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Digital Geometry Processing:A Progress Report and Research
Outlook
Peter SchröderComputer Science Department 256-80
California Institute of TechnologyPasadena, CA 91125
[email protected]
IntroductionThe Multi-Res Modeling group is working on theories,
computational representations and algorithms associated withthe new
research area of Digital Geometry Processing (DGP). Our work
touches upon and employs many elementstraditionally associated with
the areas of applied and computational mathematics, differential
geometry, computationalgeometry and topology, approximation and
operator theory, digital signal processing, information theory,
algorithms,user interface design, and even cognitive and artistic
theories of space and shape.
While the primary goal of our research is the creation of the
foundations of DGP, we do so in a context that alwaysemploys
prototype implementations and technology demonstrations to ensure
that we focus on the right bottlenecksand problems relevant in
application practice. In some cases our work pushes, in others it
pulls applications.
Synopsis of Accomplished and Future GoalsHere we briefly note
the research projects that were furthered during 2000/2001 as well
as our future plans. Publica-tions are given where applicable. More
details can be found in the subsequent sections.
Accomplishments• Fast Implementations: Approaching the
generation of subdivision surfaces from an unusual mathematical
per-
spective we were able to develop a new evaluation algorithm
which streams optimally on PIII and P4 architec-tures. See: BOLZ,
J., AND SCHRÖDER, P. Rapid Evaluation of Catmull-Clark Subdivision
Surfaces; submittedfor publication, September 2001.
• Parameterization: Processing surfaces requires that they have
a well-defined and smooth parameterization. Con-structing these
when none are given is the subject of parameterization algorithms.
We were able to developtheory and an algorithm allowing for
simultaneous, consistent parameterization of multiple,
geometrically dis-parate objects. See: PRAUN, E., SWELDENS, W., AND
SCHRÖDER, P. Consistent Mesh Parameterizations InComputer Graphics
(SIGGRAPH ’01 Proceedings), 179–184, 2001.
• Topology: Next to geometry, topology is the other major aspect
of surfaces. During 3D scanning, acquisitionerrors can lead to
“topological noise” that must be removed before further processing
can occur. See: GUSKOV,I., AND WOOD, Z. Topological Noise Removal.
Graphics Interface 2001 (2001), 19–26.
• Compression: As the disparity between transmission bandwidth
and compute power continues to increase com-pression of large
geometric datasets is more important than ever. We have recently
finished work on the best yetconnectivity encoding algorithm for
polygons, including an optimality proof regarding the entropy of
vertex anddegree symbols. See: KHODAKOVSKY, A., ALLIEZ, P.,
DESBRUN, M., AND SCHRÖDER, P. Near-OptimalConnectivity Encoding of
2-Manifold Polygon Meshes; submitted for publication, September
2001.
• Engineering Design: Subdivision surfaces are now a major
technology for free-form surface design and theyare beginning to
move into traditional engineering design areas. We have
demonstrated the feasibility of an
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integrated treatment of geometric shapes, their mechanical
response and their optimization in the case of thin,flexible
structures. One algorithmic component of such engineering
simulations is (self-)interference detectionfor which we have
developed novel theory and algorithms; accommodating trimming
operations is another thatwe developed. See: CIRAK, F., SCOTT, M.,
ANTONSSON, E., ORTIZ, M., AND SCHRÖDER, P. IntegratedModeling,
Simulation, and Design for Thin-Shell Structures using Subdivision.
Computer-Aided Design (2002).GRINSPUN, E., AND SCHRÖDER, P. Normal
Bounds for Subdivision-Surface Interference Detection.
InVisualization ’01, 2001. LITKE, N., LEVIN, A., AND SCHRÖDER, P.
Trimming for Subdivision Surfaces.Computer Aided Geometric Design
18, 5 (2001), 463–481.
• Theory: Many of the algorithms we develop are enabled by novel
theoretical insights and results. We were ableto unify all primal
and dual quadrilateral subdivision schemes in a single, very simple
algorithm and prove theirsmoothness. To justify their use in
numerical thin-shell modeling we proved that subdivision basis
functionshave square integrable curvatures. Fitting of subdivision
surfaces with the aid of quasi-interpolation operatorswas the
subject of another project. See: ZORIN, D., AND SCHRÖDER, P. A
Unified Framework for Primal/DualQuadrilateral Subdivision Schemes.
Computer Aided Geometric Design 18, 5 (2001), 429–454. REIF, U.,
ANDSCHRÖDER, P. Curvature Integrability of Subdivision Surfaces.
