Statement of Purpose

Applied Differential GeometryYiying Tong

ytong@msu.eduwww.cse.msu.edu/˜ ytong

My main research goal is to develop robust, predictive, and multi-purpose computation tools by leveragingdifferential geometric concepts and foundations. In particular, I wish to contribute a number of practical applicationsin computer graphics and related areas, such as medical imaging, biometrics, medical simulation, and visualization,by focusing on the geometrical foundations common to computational science and graphics.

IntroductionThe water sequence inside the mouth of a whale in Pixar’s Finding Nemo and the traditional flow-past-disk test inComputational Fluid Dynamics may look unrelated. Their goals are, indeed, quite different. Nevertheless, theirunderlying dynamics obey the same set of equations, namely, the Navier-Stokes equations. In fact, much could begained in both fields through exploration of the complementarity of both computational methods: their apparentantagonistic standpoints hide rich opportunities for cross-pollination.

Probably the most important goal in computer graphics applications such as simulation methods and/or photo-realistic rendering techniques is to seek real-time computations. To that purpose, many ad-hoc methods have beendeveloped. Careful examination of the mathematical foundations of these methods is, alas, often considered ir-relevant or intractable. In practice though, more and more researchers acknowledge that, on the contrary, deeperunderstanding of the theory usually leads to simpler, more robust algorithms. Thus, developing solid foundationscan in turn not only unify and generalize, but also improve existing methods. A case in point comes from computa-tional mechanics: variational integrators are now known to provide exact momentum preservation and conservativeenergy behavior— a critical criterion in providing a qualitatively-correct visual experience in simulation. Suchstrong properties are actually obtained at no additional computational cost compared to traditionally-used numer-ical time integration schemes. As one can use these integrators both explicitly or implicitly [KWT+06], there areonly benefits to adopting such robust computational framework instead of ad-hoc integrators.

Importance of GeometryIn many applications of engineering, and in particular in Computer Aided Geometric Design (CAGD), geometryevidently plays a central role. Various data structures like half-edge and constructive solid geometry tree, andalgorithms built on top of them, are intrinsically geometric in nature. But the usefulness of geometry is certainly notrestricted to CAGD: it is omnipresent in computer graphics. In fact, CAGD can be seen as an application of computergraphics to the modeling process of mechanical engineering; this is just one example of the interdisciplinary natureof computer graphics. More generally, any development in computer graphics directly benefits other areas likescientific visualization, medical applications, biometrics, entertainment, communication and education. Indeed, thecommon link between computer graphics and these areas is the reliance on geometry (including but not restrictedto low dimensional Euclidean geometry) and its applications. This means that developments in the mathematicalfoundations of graphics can and should utilize results from differential geometry, computational topology, continuummechanics, electro-magnetism and other areas that use (and are defined by) geometry. In turn, these geometricfoundations can be used to improve visualization, interactivity, and computation efficiency in all those areas.

Past ProjectsMy research interests emerged from a series of initial projects that stressed the importance of good geometric dis-cretizations in a variety of contexts.Computational Science Tools The notion of atlas of a manifold is used in the very definition of manifoldness in dif-ferential geometry. However, this notion is not as simple in computer graphics since each application has a differentset of requirements on the parameterizations of the atlas charts. One project I worked on demonstrated, for instance,that proper discretization of geodesics can lead to an intrinsic parameterization of surfaces independent of its embed-

ding in 3D Euclidean space [LTD05]. This seemingly simple example is actually quite representative of a deeperunderlying principle: the rationale behind a proper discretization of geometric concepts is not only relevant for Eu-clidean curves and surfaces, but also crucial to the treatment of more advanced, general concepts such as tangentbundles and cotangent bundles. An early attempt to discretize vector fields and the well-known notion of Helmholtz-Hodge decomposition was introduced in one of our papers [TLHD03] where we showed how a clean geometric dis-cretization leads to exact discrete differential identities such as the vanishing of the curl of any gradient vector field.

Hodge Decomposition

Since then, a more general approach called discrete exterior calculus (DEC) hasbeen proposed, based on a systematic discretization of Elie Cartan’s exterior calculus,the natural calculus on arbitrary manifolds. This novel idea was initially developedin computational electromagnetism, and used with great success. While the basic el-ement in exterior calculus is the differential form, discrete differential forms in DECare defined as mappings from all k-simplices in a mesh to real numbers, which canbe represented using simple data structures such as arrays with fixed lengths. Conse-quently, operators such as exterior derivative can be directly implemented using the transpose of adjacency matrices.DEC not only provides a proper discretization for its continuous counterpart, but also heightens our intuition sinceit greatly clarifies abstract concepts. To make these ideas accessible to other researchers and practitioners in graph-ics, my collaborators and I wrote an introductory chapter for a recently-published book on Discrete DifferentialGeometry (DDG) [DKT07].

