1 to be the set of vertices on level i , 1 affected
by v through the smoothing operator H . The sets V i can now be
[R
defined recursively starting from level i = l to i = 0:
V i,1 =
i,1 (v):
v2V i
i
,
1
The set R (v) depends on the size of the smoothing stencil and
whether v is even or odd (cf. Fig. 11). If the smoothing filter
is 1-ring, e.g., Gaussian, then Ri,1 (v) = v if v is even and
Ri,1 (m) = ve1 ; ve2 if m is odd. If the smoothing filter is 2vk 1 k K
ring, e.g., Taubin, then Ri,1 (v) = v
if v is even and Ri,1 (m) = ve1 ; ve2; vf 1 ; vf 2 if v is odd. BeV i and
cause of restriction, these vertices always exist. For v
0
i
,
1
0
v R (v) we let c(v; v ) be the coefficient in the analysis stencil. Thus
f
fg
g
f
2
H si )(v0 ) =
(
X
f g[f j g
g
2
vjv0 2Ri,1 (v)
c(v; v0 )si (v):
2
c(v; v) = (1 , ) (1 , ) + =6
c(v; vk ) = =6K
c(m; ve1 ) = ((1 , ) + (1 , ) + =3)=K
c(m; vf 1 ) = =3K;
where for each c(v; v0 ), K is the outdegree of v0 .
The algorithm first copies the old points si (v) for v 2 V i and
i l into the storage location for the detail. If then propagates
the incremental changes of the modified points from level l to the
coarser levels and adds them to the old points (saved in the detail
locations) to find the new points. Then it recomputes the detail
vectors that depend on the modified points.
We assume that before the edit, the old points sl (v) for v
V l were saved
in the detail locations. The algorithm starts out by
building V i,1 and saving the points si,1 (v) for v
V i,1 in
the detail locations. Then the changes resulting from the edit are
propagated to level i 1. Finally S si,1 is computed and used to
update the detail vectors on level i.
2
2
,
LocalAnalysis(i)
8v 2iV, i : 8v0 02 Ri, (v) :
V0 [= fv g 0
v :d[i , 1] := v :s[i , 1]
8v 20 V i : 8v0 2 Ri, 0 (v) :
v :s[i , 1] += c(v; v ) (v:s[i] , v:d[i])
8v 2 V i, : t
v:d[i] = v:F (i) (v:s[i] , subd(v; i , 1))
8m 2 N (v;i; 1) : t
m:d[i] = m:F (i) (m:s[i] , subd(m;i , 1))
1
1
1
1
1
v
5
This could be implemented by running over the v0 and each time
computing the above sum. Instead we use the dual implementation,
iterate over all v, accumulating (+=) the right amount to si (v0 ) for
v0 Ri,1 (v). In case of a 2-ring Taubin smoother the coefficients
are given by
Note that the odd points are actually computed twice. For the Loop
scheme this is less expensive than trying to compute a predicate to
avoid this. For Butterfly type schemes this is not true and one can
avoid double computation by imposing an ordering on the triangles.
The top level code is straightforward:
LocalAnalysis
8v 2 V l : v:d[l] := v:s[l]
for i := l downto 0
LocalAnalysis(i)
It is difficult to make incremental local analysis adaptive, as it is
formulated purely in terms of vertices. It is, however, possible to
adaptively clean up the triangles affected by the edit and (un)refine
them if needed.
4.5 Adaptive Rendering
The adaptive rendering algorithm decides which triangles will be
drawn depending on the rendering performance available and level
of detail needed.
The algorithm uses a flag t:draw which is initialized to false,
but set to true as soon as the area corresponding to t is drawn.
This can happen either when t itself gets drawn, or when a set of
its descendents, which cover t, is drawn. The top level algorithm
loops through the triangles starting from the level n 1. A triangle
,
is always responsible for drawing its children, never itself, unless it
is a coarsest-level triangle.
AdaptiveRender
i = n , 1 downto 0
8t 2 T i : if not t:leaf then
Render (t)
8t 2 T 0 : if not t:draw then
for
are never copied, and a boundary is needed to delineate the actual
submesh.
The algorithms we have described above make heavy use of
container classes. Efficient support for sets is essential for a fast
implementation and we have used the C++ Standard Template Library. The mesh editor was implemented using OpenInventor and
OpenGL and currently runs on both SGI and Intel PentiumPro
workstations.
displaylist:append(t)
T-vertex
Figure 12: Adaptive rendering: On the left 6 triangles from level i,
one has a covered child from level i + 1, and one has a T-vertex.
On the right the result from applying Render to all six.
The Render (t) routine decides whether the children of t have to be
drawn or not (cf. Fig.12). It uses a function edist (m) which measures the distance between the point corresponding to the edge's
middle vertex m, and the edge itself. In the when case any of the
children of t are already drawn or any of its middle vertices are far
enough from the plane of the triangle, the routine will draw the rest
of the children and set the draw flag for all their vertices and t. It
also might be necessary to draw a triangle if some of its middle
vertices are drawn because the triangle on the other side decided
to draw its children. To avoid cracks, the routine cut(t) will cut
t into 2, 3, or 4, triangles depending on how many middle vertices
are drawn.
Render (t)
9 c 2 t:child j c:draw = true
9 m 2 t:mid vertex j edist(m) > D ) then
8c 2 t:child :
if (
or
if not c:draw then
displaylist:append(c)
8v 2 c : v:draw := true
t:draw := true
else if 9 m 2 t:mid vertex j m:draw = true
8t 2 cut(t) : displaylist:append (t )
t:draw := true
0
Figure 13: On the left are two meshes which are uniformly subdivided and consist of 11k (upper) and 9k (lower) triangles. On
the right another pair of meshes mesh with approximately the same
numbers of triangles. Upper and lower pairs of meshes are generated from the same original data but the right meshes were optimized through suitable choice of S . See the color plates for a
comparison between the two under shading.
0
4.6 Data Structures and Code
The main data structure in our implementation is a forest of triangular quadtrees. Neighborhood relations within a single quadtree
can be resolved in the standard way by ascending the tree to the
least common parent when attempting to find the neighbor across a
given edge. Neighbor relations between adjacent trees are resolved
explicitly at the level of a collection of roots, i.e., triangles of a
coarsest level graph. This structure also maintains an explicit representation of the boundary (if any). Submeshes rooted at any level
can be created on the fly by assembling a new graph with some set
of triangles as roots of their child quadtrees. It is here that the explicit representation of the boundary comes in, since the actual trees
5 Results
In this section we show some example images to demonstrate various features of our system and give performance measures.
Figure 13 shows two triangle mesh approximations of the Armadillo head and leg. Approximately the same number of triangles
are used for both adaptive and uniform meshes. The meshes on the
left were rendered uniformly, the meshes on the right were rendered
adaptively. (See also color plate 15.)
Locally changing threshold parameters can be used to resolve an
area of interest particularly well, while leaving the rest of the mesh
at a coarse level. An example of this “lens” effect is demonstrated
in Figure 14 around the right eye of the Mannequin head. (See also
color plate 16.)
We have measured the performance of our code on two platforms: an Indigo [email protected] with Solid Impact graphics,
and a [email protected] with an Intergraph Intense 3D board.
We used the Armadillo head as a test case. It has approximately
172000 triangles on 6 levels of subdivision. Display list creation
took 2 seconds on the SGI and 3 seconds on the PC for the full
model. We adjusted R so that both machines rendered models at
5 frames per second. In the case of the SGI approximately 113,000
triangles were rendered at that rate. On the PC we achieved 5
frames per second when the rendering threshold had been raised
enough so that an approximation consisting of 35000 polygons was
used.
The other important performance number is the time it takes to
recompute and re-render the region of the mesh which is changing
as the user moves a set of control points. This submesh is rendered
in immediate mode, while the rest of the surface continues to be
rendered as a display list. Grabbing a submesh of 20-30 faces (a
typical case) at level 0 added 250 mS of time per redraw, at level 1
it added 110 mS and at level 2 it added 30 mS in case of the SGI.
The corresponding timings for the PC were 500 mS, 200 mS and
60 mS respectively.
Figure 14: It is easy to change S locally. Here a “lens” was applied
to the right eye of the Mannequin head with decreasing S to force
very fine resolution of the mesh around the eye.
6
Conclusion and Future Research
We have built a scalable system for interactive multiresolution editing of arbitrary topology meshes. The user can either start from
scratch or from a given fine detail mesh with subdivision connectivity. We use smooth subdivision combined with details at each
level as a uniform surface representation across scales and argue
that this forms a natural connection between fine polygonal meshes
and patches. Interactivity is obtained by building both local and
adaptive variants of the basic analysis, synthesis, and rendering algorithms, which rely on fast lazy evaluation and tree pruning. The
system allows interactive manipulation of meshes according to the
polygon performance of the workstation or PC used.
There are several avenues for future research:
Multiresolution transforms readily connect with compression.
We want to be able to store the models in a compressed format
and use progressive transmission.
Features such as creases, corners, and tension controls can easily
be added into our system and expand the users' editing toolbox.
Presently no real time fairing techniques, which lead to more
intuitive coarse levels, exist.
In our system coarse level edits can only be made by dragging
coarse level vertices. Which vertices live on coarse levels is
currently fixed because of subdivision connectivity. Ideally the
user should be able to dynamically adjust this to make coarse
level edits centered at arbitrary locations.
The system allows topological edits on the coarsest level. Algorithms that allow topological edits on all levels are needed.
An important area of research relevant for this work is generation of meshes with subdivision connectivity from scanned data
or from existing models in other representations.
Acknowledgments
We would like to thank Venkat Krishnamurthy for providing the
Armadillo dataset. Andrei Khodakovsky and Gary Wu helped beyond the call of duty to bring the system up. The research was
supported in part through grants from the Intel Corporation, Microsoft, the Charles Lee Powell Foundation, the Sloan Foundation, an NSF CAREER award (ASC-9624957), and under a MURI
(AFOSR F49620-96-1-0471). Other support was provided by the
NSF STC for Computer Graphics and Scientific Visualization.
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Chapter 7
Interpolatory Subdivision for Quad
Meshes
Author: Leif Kobbelt
Interpolatory Subdivison for Quad-Meshes
A simple interpolatory subdivision scheme for quadrilateral nets
with arbitrary topology is presented which generates C1 surfaces
in the limit. The scheme satisfies important requirements for practical applications in computer graphics and engineering. These requirements include the necessity to generate smooth surfaces with
local creases and cusps. The scheme can be applied to open nets
in which case it generates boundary curves that allow a C 0 -join of
several subdivision patches. Due to the local support of the scheme,
adaptive refinement strategies can be applied. We present a simple
device to preserve the consistency of such adaptively refined nets.
The original paper has been published in:
L. Kobbelt
Interpolatory Subdivision on Open Quadrilateral Nets with Arbitrary Topology,
Computer Graphics Forum 15 (1996), Eurographics ’96 issue, pp. 409–420
3.1 Introduction
The problem we address in this paper is the generation of smooth
interpolating surfaces of arbitrary topological type in the context of
practical applications. Such applications range from the design of
free-form surfaces and scattered data interpolation to high quality
rendering and mesh generation, e.g., in finite element analysis. The
standard set-up for this problem is usually given in a form equivalent to the following:
A net N = (V; F ) representing the input is to be mapped to a refined net N 0 = (V 0 ; F 0 ) which is required to be a sufficiently close
approximation of a smooth surface. In this notation the sets V and
V 0 contain the data points pi ; p0i 2 IR3 of the input or output respectively. The sets F and F 0 represent the topological information of
the nets. The elements of F and F 0 are finite sequences of points
sk V or s0k V 0 each of which enumerates the corners of one not
necessarily planar face of a net.
If all elements sk 2 F have length four then N is called a quadrilateral net. To achieve interpolation of the given data, V V 0 is
required. Due to the geometric background of the problem we assume N to be feasible, i.e., at each point pi there exists a plane Ti
such that the projection of the faces meeting at pi onto Ti is injective. A net is closed if every edge is part of exactly two faces. In
open nets, boundary edges occur which belong to one face only.
There are two major ‘schools’ for computing N 0 from a given N.
The first or classic way of doing this is to explicitely find a collection of local (piecewise polynomial) parametrizations (patches) corresponding to the faces of N. If these patches smoothly join at common boundaries they form an overall smooth patch complex. The
net N 0 is then obtained by sampling each patch on a sufficiently fine
grid. The most important step in this approach is to find smoothly
joining patches which represent a surface of arbitrary topology. A
lot of work has been done in this field, e.g., [16], [15], [17] . . .
Another way to generate N 0 is to define a refinement operator
S which directly maps nets to nets without constructing an explicit
parametrization of a surface. Such an operator performs both, a
topological refinement of the net by splitting the faces and a geometric refinement by determining the position of the new points
in order to reduce the angles between adjacent faces (smoothing).
By iteratively applying S one produces a sequence of nets Ni with
N0 = N and Ni+1 = S Ni . If S has certain properties then the sequence S i N converges to a smooth limiting surface and we can set
N 0 := S k N for some sufficiently large k. Algorithms of this kind
are proposed in [2], [4], [14], [7], [10], and [11]. All these schemes
are either non-interpolatory or defined on triangular nets which is
not appropriate for some engineering applications.
The scheme which we present here is a stationary refinement
scheme [9], [3], i.e., the rules to compute the positions of the new
points use simple affine combinations of points from the unrefined
net. The term stationary implies that these rules are the same on
every refinement level. They are derived from a modification of the
well-known four-point scheme [6]. This scheme refines polgons by
S : (pi ) 7! (p0i ) with
p02i
:=
pi
p02i+1
:=
8+ω
(pi + pi+1 )
16
p
ω
(pi
16
1 + pi+2 )
(11)
where 0 < ω < 2 ( 5 1) is sufficient to ensure convergence to a
smooth limiting curve [8]. The standard value is ω = 1 for which
the scheme has cubic precision. In order to minimize the number of
special cases, we restrict ourselves to the refinement of quadrilateral
nets. The faces are split as shown in Fig. 10 and hence, to complete
the definition of the operator S , we need rules for new points corresponding to edges and/or faces of the unrefined net. To generalize
the algorithm for interpolating arbitrary nets, a precomputing step
is needed (cf. Sect. 3.2).
Figure 10: The refinement operator splits one quadrilateral face into
four. The new vertices can be associated with the edges and faces
of the unrefined net. All new vertices have valency four.
The major advantages that this scheme offers, are that it has the
interpolation property and works on quadrilateral nets. This seems
to be most appropriate for engineering applications (compared to
non-interpolatory schemes or triangular nets), e.g., in finite element
analysis since quadrilateral (bilinear) elements are less stiff than triangular (linear) elements [19]. The scheme provides the maximum
flexibility since it can be applied to open nets with arbitrary topology. It produces smooth surfaces and yields the possibility to generate local creases and cusps. Since the support of the scheme is local,
adaptive refinement strategies can be applied. We present a technique to keep adaptively refined nets C0 -consistent (cf. Sect. 3.6)
and shortly describe an appropriate data structure for the implementation of the algorithm.
3.2 Precomputing: Conversion to Quadrilateral
Nets
It is a fairly simple task to convert a given arbitrary net Ñ into a
quadrilateral net N. One straightforward solution is to apply one
single Catmull-Clark-type split C [2] to every face (cf. Fig. 11).
This split operation divides every n-sided face into n quadrilaterals
and needs the position of newly computed face-points and edgepoints to be well-defined. The vertices of Ñ remain unchanged.
The number of faces in the modified net N equals the sum of the
e
lengths of all sequences sk 2 F.
The number of faces in the quadrilateralized net N can be reduced by half if the net Ñ is closed,
p by not applying C but
rather its (topological) square root C , i.e., a refinement operator
whose double application is equivalent to one application of C (cf.
Fig. 11). For this split,
p only new face-points have to be computed.
For open nets, the C -split modifies the boundary polygon in a
non-intuitive way. Hence, one would have to handle several special
cases with boundary triangles if one is interested in a well-behaved
boundary curve of the resulting surface.
3.3 Subdivision Rules for Closed Nets with Arbitrary Topology
The topological structure of any quadrilateral net after several applications of a uniform refinement operator consists of large regular regions with isolated singularities which correspond to the nonregular vertices of the initial net (cf. Fig. 12). By topological regularity we mean a tensor product structure with four faces meeting
at every vertex. The natural way to define refinement operators for
quadrilateral nets is therefore to modify a tensor product scheme
such that special rules for the vicinity of non-regular vertices are
found. In this paper we will use the interpolatory four-point scheme
[6] in its tensor product version as the basis for the modification.
α
Edge−point:
σ
µ
µ
σ
µ
ν
ν
µ
β
µ
ν
ν
µ
α
σ
µ
µ
σ
Face−Point:
β
Figure 13: Subdivision masks for regular regions with α =
+ω and σ = α2 , µ = α β, ν = β2 .
β = 816
for computing the edge-points. However, once all the edge-points
are known, there always are exactly two possibilities to choose four
consecutive edge-points when computing a certain face-point since
the net is quadrilateral. It is an important property of tensor product
schemes on regular nets that both possibilities lead to the same result (commuting univariant refinement operators). In order to modify the tensor product scheme as little as possible while generalizing
it to be applicable for nets with arbitrary topology, we want to conserve this property. Hence, we will propose a subdivision scheme
which only needs one additional rule: the one for computing edgepoints corresponding to edges adjacent to a non-regular vertex. All
other edge-points and all face-points are computed by the application of the original four-point scheme and the additional rule will be
such that both possibilities for the face-points yield the same result.
We use the notation of Fig. 14 for points in the neighborhood of
a singular vertex p. The index i is taken to be modulo n where n is
the number of edges meeting at p. Applying the original four-point
rule wherever possible leaves only the points xi and yi undefined.
If we require that both possible ways to compute yi by applying the
standard four-point rule to succeeding edge-points lead to the same
result, we get a dependence relating xi+1 to xi
xi+1 = xi +
w
(hi
8
hi+1 ) +
p02i+1;2 j :=
8+ω
(pi; j + pi+1; j )
16
ω
(p i
16
1; j + pi+2; j ):
w2
(k i
8 (4 + w)
w
(li+2
8
Figure 12: Isolated singularities in the refined net.
Consider a portion of a regular quadrilateral net with vertices
pi; j . The vertices can be indexed locally such that each face is represented by a sequence si; j = fpi; j ; pi+1; j ; pi+1; j+1 ; pi; j+1 g. The
points p0i; j of the refined net can be classified into three disjunct
groups. The vertex-points p02i;2 j := pi; j are fixed due to the interpolation requirement. The edge-points p02i+1;2 j and p02 j;2i+1 are computed by applying the four-point rule (11) in the corresponding grid
direction, e.g.,
(12)
Finally, the face-points p02i+1;2 j+1 are computed by applying the four-point rule to either four consecutive edge-points
p02i+1;2 j 2 ; : : : ; p02i+1;2 j+4 or to p02i 2;2 j+1 ; : : : ; p02i+4;2 j+1 . The resulting weight coefficient masks for these rules are shown in
Fig. 13. The symmetry of the face-mask proves the equivalence
of both alternatives to compute the face-points. From the differentiability of the limiting curves generated by the four-point scheme,
the smoothness of the limiting surfaces generated by infinitely refining a regular quadrilateral net, follows immediately. This is a
simple tensor product argument.
For the refinement of irregular quadrilateral nets, i.e., nets which
include some vertices where other than four faces meet, a consistent
indexing which allows the application of the above rules is impossible. If other than four edges meet at one vertex, it is not clear how
to choose the four points to which one can apply the above rule
ω
16 ,
li
1) +
2
ki+2 )+
4+w
(li+1
8
li );
which can be considered as compatibility condition. In the regular
case, this condition is satisfied for any tensor product rule. The
compatibility uniquely defines the cyclic differences 4xi = xi+1
xi which sum to zero (telescoping sums). Hence, there always exists
a solution and even one degree of freedom is left for the definition
of the xi .
hi
ki−1
li
y
li+1 i xi li−1
p
hi+1 xi+1
k i−2
l
ki+1 i+2
ki+2
ki
Figure 14: Notation for vertices around a singular vertex P.
The points xi will be computed by rotated versions of the same
subdivision mask. Thus, the vicinity of p will become more and
more symmetric while refinement proceeds. Hence, the distance
p
CN
N
CN
Figure 11: Transformation of an arbitrary net Ñ into a quadrilateral net N by one Catmull-Clark-split C (middle) or by its square root (right,
for closed nets).
between p and the center of gravity of the xi will be a good measure for the roughness of the net near p and the rate by which this
distance tends to zero can be understood as the ‘smoothing rate’.
The center of gravity in the regular (n = 4) case is:
1
n
n 1
∑ xi =
i=0
4+w
1
p+
8
2n
n 1
∑ li
i=0
w
8n
n 1
∑ hi
i=0
:
(13)
In the non-regular case, we have
1
n
n 1
∑ xi
i=0
=
xj +
1
n
n 2
(n
i=0
∑
1
i) 4xi+ j ;
(14)
j 2 f0; : : : ; n
1g:
Combining common terms in the telescoping sum and equating the
right hand sides of (13) and (14) leads to
xj
w
4+w
4+w
hj +
lj +
p
8
8
8
=
w
v j;
8
(15)
4
n
n 1
∑ li
i=0
(l j 1 + l j + l j+1 ) +
w
(k j
4+w
4w
(4 + w ) n
2 + k j 1 + k j + k j+1 )
(16)
n 1
∑ ki
i=0
λ1 = 1;
1 > λ2 = λ3 ;
jλ2 j jλi j 8i 4
>
;
:
Table 2 shows theses eigenvalues of the refinement matrix A for vertices with n adjacent edges in the standard case ω = 1. The computation of the spectrum can be done by exploiting the block-circulant
structure of A. We omit the details here, because the dimension of
A is k k with k = 30 n + 1.
where we define the virtual point
v j :=
applied. In the regular regions of the net (which enlarge during refinement), the smoothness of the limiting surface immediately follows from the smoothness of the curves generated by the univariate
four-point scheme. Hence to complete the convergence analysis,
it is sufficient to look at the vicinities of the finitely many isolated
singular vertices (cf. Fig. 12).