Advances in Computational Mathematics 14,2 (2001), 157–174.
SCHRÖDER, P., AND SWELDENS, W. Digital Geometry Processing. In
Sixth AnnualSymposium on Frontiers of Engineering, 41–44, 2001.
LITKE, N., LEVIN, A., AND SCHRÖDER, P. FittingSubdivision
Surfaces. In Visualization ’01, 2001.
Future Goals• Fast Implementations: We will continue our
scalable algorithm design for subdivision on the P4 and Itanium
(pending access to such hardware). A new project is aimed at
streaming delivery and display of very large terraindatasets to PDA
style devices.
• Parameterization: We have begun work on extending semi-regular
remeshing algorithms so that they can ac-commodate topology changes
during hierarchy construction to further extend their
applicability.
• Topology: Direct extraction of semi-regular meshes from
volumes requires topology filtering and discovery onthe volume
itself. These are subjects of an ongoing project.
• Compression: We are pursuing novel irregular coders for
terrain databases as well as further refinement of ourprogressive
coders for Catmull-Clark surfaces with details. On the latter we
are putting particular emphasis onvery efficient implementation on
Intel architectures. Additionally we have started to work on
compression forvery high-end scientific simulations on very large
supercomputers (ASCI machines).
• Engineering Design: Adaptive numerical solvers for the
Subdivision Element Method are the subject of both the-oretical and
algorithmic investigations in this coming year. Additionally we are
working on subdivision schemeswhich combine triangles and
quadrilaterals; novel formulations of thin-shell energies based on
differential ge-ometry operators and their discretizations on
meshes; and fast algorithms for variational surface modeling.
• Theory: Repeated averaging as a universal primitive for the
construction of large families of subdivision schemesis the subject
of one investigation, particularly focusing on
√3 schemes. For compression we are developing the-
ory for optimal algorithms in the symmetric Haussdorf distance,
which is particularly relevant for terrain com-pression.
Project DetailsIn the following sections we briefly outline some
of our ongoing research projects, both long and short term.
Moredetailed documentation on all of our work can be found on our
website at http://multires.caltech.edu and in particularon our
publications page.
Implementation and Optimization EffortsUsing subdivision
algorithms every day we often do not appreciate that many details
that appear easy and straightfor-ward to us do not appear that way
to some of our target audiences. We address this through our
extensive outreach
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efforts to various communities in the form of short courses.
Perhaps even more importantly we make many of ourcodes available.
This ranges from different subdivision to geometry compression and
remeshing codes.
Accomplishments in the past year Since in many settings ultimate
performance is an important concern we havebegun to pay particular
attention to codes that are optimized for Intel architectures. One
such code has achieved 1.2flops/cycle utilization for Catmull-Clark
subdivision on a PIII (and 1.8 flops/cycle on the P4). We achieved
this resultthrough a new surface evaluation method based on
precomputed tables. This new approach was designed to
addresscaching issues on modern CPUs. Careful attention in the
implementation finally allowed us to achieve previouslyunheard of
performance (see Figure 1). A paper on this work is currently in
preparation and will be submitted to theWeb3D conference.
Figure 1: A Catmull-Clark control mesh with tagged edges and
vertices (left). Subdivision yields the surface on theright. The
surfaces was evaluated to 77k quads with our new cache conscious
method in 14mS on a P4 (1.7GHz).
Goals for the coming year Our fast evaluation technique was
based on a particular mathematical approach whichallowed us to
precompute tables and build the actual surface at run time through
very efficient, streaming accumulationof these tables with
appropriate weights. We are now working on achieving similar
results for depth first, movingwindow recursive versions of
subdivision with details. The latter is critical in achieving
utmost performance in ourprogressive geometry decompression codes.
Further scalability studies, in particular with an eye towards the
Itanium,are also on the agenda. The codes themselves will all be
made publicly available this coming year. An explicit goal ofthis
effort is to help the largest number of people possible to achieve
optimal performance for these types of geometryon modern
architectures.
ParameterizationOne of the more fundamental problems in the
treatment of surfaces is the construction of a parameterization. In
manysituations, in particular when surfaces are created through 3D
scanning (e.g., laser scanners, consumer camera basedsystems, and
medical volumetric imaging), a given surface geometry is only known
as a set of samples (“vertices”)and their connectivity
(“triangles”). In order to apply any processing to these surfaces
beyond plain rendering, aparameterization must be found. Only then
does it make sense to speak of, for example, smoothing a surface,
orcompressing a surface, or simulating the mechanical response of a
surface. Such parameterizations are also the basisfor resampling of
geometry to imbue it with a semi-regular sampling pattern. The
latter is the best one can hope forin the arbitrary topology
setting and forms the foundation for the most powerful mathematical
processing techniques(subdivision and wavelets) we currently have
for surfaces.