Discrete Exterior Calculus

As elasticity (a model of both shape and its governing dynamics) relies heav-ily on calculus on curved objects, it is natural to apply the DEC framework inthis context. To that purpose, we developed a reformulation of continuum elas-ticity to clearly separate metric-dependent and metric-independent geometric no-tions involved in mechanics. In this formulation, the stress tensor is expressedas a covector-valued 2-form that implies the Cauchy Lemma by definition. Thisalso indicates that a proper discretization of the stress tensor should give val-ues associated with discrete facets [KAT+07], an idea we are now pursuing.

Fluid Simulation

We have also applied this same framework to fluid simulation, as fluid flow mod-els involve another central concept in exterior calculus: the notion of Lie derivatives.Our geometric discretization leads to a simple, stable circulation-preserving algo-rithm designed to keep the circulation of every discrete loop preserved at each timestep as it moves with the flow. Akin to the important preservation of angular momen-tum in a rigid body motion, preservation of vorticity is crucial to fluid simulation—inparticular visually, since this vorticity will affect the motion of smoke rings and vor-tices to which we are quite accustomed. As the incompressibility of fluids guaranteesthat the divergence is null everywhere, vorticity is the only quantity of interest when the domain is fixed. Since ouralgorithm enforces the preservation of local integrals of vorticity (equivalent to circulation along the boundary ofdiscrete facets according to Stokes’ theorem), the resulting numerical scheme does not suffer from the usual vorticitydissipation that plagues other techniques [ETK+07].

Chain Mail

Applications in Graphics One of the major applications of the above-mentionedparameterization is texture mapping, i.e., mapping pictures onto faces of the models,a commonly used tool to decorate objects with rich visual (color) details in graphics.Building an atlas of charts seamlessly stitching texture images is a difficult task thatcannot be directly achieved from examining the model alone. We proposed a novelalgorithm based on the minimization of certain compatibility functions that measurethe discontinuity in the color fields and the color gradient fields on the 2D manifoldsacross various parameterizations [ZWT+05]. The mathematical framework we used in this application facilitates

texture mapping of complex objects by drastically reducing the tedious parameter adjustments that artists typicallyhave to do to produce a visually pleasing result.

Texture mapping has also been used to replace geometric details of objects (such as grooves or bumps), sinceearly graphics hardware had very limited polygon throughput. With the recent advent of powerful graphics cards, wecan now manipulate detailed meshes with millions of triangles in realtime, enhancing the visual experience throughparallax, shadows and, other effects. Meshes are also very amenable to direct editing and animation, something thatartists are particularly looking for. However, complex details such as weaves, ivies, and chain-mails still requirea huge amount of labor to incorporate in 3D models. We recently offered a novel algorithmic way to synthesizesuch intricate details on arbitrary surfaces with the help of the classic “min-cut” algorithm and a low-distortion shellmapping [ZHW+06]: from a base mesh and a given 3D texture swatch, a geometric texture locally similar to theswatch everywhere is synthesized over the base mesh, resulting in models with intricate geometric and topologicaldetails requiring very little user time. This again shows that good mathematical foundations can help tremendouslyin the design of practical algorithms.

Quadrangulation

Current EndeavorsStimulated by our recent results, we are now developing more computational methodsby exploring structure-preserving discretizations of deeper geometric principles.Meshing One obvious application of the DDG framework in graphics and computa-tional science is the adaptive meshing of static objects, through the clean treatment oftopology. Recently, we have started studying harmonic one-forms and cohomologyfor the purpose of quadrangulation [TACSD06]. Harmonicity of one-forms is cru-cial as it guarantees local smoothness (for the quads to be well-shaped) and globalperiodicity (to avoid T-junctions). Through appropriate manipulation of the topology and geometry of the meshes,we have shown how to assemble a sparse linear system to provide an efficient solution. We are exploring ways togeneralize the method to create hexahedral meshes, which are commonly used in Finite Element Analysis. Throughprincipal component analysis of the Laplacian operator, a central notion in DDG, we seek methods that can helpthe theoretical analysis of shapes, as well as practical algorithms such as surface reconstructing from 3D scannerdata [ACSTD07].

