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Real-Time Example-Based Materials inLaplace-Beltrami Shape Space
Jing Zhao1,2
Fei Zhu3,4
Yong Tang1,2,∗
Liyou Xu3,4
Sheng Li3,4
Guoping Wang3,4,†
1College of Information Science and Engineering of Yanshan University,Qinhuangdao,China,0660042The Key Laboratory for Computer Virtual Technology and System Integration of Hebei Province,China,066004
3School of Electronics Engineering and Computer Science of Peking University,Beijing,China,1008714Beijing Engineering Research Center for Virtual Simulation and Visualization,China,100871
AbstractIn this paper, we present a new real-timesimulation framework for example-based elasticmaterials with different topology structuresbetween the object and input examples. Thegeometry descriptions in the Euclidean spaceof all input shapes are projected into a commonreduced shape space spanned by the Laplace-Beltrami eigenfunctions constructed on thevolumetric mesh. The reduced shape interpola-tion space is scale and topology independent.The shape interpolation process is computedtotally in the reduced subspace by solving anonlinear energy optimization problem which iscalculated in the model reduction subspace us-ing cubature elements. To accelerate the energysolving process, we eliminate the transmissionprocess between the reduced spaces and the Eu-clidean space by establishing a direct projectionfrom Laplace-Beltrami shape space to the modelreduction subspace. Experiments demonstratethat our method can achieve real-time efficiencywhile providing comparable simulation quality.
Keywords: deformable simulation, reducedspace, example-based material, Laplace-
∗Corresponding author.†Corresponding author.
Beltrami eigen-analysis
1 Introduction
The simulation of deformable objects is a com-mon task in physically-based animation field be-cause of its broadly used in computer graphicsand virtual reality systems. The material prop-erties tuning of the deformable behavior is achallenging and tedious work, while achievingart-directed deformation results becomes anoth-er intuitive choice.
Recently, Martin et al. [1] propose anapproach of example-based elastic materialsachieving artistic control. They construct anexample space by interpolating example posesprovided by users and generate an additionalforce which attracts the object towards the ex-ample space. It contains a time-consuming pro-cess to reconstruct a consistent geometric rep-resentation of interpolated example poses. Ad-ditionally, the input examples share the sametopology with the simulation object. Zhu etal. [2] express shapes in a common shape s-pace in which the basis are the coupled Laplace-Beltrami eigenfunctions. In the proposed shapespace, different models can be animated us-ing the same examples. Compared to Mar-
tin’s method, the efficiency has been improvedby using the reduced shape interpolation space.The whole simulation process is far from real-time efficiency since they obtain the interpola-tion shape by solving a constrained nonlinear-optimization problem. The energy computing ison the whole Euclidean space which is related tothe vertices number of the object’s surface mesheven though the process only contains several it-eration steps. We achieve example-based elasticdeformation at real-time rate, and at the sametime we keep the different topology structuresfor both the examples and the simulation objec-t using the Laplace-Beltrami shape space. Forthe details, in the simulation process, we takeuse of the nonlinear modal basis and the cuba-ture elements to compute internal force for ar-bitrary materials presented by An et al.[3]. Inthe shape interpolation process, we obtain thetarget interpolation configuration to solve a con-strained nonlinear optimization problem totallyin the subspace. We solve the energy minimiza-tion issue in the subspace by proposing a directprojection from the Laplace-Beltrami shape s-pace to the model reduction subspace.
2 Related Work
Example-Based Deformation Martin and hiscolleges [1] propose an example-based elasticmaterial with the finite element method(FEM),which allows the deformation behavior to be im-plicitly controlled by specifying a set of exam-ple poses. They employ a nonlinear Green s-train as the deformation measure and construc-t an elastic energy that pulls current configura-tion towards the preferred configuration definedby examples. It involves solving an expensivenonlinear optimization problem to reconstruc-t a consistent geometric representation of inter-polated example poses. In their following re-searches [4], they utilize an incompatible repre-sentation for input and interpolate poses to inter-polate between elements individually. The pro-posed method achieves significant performanceimprovements compared to previous work. Ex-cept for example-based elastic solid deformationmaterials, Frohlich et al. [5] extend the deforma-tion to discrete shells based on an extension ofthe discrete shell energy and Jones et al. [6] pro-
pose the example-based plastic deformation forrigid bodies based on linear blend skinning andan unmodified rigid body simulator. Koyama etal. [7] present a real-time example-based elas-tic deformation method using the shape match-ing framework and the linear interpolation of theexample poses. Their method is based on a ge-ometry framework and achieves real-time effi-ciency, while the simulation effect is not so goodas Martin’s method. The method proposed byZhang et al. [8] also obtains a real-time simu-lation rate using a subspace integration. Theyformulate a new potential using example-basedGreen strain tensor which attracts the simulationmodel to the example-based deformation featurespace. Wampler [9] introduces a novel approachfor example-based inverse kinematic mesh ma-nipulation which generates high quality defor-mations for a wide range of inputs. And the ap-proach is fast enough to run in real time.
