Discrete Exterior Calculus for Variational Problems in ...
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Discrete Exterior Calculus for Variational Problems in Computer Vision and Graphics
Mathieu DesbrunUSC CS dept, 3737 Watt Way, PH434
Los Angeles, CA 90089.email:[email protected]
Anil N. Hirani107-81, Control and Dynamical Systems
Caltech, Pasadena, CA 91125.email: [email protected]
Jerrold E. Marsden107-81, Control and Dynamical Systems
Caltech, Pasadena, CA 91125.email: [email protected]
Abstract— This paper demonstrates how discrete exteriorcalculus tools may be useful in computer vision and graphics.A variational approach provides a link with mechanics.
Many problems in computer vision, image processing andcomputer graphics can be posed naturally as variationalproblems. Examples are template matching, image restora-tion, image segmentation, computation of minimal distortionmaps and many others. See for example the special issueof the International Journal of Computer Vision (December2002, Volume 50, Issue 3).
Therefore, new mathematical and computational tools thathave been developed for solving variational problems in otherareas can be usefully employed in computer vision, imageprocessing and computer graphics.
An emerging general tool which we believe will be usefulfor the computational solution of variational problems incomputer vision and related fields is the development of acalculus on discrete manifolds. We have made some progressin this field (Hirani  and Desbrun et al ) by developinga discrete exterior calculus (DEC). Bits and pieces of such acalculus have been appearing in literature but we believe and  introduce several key new concepts in a systematicway. Our development of DEC includes discrete differentialforms, discrete vector fields and the operators acting onthese. In related work, a discrete calculus leads for example,to the proper definition of discrete divergence and curloperators which has applications such as a discrete Hodgedecomposition of 3D vector fields on irregular grids (Tonget al ).
A closely related area in which a great deal of progresshas been made, is discrete mechanics (see, e.g., Marsdenand West  and Lew et al ). The main idea thereis to discretize the variational principle itself rather than theEuler-Lagrange equations. In these and related references, thediscretization is in time only, but one of the most promisingareas in which DEC methods can be used is in spatiallyextended mechanics; that is, in classical field theory ornetworks of interconnected systems, where discrete methodshave been developed and applied; see, for instance Lew et. al.. We believe that there are important lessons to be learnedfrom discrete mechanics when solving variational problemsin computer vision and graphics.
This development of a discrete calculus, when combinedwith the methods of discrete mechanics and other recent work
(e.g., , , , , , , , ) is likely tohave promising applications in a field like computer visionwhich offers such a rich variety of challenging variationalproblems to be solved computationally. As a specific examplewe consider the problem of template matching and show hownumerical methods derived from a discrete exterior calculusare starting to play an important role in solving the equationsof averaged template matching. We also show some exam-ple applications using variational problems from computergraphics and mechanics to demonstrate that formulating theproblem discretely and using discrete methods for solutioncan lead to efficient algorithms.
II. TEMPLATE MATCHING AND DEC
We start with a concrete example of a vision problem inwhich discrete exterior calculus turns out to play an importantrole in computations. This is the problem of deformabletemplate matching. Deformable template matching is a tech-nique for comparing images with applications in computervision, medical imaging and other fields. It has been reportedon extensively in the literature. See for example, Younes, Trouve , , Grenander and Miller  and thereferences therein.
Template matching is based on computing a deformationinduced distance between two images. The “energy” requiredto do a deformation that takes one image to the other definesthe distance between them. The deformations are often takento be diffeomorphisms of the image rectangle, i.e smoothmaps with smooth inverse. The energy can be defined usingvarious metrics on the space of diffeomorphisms.
In this way of posing the problem, template matching issimilar to the way fluid mechanics is formulated. In fluidmechanics, averaged equations have been shown to have theproperty that length scales smaller than a certain parameterin the equation are averaged over correctly and don’t needto be resolved in a numerical solution. See Marsden andShkoller  for details. Motivated by this, in Hirani et al we derived the partial differential equation that we callthe Averaged Template Matching Equation (ATME). Thisequation can also be written in a div, grad, curl form orin Lie derivative form. The unknown in ATME is the timedependent vector field that makes the initial image flow tothe final image while minimizing the kinetic energy.