Let p0 ; : : : ; pk be the points from a fixed neighborhood of the singular vertex p0 . The size of the considered neighborhood depends
on the support of the underlying tensor product scheme and contains 5 ‘rings’ of faces around p0 in our case. The collection of
all rules to compute the new points p00 ; : : : ; p0k of the same ‘scaled’
(5-layer-) neighborhood of p0 = p00 in the refined net can be represented by a block-circulant matrix A such that (p0i )i = A (pi )i . This
matrix is called the refinement matrix. After [1] and [18] the convergence analysis can be reduced to the analysis of the eigenstructure
of A. For the limiting surface to have a unique tangent plane at p0
it is sufficient that the leading eigenvalues of A satisfy
:
Hence, the x j can be computed by applying (11) to the four points
h j , l j , p and v j . The formula also holds in the case n = 4 where v j =
l j+2 . Such a virtual point v j is defined for every edge and both of
its endpoints. Hence to refine an edge which connects two singular
vertices p1 and p2 , we first compute the two virtual points v1 and
v2 and then apply (11) to v1 , p1 , p2 and v2 . If all edge-points x j are
known, the refinement operation can be completed by computing
the face-points y j . These are well defined since the auxillary edgepoint rule is constructed such that both possible ways lead to the
same result.
3.4 Convergence Analysis
The subdivision scheme proposed in the last section is a stationary scheme and thus the convergence criteria of [1] and [18] can be
n
3
4
5
6
7
8
9
λ1
1:0
1:0
1:0
1:0
1:0
1:0
1:0
λ2
0.42633
0.5
0.53794
0.55968
0.5732
0.58213
0.58834
λ3
0.42633
0.5
0.53794
0.55968
0.5732
0.58213
0.58834
λi4 0.25
0.25
0.36193
0.42633
0.46972
0.5
0.52180
Table 2: Leading eigenvalues of the subdivision matrix
In addition to a uniquely defined tangent plane we also have to
have local injectivity in order to guarantee the regularity of the surface. This can be checked by looking at the natural parametrization
of the surface at p0 which is spanned by the eigenvectors of A corresponding to the subdominant eigenvalues λ2 and λ3 . The injectivity of this parametrization is a sufficient condition. The details
can be found in [18]. Fig. 15 shows meshes of ‘isolines’ of these
characteristic maps which are well-behaved.
Figure 15: Sketch of the characteristic maps in the neighborhood of singular vertices with n = 3; 5; : : : ; 9.
3.5 Boundary Curves
If a subdivision scheme is supposed to be used in practical modeling or reconstruction applications, it must provide features that
allow the definition of creases and cusps [12]. These requirements
can be satisfied if the scheme includes special rules for the refinement of open nets which yield well-behaved boundary curves that
interpolate the boundary polygons of the given net. Having such
a scheme, creases can be modeled by joining two separate subdivision surfaces along a common boundary curve and cusps result from a topological hole in the initial net which geometrically
shrinks to a single point, i.e., a face s = fp1 ; : : : ; pn g of a given net
is deleted to generate a hole and its vertices are moved to the same
location pi = p (cf. Fig. 16).
To allow a C0 -join between two subdivision patches whose initially given nets have a common boundary polygon, it is necessary
that their limiting boundary curves only depend on these common
points, i.e., they must not depend on any interior point. For our
scheme, we achieve this by simply applying the original univariate
four-point rule to boundary polygons. Thus, the boundary curve
of the limiting surface is exactly the four-point curve which is defined by the initial boundary polygon. Further, it is necessary to not
only generate smooth boundary curves but rather to allow piecewise
smooth boundary curves, e.g., in cases where more than two subdivision patches meet at a common point. In this case we have to
cut the boundary polygon into several segments by marking some
vertices on the boundary as being corner vertices. Each segment
between two corner vertices is then treated separately as an open
polygon.
When dealing with open polygons, it is not possible to refine the
first or the last edge by the original four-point scheme since rule
(11) requires a well-defined 2-neighborhood. Therefore, we have
to find another rule for the point p1m+1 which subdivides the edge
m
m
m
pm
pm
0 p1 . We define an extrapolated point p 1 := 2 p0
1 . The
m+1
point p1
then results from the application of (11) to the subm m
polygon pm 1 ; pm
0 ; p1 ; p2 . Obviously, this additional rule can be expressed as a stationary linear combination of points from the nonextrapolated open polygon:
p1m+1 :=
8
w m 8+ 2w m
p +
p1
16 0
16
w m
p
16 2
(17)
m+1
The rule to compute the point p2n
1 subdividing the last edge
m
pm
n 1 pn is defined analogously.
This modification of the original scheme does not affect the convergence to a continuously differentiable limit, because the estimates for the contraction rate of the maximum second forward difference used in the convergence proof of [6] remain valid. This
is obvious since the extrapolation only adds the zero component
42 pm1 to the sequence of second order forward differences. The
main convergence criterion of [13] also applies.
It remains to define refinement rules for inner edges of the net
which have one endpoint on the boundary and for faces including
at least one boundary vertex. To obtain these rules we use the same
heuristic as in the univariate case. We extrapolate the unrefined
net over every boundary edge to get an additional layer of faces.
When computing the egde- and face-points refining the original net
by the rules from Sect. 3.3, these additional points can be used.
To complete the refinement step, the extrapolated faces are finally
deleted.
Let q1 ; : : : ; qr be the inner points of the net which are connected
to the boundary point p then the extrapolated point will be
p := 2 p
1
r
r
∑ qi
i=1
:
If the boundary point p belongs to the face s = fp; q; u; vg and is
not connected to any inner vertex then we define p := 2 p u.
For every boundary edge p q we add the extrapolated face s =
fp ; q ; q ; p g .
Again, the tangent-plane continuity of the resulting limiting surface can be proved by the sufficient criteria of [1] and [18]. This
is obvious since for a fixed number of interior edges adjacent to
some boundary vertex p, the refinement of the extrapolated net can
be rewritten as a set of stationary refinement rules which define
the new points in the vicinity of p as linear combinations of points
from the non-extrapolated net. However the refinement matrix is
no longer block-circulant.
At every surface point lying on the boundary of a tangent plane
continuous surface, one tangent direction is determined by the tangent of the boundary curve (which in this case is a four-point curve
that does not depend on inner vertices). On boundaries, we can
therefore drop the requirement of [18] that the leading eigenvalues of the refinement matrix have to be equal. This symmetry
is only a consequence of the assumption that the rules to compute the new points around a singular vertex are identical modulo
Figure 16: Modeling sharp features (piecewise smooth boundary, crease, cusp)
rotations (block-circulant refinement matrix). Although λ2 6= λ3
causes an increasing local distortion of the net, the smoothness of
the limiting surface is not affected. This effect can be viewed as
a reparametrization in one direction. (Compare this to the distortion of a regular net which is refined by binary subdivision in one
direction and trinary in the other.)
We summarize the different special cases which occur when refining an open net by the given rules. In Fig. 17 the net to be refined
consists of the solid white faces while the extrapolated faces are
drawn transparently. The dark vertex is marked as a corner vertex.
We have to distinguish five different cases:
A
C
C
D
C
A
C
D
D C
C
C
C
C
C
D
C
C
D
C
C
C
C
D
B
E
D
D
C
B
C
C
D
A
C
C
C
gions with high curvature while ‘flat’ regions may be approximated
rather coarsely. Hence, in order to keep the amount of data reasonable, the next step is to introduce adaptive refinement features.
The decision where high resolution refinement is needed,
strongly depends on the underlying application and is not discussed
here. The major problem one always has to deal with when adaptive refinement of nets is performed is to handle or eliminate C 1 inconsistencies which occur when faces from different refinement
levels meet. A simple trick to repair the resulting triangular holes
is to split the bigger face into three quadrilaterals in an Y-fashion
(cf. Fig 18). However this Y-split does not repair the hole. Instead
it shifts the hole to an adjacent edge. Only combining several Yelements such that they build a ‘chain’ connecting two inconsistencies leads to an overall consistent net. The new vertices necessary
for the Y-splits are computed by the rules of Sect. 3.3. The fact
that every Y-element contains a singular (n = 3) vertex causes no
problems for further refinement because this Y-element is only of
temporary nature, i.e., if any of its three faces or any neighboring
face is to be split by a following local refinement adaption, then first
the Y-split is undone and a proper Catmull-Clark-type split is performed before proceeding. While this simple technique seems to
be known in the engineering community, the author is not aware of
any reference where the theoretical background for this technique
is derived. Thus, we sketch a simple proof that shows under which
conditions this technique applies.
s
Figure 17: Occurences of the different special cases.
p
r
q
A: Within boundary segments, we apply (11) to four succeeding
boundary vertices.
B: To the first and the last edge of an open boundary segment, we
apply the special rule (17).
C: Inner edge-points can be computed by application of (15). If
necessary, extrapolated points are involved.
D: For every face-point of this class, at least one sequence of four
C-points can be found to which (11) can be applied. If there are
two possibilities for the choice of these points then both lead to the
same result which is guaranteed by the construction of (15).
E: In this case no appropriate sequence of four C-points can be
found. Therefore, one has to apply (17) to a B-point and the two Cpoints following on the opposite side of the corner face. In order to
achieve independence of the grid direction, even in case the corner
vertex is not marked, we apply (17) in both directions and compute
the average of the two results.
3.6 Adaptive Refinement
In most numerical applications, the exponentially increasing number of vertices and faces during the iterative refinement only allows
a small number of refinement steps to be computed. If high acuracy
is needed, e.g., in finite element analysis or high quality rendering,
it is usually sufficient to perform a high resolution refinement in re-
Figure 18: A hole in an adaptively refined net and an Y-element to
fill it.
First, in order to apply the Y-technique we have to restrict the
considered nets to balanced nets. These are adaptively refined nets
(without Y-elements) where the refinement levels of neighboring
faces differ at most by one. Non-balanced inconsistencies can not
be handled by the Y-technique. Hence, looking at a particular face
s from the n-th refinement level, all faces having at least one vertex
in common with s are from the levels (n 1), n, or (n + 1). For
the proof we can think of first repairing all inconsistencies between
level n 1 and n and then proceed with higher levels. Thus, without
loss of generality, we can restrict our considerations to a situation
where all relevant faces are from level (n 1) or n.
A critical edge is an edge, where a triangular hole occurs due
to different refinement levels of adjacent faces. A sequence of Yelements can always be arranged such that two critical edges are
connected, e.g., by surrounding one endpoint of the critical edge
with a ’corona’ of Y-elements until another critical edge is reached
(cf. Fig. 19). Hence, on closed nets, we have to require the number
of critical edges to be even. (On open nets, any boundary edge can
stop a chain of Y-elements.) We show that this is always satisfied,
by induction over the number of faces from the n-th level within
an environment of (n 1)-faces. Faces from generations > n or
< (n
1) do not affect the situation since we assume the net to be
balanced.
Face4Typ
Face9Typ
Face4Typ
Figure 21: References between different kinds of faces.
Figure 19: Combination of Y-elements
The first adaptive Catmull-Clark-type split on a uniformly refined net produces four critical edges. Every succeeding split
changes the number of critical edges by an even number between
4 and 4, depending on the number of direct neighbors that have
been split before. Thus the number of critical edges is always even.
However, the n-faces might form a ring having in total an even
number of critical edges which are separated into an odd number
‘inside’ and an odd number ‘outside’. It turns out that this cannot happen: Let the inner region surrounded by the ring of n-faces
consist of r quadrilaterals having a total number of 4r edges which
are candidates for being critical. Every edge which is shared by
two such quadrilaterals reduces the number of candidates by two
and thus the number of boundary edges of this inner region is again
even.
The only situation where the above argument is not valid, occurs
when the considered net is open and has a hole with an odd number
of boundary edges. In this case, every loop of n-faces enclosing
this hole will have an odd number of critical edges on each side.
Hence, we have to further restrict the class of nets to which we
can apply the Y-technique to open balanced nets which have no
hole with an odd number of edges. This restriction is not serious
because one can transform any given net in order to satisfy this
requirement by applying an initial uniform refinement step before
adaptive refinement is started. Such an initial step is needed anyway
if a given arbitrary net has to be transformed into a quadrilateral one
(cf. Sect. 3.2).
It remains to find an algorithm to place the Y-elements correctly, i.e., to decide which critical edges should be connected by
a corona. This problem is not trivial because interference between
the Y-elements building the ‘shores’ of two ‘islands’ of n-faces lying close to each other, can occur. We describe an algorithm which
only uses local information and decides the orientation separately
for each face instead of ‘marching’ around the islands.
The initially given net (level 0) has been uniformly refined once
before the adaptive refinement begins (level 1). Let every vertex
of the adaptively refined net be associated with the generation in
which it was introduced. Since all faces of the net are the result
of a Catmull-Clark-type split (no Y-elements have been placed so
far), they all have the property that three of its vertices belong to
the same generation g and the fourth vertex belongs to a generation
g0 < g. This fact yields a unique orientation for every face. The
algorithm starts by marking all vertices of the net which are endpoints of a critical edge, i.e. if a (n 1)-face fp; q; : : :g meets two
n-faces fp; r; s; : : :g and fq; r; s; : : :g then p and q are marked (cf.
Fig. 18). After the marking-phase, the Y-elements are placed. Let
s = fp; q; u; vg be a face of the net where p is the unique vertex
which belongs to an elder generation than the other three. If neither
q nor v are marked then no Y-element has to be placed within this
face. If only one of them is marked then the Y-element has to be
oriented as shown in Fig. 20 and if both are marked this face has to
be refined by a proper Catmull-Clark-type split.
The correctness of this algorithm is obvious since the vertices
which are marked in the first phase are those which are common to
faces of different levels. The second phase guarantees that a corona
of Y-elements is built around each such vertex (cf. Fig. 19).
3.7 Implementation and Examples
The described algorithm is designed to be useful in practical applications. Therefore, besides the features for creating creases and
cusps and the ability to adaptively refine a given quadrilateral net,
efficiency and compact implementation are also important. Both
can be achieved by this algorithm. The crucial point of the implementation is the design of an appropriate data structure which
supports an efficient navigation through the neighborhood of the
vertices. The most frequently needed access operation to the data
structure representing the balanced net, is to enumerate all faces
which lie around one vertex or to enumerate all the neighbors of
one vertex. Thus every vertex should be associated with a linked
list of the objects that constitute its vicinity. We propose to do this
implicitely by storing the topological information in a data structure Face4Typ which contains all the information of one quadrilateral face, i.e., references to its four corner points and references
to its four directly neighboring faces. By these references, a doubly
linked list around every vertex is available.
Since we have to maintain an adaptively refined net, we need
an additional datatype to consistently store connections between
faces from different refinement levels. We define another structure Face9Typ which holds references to nine vertices and eight
neighbors. These multi-faces can be considered as ‘almost’ split
faces, where the geometric information (the new edge- and facepoints) is already computed but the topological split has not yet
been performed. If, during adaptive refinement, some n-face is
split then all its neighbors which are from the same generation are
converted into Face9Typ’s. Since these faces have pointers to
eight neighbors, they can mimic faces from different generations
and therefore connect them correctly. The Face9Typ’s are the
candidates for the placement of Y-elements in order to re-establish
consistency. The various references between the different kinds of
faces are shown in Fig. 21.
To relieve the application program which decides where to adaptively refine, from keeping track of the balance of the net, the implementation of the refinement algorithm should perform recursive
refinement operations when necessary, i.e., if a n-face s is to be refined then first all (n 1)-neighbors which have at least one vertex
in common with s must be split.
The following pictures are generated by using our experimental implementation. The criterion for adaptive refinement is a discrete approximation of the Gaussian curvature. The running time
of the algorithm is directly proportional to the number of computed
points, i.e., to the complexity of the output-net. Hence, since the
number of regions where deep refinement is necessary usually is
p
q
p
q
p
q
p
q
v
u
v
u
v
u
v
u
Figure 20: The orientation of the Y-elements depends on whether the vertices q and v are marked (black) or not (white). The status of vertices
p and u does not matter (gray).
fixed, we can reduce the space- and time-complexity from exponential to linear (as a function of the highest occurring refinement
level in the output).
References
[1] A. Ball / D. Storry, Conditions for Tangent Plane Continuity
over Recursively Generated B-Spline Surfaces, ACM Trans.
Graph. 7 (1988), pp. 83–102
[2] E. Catmull, J. Clark, Recursively generated B-spline surfaces
on arbitrary topological meshes, CAD 10 (1978), pp. 350–355
[3] A. Cavaretta / W. Dahmen / C. Micchelli, Stationary Subdivision, Memoirs of the AMS 93 (1991), pp. 1-186
[4] D. Doo / M. Sabin, Behaviour of Recursive Division Surfaces
Near Extraordinary Points, CAD 10 (1978), pp. 356–360
[5] S. Dubuc, Interpolation Through an Iterative Scheme, Jour. of
Mathem. Anal. and Appl., 114 (1986), pp. 185–204
[6] N. Dyn / J. Gregory / D. Levin, A 4-Point Interpolatory Subdivision Scheme for Curve Design, CAGD 4(1987), pp. 257–268
[7] N. Dyn / J. Gregory / D. Levin, A Butterfly Subdivision Scheme
for Surface Interpolation with Tension Controll, ACM Trans.
Graph. 9 (1990), pp. 160–169
[8] N. Dyn / D. Levin, Interpolating subdivision schemes for the
generation of curves and surfaces, Multivar. Approx. and Interp., W. Haus̈mann and K. Jetter (eds.), 1990 Birkhäuser Verlag, Basel
[9] N. Dyn, Subdivision Schemes in Computer Aided Geometric
Design, Adv. in Num. Anal. II, Wavelets, Subdivisions and
Radial Functions, W.A. Light ed., Oxford Univ. Press, 1991,
pp. 36–104.
[10] N. Dyn / D. Levin / D. Liu, Interpolatory ConvexityPreserving Subdivision Schemes for Curves and Surfaces,
CAD 24 (1992), pp. 221–216
[11] M. Halstead / M. Kass / T. DeRose, Efficient, fair interpolation using Catmull-Clark surfaces, Computer Graphics 27
(1993), pp. 35–44
[12] H. Hoppe, Surface Reconstruction from unorganized points,
Thesis, University of Washington, 1994
[13] L. Kobbelt, Using the Discrete Fourier-Transform to Analyze
the Convergence of Subdivision Schemes, Appl. Comp. Harmonic Anal. 5 (1998), pp. 68–91
[14] C. Loop, Smooth Subdivision Surfaces Based on Triangles,
Thesis, University of Utah, 1987
[15] C. Loop, A G1 triangular spline surface of arbitrary topological type, CAGD 11 (1994), pp. 303–330
[16] J. Peters, Smooth mesh interpolation with cubic patches, CAD
22 (1990), pp. 109–120
[17] J. Peters, Smoothing polyhedra made easy, ACM Trans. on
Graph., Vol 14 (1995), pp. 161–169
[18] U. Reif, A unified approach to subdivision algorithms near
extraordinary vertices, CAGD 12 (1995), pp. 153–174
[19] K. Schweizerhof, Universität Karlsruhe private communication
Figure 22: Examples for adaptively refined nets.
Chapter 8
A Variational Approach to Subdivision
Speaker: Leif Kobbelt
Variational Subdivision Schemes
Leif Kobbelt
Max-Planck-Institute for Computer Sciences
Preface
The generic strategy of subdivision algorithms which is to define
smooth curves and surfaces algorithmically by giving a set of simple rules for refining control polygons or meshes is a powerful technique to overcome many of the mathematical difficulties emerging
from (polynomial) spline-based surface representations. In this section we highlight another application of the subdivision paradigm
in the context of high quality surface generation.
From CAGD it is known that the technical and esthetic quality
of a curve or a surface does not only depend on infinitesimal properties like the Ck differentiability. Much more important seems to
be the fairness of a geometric object which is usually measured by
curvature based energy functionals. A surface is hence considered
optimal if it minimizes a given energy functional subject to auxiliary interpolation or approximation constraints.
Subdivision and fairing can be effectively combined into what is
often refered to as variational subdivision or discrete fairing. The
resulting algorithms inherit the simplicity and flexibility of subdivision schemes and the resulting curves and surfaces satisfy the sophisticated requirements for high end design in geometric modeling
applications.
The basic idea that leads to variational subdivision schemes is
that one subdivision step can be considered as a topological split
operation where new vertices are introduced to increase the number
of degrees of freedom, followed by a smoothing operation where
the vertices are shifted in order to increase the overall smoothness. From this point of view is is natural to ask for the maximum smoothness that can be achieved on a given level of refinement while observing prescribed interpolation constraints.