The limitation of the previous example-basedapproaches is circumvented that all examplesmust have identical topology with the simu-lated object. Zhu et al. [2] propose a newmethod which allows all the examples are ar-bitrary in size, similar but not identity in shapewith the object. They project all the geometrystructures to a common shape space spanned bythe Laplace-Beltrami eigenfunctions and inter-polate the examples via a weighted-energy min-imization to find the target configuration whichguides the object to a desired deformation. Thetarget configuration is solved by a nonlinearoptimization problem and the nonlinear shapeinterpolation process is time-consuming sincethe nonlinear deformation energy between twoshapes is defined on the surface mesh, whichmeans the computation efficiency is relative tothe resolution of the mesh.Subspace Simulation A lot of researches havebeen proposed to improve the speed of sim-ulating deformable models, especially for thesubspace simulation approach which has gainedpopularity in computer graphics recently [10].The modal analysis method was first proposedin graphics to build linear vibration modes as thesubspace basis [11]. To handle rotational defor-mations, Choi and Ko develope the modal warp-ing method by removing incorrect vertex defor-mation components caused by linear elasticity[12]. Barbic et al. [13] construct nonlinear de-
formation modes from modal derivatives for theSt.Venant-Kirchhoff materials, since the internalforce of the special material can be simply rep-resented as a vector of cubic polynomials in thereduced coordinates. An et al. [3] calculate thereduced elastic force in the subspace by usingcubature approximation elements. Their methodis not limited to the StVK material and reducesthe computational complexity of subspace sim-ulation to O(r3), in which r is the number ofmodes in the subspace basis. A basis augmenta-tion scheme is presented to capture local defor-mation caused by collision contact [14].Shape Interpolation Shape interpolation is animportant issue in geometry processing. Most ofthe shape interpolation schemes are based on theinterpolation for the selected geometric quanti-ties of a shape. The interpolation process is tocompute the average quantities for all examples.The reconstruction process of the shape is typ-ically done as a least-squares problem depend-ing on the quantities of the vertices positions arelinear or nonlinear interpolation. One effectiveresearch is depended on maintaining the rigidcriteria called as-rigid-as-possible [15]. Theirmethod computes preferred interpolations by lo-cal affine transformations of the local geomet-rical elements. Xu et al. [16] propose a non-linear gradient field interpolation method. Thegeometric quantities conclude the vertex coor-dinates and surface orientation, and the recon-struction is a Poisson problem. Winkler et al.[17] utilize the edge lengths and dihedral anglesof triangles of a surface mesh and Martin et al.[1] take use of the strain tensors of the tetrahe-dral mesh as the geometric interpolation quan-tities. Due to the nonlinear optimization prob-lem in the reconstruction process, the methodsrun at real-time rates only for very coarse mesh-es. Von-Tycowicz et al. [18] restrict the shapeoptimization problem to a low-dimensional sub-space that is specifically designed for the shapeinterpolation problem.
3 Background
In this session, we present an overview of thebasic example-based elastic material simulationframework and the reduced subspace simulationmethod.
3.1 Example-Based Elastic Deformation
The key idea for the simulation of the example-based elastic materials is to add an additionalelastic potential on the basic simulator of elas-tic solids, which attracts the deformable objecttowards the desirable deformation characterizedby the examples. The equations of motion of anobject discretize in space are given by
Mx+Dx+ f int +∂Wp
∂x= fext (1)
Here M is the mass matrix, x and x are thepositions and accelerations of the object’s n-odal degrees of freedom (DOFs), f int representsthe internal force, Wp = W (Xtarget, x) repre-sents the additional potential between the objec-t’s deformed configuration x and its closest tar-get configuration Xtarget on the example man-ifold spanned by input examples, fext denotesthe sum of external forces due to gravity, fric-tion and contacts.