Accomplishments in the past year Because of their tremendous
importance we have been actively working onparameterization
algorithms over the past several years and have been able to
considerably refine our basic algorithms.All previous work was
focused on individual surfaces. This past year we demonstrated an
algorithm to simultaneouslyparameterize a set of surfaces with
compatible parameterizations. The significance of this
accomplishment arises fromthe fact that this technology allows us
to apply, for the first time, numerous algorithms developed for
groups of objects.
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For example, we have demonstrated principal component analysis
on a set of surfaces (see Figure 2). Such methodsare well known
from image processing where they can be used, for example, for fast
search in large databases. Thepaper on this work was published in
Siggraph 2001.
+ + + + + + + =
Figure 2: When given a set of head models an obvious shape to
compute is their average. In general the connectivityand sampling
patterns of the models are different and computing the average is
non trivial. After computing consistentmesh parameterizations (red
patch boundaries) and remeshing, all models have the same
connectivity and samplingpattern so computing the average becomes
trivial.
Because the underlying numerical computing issues in all
parameterization codes are very challenging such codeshave in
general been very fragile. As a result many surfaces produced from
3D scanning contain “topological noise”and we have developed a
topological noise removal algorithms for the cleanup of such
geometry. A paper on this workwas published in Graphics Interface
2001.
Figure 3: Example of a hybrid mesh (left to right). Starting
with the Stanford Buddha original dataset a base domainwith genus 0
is constructed. After one regular refinement step, tunnels are
created between arms and head followedby another regular refinement
step and creation of a tunnel between the feet. After one more
refinement step threesmall tunnels are created on the sides
followed by further regular refinement. The model remains a valid
2-manifoldthroughout and the final remesh has genus six.
Goals for the coming year While semi-regular sampling of
surfaces enables a variety of very powerful processingmethods, the
underlying parameterizations are not flexible enough. For example,
if the coarsest level of a hierarchicalrepresentation has to
resolve all fine scale topological features present in the full
resolution model, the coarsest levelmay not be very coarse at all.
To address this we have begun work on constructing parameterization
algorithms thatallow for topological changes in the hierarchy.
Figure 3 shows one of these “hybrid meshes.” The resulting
meshrepresentations for surfaces are hybrid in that they are
semi-regular whenever possible and allow irregular
operations(insertion/removal of handles, boundaries, or
modification of patch layout) when required. Currently the method
reliesvery much on user intervention. In this coming year we hope
to automate this process.
Since most geometry capturing methods go through a volumetric
processing stage involving distance functions wehave in the past
developed algorithms that perform “Semi-regular Mesh Extraction
from Volumes” directly. Thesemethods are currently being extended
to allow for out of core extraction, reliable topology elision
(skipping irrelevantdetails for a given task) and robust topology
discovery. The latter is particularly important for scientific and
medicalapplications which we envision down the line.
CompressionAs digital geometry is used in more and more
applications and as the desired geometric complexity increases
(e.g.,in games, e-commerce, distance learning, tele-medicine)
compression of this geometry becomes ever more important.This is
particularly true as there is no economical solution in sight for
the “last mile” problem of broadband deliveryservices. A critical
element of such compression algorithms is that they be progressive.
We want to enable “author
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once” technology independent of the rendering capabilities of
the receiving terminal (scientific workstations to PDAsand cell
phones).
Accomplishments in the past year From a pure performance point
of view, in particular for high resolution meshesof arbitrary
topology, semi-regular compression algorithms are far superior to
any irregular compression. We havepioneered such algorithms in
“Progressive Geometry Compression” which was published in Siggraph
2000. Therewe applied compression to Loop subdivision surfaces with
details. This code has been made available to the MPEG4standards
effort and have been incorporated into the standards committee code
base (Blaxxun player).
In geometry compression we distinguish between irregular and
semi-regular compression. Irregular compressionmakes no assumptions
about the connectivity of faces in a mesh and is most appropriate
for models with small numbersof faces whose shape and placement
have been highly optimized (see Figure 4 for examples of such
coarse meshes).It is also of great importance for settings in which
very many objects, each very simple, appear in a single scene.We
just finished work on a connectivity compression algorithm for
arbitrary polyhedral surfaces, i.e., the mesh maycontain faces of
any type. The underlying algorithm is based on valence and degree
entropy coding which is provably(near-)optimal1. Surprisingly the
resulting algorithm is both simpler and more general than previous
approaches andbeats all previous methods including those
specialized to triangle and quad meshes. A paper on this work has
justbeen submitted to a special issue of GMOD.
