Efficient Mesh Deformation Using Tetrahedron Control Mesh

Transcription

1 Efficient Mesh Deformation Using Tetrahedron Control Mesh Jin Huang Lu Chen Xinguo Liu Hujun Bao State Key Lab of CAD&CG, Zhejiang University Hangzhou, P. R. China Abstract It is a challenging problem to interactively deform densely sampled complex objects. This paper proposed an easy but efficient approach to it by using coarse control meshes to embed the target objects. The control mesh can be efficiently deformed by various existing methods, and then the target object can be accordingly deformed by interpolation. One of the most simplest interpolation methods is to use the barycentric coordinates, which however generates apparent first-order discontinuity artifacts across the boundary due to its piecewise linear property. To avoid such artifacts, this paper introduced a modified barycentric interpolation (modified-bi) technique. The central idea is to add a local transformation at each control vertex for interpolation, so that we can minimize the first-order discontinuity by optimizing the local transformations. We also minimize the second order derivatives of the interpolation function to avoid undesired vibrations. While focus on deforming 3D objects embedded in tetrahedron meshes, the proposed method is applicable to 2D image objects embed in planar triangular meshes. The experimental results in both 2D and 3D demonstrated the success and advantages of the proposed method. Figure 1: Object twisting example. On the left is the original Bar shape (23K vertices) rendered with the wireframe of the control mesh (35 vertices, 78 tetrahedrons). The deformation result by using our modified barycentric interpolation (middle) is visually very smooth compared with the result of the traditional barycentric interpolation (right). CR Categories: I.3.5 [Computational Geometry and Object Modeling] Curve, surface, solid, and object representations; Keywords: Shape Modeling, Mesh Deformation, Barycentric Interpolation. 1 Introduction Highly complex models become more and more popular and bring big challenges for nowadays geometry processing algorithms. To address the problem, people create a sparse control mesh or volumetric control lattice to embed and manipulate the target model by using various interpolation techniques. Free form deformation (FFD) methods adopt spline functions to achieve smooth interpolation results [Sederberg and Parry 1986]. Such smooth interpolation can be achieved alternatively by using mean value coordinate (MVC) [Floater 2003; Ju et al. 2005], which doesn t need a volumetric control lattice. While high quality results are achieved, the evaluation procedure requires the information of a relative large stencil of neighborhoods or even the whole control mesh, which is not suitable for GPU implementation. hj@cad.zju.edu.cn chenlu@cad.zju.edu.cn xgliu@cad.zju.edu.cn corresponding author: Xinguo Liu bao@cad.zju.edu.cn Copyright 2008 by the Association for Computing Machinery, Inc. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Permissions Dept, ACM Inc., fax +1 (212) or permissions@acm.org. SPM 2008, Stony Brook, New York, June 02 04, ACM /08/0006 $ Barycentric interpolation is possibly the simplest technique widely adopted in many graphics applications. In 3D, a barycentric interpolation involves only four vertex values of a tetrahedron. In another word, the basis function corresponding to a control vertex has non-zero value only in its incident tetrahedrons. Such local support property of the barycentric interpolation enables fast computation and local control. However, it produces noticeable first-order discontinuity artifacts across the boundary when the control mesh is sparse, as illustrated by the twisting example in Figure 1(right). Increasing the resolution of the control mesh could attenuate the artifacts, but the cost for manipulating the control mesh will be increased accordingly. Manipulating highly complex objects via a control mesh/lattice actually separates the deformation problem into manipulation phase and interpolation phase. In manipulation phase, the control mesh is alternated in various ways. In interpolation phase, the embedded shape is deformed to follow the control mesh. Various mesh editing and deformation techniques can be applied to the first phase[yu et al. 2004; Sorkine et al. 2004; Botsch and Kobbelt 2004; Zayer et al. 2005; Botsch and Kobbelt 2005]. In this paper, we focus on the second phase smoothly deform the embeded target object given the manipulation results of the control mesh. The motivation of this work is to use extremely coarse control mesh to drive the deformation of highly complex objects in a simple manner such that the CPU cost for manipulating the coarse control mesh is very small and the deformation of the target fine mesh can be efficiently evaluated using modern GPUs. And we advocate tetrahedron mesh instead of hexahedron mesh for its flexibility in representing arbitrary topological structure of shape. To achieve high performance of evaluation, our method is based on the traditional barycentric interpolation (BI). The central idea is to add a local transformation on each control vertex for interpolation, so that we can optimize the local transformations to minimize the