3.2 Subspace Simulation
Model reduction is an efficient technique wide-ly used to speed up the integration. The basicsubspace simulation is to convert the displace-ment vector u ∈ R3n into a subspace spannedby a set of r(r ≪ 3n) representative deforma-tion modes. These displacement vectors can beassembled into a 3n × r matrix U , as the ba-sis for subspace simulation. As the method de-scribed in [13], we use nonlinear modal basisto construct U , in which U is mass orthogonal:UTMU = I , I is the r × r identity matrix.Then the full-space displacement u is convert-ed to u = Uq, in which q ∈ Rr represents thereduced coordinates in the subspace. By com-bining u = Uq and UTMU = I with Equation1, we obtain the governing equation in the sub-space:
q+UTDUq+UT f int(Uq)+UT ∂Wp
∂x= UT fext
(2)By using implicit integration, Equation 2 can beformulated into a dense r× r linear system, andthe entire simulation can be orders of magnitudefaster than the full-space simulation.
4 Example-Based SubspaceSimulation
4.1 Real-time simulation framework
The previous example-based material simula-tion method has the same limitation that the in-put examples are constrained to share the sametopology with the object. [2] proposed theLaplace-Beltrami eigen-analysis method to con-struct a scale and topology independent shapespace. Our method proceeds also in two stagesas: the pre-computation stage and the run-timesimulation stage. Here we outline the proce-dures in each stage:Pre-computation:
(1) Construct the non-inertial frame of thesimulation model and align examples to the lo-cal frame;
(2) Compute m leading eigenfunctions andthe corresponding eigenvalues for both the sim-ulation model and input examples, and normal-ize the eigenfunctions;
(3) Align the examples’ eigenfunctions withthe object’s top m eigenfunctions;
(4) Project the geometry of all examples ontotheir eigenfunctions and get corresponding co-efficients as the shape descriptors in the non-inertial frame;
(5) Compute the nonlinear modal basis for themodel;
(6) Compute the optimized cubature elementsusing the random forces and the input reducedmodal basis.Run-time:
(1) Obtain the non-inertial frame displace-ment of the current configuration and project thedisplacement to the Laplace-Beltrami shape s-pace for the shape description;
(2) Find the target configuration according tothe examples and object’s current configurationin the Laplace-Beltrami shape space, and theshape interpolation process is computed in thenon-inertial frame;
(3) Reconstruct the shape of the target inter-polation configuration from Laplace-Beltrami s-pace to the Euclidean space to calculate theexample-driven force, which is computed by thelinear force with the displacement of the currentand target configuration by simplicity;
(4) Perform the rigid body simulation frame-work in inertial frame;
(5) Compute the deformed simulation withthe example-driven force, the external forcesand the forces caused by the rigid motion in non-inertial frame, step the deformed simulation;
(6) Project the non-inertial frame displace-ment to the inertial frame to render the defor-mation.
We demonstrate the runtime overview in Fig-ure 1. Three spaces have been described here.In the Euclidean space, three input poses of theelastic cuboid are given as examples as geom-etry shapes, each with different number of ver-tices. All the geometry shapes are projected on-to their first m eigenfunctions in the Laplace-Beltrami interpolation space(black points). Inthe interpolation process, we build a connectionfrom Laplace-Beltrami shape space to the modelreduction subspace to compute the elastic ener-gy quickly. After the interpolation process, thetarget configuration (red point) in the Laplace-Beltrami space is reconstructed to the Euclideanspace.
Figure 1: Overview of run-time operations.