We use an energy functional as a mathematical criterion to rate
the smoothness of a polygon or a mesh. In the continuous setting,
such scalar valued fairing functionals are typically defined as an
integral over a combination of (squared) derivatives. In the discrete
setting, we approximate such functionals by a sum over (squared)
divided differences.
In the following we reproduce a few papers where this approach
is described in more detail. In the univariate setting we consider interpolatory variational subdivision schemes which perform
a greedy optimization in the sense that when computing the polygon Pm+1 from Pm the new vertices’ positions are determined by
Computer Graphics Group, Max-Planck-Institute for Computer Sciences, Im Stadtwald, 66123 Saarbrücken, Germany, [email protected]
an energy minimization process but when proceeding with Pm+2
the vertices of Pm+1 are not adjusted.
In the bivariate setting, i.e., the subdivision and optimization of
triangle meshes, we start with a given control mesh P0 whose vertices are to be interpolated by the resulting mesh. In this case it
turns out that the mesh quality can be improved significantly if we
use all the vertices from Pm n P0 for the optimization in the mth
subdivision step.
Hence the algorithmic structure of variational subdivision degenerates to an alternating refinement and (constrained) global optimization. In fact, from a different viewing angle the resulting algorithms perform like a multi-grid solver for the discretized optimization problem. This observation provides the mathematical
justification for the discrete fairing approach.
For the efficient fairing of continuous parameteric surfaces, the
major difficulties arise from the fact that geometrically meaningful
energy functionals depend on the control vertices in a highly nonlinear fashion. As a consequence we have to either do non-linear
optimization or we have to approximate the true functional by a
linearized version. The reliability of this approximation usually
depends on how close to isometric the surface’s parameterization
is. Alas, spline-patch-based surface representations often do not
provide enough flexibility for an appropriate re-parameterization
which would enable a feasible linearization of the geometric energy functional. Figure 1 shows two surfaces which are both optimal with respect to the same energy functional but for different
parameterizations.
Figure 1: Optimal surfaces with respect to the same functional and
interpolation constraints but for different parameterizations (isometric left, uniform right).
With the discrete fairing approach, we can exploit the auxiliary
freedom to define an individual local parameterization for every
vertex in the mesh. By this we find an isometric parameterization
for each vertex and since the vertices are in fact the only points
where the surface is evaluated, the linearized energy functional is a
good approximation to the original one.
The discrete fairing machinery turns out to be a powerful tool
which can facilitate the solution of many problems in the area of
surface generation and modeling. The overall objective behind the
presented applications will be the attempt to avoid, bypass, or at
least delay the mathematically involved generation of spline CADmodels whenever it is appropriate.
I
Univariate Variational Subdivision
In this paper a new class of interpolatory refinement schemes is
presented which in every refinement step determine the new points
by solving an optimization problem. In general, these schemes are
global, i.e., every new point depends on all points of the polygon
to be refined. By choosing appropriate quadratic functionals to be
minimized iteratively during refinement, very efficient schemes producing limiting curves of high smoothness can be defined. The well
known class of stationary interpolatory refinement schemes turns
out to be a special case of these variational schemes.
The original paper which also contains the omitted
proofs has been published in:
L. Kobbelt
A Variational Approach to Subdivision,
CAGD 13 (1996) pp. 743–761, Elsevier
1.1 Introduction
Interpolatory refinement is a very intuitive concept for the construction of interpolating curves or surfaces. Given a set of points p0i 2
IRd which are to be interpolated by a smooth curve, the first step of a
refinement scheme consists in connecting the points by a piecewise
linear curve and thus defining a polygon P0 = (p00 ; : : : ; p0n 1 ).
This initial polygon can be considered as a very coarse approximation to the final interpolating curve. The approximation can be
improved by inserting new points between the old ones, i.e., by subdividing the edges of the given polygon. The positions of the new
points p12i+1 have to be chosen appropriately such that the resulting (refined) polygon P1 = (p10 ; : : : ; p12n 1 ) looks smoother than the
given one in some sense (cf. Fig. 2). Interpolation of the given
points is guaranteed since the old points p0i = p12i still belong to the
finer approximation.
By iteratively applying this interpolatory refinement operation,
a sequence of polygons (Pm ) is generated with vertices becoming
more and more dense and which satisfy the interpolation condition
m+1
pm
for all i and m. This sequence may converge to a smooth
i = p2i
limit P∞ .
Many authors have proposed different schemes by explicitly givm+1
ing particular rules how to compute the new points p2i
as a func+1
tion of the polygon Pm to be refined. In (Dubuc, 1986) a simple
refinement scheme is proposed which uses four neighboring vertices to compute the position of a new point. The position is determined in terms of the unique cubic polynomial which uniformly
interpolates these four points. The limiting curves generated by this
scheme are smooth, i.e., they are differentiable with respect to an
equidistant parametrisation.
Appropriate formalisms have been developed in (Cavaretta et al.,
1991), (Dyn & Levin, 1990), (Dyn, 1991) and elsewhere that allow
an easy analysis of such stationary schemes which compute the new
points by applying fixed banded convolution operators to the original polygon. In (Kobbelt, 1995b) simple criteria are given which
can be applied to convolution schemes without any band limitation
as well (cf. Theorem 2).
(Dyn et al., 1992) and (Le Méhauté & Utreras, 1994) propose
non-linear refinement schemes which produce smooth interpolating
(C1 -) curves and additionally preserve the convexity properties of
the initial data. Both of them introduce constraints which locally
define areas where the new points are restricted to lie in. Another
possibility to define interpolatory refinement schemes is to dualize
corner-cutting algorithms (Paluszny et al., 1994). This approach
leads to more general necessary and sufficient convergence criteria.
In this paper we want to define interpolatory refinement schemes
in a more systematic fashion. The major principle is the following:
We are looking for refinement schemes for which, given a polygon
Pm , the refined polygon Pm+1 is as smooth as possible. In order
to be able to compare the “smoothness” of two polygons we define functionals E (Pm+1 ) which measure the total amount of (discrete) strain energy of Pm+1 . The refinement operator then simply
m +1
chooses the new points p2i
such that this functional becomes a
+1
minimum.
An important motivation for this approach is that in practice
good approximations to the final interpolating curves should be
achieved with little computational effort, i.e., maximum smoothness after a minimal number of refinement steps is wanted. In nondiscrete curve design based, e.g., on splines, the concept of defining
interpolating curves by the minimization of some energy functional
(fairing) is very familiar (Meier & Nowacki, 1987), (Sapidis, 1994).
This basic idea of making a variational approach to the definition of refinement schemes can also be used for the definition of
schemes which produce smooth surfaces by refining a given triangular or quarilateral net. However, due to the global dependence
of the new points from the given net, the convergence analysis of
such schemes strongly depends on the topology of the net to be
refined and is still an open question. Numerical experiments with
such schemes show that this approach is very promising. In this
paper we will only address the analysis of univariate schemes.
1.2 Known results
Given an arbitrary (open/closed) polygon Pm = (pm
i ), the difference
polygon 4k Pm denotes the polygon whose vertices are the vectors
k
4k pmi := ∑
k
j
j=0
(
1)k+ j pm
i+ j :
In (Kobbelt, 1995b) the following characterization of sequences
of polygons (Pm ) generated by the iterative application of an interpolatory refinement scheme is given:
Figure 2: Interpolatory refinement
In (Dyn et al., 1987) this scheme is generalized by introducing
an additional design or tension parameter. Replacing the interpolating cubic by interpolating polynomials of arbitrary degree leads
to the Lagrange-schemes proposed in (Deslauriers & Dubuc, 1989).
Raising the degree to (2k + 1), every new point depends on (2k + 2)
old points of its vicinity. In (Kobbelt, 1995a) it is shown that at least
for moderate k these schemes produce Ck -curves.
Lemma 1 Let (Pm ) be a sequence of polygons. The scheme by
which they are generated is an interpolatory refinement scheme
m+1
(i.e., pm
for all i and m) if and only if for all m; k 2 IN
i = p2i
the condition
k
4k pmi = ∑
j=0
k
j
4k p2im
+1
+j
holds for all indices i of the polygon 4k Pm .
Also in (Kobbelt, 1995b), the following sufficient convergence
criterion is proven which we will use in the convergence analysis in
the next sections.
Theorem 2 Let (Pm ) be a sequence of polygons generated by
the iterative application of an arbitrary interpolatory refinement
scheme. If
∞
∑ k2km 4k+l Pm k∞
<
∞
m=0
for some l 2 IN then the sequence (Pm ) uniformly converges to a
k-times continuously differentiable curve P∞ .
This theorem holds for all kinds of interpolatory schemes on
open and closed polygons. However, in this paper we will only
apply it to linear schemes whose support is global.
1.3 A variational approach to interpolatory
refinement
In this and the next two sections we focus on the refinement of
closed polygons, since this simplifies the description of the refinement schemes. Open polygons will be considered in Section 1.6.
m
Let Pm = (pm
0 ; : : : ; pn 1 ) be a given polygon. We want Pm+1 =
m+1
m+1
(p 0
;:::;p
2n 1 ) to be the smoothest polygon for which the interm+1
polation condition p2i
= pm
i holds. Since the roughness at some
m+1
vertex pi
is a local property we measure it by a an operator
K (pim+1 ) :=
∑
j=0
:
Our goal is to minimize the total strain energy over the whole
polygon Pm+1 . Hence we define
2n 1
∑
i=0
K (pim+1 )2
(1)
to be the energy functional which should become minimal. Since
m+1
the points p2i
of Pm+1 with even indices are fixed due to the inm+1
terpolation condition, the points p2i
with odd indices are the only
+1
free parameters of this optimization problem. The unique minimum
of the quadratic functional is attained at the common root of all partial derivatives:
0
B
@
β0
β2
..
.
β2
β0
..
.
0
B
@
β4
β2
..
.
β1
β3
..
.
m+1
∂p2l
+1
E (Pm+1 )
=
=
k
∂
i=0
m+1
∂p2l
+1
∑
2
k
k
i=0
j=0
∑ αi ∑ α j p2lm++11
k
=
2
m+1
K (p2l
+1+r
∑
i= k
m+1
βi p2l
+1+i
(3)
:::
:::
..
β2
β4
.
β1
β1
..
.
10
B
C
B
AB
@
β3
β1
..
.
1
C
C
C
A
p1m+1
p3m+1
..
.
m+1
p2n 1
β3
β5
:::
:::
..
.
=
10
B
C
AB
@
pm
0
pm
1
..
.
pm
n 1
1
C
C
A
(4)
m +1
βi p2l
+1+i
=
0;
l = 0; : : : ; n
1
(5)
the Euler-Lagrange-equation.
Theorem 3 The minimization of E (Pm+1 ) has a well-defined solution if and only if the characteristic polynomial α(z) for the local
measure K has no diametric roots z = ω on the unit circle with
Arg(ω) 2 π IN=n. (Proof to be found in the original paper)
Remark The set π IN=2m becomes dense in IR for increasing refinement depth m ! ∞. Since we are interested in the smoothness
properties of the limiting curve P∞ , we should drop the restriction
that the diametric roots have to have Arg(ω) 2 π IN=n. For stability
reasons we require α(z) to have no diametric roots on the unit circle
at all.
The optimization by which the new points are determined is a
geometric process. In order to obtain meaningful schemes, we have
to introduce more restrictions on the energy functionals E or on the
measures of roughness K.
For the expression K 2 (pi ) to be valid, K has to be vector valued,
i.e., the sum of the coefficients α j has to be zero. This is equivalent
to α(1) = 0. Since
k
∑
βi =
k
∑
k
∑ αi α j =
i=0 j=0
k
∑ αj
2
j=0
the sum of the coefficients βi also vanishes in this case and affine
invariance of the (linear) scheme is guaranteed because constant
functions are reproduced.
2
i)
i+ j
i = 0; : : : ; k:
;
m+1
which follows from (2) by separation of the fixed points p2i
=
pm
from
the
variables.
Here,
both
matrices
are
circulant
and
(ali
most) symmetric. A consequence of this symmetry is that the new
points do not depend on the orientation by which the vertices are
numbered (left to right or vice versa).
To emphasize the analogy between curve fairing and interpolatory refinement by variational methods, we call the equation
i= k
∂
∑ α j α j +i
Hence, the strain energy E (Pm+1 ) becomes minimal if the new
m +1
points p2i
are the solution of the linear system
+1
i= k
k
∑ αj zj
k i
j =0
∑
α j pim++j1 r :
j=0
E (Pm+1 ) :=
i = βi =
β
k
k
The coefficients α j in this definition can be an arbitrary finite sequence of real numbers. The indices of the vertices pim+1 are taken
modulo 2n according to the topological structure of the closed polygon Pm+1 . To achieve full generality we introduce the shift r such
that K (pim+1 ) depends on pim+r 1 ; : : : ; pim++k1 r . Every discrete measure of roughness K is associated with a characteristic polynomial
α(z) =
with the coefficients
(2)
1.4 Implicit refinement schemes
In the last section we showed that the minimization of a quadratic
energy functional (1) leads to the conditions (5) which determine
the solution. Dropping the variational background, we can more
generally prescribe arbitrary real coefficients β k ; : : : ; βk (with
β i = βi to establish symmetry and ∑ βi = 0 for affine invariance)
and define an interpolatory refinement scheme which chooses the
m+1
new points p2i
of the refined polygon Pm+1 such that the homo+1
geneous constraints
k
∑
i= k
m+1
βi p2l
+1+i
=
l = 0; : : : ; n
0;
1
Remark We do not consider implicit refinement schemes with
complex coefficients βi since then (6) in general has no real solutions.
Example To illustrate the statement of the last theorem we look
at the 4-point scheme of (Dubuc, 1986). This is a stationary rem+1
finement scheme where the new points p2i
are computed by the
+1
rule
9 m
m
(p + pi+1 )
16 i
1 m
m
(p
+ pi+2 ):
16 i 1
The scheme can be written in implicit form (6) with k = 3 and
β3 = 1, β2 = 0, β1 = 9, β0 = 16 since the common factor
1
The roots of β(z) are z1 = : : : = z4 = 1 and
16 is not relevant.
p
z5;6 = 2 3. From the construction of the last proof we obtain
α(z)
= (2 +
p
3)
(3 +
p
p
12) z + 3 z2 + z3
as one possible solution. Hence, the quadratic strain energy
which is minimized by the 4-point scheme is based on the local
roughness estimate
K (pi )
= (2 +
p
3 ) pi
k
Ek (Pm+1 )
(3 +
p
p
12) pi+1 + 3 pi+2 + pi+3 :
1.5 Minimization of differences
Theorem 2 asserts that a fast contraction rate of some higher differences is sufficient for the convergence of a sequence of polygons to
a (k times) continuously differentiable limit curve. Thus it is natural to look for refinement schemes with a maximum contraction
of differences. This obviously is an application of the variational
approach. For the quadratic energy functional we make the ansatz
2n 1
Ek (Pm+1 ) :=
∑ k4k pim+1 k2
i=0
:
∑
=
i=0
(7)
∂
k4k p2lm++11
m+1
∂ p2l
+1
k
=
Theorem 4 Let β k ; : : : ; βk be an arbitrary symmetric set of real
coefficients (β i = βi ). Then there always exists a (potentially complex valued) local roughness measure K such that (6) is the EulerLagrange-equation corresponding to the minimization of the energy
functional (1). (Proof to be found in the original paper)
=
∂
m+1
∂ p2l
+1
(6)
are satisfied. We call these schemes: implicit refinement schemes
to emphasize the important difference to other refinement schemes
where usually the new points are computed by one or two explicitly
given rules (cf. the term implicit curves for curves represented by
f (x; y) = 0). The stationary refinement schemes are a special case
of the implicit schemes where β2 j = δ j;0 . In general, the implicit
schemes are non-stationary since the resulting weight coefficients
m+1
by which the new points p2i
are computed depend on the number
+1
of vertices in Pm .
In (Kobbelt, 1995b) a general technique is presented which allows to analyse the smoothness properties of the limiting curve generated by a given implicit refinement scheme.
The next theorem reveals that the class of implicit refinement
schemes is not essentially larger than the class of variational
schemes.
m+1
p2i
+1
The partial derivatives take a very simple form in this case
2
∑
k +i
( 1)
i=0
=
k
i
m+1
2 ( 1)k 42k p2l
+1
i
k2
4k p2lm
+1
+1
i
k:
and the corresponding Euler-Lagrange-equation is
42k p2lm
+1
+1
k = 0;
l = 0; : : : ; n
1
(8)
where, again, the indices of the pim+1 are taken modulo 2n. The
characteristic polynomial of the underlying roughness measure K
is α(z) = (z 1)k and thus solvability and affine invariance of the
refinement scheme are guaranteed. The solution of (8) only requires
the inversion of a banded circulant matrix with bandwidth 2 b k2 c + 1.
Theorem 5 The refinement scheme based on the minimization of
Ek in (7) produces at least C k -curves. (Proof to be found in the
original paper)
In order to prove even higher regularities of the limiting curve
one has to combine more refinement steps. In (Kobbelt, 1995b)
a simple technique is presented that allows to do the convergence
analysis of such multi-step schemes numerically. Table 1 shows
some results where r denotes the number of steps that have to be
combined in order to obtain these differentiabilities.
In analogy to the non-discrete case where the minimization of the
integral over the squared k-th derivative has piecewise polynomial
C2k 2 solutions (B-splines), it is very likely that the limiting curves
generated by iterative minimization of Ek are actually in C2k 2 too.
The results given in Table 1 can be improved by combining more
than r steps. For k = 2; 3, however, sufficiently many steps have
already been combined to verify P∞ 2 C2k 2 .
k
2
3
4
5
6
r
2
11
2
7
3
diff’ty
C2
C4
C5
C7
C8
k
7
8
9
10
11
r
6
4
6
4
6
diff’ty
C10
C11
C13
C14
C16
Table 1: Lower bounds on the differentiability of P∞ generated by
iterative minimization of Ek (Pm ).
For illustration and to compare the quality of the curves generated by these schemes, some examples are given in Fig. 3. The
curves result from applying different schemes to the initial data
P0 = (: : : ; 0; 1; 0; : : :). We only show the middle part of one periodic interval of P∞ . As expected, the decay of the function becomes
slower as the smoothness increases.
Remark Considering Theorem 2 it would be more appropriate to minimize the maximum difference k4k Pm k∞ instead of
k4k Pm k2 . However, this leads to non-linear refinement schemes
which are both, hard to compute and difficult to analyse. Moreover, in (Kobbelt, 1995a) it is shown that a contraction rate of
F
E2
E3
E5
Figure 3: Discrete curvature plots of finite approximations to the curves generated by the four-point scheme F (P∞
minimization of E2 (P∞ 2 C2 ), E3 (P∞ 2 C4 ) and E5 (P∞ 2 C7 ).
k42k Pm k∞ = O(2
implies k4k Pm k∞ = O(2 m (k ε) ) for every ε > 0. It is further shown that k4k Pm k∞ = O(2 mk ) is the
theoretical fastest contraction which can be achieved by interpolatory refinement schemes. Hence, the minimization of k4k Pm k∞
cannot improve the asymptotic behavior of the contraction.
mk )
1.6 Interpolatory refinement of open polygons
The convergence analysis of variational schemes in the case of open
finite polygons is much more difficult than it is in the case of closed
polygons. The problems arise at both ends of the polygons Pm
where the regular topological structure is disturbed. Therefore, we
can no longer describe the refinement operation in terms of Toeplitz
matrices but we have to use matrices which are Toeplitz matrices almost everywhere except for a finite number of rows, i.e., except for
the first and the last few rows.
However, one can show that in a middle region of the polygon
to be refined the smoothing properties of an implicit refinement
scheme applied to an open polygon do not differ very much from
the same scheme applied to a closed polygon. This is due to the fact
that in both cases the influence of the old points pm
i on a new point
m+1
p2 j+1 decrease exponentially with increasing topological distance
ji jj for all asymptotically stable schemes (Kobbelt, 1995a).
For the refinement schemes which iteratively minimize forward
differences, we can at least prove the following.
Theorem 6 The interpolatory refinement of open polygons by iteratively minimizing the 2k-th differences, generates at least C k 1 curves. (Proof to be found in the original paper)
The statement of this theorem only gives a lower bound for the
differentiability of the limiting curve P∞ . However, the author conjects that the differentiabilities agree in the open and closed polygon
case. For special cases we can prove better results.
Theorem 7 The interpolatory refinement of open polygons by iteratively minimizing the second differences, generates at least C 2 curves. (Proof to be found in the original paper)
1.7 Local refinement schemes
By now we only considered refinement schemes which are based
on a global optimization problem. In order to construct local refinement schemes we can restrict the optimization to some local
m+1
subpolygon. This means a new point p2l
is computed by mini+1
m
mizing some energy functional over a window pm
l r ; : : : ; pl +1+r . As
the index l varies, the window is shifted in the same way.
2 C1 ) and the iterative
Let E be a given quadratic energy functional. The solution of
m
its minimization over the window pm
l r ; : : : ; pl +1+r is computed by
solving an Euler-Lagrange-equation
m+1
r
B (p2l
)
+1+2i i=
m r+1
r = C (pl +i )i= r :
(9)
The matrix of B 1 C can be computed explicitly and the weight
m+1
coefficients by which a new point p2l
is computed, can be read
+1
1
off from the corresponding row in B C . Since the coefficients
depend on E and r only, this construction yields a stationary refinement scheme.