2 first-order discontinuity. We also minimize the second order derivatives of the interpolation function to avoid undesired vibrations. There are several advantages in our modified barycentric interpolation techniques: (1) the positions of the control vertices are reproduced; (2) it can generate visually smooth interpolation results even with extremely coarse control mesh; (3) it can be locally evaluated, implemented in GPU achieving high performance. Though the first-order continuity is not guaranteed theoretically, our method greatly improves the smoothness of the deformation results as numerically validated in our experiments. The main contribution of this paper is an efficient mesh deformation scheme which uses a coarse control mesh and the novel modified barycentric interpolation technique. 2 Related Work Our work is closely related to many shape deformation techniques. Theoretically speaking, any deformation can be achieved by manually manipulating individual vertices, which is however impractical due to the complexity of meshes in practice. FFD techniques reduce the manual work by embedding the complex shape in a relatively simple control lattice [Sederberg and Parry 1986]. The user can control FFD results by various controls[milliron et al. 2002], such as lattice [Sederberg and Parry 1986; Coquillart 1990; MacCracken and Joy 1996], feature lines [Beier and Neely 1992], control curves [Barr 1984; Singh and Fiume 1998; Kho and Garland 2005], points [Hsu et al. 1992] and surfaces [Feng et al. 1996]. Then the embedded shape is deformed via interpolation. Mean value coordinates (MVC) [Ju et al. 2005] provides an alternative interpolation method, which uses a coarse control surface instead of a control lattice. However, MVC is inversely proportional to the Euclidean distance, which causes undesired artifacts in nearby components, as exemplified by Figure 2(e) in [Joshi et al. 2007]. The positive MVC avoids this problem by culling the contributions from invisible surfaces [Lipman et al. 2007]. Though the positive MVC can be fast computed in GPU, the interpolation procedure is not suitable for GPU implementation. Gradient domain deformation technique refers to a family of methods recently developed after the work of poisson mesh editing[yu et al. 2004], Laplacian mesh editing[alexa 2003; Sorkine et al. 2004] and deformation transfer [Sumner and Popović 2004], which basically reconstruct the deformed shape from the alternated gradients or Laplacian/differential coordinates, preserving surface details. By far, several new representation for encoding surface and volumetric details have been proposed, such as graph-laplacian [Zhou et al. 2005], rotation-invariant [Lipman et al. 2005] and edge-based differential coordinates [Huang et al. 2006b; Sorkine and Alexa 2007]. Gradient domain techniques were also adopted in sketch-based user interface [Zhou et al. 2005; Nealen et al. 2005], shape interpolation [Xu et al. 2005] and mesh-based IK [Sumner et al. 2005; Der et al. 2006]. In addition, several dimension reduction methods based on mean value coordinates [Huang et al. 2006c] and skeleton models [Yoshizawa et al. 2007; Weber et al. 2007; Shi et al. 2007] were developed for acceleration and robustness. Our method in this paper is also applicable to deforming 2D image objects [Beier and Neely 1992; Cohen-Or et al. 1998; Igarashi et al. 2005]. Recently, Schaefer et al. [Schaefer et al. 2006] proposed an novel image deformation algorithm using moving least squares (MLS). This technique shares some similarity with ours in using affine transformations to deform the image content as-rigid-as-possible. But ours uses a sparse control mesh, and works for both 2D and 3D objects. In the rest of this paper, we will first introduce the modified barycentric interpolation (modified-bi) method in Section 3; then present an energy minimization framework for constructing modified-bi in Section 4; then combine the modified-bi with several existing deformation techniques and present the experiments results in Section 5. Finally, we conclude this paper with some discussions on future work in Section 6. 