4.2 Space description
For the whole simulation framework, we havethree spaces and we will explain them in details.We have listed most of the variables of this paperin Table 1.Laplace-Beltrami subspace Based on Laplace-Beltrami eigen-analysis method, we obtain theeigenfunctions on the volumetric mesh. Thereare several ways to approximate the Laplace-Beltrami operator and its eigenfunctions on dis-
Table 1: Symbols. This table summarizes some of the symbols.Symbol Dimension Definitionm scalar num of aligned eigenfunctionsr scalar num of nonlinear modal basiss scalar num of optimized cubature elementsn scalar num of vertices of the objectu 3n vector displacement in the fullspaceq r vector displacement in the model reduction subspacel 3m vector shape description in the Laplace-Beltrami subspaceU 3n× r matrix the nonlinear modal basisL 3n×m matrix top m eigenfunctions aligned between the examples and the objectT m× r matrix projection from Laplace-Beltrami shape space to model reduction subspacex(q) 3n vector the vertex position in the non-inertia framev, ω 3 vector linear and angular velocities in non-inertia frame of rigid simulationv, ω 3 vector linear and angular acceleration velocities in non-inertia frame
crete representations of manifolds. We choosethe linear FEM approximation for its highly ac-curacy [19, 20]. The normalized eigenfunctionsconstruct the reduced interpolation shape space.The volumetric mesh in the Euclidean spacecan be projected to Laplace-Beltrami subspace,while the shape description of the reduced in-terpolation shape space can be reconstructed inthe Euclidean space by using eigenfunctions andeigenvalues. The eigenfunctions are normalizedand computed on the tetrahedral mesh, and thevalues of the eigenfunctions on each vertex isbetween 0 and 1. We render the eigenfunction-s with different colors on each vertex using thepiecewise linear function. We demonstrate thesecond eigenfunction for the interior of the ar-madillo model as Figure 2. The first constan-t eigenfunction is excluded from the computedeigenfunctions since it encodes only rigid mo-tion of rotation and translation.
Figure 2: Rendering interior eigenfunction ef-fect for the armadillo model.
The eigenfunction of different shapes need tobe aligned before they can be used as the com-mon basis. We align the eigenfunctions with themethod in [21]. The registration process is con-venient and can be computed in advance.Full-space Full-space is the geometry descrip-tion in Euclidean space and it is the transition s-pace between the Laplace-Beltrami shape spaceand the model reduction subspace. Also we ren-der the deformation displacement and excecutethe rigid motion in this frame.Reduced subspace constructed by modal ba-sis The subspace is constructed by the nonlinearmodal basis in the pre-computation process. Inour current implementation, we use linear modalanalysis and modal derivatives proposed in [13]to generate both linear and nonlinear deforma-tion modes in the basis.
The spaces can be converted to each other as:(1) Project to the Laplace-Beltrami space.
The geometry description of the examples andthe object in Euclidean space is projected ontothe Laplace-Beltrami interpolation subspace bytheir own eigenfunctions. The projection is donethrough volume-inner product of the shape’s ge-ometry and its Laplace-Beltrami eigenfunctions,which can be described as:
(< px, λiLi >,< py, λiLi >,< pz, λiLi >)(3)
where 1 ≤ i ≤ m, p(px, py, pz) is the geome-try vertices position in Euclidean space, λi,fi isthe i-th eigenvalue and its corresponding eigen-
function, and <,> represents the volume-innerproduct. The description in Lapalace-Beltramisubspace is a m× 3 vector.
(2) Reconstruct geometry shapes in theEuclidean space. By using the equation:m∑i=1
li1λiLi, we get a geometry description of Eu-
clidean space.(3) Project shape description from Laplace-
Beltrami space to the model reduction subspace.As Zhu et al. [2] did in each time step, the shapedescriptor is reconstructed in the Euclidean s-pace and the nonlinear elastic energy is comput-ed by a geometry deformation energy on the ob-ject’s surface mesh, which is a time-consumingprocess. To accelerate the target shape inter-polation process, we solve the elastic energyin the model reduction subspace described asequation (6) by using the optimized cubatureelements. Since we have to utilize the Eu-clidean space as the transition space between theLaplace-Beltrami shape space and the model re-duction subspace, which need to compute thelarge matrix-multiply operation twice with ma-trix L and matrix U , which are constant in thewhole simulation process and can be computedbeforehand. We propose a direct projection ma-trix from Laplace-Beltrami subspace to modelreduction subspace by computing T = LTU .
4.3 Shape interpolation
The shape interpolation process is to find the op-timized target configuration in the example man-ifold for the current configuration. In our sys-tem, we solve the shape interpolation problemtotally in reduced space.