For such local schemes the convergence analysis is independent
from the topological structure (open/closed) of the polygons to be
refined. The formalisms of (Cavaretta et al., 1991), (Dyn & Levin,
1990) or (Kobbelt, 1995b) can be applied.
Minimizing the special energy functional Ek (P) from (7) over
open polygons allows the interesting observation that the resulting
refinement scheme has polynomial precision of degree k 1. This
is obvious since for points lying equidistantly parameterized on a
polynomial curve of degree k 1, all k-th differences vanish and
Ek (P) = 0 clearly is the minimum of the quadratic functional.
Since the 2r + 2 points which form the subpolygon
m
pm
l r ; : : : ; pl +1+r uniquely define an interpolating polynomial
of degree 2r + 1, it follows that the local schemes based on
the minimization of Ek (P) are identical for k 2r + 2. These
schemes coincide with the Lagrange-schemes of (Deslauriers &
Dubuc, 1989). Notice that k 4r + 2 is necessary because higher
+1
differences are not possible on the polygon p2m(+l 1r) ; : : : ; pm
2(l +1+r)
and minimizing Ek (P) 0 makes no sense.
The local variational schemes provide a nice feature for practical purposes. One can use the refinement rules defined by the
coefficients in the rows of B 1 C in (9) to compute points which
subdivide edges near the ends of open polygons. Pure stationary
refinement schemes do not have this option and one therefore has
to deal with shrinking ends. This means one only subdivides those
edges which allow the application of the given subdivision mask
and cuts off the remaining part of the unrefined polygon.
If k 2r + 2 then the use of these auxiliary rules causes the limiting curve to have a polynomial segment at both ends. This can
be seen as follows. Let P0 = (p00 ; : : : ; p0n ) be a given polygon and
denote the polynomial of degree 2r + 1 k 1 uniformly interpolating the points p00 ; : : : ; p02r+1 by f (x).
The first vertex of the refined polygon P1 which not necessarily
lies on f (x) is p12r+3 . Applying the same refinement scheme iteratively, we see that if pm
δ is the first vertex of Pm which does not lie
m
m+1
on f (x) then pδm+1 = p2δ
is the first vertex of Pm+1 with this
m+1
m 2r 1
property. Let δ0 = 2r + 2 and consider the sequence
δm
lim
= (2r + 2)
m!∞ 2m
m
(2r + 1)
lim
m!∞
∑2
i
=
1:
i=1
Hence, the limiting curve P∞ has a polynomial segment f (x)
between the points p00 and p01 . An analog statement holds at the
opposite end between p0n 1 and p0n .
This feature also arises naturally in the context of Lagrangeschemes where the new points near the ends of an open polygon
can be chosen to lie on the first or last well-defined polynomial. It
can be used to exactly compute the derivatives at the endpoints p00
and p0n of the limiting curve and it also provides the possibility to
smoothly connect refinement curves and polynomial splines.
1.8 Computational Aspects
Since for the variational refinement schemes the computation of the
m+1
new points p2i
involves the solution of a linear system, the algo+1
rithmic structure of these schemes is slightly more complicated than
it is in the case of stationary refinement schemes. However, for the
refinement of an open polygon Pm the computational complexity is
still linear in the length of Pm . The matrix of the system that has
to be solved, is a banded Toeplitz-matrix with a small number of
pertubations at the boundaries.
In the closed polygon case, the best we can do is to solve the
circulant system in the Fourier domain. In particular, we transform
the initial polygon P0 once and then perform m refinement steps
in the Fourier domain where the convolution operator becomes a
bm is finally transformed
diagonal operator. The refined spectrum P
back in order to obtain the result Pm . The details can be found in
(Kobbelt, 1995c). For this algorithm, the computational costs are
bm which can
dominated by the discrete Fourier transformation of P
be done in O(n log(n)) = O(2m m) steps. This is obvious since the
number n = 2m n0 of points in the refined polygon Pm allows to
apply m steps of the fast Fourier transform algorithm.
The costs for computing Pm are therefore O(m) per point compared to O(1) for stationary schemes. However, since in practice
only a small number of refinement steps are computed, the constant
factors which are hidden within these asymptotic estimates are relevant. Thus, the fact that implicit schemes need a smaller bandwidth
than stationary schemes to obtain the same differentiability of the
limiting curve (cf. Table 1) equalizes the performance of both.
In the implementation of these algorithms it turned out that all
these computational costs are dominated by the ‘administrative’
overhead which is necessary, e.g., to build up the data structures.
Hence, the differences in efficiency between stationary and implicit
refinement schemes can be neglected.
References
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(1989), Symmetric iterative interpolation processes,
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scheme, Jour. of Mathem. Anal. and Appl. 114, 185–204
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subdivision schemes for the generation of curves and surfaces, in: Haußmann W. and Jetter K. eds., Multivariate Approximation and Interpolation, Birkhäuser Verlag,
Basel
[Dyn et al., 1992] Dyn, N. and Levin, D. and Liu, D. (1992), Interpolatory convexity-preserving subdivision schemes for
curves and surfaces, CAD 24, 221–216
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aided geometric design, in: Light, W. ed., Advances in
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Interpolanten, Universität Karlsruhe
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Schemes, Appl. Comp. Harmonic Anal. 5 (1998), pp. 68–
91
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Low Pass Filtering, in Daehlen, M. and Lyche, T. and
Schumaker, L. eds., Math. Meth in CAGD III
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II
Discrete Fairing
Many mathematical problems in geometric modeling are merely
due to the difficulties of handling piecewise polynomial parameterizations of surfaces (e.g., smooth connection of patches, evaluation
of geometric fairness measures). Dealing with polygonal meshes is
mathematically much easier although infinitesimal smoothness can
no longer be achieved. However, transferring the notion of fairness
to the discrete setting of triangle meshes allows to develop very
efficient algorithms for many specific tasks within the design process of high quality surfaces. The use of discrete meshes instead
of continuous spline surfaces is tolerable in all applications where
(on an intermediate stage) explicit parameterizations are not necessary. We explain the basic technique of discrete fairing and give
a survey of possible applications of this approach.
The original paper has been published in:
L. Kobbelt
Variational Design with Parametric Meshes
of Arbitrary Topology,
in Creating fair and shape preserving curves
and surfaces, Teubner, 1998
2.1 Introduction
Piecewise polynomial spline surfaces have been the standard representation for free form surfaces in all areas of CAD/CAM over the
last decades (and still are). However, although B-splines are optimal with respect to certain desirable properties (differentiability,
approximation order, locality, . . . ), there are several tasks that cannot be performed easily when surface parameterizations are based
on piecewise polynomials. Such tasks include the construction of
globally smooth closed surfaces and the shape optimization by minimizing intrinsically geometric fairness functionals [5, 12].
Whenever it comes to involved numerical computations on free
form surfaces — for instance in finite element analysis of shells —
the geometry is usually sampled at discrete locations and converted
into a piecewise linear approximation, i.e., into a polygonal mesh.
Between these two opposite poles, i.e., the continuous representation of geometric shapes by spline patches and the discrete representation by polygonal meshes, there is a compromise emerging
from the theory of subdivision surfaces [9]. Those surfaces are defined by a base mesh roughly describing its shape, and a refinement
rule that allows one to split the edges and faces in order to obtain a
finer and smoother version of the mesh.
Subdivision schemes started as a generalization of knot insertion
for uniform B-splines [11]. Consider a control mesh [ci; j ] and the
knot vectors [ui ] = [i hu ] and [vi ] = [i hv ] defining a tensor product
B-spline surface S . The same surface can be given with respect to
the refined knot vectors [ûi ] = [i hu =2] and [v̂i ] = [i hv =2] by computing the corresponding control vertices [ĉi; j ], each ĉi; j being a
simple linear combination of original vertices ci; j . It is well known
that the iterative repetition of this process generates a sequence of
meshes Cm which converges to the spline surface S itself.
The generic subdivision paradigm generalizes this concept by
allowing arbitrary rules for the computation of the new control vertices ĉi; j from the given ci; j . The generalization also includes that
we are no longer restricted to tensor product meshes but can use
rules that are adapted to the different topological special cases in
meshes with arbitrary connectivity. As a consequence, we can use
any (manifold) mesh for the base mesh and generate smooth surfaces by iterative refinement.
The major challenge is to find appropriate rules that guarantee
the convergence of the meshes Cm generated during the subdivision
process to a smooth limit surface S = C∞ . Besides the classical
stationary schemes that exploit the piecewise regular structure of
iteratively refined meshes [2, 4, 9], there are more complex geometric schemes [15, 8] that combine the subdivision paradigm with
the concept of optimal design by energy minimization (fairing).
The technical and practical advantages provided by the representation of surfaces in the form of polygonal meshes stem from
the fact that we do not have to worry about infinitesimal inter-patch
smoothness and the refinement rules do not have to rely on the existence of a globally consistent parameterization of the surface. In
contrast to this, spline based approaches have to introduce complicated non-linear geometric continuity conditions to achieve the
flexibility to model closed surfaces of arbitrary shape. This is due
to the topologically rather rigid structure of patches with triangular
or quadrilateral parameter domain and fixed polynomial degree of
cross boundary derivatives. The non-linearity of such conditions
makes efficient optimization difficult if not practically impossible.
On discrete meshes however, we can derive local interpolants according to local parameterizations (charts) which gives the freedom
to adapt the parameterization individually to the local geometry and
topology.
In the following we will shortly describe the concept of discrete
fairing which is an efficient way to characterize and compute dense
point sets on high quality surfaces that observe prescribed interpolation or approximation constraints. We then show how this approach can be exploited in several relevant fields within the area of
free form surface modeling.
The overall objective behind all the applications will be the attempt to avoid, bypass, or at least delay the mathematically involved
generation of spline CAD-models whenever it is appropriate. Especially in the early design stages it is usually not necessary to have an
explicit parameterization of a surface. The focus on polygonal mesh
representations might help to free the creative designer from being
confined by mathematical restrictions. In later stages the conversion into a spline model can be based on more reliable information
about the intended shape. Moreover, since technical engineers are
used to performing numerical simulations on polygonal approximations of the true model anyway, we also might find short-cuts that
allow to speed up the turn-around cycles in the design process, e.g.,
we could alter the shape of a mechanical part by modifying the FEmesh directly without converting back and forth between different
CAD-models.
2.2 Fairing triangular meshes
The observation that in many applications the global fairness of a
surface is much more important than infinitesimal smoothness motivates the discrete fairing approach [10]. Instead of requiring G1
or G2 continuity, we simply approximate a surface by a plain triangular C0 – mesh. On such a mesh we can think of the (discrete) curvature being located at the vertices. The term fairing in this context
means to minimize these local contributions to the total (discrete)
curvature and to equalize their distribution across the mesh.
We approximate local curvatures at every vertex p by divided
differences with respect to a locally isometric parameterization µp .
This parameterization can be found by estimating a tangent plane
Tp (or the normal vector np ) at p and projecting the neighboring
vertices pi into that plane. The projected points yield the parameter
values (ui ; vi ) if represented with respect to an orthonormal basis
feu ; ev g spanning the tangent plane
pi
p
=
ui eu + vi ev + di np :
Another possibility is to assign parameter values according to the
lengths and the angles between adjacent edges (discrete exponential
map) [15, 10].
To obtain reliable curvature information at p, i.e., second order
partial derivatives with respect to the locally isometric parameterization µp , we solve the normal equation of the Vandermonde system
VT V
h1
2
fuu ; fuv ;
1
2 f vv
iT
=
V T [di ]i
with V = [ u2i ; ui vi ; v2i ]i by which we get the best approximating
quadratic polynomial in the least squares sense. The rows of the inverse matrix (V T V ) 1V T =: [αi; j ] by which the Taylor coefficients
f of this polynomial are computed from the data [di ]i , contain the
coefficients of the corresponding divided difference operators Γ .
Computing a weighted sum of the squared divided differences is
equivalent to the discrete sampling of the corresponding continuous
fairness functional. Consider for example
Z
κ21 + κ22 d S
S
which is approximated by
∑
pi
ωi
kΓuu (p j
2 kΓuv (p j
pi )k2 +
pi )k2
+
kΓvv (p j
pi )k2
(10)
:
Notice that the value of (10) is independent of the particular choices
feu ev g for each vertex due to the rotational invariance of the func;
tional. The discrete fairing approach can be understood as a generalization of the traditional finite difference method to parametric
meshes where divided difference operators are defined with respect
to locally varying parameterizations. In order to make the weighted
sum (10) of local curvature values a valid quadrature formula, the
weights ωi have to reflect the local area element which can be approximated by observing the relative sizes of the parameter triangles in the local charts µp : pi p 7! (ui ; vi ).
Since the objective functional (10) is made up of a sum over
squared local linear combinations of vertices (in fact, of vertices
being direct neighbors of one central vertex), the minimum is characterized by the solution of a global but sparse linear system. The
rows of this system are the partial derivatives of (10) with respect
to the movable vertices pi . Efficient algorithms are known for the
solution of such systems [6].
2.3 Applications to free form surface design
When generating fair surfaces from scratch we usually prescribe a
set of interpolation and approximation constraints and fix the remaining degrees of freedom by minimizing an energy functional.
In the context of discrete fairing the constraints are given by an initial triangular mesh whose vertices are to be approximated by a fair
surface being topologically equivalent. The necessary degrees of
freedom for the optimization are obtained by uniformly subdividing the mesh and thus introducing new movable vertices.
The discrete fairing algorithm requires the definition of a local
parameterization µp for each vertex p including the newly inserted
ones. However, projection into an estimated tangent plane does not
work here, because the final positions of the new vertices are obviously not known a priori. In [10] it has been pointed out that
in order to ensure solvability and stability of the resulting linear
system, it is appropriate to define the local parameterizations (local metrics) for the new vertices by blending the metrics of nearby
vertices from the original mesh. Hence, we only have to estimate
the local charts covering the original vertices to set-up the linear
system which characterizes the optimal surface. This can be done
prior to actually computing a solution and we omit an additional
optimization loop over the parameterization.
When solving the sparse linear system by iterative methods we
observe rather slow convergence. This is due to the low-pass filter characteristics of the iteration steps in a Gauß-Seidel or Jacobi
scheme. However since the mesh on which the optimization is performed came out of a uniform refinement of the given mesh (subdivision connectivity) we can easily find nested grids which allow the
application of highly efficient multi-grid schemes [6].
Moreover, in our special situation we can generate sufficiently
smooth starting configurations by midpoint insertion which allows
us to neglect the pre-smoothing phase and to reduce the V-cycle of
the multi-grid scheme to the alternation of binary subdivision and
iterative smoothing. The resulting algorithm has linear complexity
in the number of generated triangles.
The advantage of this discrete approach compared to the classical fair surface generation based on spline surfaces is that we do not
have to approximate a geometric functional that uses true curvatures
by one which replaces those by second order partial derivatives with
respect to the fixed parameterization of the patches. Since we can
use a custom tailored parameterization for each point evaluation of
the second order derivatives, we can choose this parameterization
to be isometric — giving us access to the true geometric functional.
Figure 4 shows an example of a surface generated this way. The
implementation can be done very efficiently. The shown surface
consists of about 50K triangles and has been generated on a SGI
R10000 (195MHz) within 10 seconds. The scheme is capable of
generating an arbitrarily dense set of points on the surface of minimal energy. It is worth to point out that the scheme works completely automatic: no manual adaption of any parameters is necessary, yet the scheme produces good surfaces for a wide range of
input data.
2.4 Applications to interactive modeling
For subdivision schemes we can use any triangular mesh as a control mesh roughly describing the shape of an object to be modeled.
The flexibility of the schemes with respect to the connectivity of the
underlying mesh allows very intuitive modifications of the mesh.
The designer can move the control vertices just like for Bezierpatches but she is no longer tied to the common restrictions on the
connectivity which is merely a consequence of the use of tensor
product spline bases.
When modeling an object by Bezier-patches, the control vertices
are the handles to influence the shape and the de Casteljau algorithm
associates the control mesh with a smooth surface patch. In our
more general setting, the designer can work on an arbitrary triangle
mesh and the connection to a smooth surface is provided by the
discrete fairing algorithm. The advantages are that control vertices
are interpolated which is a more intuitive interaction metaphor and
the topology of the control structure can adapt to the shape of the
object.
Figure 5 shows the model of a mannequin head. A rather coarse
triangular mesh allows already to define the global shape of the
head (left). If we add more control vertices in the areas where more
detail is needed, i.e., around the eyes, the mouth and the ears, we
can construct the complex surface at the far right. Notice how the
discrete fairing scheme does not generate any artifacts in regions
where the level of detail changes.
2.5 Applications to mesh smoothing
In the last sections we saw how the discrete fairing approach can be
used to generate fair surfaces that interpolate the vertices of a given
triangular mesh. A related problem is to smooth out high frequency
noise from a given detailed mesh without further refinement. Consider a triangulated surface emerging for example from 3D laser
scanning or iso-surface extraction out of CT volume data. Due to
measurement errors, those surfaces usually show oscillations that
do not stem from the original geometry.
Figure 4: A fair surface generated by the discrete fairing scheme. The flexibility of the algorithm allows to interpolate rather complex data by
high quality surfaces. The process is completely automatic and it took about 10 sec to compute the refined mesh with 50K triangles. On the
right you see the reflection lines on the final surface.
Figure 5: Control meshes with arbitrary connectivity allow to adapt the control structure to the geometry of the model. Notice that the
influence of one control vertex in a tensor product mesh is always rectangular which makes it difficult to model shapes with non-rectangular
features.
Constructing the above mentioned local parameterizations, we
are able to quantify the noise by evaluating the local curvature.
Shifting the vertices while observing a maximum tolerance can reduce the total curvature and hence smooth out the surface. From a
signal processing point of view, we can interpret the iterative solving steps for the global sparse system as the application of recursive
digital low-pass filters [13]. Hence it is obvious that the process
will reduce the high frequency noise while maintaining the low frequency shape of the object.
Figure 6 shows an iso-surface extracted from a CT scan of an
engine block. The noise is due to inexact measurement and instabilities in the extraction algorithm. The smoothed surface remains
within a tolerance which is of the same order of magnitude as the
diagonal of one voxel in the CT data.
2.6 Applications to surface interrogation
Deriving curvature information on a discrete mesh is not only useful for fair interpolation or post-processing of measured data. It can
also be used to visualize artifacts on a surface by plotting the color
coded discrete curvature directly on the mesh. Given for example
the output of the numerical simulation of a physical process: since
deformation has occurred during the simulation, this output typically consists merely of a discrete mesh and no continuous surface
description is available.
Figure 6: An iso-surface extracted from a CT scan of an engine
block. On the left, one can clearly see the noise artifacts due to
measurement and rounding errors. The right object was smoothed
by minimizing the discrete fairing energy. Constraints on the positional delocation were imposed.
Using classical techniques from differential geometry would require to fit an interpolating spline surface to the data and then visualize the surface quality by curvature plots. The availability of
samples of second order partial derivatives with respect to locally
isometric parameterizations at every vertex enables us to show this
information directly without the need for a continuous surface.
Figure 7 shows a mesh which came out of the FE-simulation of
a loaded cylindrical shell. The shell is clamped at the boundaries
and pushed down by a force in normal direction at the center. The
deformation induced by this load is rather small and cannot be detected by looking, e.g., at the reflection lines. The discrete mean
curvature plot however clearly reveals the deformation. Notice that
histogram equalization has been used to optimize the color contrast
of the plot.
2.7 Applications to hole filling and blending
Another area where the discrete fairing approach can help is the
filling of undefined regions in a CAD model or in a measured data
set. Of course, all these problems can be solved by fairing schemes
based on spline surfaces as well. However, the discrete fairing approach allows one to split the overall (quite involved) task into simple steps: we always start by constructing a triangle mesh defining
the global topology. This is easy because no G1 or higher boundary conditions have to be satisfied. Then we can apply the discrete
fairing algorithm to generate a sufficiently dense point set on the objective surface. This part includes the refinement and energy minimization but it is almost completely automatic and does not have to
be adapted to the particular application. In a last step we fit polynomial patches to the refined data. Here we can restrict ourselves
to pure fitting since the fairing part has already been taken care of
during the generation of the dense data. In other words, the discrete
fairing has recovered enough information about an optimal surface
such that staying as close as possible to the generated points (in a
least squares sense) is expected to lead to high quality surfaces. To
demonstrate this methodology we give two simple examples.
First, consider the point data in Figure 8. The very sparsely
scattered points in the middle region make the task of interpolation
rather difficult since the least squares matrix for a locally supported
B-spline basis might become singular. To avoid this, fairing terms
would have to be included into the objective functional. This however brings back all the problems mentioned earlier concerning the
possibly poor quality of parameter dependent energy functionals
and the prohibitive complexity of non-linear optimization.
Alternatively, we can connect the points to build a spatial triangulation. Uniform subdivision plus discrete fairing recovers the
missing information under the assumption that the original surface
was sufficiently fair. The un-equal distribution of the measured data
points and the strong distortion in the initial triangulation do not
cause severe instabilities since we can define individual parameterizations for every vertex. These allow one to take the local geometry
into account.
Another standard problem in CAD is the blending or filleting
between surfaces. Consider the simple configuration in Figure 9
where several plane faces (dark grey) are to be connected smoothly.