3 Modified Barycentric Interpolation (modified-bi) Let Ω be an object in R d, d = 2or3,andΦ =(U,T ) be a finite element tessellation of Ω, whereu =(u 1,...,u n ) denotes the vertex set and T =(t 1,...,t m ) denotes the element set. we call Φ the control mesh of the object, and U the control vertices. Φ is a planar triangle mesh for d = 2, and a tetrahedron mesh for d = 3. Consider a deformation function x(u): Ω R d, with known values at the vertices x(u i )=x i, u i U. The simplest way to define the function value for the interior of an element is by using barycentric interpolation: n x(u)= φ i (u)x i, (1) i=1 where φ i (u) is the barycentric coordinate basis function, which satisfies n i=1 φ i(u) 1andφ i (u j )=δ ij. The above barycentric interpolation is simple and has many good properties, such as local support and linear reproduction. It however often generates non-smooth (first-order discontinuity) artifacts across the element boundaries, as shown in Figure 1 (right). To address the problem, we introduce an affine transformation at each control vertex, which can change the deformation gradient to be as close as possible to each other at the boundaries. This modified barycentric interpolation (modified-bi) is formulated as follows: n x(u)= φ i (u)(x i + M i (u u i )), (2) i=1 where M i is the transformation matrix associated with vertex u i to be determined. It is worth pointing out that the traditional barycentric interpolation is a special case of the modified-bi with all M i = 0. An important property of our modification is that, for any set of {M i }, it still follows the control values, i.e. n x(u k )= φ i (u k )(x i + M i (u k u i )) = x k, u k U. i=1 This property is critical for deformation application. By the linear reproduction and partition of unit property of the barycentric coordinate φ i (u), the modified-bi function x(u) doesn t change if we add an arbitrary local transformation T to all M i, shown as follows: i φ i (u)(x i +(M i + T)(u u i )) = x(u)+ i φ i (u)t(u u i ) = x(u)+t( i φ i (u)u i φ i (u)u i ) = x(u)+t(u u) = x(u). For later convenience, we arrange all M i continuously into a d d n dimension vector m, all control value x i into a d n dimension (3) 242

3 (a) Original (c) s = 1, θ = 60 (b) BI (d) s = 0.8, θ = 26 (e) s = 0.5, θ = 0 Figure 2: Function of the local transformation in 2D. (a) original image and the control mesh shown in red wires; (b) deformation result by manipulating the control points in the second outer loop and using barycentric interpolation (BI); (c) & (d) & (e): deformation results by adding a local transformation M to the central control point marked by a big white dot and using modified-bi. M takes the form M = s Rθ I2 2, where Rθ denotes a 2D rotation of angle θ. 4.2 vector x, and rewrite the modified interpolation function x(u) in the following matrix form: x(u) = A(u)m + B(u)x, In the literatures of fair curve/surface design, a simple but effective measurement for the vibration is the thin-plate energy [Moreton and Se quin 1992; Welch and Witkin 1992]. Take a parametric surface S(u, v) as example, the thin-plate energy is defined as: Suv + Svv )ds. Ethin (S) = (Suu (4) where A(u) and B(u) are respectively d (d d n) matrix and d (d n) matrix. A(u) s elements are u s second-order polynomials formed by the ui, φi (u) and u, while B(u) s elements are u s first-order polynomials formed by φi (u). Therefore, the second or- S 2 A(u) u u form a constant order-4 tensor, the first order B(u) derivatives u form a constant order-3 tensor, and the second 2 B(u) order derivatives u u are all zero S2 + S2 is actually the squared Frobenius norm Note that Suu uv vv of the Hessian matrix of S(u, v). der derivatives Inspired by the thin-plate energy, we measure the vibration of the 2 x(u) deformation function x(u) by its Hessian matrix u u as follows: x(u) 2 dτ = x(u) t. (7) Evibr (x) = u u Ω u u F t F t T In Equation (2) of modified-bi, Mi is a local transformation in the vicinity of vertex ui. The 2D examples in Figure 2 demonstrate how the local transformations affect the interpolation results. Only one local transformation is added at the center point of the control mesh for clarity. The visually best interpolation result is achieved in Figure 2(c). In the next section, we propose an optimization method to find the optimal local transformations that produce visually plausible smooth interpolation results. 4 Vibration Energy where F denotes the Frobenius norm, and t denotes the volume/area of tetrahedron/triangle t. Here we have taken advantage 2 x(u) of the property that the Hessian matrix u u is constant in the interior of each element. Optimization 4.3 Total Energy and Minimization We first introduce our energy terms used for optimizing the local transformations. The first one measures the gradient discontinuity (first-order) while the second one measures the vibration (secondorder) of the deformation. With the discontinuity energy and vibration energy defined in above ( Equation (7) and Equation (6)), we formulate the following total energy function for optimizing the deformation function x(u): 4.1 where α and β are two weighting coefficients. Etotal (x) = α Evibr (x) + β Edisc (x), Discontinuity Energy Now, the optimal local transformations Mi in the modified-bi can be obtained by minimizing the total energy Etotal (x): Recall that φi (u) are piecewise linear basis functions of u, the deformation gradient of x(u) may not be continuous