The deformable object is represented by atetrahedral mesh with n vertices. The input ex-amples are consist of k poses, represented as m-dimensional points ei(1 ≤ i ≤ k) in the shapespace spanned by the eigenfunctions. We inter-polate the examples to get the target configura-tion that guides the deformation of the object.Similar as [2] did, we also use nonlinear inter-polation approach. The target configuration tis defined as the one that minimizes the sum ofweighted deformation energies to all examples:
mint
k∑i=1
ωiE(t, ei) (4)
where E(·, ·) is the non-linear deformation en-ergy between two shapes, ωi(1 ≤ i ≤ k) mea-sures the guiding strength of the examples andk∑
i=1ωi = 1. The interpolation weight for each
example weight according to its closeness withthe object’s configuration c in each time step:
minωi
1
2
∥∥∥∥∥k∑
i=1
ωiei − c
∥∥∥∥∥2
F
(5)
In their method, the geometry energy is com-posed of three terms: stretching term, bendingterm and volume preservation term. Their ge-ometry energy measurement is relative to thegeometry description, and the resolution of themesh will definitely decide the solving efficien-cy. To obtain a target interpolation configura-tion for the current deformation of the objec-t, the nonlinear interpolation process is a non-linear optimization which contains several iter-ation steps. It is time-consuming to computethe energy and its gradient in the Euclidean s-pace at each iteration step. We accelerate theprocess by computing the energy with the cuba-ture integration method provided by [3] in themodel reduction subspace. As noted in 4.2, wehave a direct projection from Laplace-Beltramishape subspace to the model reduction subspace.With the projection, the elastic energy and itscorresponding energy gradient can be comput-ed easily for each target configuration. Accord-ing to the method, the elastic force can be for-mulated in terms of a potential energy function,E(q) : Rr → R, given by the domain integral,
E(t, ei) =
∫ΩΨ(Xei ; qei)dΩX
=
s∑j=1
ωjΨj(Xei ; qei)(6)
where the undeformed shape Xei is the ith ge-ometry shape in the Euclidean space recon-structed from the Laplace-Beltrami space withthe object’s eigenfunction, Ψ(Xei ; qei) is thenonnegative strain energy density at materialpoint Xei in the undeformed material domain Ω,s is the number of cubature elements and qei de-scribes the displacement between the target con-figuration and example ei in the model reductionspace. The optimizing cubature elements and
the corresponding weights obtained in the pre-computation process.
5 Results
We test our system on an Intel Core i5 2.8GHzprocessor. We use the OpenMp library to paral-lelize our simulation steps. Time step size is uni-formly set to 0.001s for all simulations. Current-ly, we choose the StVK material model, whileour proposed method is not limited to the usingmaterial. We use two input examples for all theexperiments in the paper.
As shown in Figure 3, the first picture is theinitial configuration, the second bar deformedunder a twisted force on the bottom simulated inthe full space, the third twisted bar is deformedwith the same topology between the input exam-ples and the object in the full space, the fourthone demonstrates the twisted bar deformed bydifferent topology examples(cylinder examples)in the full space, and the last picture shows thetwisted bar deformed on the same situation asthe fourth one which is performed in the reducedspace with our method. The last four picturesshow similar deformation effects with differen-t conditions, and our method with different in-put examples computes the interpolation shapein the model reduction space which achieves e-qually animation effect as in the full space.
Figure 3: Bar deformed with differentsituations.
Without constrains As [22] pointed out, thesimulation method in reduced subspace is notsuitable for simulating the rigid motion of adeformable object, since incorporating rigidmodes into the basis U will cause U to be time-dependent and we cannot afford updating U overtime. We simulate the rigid motion separatelyfrom subspace deformation by rigid body dy-namics, as did in [13, 23, 24]. The free motionsimulation object contains the rigid motion anddeformed motion. The shape interpolation pro-
cess and the simulation process in the reducedspace are computed in the non-inertial frame.Figure 4 and Figure 5 show the simulation ef-fects for deformable objects without constrains.
Figure 4: An elastic bar deforms under gravityusing a S-cuboid shaped example anda S-cylinder shaped example.
Figure 5: A car model hits the wall and re-acts in diverse ways with no ex-ample(top left), an arc-shaped ex-ample(top right), a s-shape exam-ple(bottom left) and a twisted exam-ple(bottom right).