Mesh Puppetry

Deformation Models A natural extension to the static case is the modeling of shapeschanging over time. There are traditionally two types of discretization for deformableobjects, namely, the Eulerian and the Lagrangian approaches. In the Lagrangian ap-proach, sample points move with the object, resulting in effecient representation andupdate rules necessary for most real-time applications. We are exploring ways to fur-ther improve the efficiency, for example, through combination of inverse kinematicsand preservation of local differential geometric quantities like mean curvature nor-mal [SZT+07]. In the Eulerian approach, sample points are fixed in space. Although relatively less efficient, ithandles topological changes smoothly. We are investigating remedies for known problems in this approach likethe lack of volume control. Our initial results can already benefit applications such as simultaneous smoothing ofsurfaces in medical datasets, handling of high-genus surfaces, and incompressible fluid simulation [MMTD07].

Eulerian Processing

Applications in Biochemistry, Medical Imaging/Simulation, and BiometricsThe above modeling tools can find an abundance of applications. We wish to inves-tigate the application of a mixture of Eulerian and Lagrangian methods to deal withmolecular surfaces, as both the capability of dealing with a large number of modelsand robust handling of topological alterations are necessary in problems like proteinfolding and docking.

Another direction we wish to explore is the application of aforementioned eigen-analysis based on discrete Laplacian operator. For example, we plan to extract fiber

tracks in the white matter of brains through Laplacian-based spectral analysis applied to the tensor fields obtainedthrough Diffusion Tensor Imaging, a magnetic resonance imaging technique. Similarly, we also plan to do spec-tral analysis of the finite-time Lyapunov exponent field for robust extraction of Lagrangian Coherent Structure, aparticularly pertinent concept in the analysis of time-varying dynamical systems.

Cortical Surface Fairing

As 3D techniques are increasingly accessible and utilized in biometrics, we alsoplan to employ 3D deformable objects for robust identification and verification offaces or other biometrics of humans under the influences of natural variations likeaging and weight gain/loss. Other areas of interest include template matching andsurface fairing for brain imaging [ETKD07].Simulation For simulation of elastic objects, another interesting avenue to exploreis time integration through discretization of variational principles such as the stationary action principle. The re-sulting integrators are the variational integrators we mentioned earlier, that have now been proven to be vastly morerobust, stable, and globally conservative than the integrators resulting from arbitrary discretization of the equationsof motion. We have recently introduced a more general variational integrator based on the Pontryagin-Hamiltonprinciple. With the additional degrees of freedom introduced by the Pontryagin principle, we can substitute thenon-linear solve of the discrete Euler-Poincare equations by a minimization procedure to make the numerical im-plementation truly variational [KWT+06]. In addition, this procedure gives us enough leeway to design motioninterpolation techniques based on subdivision of the animation in space time and adaptive sampling of spatial andtemporal discretization during simulation—ideas that we are currently developing.

Variational Integrator

A variational integration of fluid mechanics is another interesting, yet mostlyunexplored approach to preserving the defining invariants such as circulation andenergy. According to the continuous theory, a correct treatment of these fluid dynam-ics equations should respect the particle-relabeling symmetry: indeed, according toNoether’s Theorem, this symmetry would automatically guarantee the preservationof vorticity, responsible for the intricate visual details evident in fluid motion. Thissymmetry also allows the use of reduction theory, greatly reducing the dimensional-ity of the problem. In order to achieve such a symmetry-preserving discretization, weare working on a discretization of the Lie groups (more precisely, of the volume-preserving diffeomorphisms) andthe corresponding Lie algebras involved in the continuous case. Once we identify the proper discrete counterpartof the continuous groups and algebras of the continuous theory, we can “translate” the corresponding continuousprinciple into a discrete, yet physically accurate language. We plan to continue this research direction, as it seemspromising and rich in consequences.

In addition to variation with respect to time, the discretization of variation of action as integral of the Lagrangiandensity in field theory with respect to spatial coordinates gives multisymplectic integrators, which preserve themomentum map and are potentially crucial in the aforementioned spatial adaptive sampling. Initial tests have beendone in electromagnetism, numerically exhibiting valid global energy behavior [STDM08].Extension of DDG We are also working towards a more general DDG framework through, in particular, the in-troduction of smoother basis functions (for more accurate numerics) and the construction of basis functions over(non-simplicial) cell complex (e.g, for computations on dual meshes). We have produced an initial result on 1-formson curved 2D surfaces through subdivision [WWT+06]. We are also seeking discretization methods for true vectorfields, metric tensors and connection forms, which are consistent with the existing DEC framework, as they arenecessary to carry out full-blown tensor analysis.