Goals for the coming year We have begun to extend our
progressive geometry compression work to quadrilateralmeshes
(Catmull-Clark surfaces with details), with a particular emphasis
on high performance, cache sensitive algo-rithms. We will release
this work in the form of a web browser plugin. The latter work also
builds the basis for workwe have started on compression of hybrid
meshes (see Figure 3). We are particularly eager to deploy the
decompressoron PDA form factors. The latter will require further
mathematical research on integer versions of the surface
basedwavelet transforms. Because we must use local frames the
latter are as yet wholly unexplored, and computationallymuch more
involved than ordinary image based wavelet transforms.
In general, irregular mesh compression is much harder since
geometry information and parameter information aretightly linked in
this setting. While geometry information compresses well, parameter
information does not (henceour efforts at smooth remeshing). In the
specialized setting of digital terrain elevation data (DTED), which
is of greatinterest to mapping agencies, for example, it appears to
be possible to overcome this hurdle. We have just begun
acollaboration to build rate/distortion optimal irregular
transmission schemes for such DTEDs which do not need tosubmit any
parameter information. This should significantly decrease bandwidth
needs and make it possible to buildstreaming DTED servers and
clients that can handle planet sized datasets, an explicit long
term goal of this researcheffort, on large and small devices. One
goal for this coming year is to demonstrate these algorithms on a
PDA.
At the “large” computing end of the spectrum we have just begun
a project to apply progressive geometry com-pression to the results
of complex scientific simulations such as those performed as part
of the Advanced StrategicComputing Initiative and other grand
challenge scientific simulation projects (see Figure 5). The
motivation for thiswork arises from the great disparity between
available compute power and I/O bandwidth on giant cluster
computers.Using extra cycles to significantly decrease I/O needs
will make a significant contribution in these settings. The
un-derlying algorithms rely on our earlier work on “Fitting
Subdivision Surfaces (to appear in Visualization 2001)
withquasi-interpolation operators. The resulting meshes will
concurrently be compressed with our “Progressive
GeometryCompression” algorithms (published in Siggraph 2000).
Particular challenges that this project will address are thetime
dependent aspects of evolving geometry and topology (e.g., the
interface between two mixing fluids of differingdensity) and how to
exploit coherence from time step to time step for effective
compression.
Physical Simulation for CAGD, Engineering Design and
AnimationSurfaces which are just static objects with no physical
properties only go so far. In many applications we are
also(primarily?) interested in the physical behavior or mechanical
response of the objects whose geometry was scannedor modeled. In
some cases the shape itself may be the solution that minimizes a
physical energy.
Accomplishments in the past year A recent collaboration with
colleagues from mechanical and aeronautical en-gineering has led to
the creation of a new paradigm for integrated geometric modeling,
mechanical simulation and
1Certain technical conditions force the “near” optimality. These
are of little relevance in practice though.
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Figure 4: Examples of meshes consisting of arbitrary faces. Such
meshes require the most general irregular connec-tivity encoders
for compression.
engineering design of free form surfaces, e.g., class A surfaces
(outer body panels) in the car industry, based on sub-division. The
Subdivision Element Method (SEM) uses the same representation for
geometry and simulation of itsmechanical response (the paper on
this work will appear in CAD). Our approach overcomes a traditional
and signifi-cant hurdle in engineering design: the need to convert
between different representations for geometric modeling andfinite
element simulation (and the associated losses and
inefficiencies).
Moving these ideas into engineering practice requires that we
address a number of important algorithmic com-ponents. Collision
detection for subdivision surfaces is one such area. Using interval
analysis techniques we wereable to derive bounds on the curvatures
of subdivision surfaces near irregular vertices, which allowed us
to design avery efficient algorithm for (self-)interference
detection on subdivision surface. A paper on this work will appear
inVisualization 2001.
Goals for the coming year Using tools from the numerical
simulation of thin-shells we are investigating methods toconstruct
variationally optimal surfaces which are known to be of the highest
quality among all surfaces. Using somenovel ideas concerning the
closed form expression of certain surface energies, we hope to
achieve computation speedsmaking these surfaces practical for
interactive modeling.
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Figure 5: Visualization of a fluid simulation. a) A mesh
constructed from an initial set of 109 sample points. b)
Oursurface, generated by fitting to the same sample points.