We first close the gap by a simple coarse triangular mesh. Such
a mesh can easily be constructed for any reasonable configuration
with much less effort than constructing a piecewise polynomial representation. The boundary of this initial mesh is obtained by sampling the surfaces to be joined.
We then refine the mesh and, again, apply the discrete fairing
machinery. The smoothness of the connection to the predefined
parts of the geometry is guaranteed by letting the blend surface
mesh overlap with the given faces by one row of triangles (all necessary information is obtained by sampling the given surfaces). The
vertices of the triangles belonging to the original geometry are not
allowed to move but since they participate in the global fairness
functional they enforce a smooth connection. In fact this technique
allows to define Hermite-type boundary conditions.
Figure 8: The original data on the left is very sparse in the middle region of the object. Triangulating the points in space and discretely fairing the iteratively refined mesh recovers more information which makes least squares approximation much easier. On the
right, reflection lines on the resulting surface are shown.
2.8 Conclusion
In this paper we gave a survey of currently implemented applications of the discrete fairing algorithm. This general technique can
be used in all areas of CAD/CAM where an approximation of the
actual surface by a reasonably fine triangular mesh is a sufficient
representation. If compatibility to standard CAD formats matters, a
spline fitting post-process can always conclude the discrete surface
generation or modification. This fitting step can rely on more information about the intended shape than was available in the original
setting since a dense set of points has been generated.
As we showed in the previous sections, mesh smoothing and hole
filling can be done on the discrete structure before switching to a
continuous representation. Hence, the bottom line of this approach
is to do most of the work in the discrete setting such that the mathematically more involved algorithms to generate piecewise polynomial surfaces can be applied to enhanced input data with most
common artifacts removed.
We do not claim that splines could ever be completely replaced
by polygonal meshes but in our opinion we can save a considerable
amount of effort if we use spline models only where it is really
necessary and stick to meshes whenever it is possible. There seems
to be a huge potential of applications where meshes do the job if we
find efficient algorithms.
The major key to cope with the genuine complexity of highly
detailed triangle meshes is the introduction of a hierarchical structure. Hierarchies could emerge from classical multi-resolution techniques like subdivision schemes but could also be a by-product of
mesh simplification algorithms.
An interesting issue for future research is to find efficient and
numerically stable methods to enforce convexity preservation in the
fairing scheme. At least local convexity can easily be maintained
by introducing non-linear constraints at the vertices.
Prospective work also has to address the investigation of explicit
and reliable techniques to exploit the discrete curvature information
for the detection of feature lines in the geometry in order to split a
given mesh into geometrically coherent segments. Further, we can
try to identify regions of a mesh where the value of the curvature
is approximately constant — those regions correspond to special
geometries like spheres, cylinders or planes. This will be the topic
of a forthcoming paper.
References
[1] E. Catmull, J. Clark, Recursively generated B-spline surfaces
on arbitrary topological meshes, CAD 10 (1978), pp. 350–355
Figure 7: Visualizing the discrete curvature on a finite element mesh allows to detect artifacts without interpolating the data by a continuous
surface.
Figure 9: Creating a “monkey saddle“ blend surface to join six planes. Any blend surface can be generated by closing the gap with a triangular
mesh first and then applying discrete fairing.
[2] Celniker G. and D. Gossard, Deformable curve and surface
finite elements for free-form shape design, ACM Computer
Graphics 25 (1991), 257–265.
[10] Kobbelt L., Discrete fairing, Proceedings of the Seventh IMA
Conference on the Mathematics of Surfaces, 1997, pp. 101–
131.
[3] D. Doo and M. Sabin, Behaviour of Recursive Division Surfaces Near Extraordinary Points, CAD 10 (1978), pp. 356–
360
[11] J. Lane and R. Riesenfeld, A Theoretical Development for
the Computer Generation and Display of Piecewise Polynomial Surfaces , IEEE Trans. on Pattern Anal. and Mach. Int.,
2 (1980), pp. 35–46
[4] N. Dyn, Subdivision Schemes in Computer Aided Geometric
Design, Adv. Num. Anal. II, Wavelets, Subdivisions and Radial Functions, W.A. Light ed., Oxford Univ. Press, 1991, pp.
36–104.
[12] Moreton H. and C. Séquin, Functional optimization for fair
surface design, ACM Computer Graphics 26 (1992), 167–
176.
[5] Greiner G., Variational design and fairing of spline surfaces,
Computer Graphics Forum 13 (1994), 143–154.
[13] Taubin G., A signal processing approach to fair surface design, ACM Computer Graphics 29 (1995), 351–358
[6] Hackbusch W., Multi-Grid Methods and Applications,
Springer Verlag 1985, Berlin.
[14] Welch W. and A. Witkin, Variational surface modeling, ACM
Computer Graphics 26 (1992), 157–166
[7] Hagen H. and G. Schulze, Automatic smoothing with geometric surface patches, CAGD 4 (1987), 231–235.
[15] Welch W. and A. Witkin, Free-form shape design using triangulated surfaces, ACM Computer Graphics 28 (1994), 247–
256
[8] Kobbelt L., A variational approach to subdivision , CAGD 13
(1996), 743–761.
[9] Kobbelt L., Interpolatory subdivision on open quadrilateral
nets with arbitrary topology, Comp. Graph. Forum 15 (1996),
409–420.
Chapter 9
Subdivision Cookbook
Speaker: Joe Warren
A Cookbook for Variational Subdivision
Joe Warren Henrik Weimer
Department of Computer Science
Rice University
Abstract
These course notes will attempt to answer the following questions: What is
subdivision? How can the rules for subdivision schemes be derived in a systematic
manner? How can these rules be extended to handle special topological features
such as extraordinary points or creases? We will argue that most subdivision
schemes correspond to a special type of multigrid method that generates shapes
which solve (or nearly solve) a variational problem. These subdivision schemes
are influenced by two factors: the variational functional and the local grid topology. Using a recipe based on this observation, we will build subdivision schemes
for several interesting examples including B-splines, minimum energy curve networks, membrane splines and fluid flow.
1
Introduction
Computer graphics intrinsically depends on the mathematical and algorithmic representation of shape. Traditionally, smooth shapes have been represented using parametric representations such as B-splines or implicit representations such as algebraic
surfaces. During the last two decades, subdivision has evolved as a simple yet flexible
method for the modeling of geometric shapes. Starting with a coarse polyhedron, a
subdivision scheme defines a sequence of increasingly dense polyhedra that converge
to a smooth limit shape.
Modeling with subdivision is simple and easy to implement since only discrete
geometric entities such as points, edges and polygons are involved. Furthermore, the
transformation rules that yield the next finer shape are simple weighted averages of the
vertices of the coarser shape. Thus these transformations can be implemented easily
and computed very efficiently. Yet, subdivision schemes are flexible since the actual
subdivision rules can be chosen very generally. Subdivision schemes can be evaluated
locally, that is to say if only some part of the whole model shape is of interest, computational effort has only to be expended on the evaluation of the shape in that localized
area. Finally, modeling with subdivision gives us a multiresolution representation of
the shapes for granted: The control polyhedra at coarser levels approximate the limit
shape well and can be refined arbitrarily. Throughout the last few years, multiresolution surface modeling systems have been presented that exploit this idea (see [5], [13]).
1
With these systems the user is allowed to modify the polygonal shape at any level of
subdivision and therefore has control over the shape of the object at any scale.
Much work has been dedicated to analyzing a subdivision scheme given as a set
of known subdivision rules. However, relatively little work has been done on methods
for systematically generating interesting subdivision schemes. Two main techniques
for generating subdivision schemes are known: Box-splines (see deBoor et. al [2]) are
based on the idea of repeated convolution and possess relatively simple subdivision
schemes. Unfortunately, box-splines are restricted to uniform grids. Extensions of
box-splines to topologically non-uniform grids (i.e. polyhedra) has traditionally been
done in a somewhat ad-hoc manner that defies generalization (see Loop [7]).
Variational methods (Kobbelt [4], Mallet [8], Warren and Weimer [10],[11]) generate subdivision schemes as the solution process to certain types of variational problems.
Given an initial shape, the subdivision scheme generates a sequence of shapes converging to a limit shape that follows the initial shape and minimizes a functional associated
with the variational problem. These notes will show that the actual rules for these
variational subdivision schemes are determined by two factors:
the variational functional, and,
the local grid topology.
Variational methods have been very successful in Computer Aided Geometric Design because these methods yield fair, smooth shapes. The beauty of the variational
approach to subdivision is that the known box-splines can be subsumed as a special
case (see Warren and Weimer [11]) with non-uniform grids being handled in a systematic manner. In these notes, we will demonstrate a simple ”recipe” that allows one to
systematically design customized subdivision schemes. This method can easily handle
special topological features such as extraordinary points and creases. First, we review
some common methods for solving variational problems. As we shall see, these methods are intrinsically related to variational subdivision.
2
Multigrid
At its most basic level, variational modeling entails finding a shape defined over a
given domain that minimizes a given continuous functional. Due to the difficulties of
manipulating continuous functionals, a standard approach is to discretize the domain
and to convert the problem into one of minimizing a corresponding discrete functional.
By splitting the problem domain into a discrete grid T , the corresponding variational
solution can be approximated by a discrete coefficient vector p with one value per grid
point in T . The beauty of this approach is that many important variational problems
can now be expressed as the minimization of a quadratic form pT E p where E is a
symmetric, positive definite matrix whose entries depend on the variational functional.
This matrix E is often referred to as the energy matrix associated with the variational problem. E expresses a discrete approximation to the continuous variational
functional over T . Using basic calculus, it is easy to show that pT E p is minimized
if and only if E p = 0. Based on this observation, the rows of E are often viewed as
2
discrete differences that approximate the partial differential equation associated with
the variational problem (via the Euler-Lagrange equations). Since pT E p is trivially
minimized by p = 0, extra conditions are imposed on the minimization to ensure a
non-trivial solution. As we shall see, the choice of these extra conditions is very important. For now, we simply consider systems of the form
Ep
=
b;
where the entries of b capture the associated boundary conditions.
This system of linear equations can be solved using a direct solver, such as Gaussian elimination. However, in many cases, the grid T and its associated energy matrix
E is so large that using a direct solver is impractical due to running time and space
requirements. Fortunately, the matrices E associated with most variational problems
are very sparse. Traditional iterative solution methods such as the Jacobi or GaussSeidel iteration can solve such systems of moderate size. Given an initial guess, these
methods produce increasingly accurate solutions using localized, iterative update operations. See Varga [9] for an introduction to these techniques.
However, in practice, these iterative solution methods exhibit a very curious behavior: high-frequency errors are eliminated very fast, within a few iterations. However,
low-frequency errors, governing the global appearance and nature of the solution, are
eliminated only very slowly. In fact, one often observes that iterative solvers of this
type ”stall” after a certain number of iterations and the solution only improves very
marginally once high-frequency errors have been eliminated. Multigrid methods try to
circumvent this problem of ”stalling”. Using a sequence of denser and denser grids,
multigrid computes solutions in a hierarchical fashion. If we denote the sequence of
domain grids by Ti and the corresponding energy matrices by Ei , then the multigrid
method attempts to find a sequence of vectors pi satisfying the equation
Ei pi = bi
(1)
for increasing i, i.e. for increasingly dense grids. Typically, the solution process at
level i consists of three steps (see Briggs [1] for more details):
1. Prediction: Compute an initial guess p0i to the exact solution pi . This initial
guess is derived from the solution pi,1 on the next coarser grid using some type
of linear prediction function. In terms of matrix notation, this prediction step can
be written as
p0i = Si,1 pi,1
where Si,1 is a matrix that maps vectors over Ti,1 to vectors over Ti . In multigrid
terminology, Si,1 is a prolongation operator. A common choice for this prediction operation in a typical multigrid method is piecewise linear interpolation.
2. Smoothing: Improve the guess p0i using k rounds of an iterative method such as
Jacobi or Gauss-Seidel iteration. These iterative methods have the form:
j +1
pi
=
3
j
Ai pi + Bi bi
(2)
where Ai and Bi are matrices that depend on Ei and the type of smoothing method
j
j
chosen. If we let ri denote the residual Ei pi , bi , then this residual decreases
according to the equation:
j +1
j
ri = Ai ri :
3. Coarse grid correction: Restrict the residual rik to the next coarser grid Ti,1 by
eliminating all entries in rik which do not correspond to a gridpoint in Ti,1 . Given
this restricted residual ri,1 , solve the equation Ei,1 ei,1 = ri,1 on Ti,1 . Finally,
add the correction term ei = Si,1 ei,1 to the current approximation pki .
Note the interaction of these three steps: At a given level of the multigrid process,
high frequency errors are eliminated very rapidly due to the smoothing in step two. Of
course, low frequency errors still remain after smoothing. However, restricting these
low frequency errors to coarser grids eventually turns the errors into high frequency
errors which are quickly eliminated by the smoothing operation on the coarse grid. In
practice, multigrid has proven very effective at solving systems of linear equations of
this type. As we shall see in the next section, multigrid and subdivision are intrinsically
related.
3
Subdivision
For those familiar with subdivision, it is clear that subdivision and multigrid share some
striking similarities. Each method produces a sequence of vectors pi that converge to
some ”interesting” limit shape. In fact, if we closely examine the structure of multigrid,
we can express subdivision as a special case of multigrid. Consider a version of multigrid in which only the prediction step is carried out at each level (i.e. the smoothing
and coarse grid correction steps are omitted). This process would define a sequence of
vectors p0i satisfying
p0i = Si,1 p0i,1 :
This process is exactly subdivision! The predictor Si,1 is simply a subdivision
matrix. Of course, the limit shape produced by this process depends on the type of predictor chosen. For a typical multigrid predictor, such as piecewise linear interpolation,
the limit of the p0i as i ! ∞ is just a piecewise linear shape. To have any hope that this
subdivision process might produce a reasonable solution to the actual variational problem, we must base the predictor Si very carefully on the particular variational problem.
Note that if the predictor Si happens to produce a solution p0i whose corresponding
residual ri0 is always zero, then the smoothing and coarse grid correction steps could be
omitted without difficulty. Predictors Si which produce zero residual are perfect predictors in the sense that they produce perfectly accurate solutions. Subdivision using
perfect predictors produces shapes that exactly satisfy the variational problem. We next
give a simple characterization for a general class of perfect predictors Si in terms of the
energy matrices Ei .
4
3.1
Perfect predictors
By definition, the solution vectors pi satisfy the equation Ei pi = bi where the matrix
Ei is a discrete version of the variational functional on Ti . The vector bi characterized
the boundary conditions of the variational problem on Ti . The key in our derivation of
perfect predictors is the structure of the right-hand side bi . In particular, we suggest
a specific structure for the bi that results in a multigrid scheme with perfect predictors whose rows are either locally supported or highly localized. In particular, this
choice replicates many known subdivision schemes such as those for B-splines and
box-splines.
Let Ui be the upsampling matrix that takes a vector over Ti into a vector over the
next finer grid Ti+1. Ui maps the entries of these vectors as follows: entries corresponding to grid points in Ti are replicated; entries corresponding to grid points in Ti+1 , Ti
are set to zero. Thus, the upsampling matrix Ui consists of rows that are either zero (for
new grid points) or a standard unit vector (for old grid points). Now, if we choose bi to
have the form
bi = Ui,1Ui,2 :::U0E0 p0 ;
then equation 1 becomes
Ei pi = Ui,1 :::U0E0 p0 :
(3)
Standard interpolatory methods force minimization where the solutions pi interpolate
the values of p0 on the initial grid T0 . Instead, equation 3 forces minimization where
the differences Ei pi interpolate the differences E0 p0 at grid points of T0 . Repeated
upsampling forces the remaining differences of Ei pi at grid points in Ti , T0 to be zero,
thus ensuring minimization of pT E p (see Warren and Weimer [11] for more details).
Given this framework, we can now characterize perfect predictors for the multigrid
scheme defined by equation 3.
Theorem: Given a solution pi,1 to equation 3, let pi be the guess produced by the
predictor Si,1 (i.e. pi = Si,1 pi,1). If Si,1 satisfies the equation
Ei Si,1 = Ui,1 Ei,1 ;
(4)
then Si,1 is a perfect predictor.
Proof: Multiply both sides of equation 4 by pi,1 from the right. Next, apply the
right-hand side of equation 3 to Ei,1 pi,1. Since pi = Si,1 pi,1 , we have shown that pi
also satisfies equation 3.
3.2
Example: Minimum energy curves
Historically, splines were a commonly used drafting tool in mechanical and engineering design, before the advent of computer aided design systems. Using a thin, flexible
strip of metal or wood a designer could draw smooth curves by first anchoring the strip
to a sequence of n + 1 points on the drafting table and then letting the strip slide freely
into a minimum energy configuration. Mathematically, this strip of metal or wood
could be modeled by a curve p[t ] where t 2 [0; n] (as a convention throughout these
5
notes, continuous functions are denoted by bold face letters). The bending energy of
p[t ] can be approximated by the continuous functional
Z n
2
E[p[t ]] =
p(2) [t ] dt :
(5)
0
The associated variational problem is to minimize E[p[t ]] subject to p[ j ] interpolating the jth anchor point. The minimizing functions for this particular problem are
well-known: p[t ] is a C2 piecewise cubic function known as a natural cubic spline.
Cubic splines are one of the fundamental tools of geometric design. In particular, they
possess a locally supported basis, the B-spline basis, with a number of remarkable
properties including a particularly simple subdivision scheme. Our goal in this section
is to discretize cubic splines, convert the problem to the multigrid setting and then to
systematically derive the subdivision scheme for cubic B-splines as a consequence of
equation 3. Later throughout these notes we will generalize this methodology to derive
subdivision schemes for other interesting variational problems.
A typical discretization of the problem from equation 5 is to replace p[t ] by a polygon p = ( p0 ; p1 ; ::: p2n). The even index vertices of p are placed at the n + 1 anchor
points. The remaining, odd index vertices of p are positioned such that the discrete
h i
bending energy of p is minimized. Thus, the vertex p j should approximate p 2j . In
this case, the discrete analog of E[p[t ]] is the functional
E[ p] = 8
2n,1
∑
j =1
( p j ,1
, 2p j + p j
2
+1 ) :
(6)
The term p j ,1 , 2p j + p j +1 is a discrete approximation to the continuous expression
p(2) [t ]. The summation is the discrete analog of integration. The constant 8 arises due
to the half integer grid spacing. This spacing causes the second difference term to be
normalized by a factor of 4. Squaring these terms raises this factor to 16. This spacing
also introduces an extra factor of 12 into the summation. Thus, the total effect of halving
the grid spacing is to multiply the energy functional by a factor of 8.
Moving this process into a multigrid setting, the polygon p is replaced by a sequence of polygons pi each of whom have n 2i vertices. Each polygon pi has those
vertices with indices that are multiples of 2i fixed at the anchor points. The remaining
vertices are positioned to minimize the functional
E[ pi ] = 8i 2i n,1
∑
j =1
(( pi ) j ,1
, 2( pi ) j + ( pi ) j
+1
)
2
:
Since these discrete functionals E[ pi] converge to the continuous functional E[p[t ]]
as i ! ∞, the polygons pi converge to p[t ] as i ! ∞. Using the energy matrices Ei
associated with the discrete functional we can write
E[ pi ] = pi T Ei pi
where, away from the boundary, the rows of the matrices Ei are all shifts of the single
sequence 8i (1; ,4; 6; ,4; 1). Note that this sequence encodes the mask for a fourth
difference.
6
To employ equation 3, we distinguish two kinds of local grid configurations. The
interior of the polygon pi is topologically uniform with all grid points looking similar.
Near the boundary of the grid, i.e. at the endpoints of pi , we distinguish among the
grid points based on their distance from the actual boundary. Our goal is now to find
two perfect predictors, one for the interior of the polygon and one for the endpoints
of the polygon. For the interior, we note that upsampling consists in splitting coarse
coefficients apart by inserting a 0 inbetween any two adjacent coefficients. In terms of
matrices, the condition Ei Si,1 = Ui,1 Ei,1 can be written as
0
BB
BB
8B
BB
[email protected]
0
BB
BB
BB
BB
@
:
:
:
:
1
0
0
0
0
:
:
:
:
:
:
:
,4
:
6
,4
1
0
0
6
,4
1
0
:
:
,4
1
0
0
0
:
:
:
:
1
0
1
,4
6
,4
0
0
1
,4
6
0
0
0
1
,4
0
0
0
0
1
:
:
:
:
:
:
:
:
:
:
,4
:
:
:
:
:
:
:
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
:
:
0
0
0
0
0
:
:
:
:
:
:
:
:
:
6
,4
1
:
:
:
,4
:
:
:
:
:
:
:
10
B
C
B
C
B
C
B
C
B
C
B
C
B
C
C
@
AB
:
6
,4
:
1
0
0
6
,4
1
0
:
:
:
:
:
:
1
C
C
C
C
C
Si,1 =
C
C
C
A
:
:
:
:
:
:
:
:
:
1
0
1
,4
6
,4
0
0
1
,4
6
:
:
:
:
,4
6
,4
1
:
:
:
:
:
1
C
C
C
C
C
C
C
C
A
:
Here, the dots indicate that only a small portion of the matrices is shown. However,
since the rows in the system repeat, this portion is sufficient to determine a perfect
predictor for the interior of the grid. Since this system is rank deficient, there is more
than one possible solution to the system above. However, there is one particularly nice
solution with minimal support. This predictor has the form
0
B
B
B
B
B
B
B
1B
Si,1 = B
8B
B
B
B
B
B
B
@
:
:
:
:
:
:
:
:
1
4
6
4
1
0
0
0
0
0
0
1
4
6
4
1
0
0
0
0
0
0
1
4
6
4
1
0
0
0
0
0
0
1
4
6
:
:
6
4
1
0
0
0
0
0
0
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
:
This matrix exactly encodes the subdivision scheme for uniform cubic B-splines due
to Lane and Riesenfeld [6]. Note the simple structure of Si,1 : all columns contain the
same sequence of coefficients (1; 4; 6; 4; 1) and the matrix has a 2-to-1-slant, i.e. as we
go over one column to the right, the column is shifted down by two. The action of
Si,1 on a polygon pi,1 is as follows: vertices of pi are positioned to lie either on the
midpoint of edges of pi,1 or near vertices of pi,1 (by taking 68 of a vertex plus 18 of
both of its neighbors).