Our hope in deriving the ATME was that it would allowmatching while ignoring features smaller than a fixed size.This property has not yet been verified but some progress
has been made in the analysis of the equation in one and twospatial dimensions. For example, in  we show how naturalboundary conditions leads to the reduction of the boundaryvalue problem of template matching into a parameterizedinitial value formulation. Specifically we derive the formthat the initial velocity must take to distort one image tothe other while satisfying the ATME. This initial conditionis a piecewise smooth, continuous function with a jump inthe derivative at edges of the image.
Holm’s group has analyzed and computed the solutionsof the ATME and related equations in one and two-spatialdimensions. In 1D, Holm et al , ,  found thatthe initial condition that they called a peakon leads to stablesolutions in which the initial peakons move like solitons.Other initial conditions broke up into peakons that movedaround and collided elastically. Interestingly, the initial con-dition for the 1D problem that we derived from the naturalboundary conditions is a peakon. Recently Holm’s group hasdiscovered solutions to the two spatial dimension case. Theseare collections of peakons along one dimensional curves thatmove and collide in very interesting soliton-like ways. In the2D case (see also ), the crucial step in the numericalsolution was the use of mimetic discretization of the ATMEwritten using div, grad and curl. Mimetic discretization (see,for example, ) is related to a basic form of discreteexterior calculus involving discrete forms. This suggests tous that DEC should be highly relevant to template matching.
III. DISCRETE EXTERIOR CALCULUS
In  and  we present a theory of discrete exteriorcalculus motivated by potential applications in computationalmethods for field theories (elasticity, fluids, electromag-netism) as well as in areas such as vision and graphics. Thistheory has a long history but we have aimed at a comprehen-sive, systematic, applicable, treatment. Many previous works,are incomplete both in terms of the objects that they treat aswell as the types of meshes that they allow. For more detailson the large body of work on DEC, see .
Our vision of this theory is that it should proceed abinitio as a discrete theory that parallels the continuous one.General views of the subject area of DEC are common in theliterature, but they usually stress the process of discretizinga continuous theory and the overall approach is tied to thisgoal. However, if one takes the point of view that the discretetheory can stand in its own right, then the range of applicationareas is naturally enriched and increases. Applications tographics will illustrate this point.Applications to Variational Problems. A major applica-tion areas we envision is to variational problems, be theyin mechanics, optimal control, vision or graphics. A keyingredients in this direction that should play a key role is thatof AVI’s (asynchronous variational integrators) designed forthe numerical integration of mechanical systems, as in Lew etal . These integration algorithms respect key features of
the continuous theory, such as their (multi)symplectic natureand exact conservation laws. They do so by discretizingthe underlying variational principles of mechanics ratherthan discretizing the equations. It is well-known (see thereference just mentioned for some of the literature) thatvariational problems come equipped with a rich exteriorcalculus structure and so on the discrete level, such structureswill be enhanced by the availability of a discrete exteriorcalculus.
There are other variational problems that motivate DEC.For instance, in many problems one requires a hierarchicaland network structure for solving large systems (eg, to simu-late a swarm of complex agents or power or internet systems)to enable one to simulate at a variety of resolutions andalso to make use of parallel strategies together with messageand information passing. In such a network setting, manyproblems involve optimization (such as optimal throughputin an internet setting) and optimization problems are alsovariational and so have an associated DEC structure thatcomes along with the problem. In such a spatially distributedproblem, one has a discrete system ab initio, and so it isnatural to begin with the discrete problem from the start.We have already mentioned the variational nature of severalproblems in computer vision and graphics in the Introduction.
Structured Constraints. Many constraints in numerical al-gorithms involve differential forms, such as the divergenceconstraint for incompressibility of fluids as well as the factthat differential forms are naturally the fields in electromag-netism and some of Maxwell’s equations are expressed interms of the divergence and curl operations on these fields.Preserving, as in the mimetic differencing literature, suchfeatures directly on the discrete level is another one of thegoals, overlapping with our goals for variational problems.
The Objects in DEC. To develop a discrete theory, one mustdefine discrete differential forms along with vector fields andoperators involving these. We define these on discrete man-ifolds that are piecewise affine simplicial complexes (suchas a triangle mesh embedded in 3D). Once discrete formsand vector fields are defined, a calculus can be developed bydefining the discrete exterior derivative (d), codifferential (δ)and Hodge star (∗) for operating on forms, discrete wedgeproduct (∧) for combining forms, discrete flat () and sharp() operators for going between vector fields and one formsand discrete contraction operator (iX ) for combining formsand vector fields. Once these are done one can then defineother useful operators. For example a discrete Lie derivative(£X ) can be defined by requiring that the Cartan magic(or homotopy) formula hold. A discrete divergence in anydimension can also be defined. A discrete Laplace-deRhamoperator (∆) can be defined using the usual definition ofdδ + δd. When applied to functions this is the same as thediscrete Laplace-Beltrami operator (∇2) which is the definedas div curl. We define all these operators in DEC.
