Digital Geometry Processing:A Progress Report and Research Outlook

Peter SchröderComputer Science Department 256-80

California Institute of TechnologyPasadena, CA 91125

ps@cs.caltech.edu

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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