7
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
Figure 1: Subdivision for cubic B-splines.
The second type of subdivision rule arises at the endpoints of the polygons. There,
the matrices Ei have a slightly more complicated structure. Equation 4 has in this case
the form
0
1
,2
1
1
0
0
0
BB ,2 5 ,4 1 0 0
B 1 ,4 6 ,4 1 0
8B
BB 0 1 ,4 6 ,4 1
@ 0 0 1 ,4 6 ,4
:
0
BB
BB
BB
@
:
:
:
:
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
C
C
C
C
Si,1 =
C
C
A
:
1 0 1 ,2 1
C
B
,2 5 ,4
C
B
C
B
1
,4 6
C
B
C
B
0
1
,4
C
B
[email protected]
:
:
:
:
:
:
:
:
:
0
0
1
:
:
:
:
:
It is now easy to verify that Si,1 has the form
0
B
B
B
B
1B
Si,1 = B
8B
B
B
B
@
0
0
0
0
1
8
4
1
0
0
0
0
0
4
6
4
1
0
0
0
0
1
4
6
4
1
:
:
:
:
:
:
:
:
:
:
:
1
C
C
C
C
C
C
C
C
C
C
A
1
C
C
C
C
C
C
A
:
:
Again, Si,1 has a simple structure. Its first row forces the endpoint of pi to interpolate
the endpoint of pi,1. The remaining rows agree with interior subdivision rules. In fact,
this rule produces cubic splines satisfying the natural boundary condition.
These predictors (subdivision rules) Si can easily generate arbitrarily dense polygons pi . Because these predictors are perfect (i.e. satisfy equation 4), the polygons pi
converge to the natural cubic spline p[t ]. Figure 1 shows an example of three rounds
8
of prediction (i.e. subdivision) applied to the initial polygon f(0; 4), (1; 12), (2; 10),
(3; 5), (4; 8)g.
4
A recipe for variational subdivision
The previous section argued that subdivision can be viewed as a multigrid method that
employs a perfect predictor. In every round of the multigrid process, the predictor produces an error free guess for the next solution. Thus, smoothing and coarse grid error
correction are rendered unnecessary. As an example, we derived perfect predictors
for cubic B-splines and noted that the resulting matrices have a very simple structure
with local support. This section presents a general ”recipe” for creating a variational
subdivision scheme based on these ideas. The recipe is outlined in figure 2 below.
1. Define a discrete variational functional,
2. Generate a catalog of simple mesh types arising in the
problem,
3. For each mesh type, solve for the locally supported predictor Si that minimizes jjEi Si,1 , Ui,1 Ei,1 jj.
Figure 2: A recipe for custom subdivision schemes.
The minimization process in step 3 is necessary because for many important problems a perfect predictor with local support does not exist. However, by restricting Si,1
to have fixed support and minimizing the equation jjEi Si,1 , Ui,1 Ei,1 jj, we can compute predictors that produce a reasonably accurate approximation to the exact solution.
The corresponding subdivision schemes are locally supported and produce limit shapes
that are good qualitative approximations to the exact solution. In fact, by increasing
the support of Si,1 the solution can be made arbitrarily precise. Of course, highly
local schemes cannot match the accuracy of a full multigrid method. However, at a
minimum, they provide an excellent initial guess for the smoothing and coarse grid
correction portions of full multigrid.
4.1
Relation to other work
Subdivision has been studied significantly over the course of the last decades. The
specific link between subdivision and variational problems has also been investigated
by Kobbelt [4] and Mallet [8]. Each proposes subdivision schemes consisting of a
simple predictor (perhaps piecewise linear interpolation or some known subdivision
scheme) followed by some number of smoothing steps. (In multigrid terminology, this
process is referred to as nested iteration.) Both authors neglect the possibility that
a predictor customized to the particular variational problem might produce superior
results.
9
Both Kobbelt [4] and Mallet [8] propose interpolatory schemes. The schemes proposed here are approximating in the sense that the limit shape follows an initial shape
while minimizing the variational problem. Traditionally, approximating functions,
such as B-splines, have been more successful in geometric design applications than
their interpolating counterparts because the resulting solutions typically oscillate less
and have more localized support. In some cases (e.g. B-splines and box-splines), there
exist exact, approximating schemes with locally supported bases whose interpolating
counterparts have globally supported bases. Finally, there exist variational problems
whose interpolating schemes do not converge in any reasonable sense, while their approximating counterparts do. One such example are the membrane splines discussed
later in the paper.
5
Examples
This section applies the recipe of figure 2 to several interesting example problems.
First, we generalize the subdivision scheme for minimum energy curves (B-splines)
given in the previous section to a network of curves. Next, we derive a subdivision
scheme for surfaces that mimic an elastic membrane. Then, we combine these two
problems and produce a subdivision scheme for a minimum energy curve network
connected by elastic membranes. Finally, we conclude with some ideas on further
applications of the recipe.
5.1
Minimum energy curve networks
In the case of minimum energy curves, our analysis was carried out entirely in the
discrete domain using a sequence of polygons. Here, we take a similar approach for
curve networks. A polygon network consists of a set of vertices connected by a set
of edges. Note that more than two edges in the network may be incident on a single
vertex of the network. Topologically, subdivision of a polygon network corresponds to
splitting each edge into two connected edges. In this framework, a curve network is the
limit of a sequence of increasingly dense polygonal networks produced by subdivision.
Our goal is to construct subdivision rules that position vertices on the polygon network
such that the limit of the process is a minimum energy curve network.
5.1.1
Define the energy functional
We first need to generalize the functional of equation 6 to the case of polygon networks. The crux of the functional in equation 6 was the term p j ,1 , 2p j + p j +1 . This
expression measured the amount that the polygon ( p j ,1; p j ; p j +1) ”bends”. In the case
of polygon networks, several curves may pass through a vertex. Thus, several terms
of this form may be necessary for each vertex. We suggest tagging the jth vertex of
the network with a list a[ j ] of neighbor pairs. This tagging scheme allows the user a
great deal of flexibility in creating interesting polygon networks. The discrete energy
functional for the polygon network can be written as:
10
j pi j ja[ j]j
E[ pi ] = 8i ∑
∑ (( pi )a j k 1 , 2( pi)k + ( pi )a j k 2 )2
j =1 k =i
[
; ;
]
[
; ;
]
:
(7)
Inheritance of these tags during subdivision is straightforward. A new vertex is inserted
between a pair of adjacent old vertices and is assigned a single tag pair consisting of
the two old vertices. Neighbor tags for old vertices are updated by replacing an old
adjacent vertex with the vertex inserted on the corresponding edge.
5.1.2
Generate a catalog of simple mesh types
We next identify the various types of local configurations that are possible in a polygon network. In theory, there are an infinite number of possible configurations. For
example, any number of curves can pass through a common vertex! In practice, we
focus on a few canonical vertex configurations and analyze those. The simplest two
vertex configurations have already been identified and analyzed. For vertices on the
interior of a curve and at the isolated endpoint of a curve, the energy functional of
equation 7 degenerates into the energy functional of equation 6. Since the analysis of
the previous section was entirely local, the subdivision rules for these two vertex configurations remain unchanged. Due to the interpolatory nature of the endpoint rule, the
rule for several curves terminating at a common vertex remains simple interpolation.
We now focus on three important types of vertex configurations. Other types of vertex
configurations can typically be expressed as variants or combinations of these three
configurations.
5.1.3
X vertex: Two curves smoothly intersecting at a vertex.
Y vertex: One curve smoothly branching into two curves at a vertex.
T vertex: One curve terminating at a vertex on a smooth curve.
Solve for locally supported predictors
To apply equation 4 for a particular vertex configuration, we first construct the matrices
Ei appropriate for the configuration. As in the previous case, all polygons incident
on the vertex are assumed to extend infinitely. Note that this assumption causes the
topology of the network to be locally invariant under subdivision. As a result, all
matrices Ei are multiples of a single common, infinite matrix E. Given this matrix
E, we next construct the predictor S defining the subdivision scheme at the vertex.
First, we fixed the support of the rows of S to have an appropriate size (in this case,
support equivalent to that of cubic splines). Next, we construct the matrix S with nonzero entries expressed as indeterminates. Extra conditions such as constant precision,
linear precision or symmetry can also be enforced in terms of constraints on these
indeterminates.
Given the matrices E and S, we are now ready to minimize the matrix expression
jj8E S , U E jj. This expression can be minimized with respect to a variety of norms.
In general, we recommend using the ∞-norm (although we have also used the 2-norm).
11
Due to the repetition of rows in E, minimizing this norm leads to a linear program
in a small number of variables that can be solved in a reasonably efficient manner
(see [10] for more details). Unfortunately, these vertex configurations do not admit
perfect predictors with local support, i.e. the matrix norm cannot be forced to zero.
However, they do have simple, locally supported rules that provide a quite reasonable
approximation to the true minimizer. Figure 3 summarizes these rules.
1
1
1
12
1
1
2
12
2
12
1
2
0
Figure 3: Subdivision rules for various vertex configurations
Figure 4 shows three curve networks produced by these rules and compares them
to the true minimum energy curve networks. The top row shows an example using
the X rule. The middle row shows an example using the Y rule. Note that the locally
supported subdivision rule for a Y vertex fails to produce the smooth branching of the
exact minimum energy curve network because the exact solution is globally supported.
The bottom row shows an example using the T rule. Based on experimentation with a
wide range of vertex configurations, we propose a general rule for subdivision at vertex
i of the original network. Let a[i] be the list of neighbor pairs associated with vertex
i: weight vertex i by 34 ; weight the remaining neighbors of vertex i by the number of
times that they appear in a[i] divided by 8 ja[i]j. Note that this general rule reproduces
the special rules given in figure 3.
5.2
Membrane splines
This section develops a subdivision scheme producing spline surfaces that behave like
an elastic membrane. Given a triangulated polyhedron, the scheme produces a sequence of increasingly faceted polyhedra using the standard 4-1 triangular face split.
The resulting sequence of triangulated polyhedra converges to a surface that follows
the initial polyhedron and approximately minimizes a simple notion of discrete elastic energy. This scheme is essentially a generalization of the functional subdivision
scheme based on Laplace’s equation that appeared in Warren and Weimer [11].
5.2.1
Define the energy functional
Given a polyhedron p with triangular faces, let a[i] denote a list containing the indices
of those vertices adjacent to vertex i. One reasonable measure of the elastic energy of
12
Figure 4: Initial networks (left), minimum energy curve networks via subdivision (middle), and exact solutions (right).
p is the sum of the squares of the edge lengths in p,
E[ p] =
1 j pj ja[i]j
∑ ( pi , pa[i; j])2:
2 i∑
=1 j =i
(8)
If E[ p] is expressed in matrix form as pT E p, then E is simply the Laplacian of the polyhedron p. In the case of polygon networks, a normalizing constant of 8i was necessary
when generalizing the functional to a sequence of polygon networks pi produced by
subdivision. For membrane energy, this constant conveniently works out to be 1: The
factor 4 induced by doubling the edge length is canceled by the factor 14 for summing
over a two-dimensional mesh. Therefore, the functional of equation 8 can be applied
independent of the level of subdivision.
5.2.2
Generate a catalog of simple mesh types
Cataloging different types of local topologies for triangular meshes is straightforward.
The standard approach is to classify the mesh locally based on the valence of a vertex.
In the uniform case, all vertices of a triangular mesh have valence 6. Vertices with
valence other than 6 are classified as extraordinary points. Since subdivision leaves
the valence of these vertices unchanged while introducing only new vertices of valence
six, the meshes in our catalog consist of a single extraordinary point surrounded by an
infinite uniform triangular mesh. Note that the infinite energy matrix E associated with
such a mesh is again independent of the level of subdivision.
13
5.2.3
Solve for locally supported predictors
Our approach follows that for polygon networks. For each mesh type, construct the
associated energy matrix E and fix a support for the predictor S. (For membranes, we
suggest the same support as used in Loop’s scheme [7]). Then, minimize the expression
jjE S , U E jj where the entries of S are treated as unknowns. Typically, the rules of S
are first computed in the uniform case (i.e. valence 6). These uniform rules can then
be used to pad the matrix S that expresses the subdivision rules at extraordinary points.
As in the case of minimum energy curve networks, it is impossible to satisfy
E S = U E exactly. Therefore, there does not exist a locally supported subdivision
scheme that behaves exactly like a membrane. However, there do exist simple, locally
supported subdivision rules that provide a reasonable approximation to the behavior of
a membrane. Figure 5 summarizes these rules. Note that these rules are not the exact
minimizer of jjE S , U E jj, but have the property that they are simple and close to the
true minimizer.
-2
1
-2
-2
-2
4
4
7n
-2
-2
1
-2
-2
Figure 5: Subdivision rules for membrane splines.
The figure 6 shows an example application of this subdivision scheme. A coarse
octahedron is shown on the top with increasing levels of subdivision below. Note
that the subdivision scheme produces a very ”spiky” shape. This effect models the
behavior of an elastic membrane. The surface spikes at coarse vertices and stretches
into a narrow tube to produce low elastic energy. In general, the subdivision scheme
yields a good low frequency approximation of the exact minimizer, in the sense that the
overall shape of the resulting objects is accurate. However, there are also smaller spikes
scattered over the surfaces away from the original coarsest vertices. These extra spikes
are a result of the fact that the approximate S used in the subdivision scheme does not
exactly satisfy E S = U E. Specifically, those differences of E S corresponding to new
vertices are not exactly zero as required by U E.
At this point, we have two options. The first option is to increase the support
of the predictor S. This change will produce a better predictor and a more accurate
subdivision scheme. However, the resulting shapes will still exhibit the same problem
(although at a much reduced scale). Also, manipulating subdivision rules over larger
14
Figure 6: Membrane spline defined via subdivision. Some of the large spikes were
truncated to focus on the interesting parts of the figure.
supports can be difficult due to the interactions of extraordinary points on the initial
coarse grid.
The second possibility is to employ several rounds of smoothing after each round
of subdivision (as proposed by Kobbelt [4] and Mallet [8]). After each round of subdivision, we suggest running several rounds of smoothing with one crucial modification.
Recalling equation 2 (one round of subdivision with bi = Ui,1 Ei,1 pi,1), we can express standard smoothing as
j +1
pi
=
j
Ai pi + Bi (Ui,1 Ei,1 pi,1 )
(9)
where Ai and Bi depend on Ei . Since pi,1 has been produced by an inexact subdivision
scheme, many of the differences in Ei,1 pi,1 that are zero for the exact scheme are
actually non-zero for the inexact scheme. Instead of propagating these non-zero differences, we suggest modifying the upsampling matrix Ui,1 in equation 9 to upsample
only those differences whose corresponding exact differences are non-zero. Typically,
this change causes only differences corresponding to a subset of the vertices in p0 to
be upsampled. (Some entries in E0 p0 may also be exactly zero.)
15
Figure 7: Membrane spline defined via subdivision plus smoothing (3 rounds per level).
Some of the large spikes were truncated to focus on the interesting parts of the figure.
This modified smoothing effectively eliminates the extraneous spikes of figure 6
due to their high frequency nature. Combining the low frequency accuracy of S with
the high frequency correction of smoothing yields a subdivision scheme that is visually accurate. Note that because the overall, low frequency shape yielded by the local
subdivision scheme S is very close to the exact solution, only a few rounds of smoothing are necessary to eliminate all high frequency noise. Figure 7 shows this combined
scheme (employing 3 rounds of Jacobi smoothing) applied to the coarse octahedron.
The surface at the bottom of figure 7 is the exact solution.
Note that this subdivision scheme for membrane splines is approximating, not interpolating. In particular, the spikes of pi do not interpolate the vertices of the coarse
polyhedron p0 . Instead, the spikes are determined by the differences of E0 p0 . If one
of these initial differences is zero, then the corresponding limit surface is smooth at
this vertex. Figure 8 illustrates a modification of the octahedron in which the bottom
vertex has been pushed upward to form a square pyramid. Again, the exact solution
is on the bottom right. Since the base of this pyramid is now planar, the difference in
16
E0 p0 corresponding to this vertex is zero and, thus, the limit surface is smooth at this
vertex.
Figure 8: A membrane spline for a square pyramid (3 rounds of smoothing per level).
Finally, note that the lengths of the spikes in figures 7 and 8 diverge very gradually
as the number of rounds of subdivision increases. Fortunately, the exact scheme is
convergent everywhere except at the tips of these spikes. If instead one attempts to
force interpolation at the tips of the spikes, i.e. interpolation of the coarsest control
points p0 , the resulting scheme converges to a limit surface consisting of a collection
of infinitesimally thin spikes joined at the centroid of p0 .
5.3
Membrane splines and minimum energy curve networks
The final example combines the subdivision schemes for minimum energy curve networks and membrane splines. The resulting scheme produces surfaces that mimic an
elastic membrane connecting a network of minimum energy curves. The input to the
scheme is an initial triangulated polyhedron p0 and a subset n[ p0] of the edges of p0
that forms a discrete polygon network. Subsequent polyhedra pi are defined by 4-1
triangular face splits with the polygon network tags being inherited as usual.
17
5.3.1
Define the energy functional
If En and Em are the energy functionals for curve networks and membranes respectively, then the functional for the combined scheme can be written as:
E[ p] = λ En [n[ p]]
+
Em [ p]:
(10)
As λ increases, E[ p] is dominated by the energy of the curve network En [n[ p]]. In the
limit, as λ ! ∞, the surface behaves like a minimum energy curve network on n[ p]
and an elastic membrane that interpolates the curve network on the rest of p. If E[ p] is
written in matrix form, then
E[ p] = pT (λ En + Em ) p:
If the rows of En and Em are viewed as defining two sets of differences, one set characterizing curve networks and the other one characterizing membranes, then these matrices can be merged into a single difference matrix E that captures the behavior of
the functional as λ ! ∞: For vertices in n[ p], place the corresponding rows of En into
E. For the remaining vertices of p, place the corresponding rows of Em into E. The
resulting non-symmetric difference matrix E characterizes surfaces consisting of a set
of elastic membranes interpolating a minimum energy curve network.
5.3.2
Generate a catalog of simple mesh types
For vertices on the curve network, no new cataloging is necessary since the membrane
energy has no effect on the curve network. As a first attempt, we distinguish vertices
on the interior of the membrane by their valence. Later, we will see that this catalog is
insufficient.
5.3.3
Solve for locally supported predictors
Since the catalog of simple mesh types for this combined scheme is simply the union
of the two previous catalogs, the subdivision rules for minimum energy curve networks
and membrane splines can be reused. The middle portion of figure 9 depicts several
rounds of this combined subdivision scheme (without smoothing) applied to an initial
octahedron (left) with two red curve loops lying on its surface.
Figure 9: Base octahedron (left) basic subdivision scheme (center left) , augmented
subdivision scheme (center right) and subdivision with three rounds of smoothing at
every level (right).
18
Note that the resulting surface fails to behave like a membrane away from the curve
network. In particular, it bulges outward and exhibits large spikes at grid points that
should actually be smooth. The reason for this bad behavior is that the subdivision
matrices Si produced by the combined scheme are a poor solution to the equation
Ei Si,1 = Ui,1 Ei,1 . The large residuals that result cause the spikes in the center left
of figure 9.
Figure 10: Several curve loops on a octahedron connected by elastic membrane subdivision (left) and exact solution (right).
There are two solutions to this problem. First, we could augment our catalog of
local mesh topologies and design specialized subdivision rules for membrane vertices
that are employed near the curve network. These rules can be derived using a straightforward application of our standard recipe. The center right portion of figure 9 shows
an example application of these custom rules (with the same support as before) applied
to the base octahedron. Note that the large spikes are absent and the surface behaves
more like a membrane. However, the smaller high frequency spikes that plagued the
pure membrane scheme are still present. Another obvious drawback of this solution is
19
Figure 11: A catenoid defined through subdivision and the exact solution (lower right).
that the resulting catalog of custom subdivision rules would be immense. The second
solution is more passable. We simply apply several rounds of the modified smoothing used in conjunction with membrane splines. The smoothing eliminates the high
frequency spikes that result from the incompatibility of the membrane rules with the
curve network as shown in the right of figure 9.