Finally, we also plan on studying new applications of DDG in modeling, real-time animation, and other ar-eas related to graphics through the continued development of more powerful geometric techniques offering novelcomputational abilities.

References[ACSTD07] Pierre Alliez, David Cohen-Steiner, Yiying Tong, and Mathieu Desbrun. Voronoi-based Variational Reconstruc-

tion of Unoriented Point Sets. In Symposium on Geometry Processing, July 2007.

[DKT07] Mathieu Desbrun, Eva Kanso, and Yiying Tong. Discrete Differential Forms for Computational Modeling. InA. Bobenko and P. Schroder, editors, Discrete Differential Geometry. Springer, 2007.

[ETK+07] Sharif Elcott, Yiying Tong, Eva Kanso, Peter Schroder, and Mathieu Desbrun. Stable, Circulation-Preserving,Simplicial Fluids. ACM Trans. on Graphics, 26(1):4, January 2007.

[ETKD07] I. Eckstein, Y. Tong, C.-C. J. Kuo, and M. Desbrun. Volume-controlled surface fairing. In SIGGRAPH ’07: ACMSIGGRAPH 2007 sketches, page 8, New York, NY, USA, 2007. ACM.

[KAT+07] Eva Kanso, Marino Arroyo, Yiying Tong, Arash Yavari, Jerrold E. Marsden, and Mathieu Desbrun. On theGeometric Character of Stress in Continuum Mechanics. Z. angew. Math. Phys., 58:1–14, 2007.

[KWT+06] Liliya Kharevych, Weiwei, Yiying Tong, Eva Kanso, Jerrold E. Marsden, Peter Schroder, and Mathieu Desbrun.Geometric, Variational Integrators for Computer Animation. In ACM/EG Symposium on Computer Animation,pages 43–51, July 2006.

[LTD05] Haeyoung Lee, Yiying Tong, and Mathieu Desbrun. Geodesics-based One-to-One Parameterization of 3D Trian-gle Meshes. IEEE Multimedia, 12(1):27–33, January 2005.

[MMTD07] Patrick Mullen, Alexander McKenzie, Yiying Tong, and Mathieu Desbrun. A Variational Approach to EulerianGeometry Processing. ACM Trans. on Graphics (SIGGRAPH), July 2007.

[STDM08] A. Stern, Y. Tong, M. Desbrun, and J. E. Marsden. Variational integrators for maxwell’s equations with sources.In Progress in Electromagnetics Research Symposium (PIERS), 2008. to appear.

[SZT+07] Xiaohan Shi, Kun Zhou, Yiying Tong, Mathieu Desbrun, Hujun Bao, and Baining Guo. Mesh Puppetry: Cascad-ing Optimization of Mesh Deformation with Inverse Kinematics. ACM Trans. on Graphics (SIGGRAPH), July2007.

[TACSD06] Yiying Tong, Pierre Alliez, David Cohen-Steiner, and Mathieu Desbrun. Designing Quadrangulations with Dis-crete Harmonic Forms. In ACM/EG Symposium on Geometry Processing, pages 201–210, July 2006.

[TLHD03] Yiying Tong, Santiago Lombeyda, Anil Hirani, and Mathieu Desbrun. Discrete Multiscale Vector Field Decom-position. In ACM Trans. on Graphics (SIGGRAPH), volume 22, pages 445–452, June 2003.

[WWT+06] Ke Wang, Weiwei, Yiying Tong, Mathieu Desbrun, and Peter Schroder. Edge Subdivision Schemes and theConstruction of Smooth Vector Fields. ACM Trans. on Graphics (SIGGRAPH), 25(3):1041–1048, July 2006.

[ZHW+06] K. Zhou, X. Huang, X. Wang, Y. Tong, M. Desbrun, B. Guo, and H.-Y. Sheum. Mesh Quilting For GeometricTexture Synthesis. ACM Trans. on Graphics (SIGGRAPH), 25(3):690–697, July 2006.

[ZWT+05] Kun Zhou, Xi Wang, Yiying Tong, Mathieu Desbrun, Baining Guo, and H.-Y. Shum. TextureMontage: SeamlessTexturing of Surfaces From Multiple Images. ACM Trans. on Graphics (SIGGRAPH), 24(3):1148–1155, July2005.