IV. DISCRETE MECHANICS
Many standard integrators used for simulating mechan-ical systems fail to respect the mechanical structure (likemomenta, energy, symplectic form etc.) and often havemisleading numerical dissipation. Recently progress has beenmade in the development of variational discrete mechanics,both in the fundamental theory and in the applications tochallenging problems. These include collision algorithmsand the development of AVI’s (Asynchronous VariationalIntegrators). Variational integrators are based on a discretiza-tion of Hamilton’s principle (or the Lagrange-d’Alembertprinciple if there is dissipation or external forces present),which underlies essentially all of mechanics, from particlemechanics to continuum mechanics. Preserving the basicvariational structure in the algorithm retains the structureof mechanics (such as conservation laws) at the algorith-mic level. This avoids many of the problems with existingintegrators, such as spurious dissipation, which may takevery expensive runs to eliminate by standard techniques.With an appropriate development of the connection betweenmechanics and geometry in the discrete setting, one will beable to use for geometry the technology described in thissection. The theory of variational integrators has its roots
q(t) varied curve
qi varied point
Fig. 1: Continuous vs. discrete variational principles.
in Hamilton-Jacobi theory and parts of the basic theory goback to, e.g., Moser and Veselov and its numerical analysisis due to a variety of groups, such as Suris, Marsden andWendtlandt. The strategy is to start with a Lagrangian L(q, q)and requires Hamilton’s principle, namely to make the actionintegral stationary: δ
aL(q, q) dt = 0 for fixed endpoints,
which leads to the Euler-Lagrange equations. This theory hasa PDE counterpart in which one makes a space-time integralstationary which is appropriate for elasticity and fluids, forexample.
In Discrete Mechanics, one approximates the action in-tegral for time ∆t which gives the discrete Lagrangian:Ld(q0, q1,∆t). The discrete action sum, defined by Sd =∑N−1
k=0 Ld (qk, qk+1,∆tk) is required to satisfy discrete vari-ational principle: Extremize Sd given fixed end points, q0
and qN . This yields the DEL (discrete Euler–Lagrange)equations: D2Ld (qi−1, qi,∆ti−1) + D1Ld (qi, qi+1,∆ti) =0 (where D2 denotes the second slot derivative, etc.). Thisgives an update rule (qi−1, qi) → (qi, qi+1), which is calleda variational integrator. To get energy conservation, onecan vary the time steps and get: D3Ld (qi−1, qi,∆ti−1) −
D3Ld (qi, qi+1,∆ti) = 0, which encodes the discrete versionof conservation of energy.
For example, let L(q, q) = 12 qT Mq−V (q), where M is a
symmetric positive-definite mass matrix and V is a potentialfunction; the Euler-Lagrange equations are the standard ones:Mq = −∇V (q). Approximating the action integral
using the rectangle rule yields the finite difference equationM
((qk+1 − 2qk + qk−1)/∆t2
)= −∇V (qk) while another
choice gives the well-known Newmark method for the pa-rameters γ = 1
2 and β = 0. Thus, this simple example isclosely connected with simple symplectic integrators such asthe mid-point rule and the Newmark algorithm. More sophis-ticated quadrature rules lead to more accurate integrators.
0 200 400 600 800 1000 1200 1400 16000
0 100 200 300 400 500 600 700 800 900 10000
(a) Conservative mechanics (b) Dissipative mechanics
Fig. 2: Energies: (a) non-dissipative system computed usingvariational and non-variational integrators vs a benchmarkcalculation; variational methods do not artificially dissipateenergy ; (b) dissipative system computed with variationalmidpoint and Runge-Kutta – note the accurate energy be-havior of the variational method, vs the higher order Runge-Kutta method (with artificial dissipation).