Figure 10 shows an example of this scheme applied to an initial octahedron with
one, two and three minimum energy curve loops on its surface. The left column depicts
the result of subdivision combined with three rounds of smoothing per level; the right
column depicts the exact solution. Figure 11 shows a classical shape from surface
modeling, a catenoid. A cateniod is an elastic membrane that joins two parallel, circular
rings. The main characteristic of this shape is an inward pinching of the membrane to
minimize its elastic energy. The upper left portion of the figure depicts the initial
polygon p0 for the cateniod. Vertices of p0 on the interior of the membrane were
positioned such that E0 p0 is zero for these vertices, thus forcing the membrane to be
smooth on its interior. The remaining portions of the figure depict subdivision plus
smoothing. The resulting smooth subdivision surface is shown in the lower left and the
exact solution on the lower right.
20
5.4
5.4.1
Other applications
Subdivision scheme for lofted curve networks
Given a curve network, one typical problem is lofting: produce a surface that interpolates the curve network smoothly. The beauty of equation 10 is that the surface
functional used in place of Em can be very general. If this functional penalizes surface bending across edges in the curve network, the resulting surfaces will tend to
interpolate the curve network smoothly. The top row of figure 12 depicts an example of a smooth subdivision scheme that interpolates the red minimum energy curve
network. Note that the surface has a discontinuous tangent plane across the upper red
curve. The bottom row depicts the same scheme adjusted to produce a surface that lofts
(i.e.smoothly interpolates) the curve network. The scheme is based on the following
discrete functional for triangulated polyhedra: edges of adjacent triangles contribute
energy based on their deviation from forming a parallelogram. An interesting problem for future work is developing a set of subdivision rules for a lofted version of the
scheme of Loop [7].
Figure 12: A subdivision surface smoothly interpolating a simple curve network
5.4.2
Subdivision scheme for fluid flow
Our last example does not involve modeling surfaces. Instead, we consider modeling
vector fields. In this case, the matrices Ei are derived from the Navier-Stokes
equations
ui
satisfies these
that characterize the behavior of a fluid. A discrete vector field
vi
ui
equation if Ei
= 0. Our approach is to use equation 4 to derive a set of subvi
division rules Si,1 such that Ei Si,1 ' Ui,1 Ei,1 . The resulting subdivision scheme for
vector fields has the form:
ui
vi
= S i ,1
ui,1
vi,1
:
As i ! ∞, the limitof these
discrete fields is a continuous vector field that follows the
u0
initial vector field
and nearly satisfies the underlying Navier-Stokes equations.
v0
21
Weimer and Warren [12] describe this scheme in complete detail. Figure 13 gives an
example application of this scheme in three dimensions. Shown are a coarse, userdefined vector field around a cylinder (left) and two rounds of subdivision (center left
and center right). The right portion of the figure depicts particles being traced through
the resulting smooth flow field.
Figure 13: Subdivision for fluid flow.
6
Summary
These course notes presented a recipe for the derivation of custom subdivision schemes.
Following a few simple steps, this recipes allows one to derive schemes that model a
range of interesting objects described in terms of discrete energy functionals. The
examples demonstrated how subdivision rules for cubic B-splines, minimum energy
curve networks and membrane splines can be derived using this recipe. Finally, a
combination of energy functionals was used to derive a scheme that merges minimum
energy curve networks with membranes.
Acknowledgments
This work has been supported in part under National Science Foundation grants CCR9500572 and CCR-9732344.
22
References
[1] W. L. Briggs: A Multigrid Tutorial. Society for Industrial and Applied Mathematics, 1987.
[2] C. deBoor, K. Hollig, S. Reimenschneider: Box splines. Springer Verlag, 1993.
[3] H. Hoppe, T. DeRose, T. Duchamp, M. Halstead, H. Jin, J. McDonald, J. Schweitzer, W.
Stuetzle: Piecewise Smooth Surface Reconstruction. In SIGGRAPH 94, Computer Graphics Proceedings, Annual Conference Series, 1994, pp. 295-300.
[4] L. Kobbelt: Fairing by Finite Difference Methods. In: M. Daehlen, T. Lyche, L. L. Schumaker: Mathematical Methods for Curves and Surfaces II, Vanderbilt University Press,
1998.
[5] L. Kobbelt, S. Campagna, J. Vorsatz, H.-P. Seidel: Interactive multi-resolution modeling
on arbitrary meshes. Proceedings of SIGGRAPH 1998. In Computer Graphics Proceedings, Annual Conference Series, pp. 105-114, 1998.
[6] J. Lane, R. Reisenfeld. A theoretical development for the computer generation and display
of piecewise polynomial surfaces. IEEE Transactions on Pattern Analysis and Machine
Intelligence 2, 1, pp. 35-46, 1980.
[7] C. T. Loop: Smooth subdivision surfaces based on triangles. Master’s Thesis, Department
of Mathematics, University of Utah, August 1987.
[8] J.-L. Mallet: Discrete Smooth Interpolation. ACM Transactions on Graphics, Vol 4 No. 2,
1985, pp. 74-123.
[9] R. Varga: Matrix Iterative Analysis. Academic Press, New York, 1962.
[10] H. Weimer, J. Warren: Subdivision schemes for thin plate splines. Computer Graphics
Forum 17, 3, pp. 303-313 & 392, 1998.
[11] J. Warren, H. Weimer: Subdivision schemes for variational problems. Submitted to
Computer-Aided Geometric Design, available online under http://www.cs.rice.edu/ henrik/publications.html.
[12] H. Weimer, J. Warren: Subdivision schemes for fluid flow. In SIGGRAPH 99, Computer
Graphics Proceedings, Annual Conference Series, 1999.
[13] D. Zorin, P. Schröder, W. Sweldens: Interactive multiresolution mesh editing, Proceedings
of SIGGRAPH 1997. In Computer Graphics Proceedings, Annual Conference Series, pp.
259-168, 1997.
converted from Mathematica with toLATEX by Henrik Weimer (http://www.cs.rice.edu/˜henrik)
23
Chapter 10
Subdivision Surfaces in the Making of
Geri’s Game
Speaker: Tony DeRose
Subdivision Surfaces in Character Animation
Tony DeRose
Michael Kass
Tien Truong
Pixar Animation Studios
Figure 1: Geri.
Abstract
The creation of believable and endearing characters in computer
graphics presents a number of technical challenges, including the
modeling, animation and rendering of complex shapes such as
heads, hands, and clothing. Traditionally, these shapes have been
modeled with NURBS surfaces despite the severe topological restrictions that NURBS impose. In order to move beyond these restrictions, we have recently introduced subdivision surfaces into our
production environment. Subdivision surfaces are not new, but their
use in high-end CG production has been limited.
Here we describe a series of developments that were required
in order for subdivision surfaces to meet the demands of high-end
production. First, we devised a practical technique for construct-
ing provably smooth variable-radius fillets and blends. Second, we
developed methods for using subdivision surfaces in clothing simulation including a new algorithm for efficient collision detection.
Third, we developed a method for constructing smooth scalar fields
on subdivision surfaces, thereby enabling the use of a wider class
of programmable shaders. These developments, which were used
extensively in our recently completed short film Geri’s game, have
become a highly valued feature of our production environment.
CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.3 [Computer Graphics]: Picture/Image Generation.
1 Motivation
The most common way to model complex smooth surfaces such
as those encountered in human character animation is by using a
patchwork of trimmed NURBS. Trimmed NURBS are used primarily because they are readily available in existing commercial
systems such as Alias-Wavefront and SoftImage. They do, however, suffer from at least two difficulties:
1. Trimming is expensive and prone to numerical error.
2. It is difficult to maintain smoothness, or even approximate
smoothness, at the seams of the patchwork as the model is
(a)
(b)
(c)
(d)
Figure 2: The control mesh for Geri’s head, created by digitizing a
full-scale model sculpted out of clay.
animated. As a case in point, considerable manual effort was
required to hide the seams in the face of Woody, a principal
character in Toy Story.
Subdivision surfaces have the potential to overcome both of these
problems: they do not require trimming, and smoothness of the
model is automatically guaranteed, even as the model animates.
The use of subdivision in animation systems is not new, but for a
variety of reasons (several of which we address in this paper), their
use has not been widespread. In the mid 1980s for instance, Symbolics was possibly the first to use subdivision in their animation
system as a means of creating detailed polyhedra. The LightWave
3D modeling and animation system from NewTek also uses subdivision in a similar fashion.
This paper describes a number of issues that arose when we
added a variant of Catmull-Clark [2] subdivision surfaces to our
animation and rendering systems, Marionette and RenderMan [17],
respectively. The resulting extensions were used heavily in the creation of Geri (Figure 1), a human character in our recently completed short film Geri’s game. Specifically, subdivision surfaces
were used to model the skin of Geri’s head (see Figure 2), his hands,
and his clothing, including his jacket, pants, shirt, tie, and shoes.
In contrast to previous systems such as those mentioned above,
that use subdivision as a means to embellish polygonal models, our
system uses subdivision as a means to define piecewise smooth surfaces. Since our system reasons about the limit surface itself, polygonal artifacts are never present, no matter how the surface animates
or how closely it is viewed.
The use of subdivision surfaces posed new challenges throughout the production process, from modeling and animation to rendering. In modeling, subdivision surfaces free the designer from
worrying about the topological restrictions that haunt NURBS modelers, but they simultaneously prevent the use of special tools that
have been developed over the years to add features such as variable
radius fillets to NURBS models. In Section 3, we describe an approach for introducing similar capabilities into subdivision surface
models. The basic idea is to generalize the infinitely sharp creases
of Hoppe et. al. [10] to obtain semi-sharp creases – that is, creases
whose sharpness can vary from zero (meaning smooth) to infinite.
Once models have been constructed with subdivision surfaces,
the problems of animation are generally easier than with corresponding NURBS surfaces because subdivision surface models are
seamless, so the surface is guaranteed to remain smooth as the
model is animated. Using subdivision surfaces for physically-based
Figure 3: Recursive subdivision of a topologically complicated
mesh: (a) the control mesh; (b) after one subdivision step; (c) after
two subdivision steps; (d) the limit surface.
animation of clothing, however, poses its own difficulties which we
address in Section 4. First, it is necessary to express the energy
function of the clothing on subdivision meshes in such a way that
the resulting motion does not inappropriately reveal the structure
of the subdivision control mesh. Second, in order for a physical
simulator to make use of subdivision surfaces it must compute collisions very efficiently. While collisions of NURBS surfaces have
been studied in great detail, little work has been done previously
with subdivision surfaces.
Having modeled and animated subdivision surfaces, some
formidable challenges remain before they can be rendered. The
topological freedom that makes subdivision surfaces so attractive
for modeling and animation means that they generally do not
admit parametrizations suitable for texture mapping. Solid textures [12, 13] and projection textures [9] can address some production needs, but Section 5.1 shows that it is possible to go a good
deal further by using programmable shaders in combination with
smooth scalar fields defined over the surface.
The combination of semi-sharp creases for modeling, an appropriate and efficient interface to physical simulation for animation,
and the availability of scalar fields for shading and rendering have
made subdivision surfaces an extremely effective tool in our production environment.
2 Background
A single NURBS surface, like any other parametric surface, is limited to representing surfaces which are topologically equivalent to
a sheet, a cylinder or a torus. This is a fundamental limitation for
any surface that imposes a global planar parameterization. A single
subdivision surface, by contrast, can represent surfaces of arbitrary
topology. The basic idea is to construct a surface from an arbitrary
polyhedron by repeatedly subdividing each of the faces, as illustrated in Figure 3. If the subdivision is done appropriately, the limit
of this subdivision process will be a smooth surface.
Catmull and Clark [2] introduced one of the first subdivision
schemes. Their method begins with an arbitrary polyhedron called
the control mesh. The control mesh, denoted M 0 (see Figure 3(a)),
is subdivided to produce the mesh M 1 (shown in Figure 3(b)) by
splitting each face into a collection of quadrilateral subfaces. A
face having n edges is split into n quadrilaterals. The vertices of
M 1 are computed using certain weighted averages as detailed below. The same subdivision procedure is used again on M 1 to produce the mesh M 2 shown in Figure 3(c). The subdivision surface is
defined to be the limit of the sequence of meshes M 0 ; M 1 ; ::: created
by repeated application of the subdivision procedure.
To describe the weighted averages used by Catmull and Clark it
is convenient to observe that each vertex of M i+1 can be associated
with either a face, an edge, or a vertex of M i ; these are called face,
edge, and vertex points, respectively. This association is indicated
in Figure 4 for the situation around a vertex v0 of M 0 . As indicated
in the figure, we use f ’s to denote face points, e’s to denote edge
points, and v’s to denote vertex points. Face points are positioned
at the centroid of the vertices of the corresponding face. An edge
point eij+1 , as indicated in Figure 4 is computed as
eij+1 =
1
i+1
vi + eij + f ij+
,1 + f j
4
Figure 5: Geri’s hand as a piecewise smooth Catmull-Clark surface.
Infinitely sharp creases are used between the skin and the finger
nails.
(1)
;
where subscripts are taken modulo the valence of the central vertex
v0 . (The valence of a vertex is the number of edges incident to it.)
Finally, a vertex point vi is computed as
vi+1 =
n,2 i 1
v+ 2
n
n
∑ eij + n2 ∑ f ij+1
1
j
(2)
j
Vertices of valence 4 are called ordinary; others are called extraordinary.
v0
e 13
e 0n
f 1n
v1
e 11
f 11
e 01
e 03
f 12
e 12
e 02
Figure 4: The situation around a vertex v0 of valence n.
These averaging rules — also called subdivision rules, masks, or
stencils — are such that the limit surface can be shown to be tangent
plane smooth no matter where the control vertices are placed [14,
19].1
Whereas Catmull-Clark subdivision is based on quadrilaterals,
Loop’s surfaces [11] and the Butterfly scheme [6] are based on triangles. We chose to base our work on Catmull-Clark surfaces for
two reasons:
1. They strictly generalize uniform tensor product cubic Bsplines, making them easier to use in conjunction with existing in-house and commercial software systems such as AliasWavefront and SoftImage.
2. Quadrilaterals are often better than triangles at capturing the
symmetries of natural and man-made objects. Tube-like surfaces — such as arms, legs, and fingers — for example, can
be modeled much more naturally with quadrilaterals.
1 Technical
Figure 6: A surface where boundary edges are tagged as sharp and
boundary vertices of valence two are tagged as corners. The control
mesh is yellow and the limit surface is cyan.
caveat for the purist: The surface is guaranteed to be smooth
except for control vertex positions in a set of measure zero.
Following Hoppe et. al. [10] it is possible to modify the subdivision rules to create piecewise smooth surfaces containing infinitely
sharp features such as creases and corners. This is illustrated in
Figure 5 which shows a close-up shot of Geri’s hand. Infinitely
sharp creases were used to separate the skin of the hand from the
finger nails. Sharp creases can be modeled by marking a subset
of the edges of the control mesh as sharp and then using specially
designed rules in the neighborhood of sharp edges. Appendix A
describes the necessary special rules and when to use them.
Again following Hoppe et. al., we deal with boundaries of the
control mesh by tagging the boundary edges as sharp. We have also
found it convenient to tag boundary vertices of valence 2 as corners,
even though they would normally be treated as crease vertices since
they are incident to two sharp edges. We do this to mimic the behavior of endpoint interpolating tensor product uniform cubic B-spline
surfaces, as illustrated in Figure 6.
3 Modeling fillets and blends
As mentioned in Section 1 and shown in Figure 5, infinitely sharp
creases are very convenient for representing piecewise-smooth surfaces. However, real-world surfaces are never infinitely sharp. The
corner of a tabletop, for instance, is smooth when viewed sufficiently closely. For animation purposes it is often desirable to capture such tightly curved shapes.
To this end we have developed a generalization of the Catmull-
Clark scheme to admit semi-sharp creases – that is, creases of controllable sharpness, a simple example of which is shown in Figure 7.
(a)
(b)
(c)
(d)
arbitrary number of subdivision steps, followed by another set of
rules that are applied to the limit. Smoothness therefore depends
only on the second set of rules. Hybrid subdivision can be used to
obtain semi-sharp creases by using infinitely sharp rules during the
first few subdivision steps, followed by use of the smooth rules for
subsequent subdivision steps. Intuitively this leads to surfaces that
are sharp at coarse scales, but smooth at finer scales.
Now the details. To set the stage for the general situation where
the sharpness can vary along a crease, we consider two illustrative
special cases.
Case 1: A constant integer sharpness s crease: We subdivide
s times using the infinitely sharp rules, then switch to the smooth
rules. In other words, an edge of sharpness s > 0 is subdivided using the sharp edge rule. The two subedges created each have sharpness s , 1. A sharpness s = 0 edge is considered smooth, and it
stays smooth for remaining subdivisions. In the limit where s ! ∞
the sharp rules are used for all steps, leading to an infinitely sharp
crease. An example of integer sharpness creases is shown in Figure 7. A more complicated example where two creases of different
sharpnesses intersect is shown in Figure 8.
(a)
(b)
(c)
(d)
(e)
Figure 7: An example of a semi-sharp crease. The control mesh for
each of these surfaces is the unit cube, drawn in wireframe, where
crease edges are red and smooth edges are yellow. In (a) the crease
sharpness is 0, meaning that all edges are smooth. The sharpnesses
for (b), (c), (d), and (e) are 1, 2, 3, and infinite, respectively.
One approach to achieve semi-sharp creases is to develop subdivision rules whose weights are parametrized by the sharpness s of
the crease. This approach is difficult because it can be quite hard
to discover rules that lead to the desired smoothness properties of
the limit surfaces. One of the roadblocks is that subdivision rules
around a crease break a symmetry possessed by the smooth rules:
typical smooth rules (such as the Catmull-Clark rules) are invariant
under cyclic reindexing, meaning that discrete Fourier transforms
can be used to prove properties for vertices of arbitrary valence (cf.
Zorin [19]). In the absence of this invariance, each valence must
currently be considered separately, as was done by Schweitzer [15].
Another difficulty is that such an approach is likely to lead to a
zoo of rules depending on the number and configuration of creases
through a vertex. For instance, a vertex with two semi-sharp creases
passing through it would use a different set of rules than a vertex
with just one crease through it.
Our approach is to use a very simple process we call hybrid subdivision. The general idea is to use one set of rules for a finite but
Figure 8: A pair of crossing semi-sharp creases. The control mesh
for all surfaces is the octahedron drawn in wire frame. Yellow denotes smooth edges, red denotes the edges of the first crease, and
magenta denotes the edges of the second crease. In (a) the crease
sharpnesses are both zero; in (b), (c), and (d) the sharpness of the
red crease is 4. The sharpness of the magenta crease in (b), (c), and
(d) is 0, 2, and 4, respectively.
Case 2: A constant, but not necessarily integer sharpness s: the
main idea here is to interpolate between adjacent integer sharpnesses. Let s# and s" denote the floor and ceiling of s, respectively.
Imagine creating two versions of the crease: the first obtained by
subdividing s# times using the sharp rules, then subdividing one additional time using the smooth rules. Call the vertices of this first
version v#0 ; v#1 ; :::. The second version, the vertices of which we
denote by v"0 ; v"1 ; :::, is created by subdividing s" times using the
sharp rules. We take the s"-times subdivided semi-sharp crease to
Figure 9: A simple example of a variable sharpness crease. The
edges of the bottom face of the cubical control mesh are infinitely
sharp. Three edges of the top face form a single variable sharpness
crease with edge sharpnesses set to 2 (the two magenta edges), and
4 (the red edge).
Figure 10: A more complex example of variable sharpness creases.
This model, inspired by an Edouard Lanteri sculpture, contains numerous variable sharpness creases to reduce the size of the control
mesh. The control mesh for the model made without variable sharpness creases required 840 faces; with variable sharpness creases the
face count dropped to 627. Model courtesy of Jason Bickerstaff.
s"
have vertex positions vi computed via simple linear interpolation:
vsi" = (1 , σ)v#i +σv"i
(3)
where σ = (s , s#)=(s" ,s#). Subsequent subdivisions are done using the smooth rules. In the case where all creases have the same
non-integer sharpness s, the surface produced by the above process
is identical to the one obtained by linearly interpolating between
the integer sharpness limit surfaces corresponding to s# and s". Typically, however, crease sharpnesses will not all be equal, meaning
that the limit surface is not a simple blend of integer sharpness surfaces.
The more general situation where crease sharpness is non-integer
and varies along a crease is presented in Appendix B. Figure 9 depicts a simple example. A more complex use of variable sharpness
is shown in Figure 10.
4 Supporting cloth dynamics
The use of simulated physics to animate clothing has been widely
discussed in the literature (cf. [1, 5, 16]). Here, we address the
issues that arise when interfacing a physical simulator to a set of
geometric models constructed out of subdivision surfaces. It is not
our intent in this section to detail our cloth simulation system fully
– that would require an entire paper of its own. Our goal is rather to
highlight issues related to the use of subdivision surfaces to model
both kinematic and dynamic objects.
In Section 4.1 we define the behavior of the cloth material by
constructing an energy functional on the subdivision control mesh.
If the material properties such as the stiffness of the cloth vary over
the surface, one or more scalar fields (see Section 5.1) must be defined to modulate the local energy contributions. In Section 4.2 we
describe an algorithm for rapidly identifying potential collisions involving the cloth and/or kinematic obstacles. Rapid collision detection is crucial to achieving acceptable performance.
4.1 Energy functional
For physical simulation, the basic properties of a material are generally specified by defining an energy functional to represent the
attraction or resistance of the material to various possible deformations. Typically, the energy is either specified as a surface integral
or as a discrete sum of terms which are functions of the positions of
surface samples or control vertices. The first type of specification
typically gives rise to a finite-element approach, while the second
is associated more with finite-difference methods.