With constrains The object simulation withconstrains is much more easier than the freemotion one since the deformation excluded therigid motion. As shown in Figure 6, the teddymodel is fixed on the back. The first picture isthe rest configuration, the second one shows thedeformation of the left part of the model underthe drag force enforced on the left leg, and thethird one shows the global deformation between
the two legs and two arms.
Figure 6: An elastic teddy deformed with localand global examples.
Local examples The method is easily extend-ed to the local examples. We partition the en-tire simulation object into separate regions, eachpart is influenced by the corresponding exampleregions independently. We project each regionon the entire object’s Laplace-Beltrami eigen-functions. The rest vertices displacement of theobject excluded each region are set to zero. Eachlocal region is handled independently, thus weachieve complex behaviors for different localexamples. We demonstrate the local examplesas Figure 7. The top three figures show differ-ent examples as the twist shape, the horizonaldirection s-bend shape, and the vertical direc-tion s-bend shape, respectively. The deforma-tion result shows on the second line, the first pic-ture is the initial configuration, the second pic-ture shows the left region and the right regionof the cross deformed with the first and secondexamples respectively, and the last one showsthe middle region of the cross influenced by thethird example. In Table 1, we list the detailedparameter settings and the performance data forall the examples presented in the paper. The re-sults in this paper are produced with practicalcomputation time. We can find out the time ef-ficiency is obviously better than the method in[2]. We achieve a real-time simulation effect s-ince we solve the energy minimization problemin the reduced subspace and we utilize the opti-mized cubature elements to find the target con-figuration. The optimized cubature elements canbe computed in advance with the method pro-posed in [3, 25, 26].
6 Conclusions
We present a real-time framework for example-based elastic materials with different topology
Figure 7: Eigenfunctions of the three local re-gions on the cross shape are comput-ed, respectively.
structures between the simulation object and in-put examples with FEM. We compute the eigen-functions on the tetrahedral mesh with the linearFEM approximation for the Laplace-BeltramiOperator eigen-analysis method to construct theshape interpolation subspace. We propose thedirect projection from the Laplace-Beltrami sub-space to the model reduction subspace, which isconvenient to compute the elastic energy withthe optimized cubature elements. Finally, weseparate the rigid motion from the deformablemotion with the inertia frame and the non-inertial frame. Our experiments demonstrate theflexibility and the highly simulation efficiencyfor the example-based materials to some extend.
In the subspace simulation method for de-formable objects, an important question iswhether collisions can be handled in the sub-space as well. We do not take much time on thecollision problem. For simplicity, we choose thepenalty force for collision force in the full-spacewhich provided some collision-artifact. Thisproblem can be extended in the subspace to im-prove the efficiency as [27, 28] did in the future.Finally, although our system separates rigid mo-tions from non-rigid motions, it is not strictlycorrect and the result can be different from theresult of full-space simulation. Sometimes thesystem will unstable on the rotation caused bythe collision force and example force.
Model DOFs Tets Ex. m r sZhu et.al[2] our method
tinterp ttotal tinterp ttotalbar-cuboid 981 1053 327 10 30 60 51.2 55.5 35.1 41.5bar-cylindar 981 1053 2478 10 30 60 266.8 271.6 32.7 39.4cross 4683 4912 3172 10 42 21 564.1 592.4 11.9 18.3car 2181 2369 1019 6 20 23 207.7 221.6 16.7 21.2teddy-global 1773 1755 11193 6 30 30 70.3 83.5 28.7 33.4teddy-local 1773 1755 11193 6 30 30 302.8 313.7 33.2 38.4
Table 2: Summary of all results presented in the paper. The columns indicate the number of DOFs ofthe object, the number of tetrahedral elements of the object, the vertices number of the lastexample, the number of the selected eigenfunctions, the number of modal basis, the numberof cubature elements, average time(in milliseconds) for example interpolation and total timefor one simulation step of [2] and our method, respectively.
Acknowledgements
We would like to thank Steven S. An for shar-ing his cubature source code and the anony-mous reviewers for their comments and sugges-tions. This work was partially supported byGrant No. 2015 BAK01B06 from The Nation-al Key Technology Research and Developmen-t Program of China, and Grant No. 61232014,61421062, 61170205, 61472010 from NationalNatural Science Foundation of China.
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