Additional detail is added to the surface after c) 1609 samplesand
d) 6353 samples using quasi-interpolation operators.
In a new project we have set as a goal the realistic animation
of the surface of the hand. A subdivision surfacemodel of a hand is
currently being built and we expect to incorporate ideas from
physical simulation as well as socalled “combined subdivision”
schemes directly into the surface construction to greatly simplify
this traditionally veryhard problem.
All the efforts described above are geared towards truly
predictive engineering quality simulations. Because theyare of high
accuracy, and involving sophisticated material models they tend to
be very expensive in terms of com-putation time. In a new effort we
are studying ways to simplify the equations and their simulation by
going back tobasic differential geometry principles. This work is
highly relevant for the integration of such capabilities into
inter-active environments, from online content (e.g., Shockwave
authored media) to interactive games. We are hoping toapply these
ideas to the simulation of cloth and thin-shells, using only meshes
(instead of finite element spaces) andmanifold invariants,
leveraging (and driving forward) some of the theoretical work
concerning operator discretizationsmentioned above.
Theory of Subdivision and Discretization of OperatorsSubdivision
is at the very heart of much of the mathematical machinery that we
bring to bear on the processing ofdigital geometry. In this context
subdivision may be understood as the appropriate generalization of
the standarddigital signal processing approach of “upsampling
followed by low pass filtering,” to the arbitrary topology
surfacesetting. Because of its recursive nature subdivision is also
at the heart of wavelet constructions for surfaces. To advancethe
very foundations of digital geometry processing we have therefore
invested considerable effort in the theoreticalanalysis of
subdivision schemes.
Accomplishments in the past year In the past year we have been
able to introduce a repeated averaging frameworkfor quadrilateral
schemes, which unifies both the theory and implementation of primal
and dual quadrilateral schemesof arbitrary order. A paper on this
work just appeared in CAGD. The associated software is also
available.
An important theoretical underpinning of our work with the
Subdivision Element method is the requirement thatsubdivision basis
functions are square integrable. We were able to prove this
property for essentially all knownsubdivision schemes. The paper on
this work just appeared in AiCM;
Goals for the coming year We are currently working on a new
generalized averaging construction and its analysisfor a large
family of increasingly smooth
√3 subdivision schemes. Such schemes were recently introduced
and are
attracting a fair amount of attention. All constructions so far
have been ad hoc, however.Advancement of so called “combined
subdivision” schemes, which allow the elegant and efficient
incorporation of
arbitrary boundary conditions, is another goal for the coming
year. In particular we hope to build subdivision schemeswhich work
for both triangles and quadrilaterals simultaneously. This is an
open problem in modeling applicationswhere such mixtures are
unavoidable.
In our effort to understand meshes as discretizations of
surfaces we have studied discretizations of operators onsuch meshes
intensively. Examples include mean and Gaussian curvature, Laplace
and Laplace-Beltrami as well as
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generalized second difference operators. These form the
foundation for particularly robust and highly performant
al-gorithms for denoising and enhancement of geometry as well as
physical simulation, such as cloth and shells, amongmany others.
This study also shed new light on subdivision schemes as particular
instances of gradient and cur-vature flow operators. The latter
offer the tantalizing hope that we may finally succeed in designing
(most likelynon-stationary) subdivision schemes with small stencils
which are C2 everywhere. Other benefits of these studies hasbeen a
better understanding of parameterizations, leading directly to
better compression results and a deeper under-standing of manifold
invariants and their discretization. These are now at the heart of
a new project to exploit them formore efficient parameterization
computations as well as much simplified yet more robust simulations
of the physicalresponse of surfaces.
Because subdivision basis functions overlap each other beyond a
so-called 1-ring of support, standard finite ele-ment refinement
techniques used for the construction of adaptive solvers do not
apply to the SEM. This has challengedus to come up with an entirely
new refinement paradigm for adaptive finite element simulations.
This project has justyielded very exciting results. We have
designed a new approach towards adaptive solvers which is based on
refine-ment of basis functions rather than elements. Consequently
all the usual troubles associated with adaptivity such ashanging
nodes and incompatible elements are eliminated. The approach is
based on a small set of invariants which areindependent of
dimension. As a result the code itself is very simple, carries its
own proof of correctness, and worksin any dimension. The latter was
demonstrated when one of us debugged the first version of the code
in 1D takingapproximately two days of programmer time and then made
the code run for 2D problems in a mere two hours andfor 3D problems
in an additional two hours of effort. We expect this work to make a
major splash in the finite elementand numerical methods for
engineering communities.
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