Variational integrators preserve the symplectic structure, aclassical property of mechanical systems. It is believed thatthis preservation is related to the good numerical propertiesof these integrators. These integrators also have a naturaldiscrete Noether’s theorem (preserve momenta) for systemswith symmetry. They also have excellent energy behavior formechanical systems (see, eg., ), even with some dissipa-tion added, compared to conventional schemes. Figure 2 (a)and (b) show that the energy behavior of even low ordervariational integrators can be better than that of higher orderconventional integrators for both conservative and dissipativemechanical systems. More relevant for shells and continua,is the development of AVI’s or Asynchronous variationalintegrators (), that allow one to take different time stepsat different points, which is important for efficiency. Figure2 (c) shows an example of energy behavior for a long-timerun in a 3D dynamic elasticity calculation; one sees that theenergy behavior is still excellent; this is true for both theglobal energy and the energy balance between modes.
V. OTHER APPLICATIONS OF DISCRETE MECHANICS
AND EXTERIOR CALCULUS
Parametrization is a central issue in graphics. Parameteriz-ing a 2D mesh in 3D amounts to computing a correspondence
Fig. 3: Many parametrization techniques (b-c) are not robust to mesh irregularity, exhibiting large distortions for irregular, yetgeometrically smooth meshes, as in (a). Nonlinear techniques (d) achieve better results, but often require several minutes ofcomputational time. With the same boundary conditions, our newly-developed, variationally-based technique quickly generatessmooth parametrizations, regardless of the mesh irregularity (sampling quality) as shown by the two texture-mapped members(e-f) of the novel parametrization family (denoted Intrinsic Parametrizations) .
between a discrete surface patch and an isomorphic planarmesh through a piecewise linear mapping which assigns eachmesh node a pair of coordinates (u, v) referring to its positionon the planar region. Such a one-to-one mapping provides aflat parametric space, allowing one to perform any complexoperation directly on the flat domain rather than on the curvedsurface. This facilitates most forms of mesh processing,such as surface fitting, texture mapping , or remeshing.This last application, for instance, is widely used both inthe graphics and the mesh generation community interestedin finite element computations: while graphics applicationstends to minimizes the number of triangles or quadranglesused to represent a geometry, physical computations oftenrequire special element shape quality to ensure optimalaccuracy. A good parametrization of a given (often scanned-in) geometry has been proven crucial for the efficiency ofboth isotropic and anisotropic remeshing , .
Much work on parametrization has been published overthe last ten years. Almost all techniques explicitly aim atproducing least-distorted parametrizations, and vary only bythe distortions considered and the minimization processesused. Early work used the notion of flattening to obtainan isomorphic planar triangulation (for instance, ), min-imizing ratio of angles between the 3D triangles and theirassociated 2D versions. Others (see, for instance, ) con-sidered spring-like energies that can be quickly minimizedby a linear system solver when the boundary has been fixedto an arbitrary contour (with the noticeable exception of where only a few internal points need to be fixed by theuser). More recently, a number of authors have proposednonlinear formulations to define optimal parametrizations.The MIPS  method for instance finds a “natural bound-ary” that minimizes their highly non-linear energy .Unfortunately, this requires considerable computational effort(even if hierarchical solvers can be used). Sander and co-authors  proposed another nonlinear energy for the veryspecific problem of texture stretch distortion.
Most of these techniques proposed to minimize a con-
tinuous energy over a piecewise linear surface. However,the choice of the energy and its discretization seems veryarbitrary, and most of these techniques may visually resultin non-smooth parametrizations and therefore non-smoothtextured meshes as in Figure 3(b-c). It thus seems thathere again, a discrete variational approach would be veryappropriate and would provide the necessary robustness andguaranteed smoothness quality we need in parametrizations.It turns out that a number of independent authors have pro-posed a discrete harmonic parametrization (see, for instance) who derived the same linear condition for harmonic-ity either using differential geometry, harmonic maps, orfinite elements. A boundary condition is still needed beforecomputing a discrete conformal mapping. We have recentlygeneralized this approach and proposed a new, variationally-based family of parametrizations, based on Minkowski func-tionals to measure discrete distortion measures. Our results,depicted in Figure 3(e&f), seem to prove the validity of ourapproach. Unlike the previous techniques, we can find verysmooth parametrizations independently of the mesh regular-ity. Furthermore, we have also experimented with naturalboundary conditions to provide a natural, optimal conformalparametrization that does not require any specification ofa boundary mapping as we present in . Numerical ex-periments confirm the importance of this discrete variationalapproach for conformal geometry compared to, for instance,much more computation-intensive circle packing techniques.Geometric Flows and Mesh Smoothing. These parametriza-tion results using simple, additive, geometric invariants alsohave direct applications to mesh smoothing, and more gen-erally, to geometric flows. We have partially studied thistopic  by proposing a variational approach to smoothingthat is equivalent to the mean curvature flow. But this time,we did use a direct discrete formulation, by simply statingthe process as an area minimization. The results, comparedto previous fairing methods, have again proven once againthe importance of finding the “right” discretization.Simple Constitutive Laws for Simulation. Several complex
Fig. 4: Discrete Constitutive Equations for Thin-Shells: (a) Real footage vs. Simulation of a Hat: top, a real hat is droppedon a table; bottom, our shell simulation captures the bending of the brim. (b) We can also model a curled, creased, andpinned sheet of paper. By altering dihedral angles of the flat reference configuration, we effect plastic deformation. Whilethe rendering is texture-mapped, we use flat-shading of the triangles to emphasize the discrete structure of the underlyingmesh. The final shape is fully simulated: the artist only indicates the curl radius, the crease sharpness, and the pin positions.