Finite-element approaches are possible with subdivision surfaces, and in fact some relevant surface integrals can be computed
analytically [8]. In general, however, finite-element surface integrals must be estimated through numerical quadrature, and this
gives rise to a collection of special cases around extraordinary
points. We chose to avoid these special cases by adopting a finitedifference approach, approximating the clothing with a mass-spring
model [18] in which all the mass is concentrated at the control
points.
Away from extraordinary points, Catmull-Clark meshes under
subdivision become regular quadrilateral grids. This makes them
ideally suited for representing woven fabrics which are also generally described locally by a gridded structure. In constructing the
energy functions for clothing simulation, we use the edges of the
subdivision mesh to correspond with the warp and weft directions
of the simulated woven fabrics.
Since most popular fabrics stretch very little along the warp
or weft directions, we introduce relatively strong fixed rest-length
springs along each edge of the mesh. More precisely, for each edge
from p1 to p2 , we add an energy term ks Es ( p1 ; p2 ) where
Es ( p1 ; p2 ) =
1
2
j p1 , p2 j , 12
j p1 , p2 j
:
(4)
Here, p1 and p2 are the rest positions of the two vertices, and ks is
the corresponding spring constant.
With only fixed-length springs along the mesh edges, the simulated clothing can undergo arbitrary skew without penalty. One way
to prevent the skew is to introduce fixed-length springs along the
diagonals. The problem with this approach is that strong diagonal
springs make the mesh too stiff, and weak diagonal springs allow
the mesh to skew excessively. We chose to address this problem
by introducing an energy term which is proportional to the product
of the energies of two diagonal fixed-length springs. If p1 and p2
are vertices along one diagonal of a quadrilateral mesh face and p3
and p4 are vertices along the other diagonal, the energy is given by
kd Ed ( p1 ; p2 ; p3 ; p4 ) where kd is a scalar parameter that functions
analagously to a spring constant, and where
Ed ( p1 ; p2 ; p3 ; p4 ) = Es ( p1 ; p2 )Es ( p3 ; p4 ):
(5)
The energy Ed ( p1 ; p2 ; p3 ; p4 ) reaches its minimum at zero when
either of the diagonals of the quadrilateral face are of the original
rest length. Thus the material can fold freely along either diagonal, while resisting skew to a degree determined by kd . We sometimes use weak springs along the diagonals to keep the material
from wrinkling too much.
With the fixed-length springs along the edges and the diagonal
contributions to the energy, the simulated material, unlike real cloth,
can bend without penalty. To add greater realism to the simulated
cloth, we introduce an energy term that establishes a resistance to
bending along virtual threads. Virtual threads are defined as a sequence of vertices. They follow grid lines in regular regions of the
mesh, and when a thread passes through an extraordinary vertex of
valence n, it continues by exiting along the edge bn=2c-edges away
in the clockwise direction. If p1 ; p2 ; and p3 are three points along
a virtual thread, the anti-bending component of the energy is given
by k p E p ( p1 ; p2 ; p3 ) where
E p ( p1 ; p2 ; p3 ) =
1
[C( p1 ; p2 ; p3 ) , C( p1 ; p2 ; p3 )]2
2
(6)
p3 , p2
p2 , p1 j p3 , p2 j , j p2 , p1 j (7)
C( p1 ; p2 ; p3 ) = and p1 ; p2 ; and p3 are the rest positions of the three points.
By adjusting ks , kd and k p both globally and locally, we have
been able to simulate a reasonably wide variety of cloth behavior. In
the production of Geri’s game, we found that Geri’s jacket looked a
great deal more realistic when we modulated k p over the surface of
the jacket in order to provide more stiffness on the shoulder pads, on
the lapels, and in an area under the armpits which is often reinforced
in real jackets. Methods for specifying scalar fields like k p over a
subdivision surface are discussed in more detail in section 5.1.
4.2 Collisions
The simplest approach to detecting collisions in a physical simulation is to test each geometric element (i.e. point, edge, face) against
each other geometric element for a possible collision. With N geometric elements, this would take N 2 time, which is prohibitive for
large N. To achieve practical running times for large simulations,
the number of possible collisions must be culled as rapidly as possible using some type of spatial data structure. While this can be done
in a variety of different ways, there are two basic strategies: we
can distribute the elements into a two-dimensional surface-based
data structure, or we can distribute them into a three-dimensional
volume-based data structure. Using a two-dimensional structure
has several advantages if the surface connectivity does not change.
First, the hierarchy can be fixed, and need not be regenerated each
time the geometry is moved. Second, the storage can all be statically allocated. Third, there is never any need to rebalance the tree.
Finally, very short edges in the surface need not give rise to deep
branches in the tree, as they would using a volume-based method.
It is a simple matter to construct a suitable surface-based data
structure for a NURBS surface. One method is to subdivide the
(s; t ) parameter plane recursively into an quadtree. Since each node
in the quadtree represents a subsquare of the parameter plane, a
bounding box for the surface restricted to the subsquare can be
constructed. An efficient method for constructing the hierarchy of
boxes is to compute bounding boxes for the children using the convex hull property; parent bounding boxes can then be computed in a
bottom up fashion by unioning child boxes. Having constructed the
quadtree, we can find all patches within ε of a point p as follows.
We start at the root of the quadtree and compare the bounding box
of the root node with a box of size 2ε centered on p. If there is
no intersection, then there are no patches within ε of p. If there is
an intersection, then we repeat the test on each of the children and
recurse. The recursion terminates at the leaf nodes of the quadtree,
where bounding boxes of individual subpatches are tested against
the box around p.
Subdivision meshes have a natural hierarchy for levels finer than
the original unsubdivided mesh, but this hierarchy is insufficient
because even the unsubdivided mesh may have too many faces to
test exhaustively. Since there is there is no global (s; t ) plane from
which to derive a hierarchy, we instead construct a hierarchy by
“unsubdividing” or “coarsening” the mesh: We begin by forming
leaf nodes of the hierarchy, each of which corresponds to a face
of the subdivision surface control mesh. We then hierarchically
merge faces level by level until we finish with a single merged face
corresponding to the entire subdivision surface.
The process of merging faces proceeds as follows. In order to
create the `th level in the hierarchy, we first mark all non-boundary
edges in the ` , 1st level as candidates for merging. Then until all
candidates at the `th level have been exhausted, we pick a candidate
edge e, and remove it from the mesh, thereby creating a “superface”
f by merging the two faces f1 and f2 that shared e: The hierarchy
is extended by creating a new node to represent f and making its
children be the nodes corresponding to f1 and f2 . If f were to
participate immediately in another merge, the hierarchy could become poorly balanced. To ensure against that possibility, we next
remove all edges of f from the candidate list. When all the candidate edges at one level have been exhausted, we begin the next level
by marking non-boundary edges as candidates once again. Hierarchy construction halts when only a single superface remains in the
mesh.
The coarsening hierarchy is constructed once in a preprocessing
phase. During each iteration of the simulation, control vertex positions change, so the bounding boxes stored in the hierarchy must be
updated. Updating the boxes is again a bottom up process: the current control vertex positions are used to update the bounding boxes
at the leaves of the hierarchy. We do this efficiently by storing with
each leaf in the hierarchy a set of pointers to the vertices used to
construct its bounding box. Bounding boxes are then unioned up
the hierarchy. A point can be “tested against” a hierarchy to find
all faces within ε of the point by starting at the root of the hierarchy and recursively testing bounding boxes, just as is done with the
NURBS quadtree.
We build a coarsening hierarchy for each of the cloth meshes, as
well as for each of the kinematic obstacles. To determine collisions
between a cloth mesh and a kinematic obstacle, we test each vertex
of the cloth mesh against the hierarchy for the obstacle. To determine collisions between a cloth mesh and itself, we test each vertex
of the mesh against the hierarchy for the same mesh.
5 Rendering subdivision surfaces
In this section, we introduce the idea of smoothly varying scalar
fields defined over subdivision surfaces and show how they can be
used to apply parametric textures to subdivision surfaces. We then
describe a collection of implementation issues that arose when subdivision surfaces and scalar fields were added to RenderMan.
5.1 Texturing using scalar fields
NURBS surfaces are textured using four principal methods: parametric texture mapping, procedural texture, 3D paint [9], and solid
texture [12, 13]. It is straightforward to apply 3D paint and solid
texturing to virtually any type of primitive, so these techniques
can readily be applied to texture subdivision surfaces. It is less
clear, however, how to apply parametric texture mapping, and more
generally, procedural texturing to subdivision surfaces since, unlike
NURBS, they are not defined parametrically.
With regard to texture mapping, subdivision surfaces are more
akin to polygonal models since neither possesses a global (s; t )
parameter plane. The now-standard method of texture mapping
a polygonal model is to assign texture coordinates to each of the
vertices. If the faces of the polygon consist only of triangles and
quadrilaterals, the texture coordinates can be interpolated across
the face of the polygon during scan conversion using linear or bilinear interpolation. Faces with more than four sides pose a greater
challenge. One approach is to pre-process the model by splitting
such faces into a collection of triangles and/or quadrilaterals, using some averaging scheme to invent texture coordinates at newly
introduced vertices. One difficulty with this approach is that the
texture coordinates are not differentiable across edges of the original or pre-processed mesh. As illustrated in Figures 11(a) and (b),
these discontinuities can appear as visual artifacts in the texture,
especially as the model is animated.
Fortunately, the situation for subdivision surfaces is profoundly
better than for polygonal models. As we prove in Appendix C,
smoothly varying texture coordinates result if the texture coordinates (s; t ) assigned to the control vertices are subdivided using
the same subdivision rules as used for the geometric coordinates
(x; y; z). (In other words, control point positions and subdivision can
be thought of as taking place in a 5-space consisting of (x; y; z; s; t )
coordinates.) This is illustrated in Figure 11(c), where the surface
is treated as a Catmull-Clark surface with infinitely sharp boundary edges. A more complicated example of parametric texture on a
subdivision surface is shown in Figure 12.
As is generally the case in real productions, we used a combination of texturing methods to create Geri: the flesh tones on his
head and hands were 3D-painted, solid textures were used to add
fine detail to his skin and jacket, and we used procedural texturing
(described more fully below) for the seams of his jacket.
The texture coordinates s and t mentioned above are each instances of a scalar field; that is, a scalar-valued function that varies
over the surface. A scalar field f is defined on the surface by assigning a value fv to each of the control vertices v. The proof sketch
in Appendix C shows that the function f ( p) created through subdivision (where p is a point on the limit surface) varies smoothly
wherever the subdivision surface itself is smooth.
Scalar fields can be used for more than just parametric texture
mapping — they can be used more generally as arbitrary parameters
to procedural shaders. An example of this occurs on Geri’s jacket.
A scalar field is defined on the jacket that takes on large values for
points on the surface near a seam, and small values elsewhere. The
procedural jacket shader uses the value of the this field to add the
apparent seams to the jacket. We use other scalar fields to darken
Geri’s nostril and ear cavities, and to modulate various physical
parameters of the cloth in the cloth simulator.
We assign scalar field values to the vertices of the control mesh
in a variety of ways, including direct manual assignment. In some
cases, we find it convenient to specify the value of the field directly
at a small number of control points, and then determine the rest by
interpolation using Laplacian smoothing. In other cases, we specify the scalar field values by painting an intensity map on one or
more rendered images of the surface. We then use a least squares
solver to determine the field values that best reproduce the painted
intensities.
(a)
(b)
(a)
(c)
(d)
Figure 11: (a) A texture mapped regular pentagon comprised of
5 triangles; (b) the pentagonal model with its vertices moved; (c)
A subdivision surface whose control mesh is the same 5 triangles
in (a), and where boundary edges are marked as creases; (d) the
subdivision surface with its vertices positioned as in (b).
(b)
Figure 12: Gridded textures mapped onto a bandanna modeled using two subdivision surfaces. One surface is used for the knot, the
other for the two flaps. In (a) texture coordinates are assigned uniformly on the right flap and nonuniformly using smoothing on the
left to reduce distortion. In (b) smoothing is used on both sides and
a more realistic texture is applied.
5.2 Implementation issues
We have implemented subdivision surfaces, specifically semi-sharp
Catmull-Clark surfaces, as a new geometric primitive in RenderMan.
Our renderer, built upon the REYES architecture [4], demands
that all primitives be convertible into grids of micropolygons (i.e.
half-pixel wide quadrilaterals). Consequently, each type of primitive must be capable of splitting itself into a collection of subpatches, bounding itself (for culling and bucketing purposes), and
dicing itself into a grid of micropolygons.
Each face of a Catmull-Clark control mesh can be associated
with a patch on the surface, so the first step in rendering a CatmullClark surface is to split it in into a collection of individual patches.
The control mesh for each patch consists of a face of the control
mesh together with neighboring faces and their vertices. To bound
each patch, we use the knowledge that a Catmull-Clark surface lies
within the convex hull of its control mesh. We therefore take the
bounding box of the mesh points to be the bounding box for the
patch. Once bounded, the primitive is tested to determine if it is
diceable; it is not diceable if dicing would produce a grid with too
many micropolygons or a wide range of micropolygon sizes. If
the patch is not diceable, then we split each patch by performing a
subdivision step to create four new subpatch primitives. If the patch
is diceable, it is repeatedly subdivided until it generates a grid with
the required number of micropolygons. Finally, we move each of
the grid points to its limit position using the method described in
Halstead et. al. [8].
An important property of Catmull-Clark surfaces is that they
give rise to bicubic B-splines patches for all faces except those in
the neighborhood of extraordinary points or sharp features. Therefore, at each level of splitting, it is often possible to identify one or
more subpatches as B-spline patches. As splitting proceeds, more
of the surface can be covered with B-spline patches. Exploiting
this fact has three advantages. First, the fixed 4 4 size of a Bspline patch allows for efficiency in memory usage because there
is no need to store information about vertex connectivity. Second,
the fact that a B-spline patch, unlike a Catmull-Clark patch, can be
split independently in either parametric direction makes it possible
to reduce the total amount of splitting. Third, efficient and well
understood forward differencing algorithms are available to dice Bspline patches [7].
We quickly learned that an advantage of semi-sharp creases over
infinitely sharp creases is that the former gives smoothly varying
normals across the crease, while the latter does not. This implies
that if the surface is displaced in the normal direction in a creased
area, it will tear at an infinitely sharp crease but not at a semi-sharp
one.
6 Conclusion
Our experience using subdivision surfaces in production has been
extremely positive. The use of subdivision surfaces allows our
model builders to arrange control points in a way that is natural
to capture geometric features of the model (see Figure 2), without
concern for maintaining a regular gridded structure as required by
NURBS models. This freedom has two principal consequences.
First, it dramatically reduces the time needed to plan and build an
initial model. Second, and perhaps more importantly, it allows the
initial model to be refined locally. Local refinement is not possible with a NURBS surface, since an entire control point row, or
column, or both must be added to preserve the gridded structure.
Additionally, extreme care must be taken either to hide the seams
between NURBS patches, or to constrain control points near the
seam to create at least the illusion of smoothness.
By developing semi-sharp creases and scalar fields for shading,
we have removed two of the important obstacles to the use of subdivision surfaces in production. By developing an efficient data structure for culling collisions with subdivisions, we have made subdivision surfaces well suited to physical simulation. By developing a
cloth energy function that takes advantage of Catmull-Clark mesh
structure, we have made subdivision surfaces the surfaces of choice
for our clothing simulations. Finally, by introducing Catmull-Clark
subdivision surfaces into our RenderMan implementation, we have
shown that subdivision surfaces are capable of meeting the demands
of high-end rendering.
A Infinitely Sharp Creases
Hoppe et. al. [10] introduced infinitely sharp features such as
creases and corners into Loop’s surfaces by modifying the subdivision rules in the neighborhood of a sharp feature. The same can
be done for Catmull-Clark surfaces, as we now describe.
Face points are always positioned at face centroids, independent
of which edges are tagged as sharp. Referring to Figure 4, suppose
the edge vi eij has been tagged as sharp. The corresponding edge
point is placed at the edge midpoint:
eij+1 =
vi + eij
2
(8)
:
The rule to use when placing vertex points depends on the number
of sharp edges incident at the vertex. A vertex with one sharp edge
is called a dart and is placed using the smooth vertex rule from
Equation 2. A vertex vi with two incident sharp edges is called a
crease vertex. If these sharp edges are eij vi and vi eik , the vertex point
vi+1 is positioned using the crease vertex rule:
vi+1 =
eij + 6vi + eik
8
:
(9)
The sharp edge and crease vertex rules are such that an isolated
crease converges to a uniform cubic B-spline curve lying on the
limit surface. A vertex vi with three or more incident sharp edges
is called a corner; the corresonding vertex point is positioned using
the corner rule
(10)
vi+1 = vi
meaning that corners do not move during subdivision. See
Hoppe et. al. [10] and Schweitzer [15] for a more complete discussion and rationale for these choices.
Hoppe et. al. found it necessary in proving smoothness properties of the limit surfaces in their Loop-based scheme to make further
distinctions between so-called regular and irregular vertices, and
they introduced additional rules to subdivide them. It may be necessary to do something similar to prove smoothness of our CatmullClark based method, but empirically we have noticed no anamolies
using the simple strategy above.
B General semi-sharp creases
Here we consider the general case where a crease sharpness is allowed to be non-integer, and to vary along the crease. The following procedure is relatively simple and strictly generalizes the two
special cases discussed in Section 3.
We specify a crease by a sequence of edges e1 ; e2 ; ::: in the control mesh, where each edge ei has an associated sharpness ei :s. We
associate a sharpness per edge rather than one per vertex since there
is no single sharpness that can be assigned to a vertex where two or
more creases cross.2
2 In
our implementation we do not allow two creases to share an edge.
exists a parametrization S(s; t ) for the surface in the neighborhood
of p such that S(0; 0) = p, and such that the function f (s; t ) is differentiable and the derivative varies continuously in the neighborhood
of (0; 0).
The characteristic map, introduced by Reif [14] and extended by
Zorin [19], provides such a parametrization: the characteristic map
allows a subdivision surface S in three space in the neighborhood
of a point p on the surface to be written as
eb
ea
eab
ec
ebc
Figure 13: Subedge labeling.
S(s; t ) = (x(s; t ); y(s; t ); z(s; t ))
During subdivision, face points are always placed at face centroids. The rules used when placing edge and vertex points are
determined by examining edge sharpnesses as follows:
An edge point corresponding to a smooth edge (i.e, e:s = 0) is
computed using the smooth edge rule (Equation 1).
An edge point corresponding to an edge of sharpness e:s >= 1
is computed using the sharp edge rule (Equation 8).
An edge point corresponding to an edge of sharpness e:s < 1 is
computed using a blend between smooth and sharp edge rules:
specifically, let vsmooth and vsharp be the edge points computed
using the smooth and sharp edge rules, respectively. The edge
point is placed at
(1
, e s)vsmooth + e svsharp
:
:
:
(11)
A vertex point corresponding to a vertex adjacent to zero or
one sharp edges is computed using the smooth vertex rule
(Equation 2).
A vertex point corresponding to a vertex v adjacent to three
or more sharp edge is computed using the corner rule (Equation 10).
A vertex point corresponding to a vertex v adjacent to two
sharp edges is computed using the crease vertex rule (Equation 9) if v:s 1, or a linear blend between the crease vertex
and corner masks if v:s < 1, where v:s is the average of the
incidence edge sharpnesses.
When a crease edge is subdivided, the sharpnesses of the resulting subedges is determined using Chaikin’s curve subdivision algorithm [3]. Specifically, if ea , eb , ec denote three adjacent edges of
a crease, then the subedges eab and ebc as shown in Figure 13 have
sharpnesses
eab :s
=
ebc :s
=
ea :s + 3eb :s
, 1; 0)
4
3e :s + ec :s
max( b
, 1; 0)
4
max(
A 1 is subtracted after performing Chaikin’s averaging to account for the fact that the subedges (eab ; ebc ) are at a finer level than
their parent edges (ea ; eb ; ec ). A maximum with zero is taken to
keep the sharpnesses non-negative. If either ea or eb is infinitely
sharp, then eab is; if either eb or ec is infinitely sharp, then ebc
is. This relatively simple procedure generalizes cases 1 and 2 described in Section 3. Examples are shown in Figures 9 and 10.
C Smoothness of scalar fields
In this appendix we wish to sketch a proof that a scalar field f is
smooth as a function on a subdivision surface wherever the surface
itself is smooth. To say that a function on a smooth surface S is
smooth to first order at a point p on the surface is to say that there
(12)
where S(0; 0) = p and where each of x(s; t ), y(s; t ), and z(s; t ) is
once differentiable if the surface is smooth at p. Since scalar fields
are subdivided according to the same rules as the x; y, and z coordinates of the control points, the function f (s; t ) must also be smooth.
Acknowledgments
The authors would like to thank Ed Catmull for creating the Geri’s
game project, Jan Pinkava for creating Geri and for writing and directing the film, Karen Dufilho for producing it, Dave Haumann and
Leo Hourvitz for leading the technical crew, Paul Aichele for building Geri’s head, Jason Bickerstaff for modeling most of the rest of
Geri and for Figure 10, and Guido Quaroni for Figure 12. Finally,
we’d like to thank the entire crew of Geri’s game for making our
work look so good.
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