Fig. 5: Denoising of Geometry: we initially developed avariational approach to smoothing of arbitrary meshes inorder to remove the spurious noise from scanned mesheseven when the original sampling is highly irregular and verycoarse . Recent results from our parametrization workmentioned above have made clear that another family ofdiscrete smoothing flows exists.
physical simulations such as thin-shell or cloth animation areoverly computationally-intensive for computer animation. Nosimpler method is known to obtain similar effects (buckling,wrinkles) at low cost. Our initial work on surface invariantshelped us to design new plausible constitutive laws, withvery good numerical properties (since they derive fromvariational principles). For instance, a fabric could be definedas behaving like a weighted mix between an area-preservingand a conformal surface – properties that are actually veryclose to reality for a number of fabrics such as denimor leather. Similarly, the Kirchhoff-Love theory of thin-shells can be substituted by a simpler constitutive law, veryattractive for Computer Graphics as having similar properties(buckling for instance) but with better numerical propertiesand therefore significantly reduced computation times. Thesimplest expression of flexural energy for piecewise-lineargeometry that we have considered is the square trace of thedifference between the shape operator on the undeformedsurface and the pullback of the one on the current, deformed
surface . As depicted in Figure 4, the visual impact ofthis extremely simple model matches the expected behavior.Movie clips showing an animation of the falling hat canbe found at http://multires.caltech.edu/pubs/DS-CDROM. Weintend to analyze the convergence properties, as well as todevelop further, more complex discrete models of thin-shelland other deformable objects.Discrete Hodge Decomposition. To develop a discrete 3DHelmholtz-Hodge decomposition, we followed the 2D ap-proach of Polthier  and defined the curl-free part of avector field ξ as the critical point of the functional:
ξ)2 dV and the divergence-free part as the critical point of thefunctional:
∫Ω(∇×v−ξ)2 dV , which matches its differential
analog. Use is made of two discrete operators Div andCurl. We have also introduced a multiscale representation ofthe projected fields, where fine-scale details are successivelysuppressed while main features are preserved. Such a hier-archical decomposition is interesting for numerical purposesas well as visualization. Our resulting multiscale vector fielddecomposition is a versatile computational tool: we havealready explored several applications as presented in ,from a vector field processing and visualization toolbox, tothe animation of fluids and elastic objects on irregular grids(Figure 6).
The intent of this paper is to demonstrate that a discreteexterior calculus can be a powerful and versatile tool, notonly for mechanics, but also in graphics, visualization andcomputer vision. It has close connections with mimeticdifferencing and other techniques that are known to givegood numerical results, such as those involving constraintsthat can be written in terms of differential forms, such as adivergence constraint. Even in classical field theories, such aselectromagnetism, there is much activity in applying DEC-like ideas. We expect that a further development of DEC and
Fig. 6: Even on curved 2-or 3-manifolds, the variationally-based vector field decomposition separates the curl-free andthe divergence-free components of a vector field. The bottomfigure shows the vector field of a 3D dataset (turbulencebehind a moving car), decomposed using our technique.Represented is the divergence-free part ∇× v of the vectorfield, evidenced by particle tracing and a LIC cross-sectionof this component; the potential v is also represented onthe right side. The false color cross-sections represent themagnitudes of these fields.
its applications will also be of continued use in graphics andvision problems; we outlined a few possible application areasin this paper.
Acknowledgments. The work of Desbrun was supported byNSF grants CCR-0133983, DMS-0221666, DMS-0221669,and EEC-9529152 and that of Hirani and Marsden byNSF/ITR grant ACI-0204932.
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