across the boundary of two adjacent elements. Consider two adjacent elements, Ti and T j, which share a boundary i j. We measure the discontinuity by integrating the squared difference of the deformation gradient over boundary i j as follows: Ei j (x) = i j 2 x(u) i j x(u) j i dσ, F (8) (M1,..., Mn ) = arg min Etotal (x). m (9) Recall that the modified-bi remains unchanged when adding any arbitrary transformation to all the control vertices (Equation (3) in Section 3). We simply set the transformation of the first control vertex as zero to avoid the singularity in solving the above energy minimization problem. (5) where i j denotes the boundary at the side of Ti, j i denotes the boundary at the side of T j. Note that Etotal (x), Evibr (x) and Edisc (x) are all quadratic energies with respect to both m and x, since x(u) = A(u)m + B(u)x (see Equation (4)) and integration in Equation (6) and Equation (7) will eliminate u. Therefore, they have constant Hessian matrices, and we need only solve a sparse linear system when minimizing the total energy Etotal (x), which can be performed efficient enough for interactive applications because the coarse control mesh usually has a few hundreds of vertices ( in our examples). Summing Ei j (x) for all neighboring element pairs in Ω, we have the following energy term measuring the first order discontinuity of x(u): Edisc (x) = Ei j (x). (6) for all adjacent Ti and T j 243

4 Mean 1 (a) original image object StdDev (b) α = 100 Max Bar Dragon Pho Bunny Dinosor StdDev Standard Deviation. 2 Pho Phonograph. (c) α = 0.25 Table 2: Numerical results of the discontinuity energy. In each cell, the values from top to down are respectively values of barycentric interpolation, modified-bi and their ratio. (d) α = 0.01 Figure 3: Comparison of deformation results with various user specified α values in Equation (8). 4.4 tinuity energies Edisc (x) (see Equation (5)) for all shared boundaries on the coarse control mesh. For this twisting example, The Max/Mean/StdDev(Standard Deviation) of the energies for the barycentric interpolation and modified-bi, and their ratios are shown in the Bar column of Table 2. The energy ratios showed that the modified-bi reduced the discontinuity down to less than 1/150. The other columns in Table 2 for different object and deformation poses also demonstrate that our method improves the deformation results greatly. Weighting Scheme We can manipulate the importance of the vibration and discontinuity energy by adjusting the weighting coefficients in Equation (8). We first set α = 1 and automatically choose a value for β such that Evibr (x) and β Edisc (x) are comparable, i.e., their Hessian matrices have the same Frobenius norm. Then the user is allowed to manually adjust the value of α to balance between fairness and continuity of the interpolation result. In Figure 3 we use a 2D example to demonstrate the affections of different α values. As expected, when we increase α, the result is more like a BI result as shown in Figure 3(b) since BI have the smallest fairness energy (= 0); when we decrease α, the result becomes more smooth across the control mesh s element boundary as shown in Figure 3(c); but vibration occurs when α approaches to zero as shown in Figure 3(d). We empirically found that α = 0.2 and α = 0.1 work well respectively for 2D objects and 3D objects. 5 Implementation and Results We have implemented a deformation system using the modifiedbi for interpolation and several existing techniques for manipulating the coarse control mesh. We generate the coarse tetrahedron meshes for the target objects using an automatic mesh generator NETGEN ( We extrude the surface outward so that the generated tetrahedron mesh fully envelops the target object. To accelerate minimizing the total energy Etotal (x) of the modifiedbi, we precomputed a Cholesky factorization for the matrix of the resultant linear system. For the Stanford Bunny object, the precomputing step (calculating the Hesse matrix about m and x) takes about less than 1s, the run-time step takes about 4ms for solving the linear systems and 17ms for interpolating the fine mesh respectively. Taking advantage of the local support property of the modified-bi, we implement interpolation in GPU, which achieves about 6 9 times of acceleration. A detailed performance list is summarized in Table 1. Figure 4: Deformation results. Top: the original 3D model embedded in a coarse tetrahedron mesh; Middle: results by modified-bi; Bottom: results by barycentric interpolation, where apparent firstorder discontinuity artifacts exist. Figure 4 also compares the results with those of the traditional barycentric interpolation. These comparisons show that modifiedbi produces visually smooth warping results, while barycentric interpolation produces undesired crease artifacts due to its piecewise linear property. More deformation results on the Phonograph model are shown in Figure 5. In Figure 1 and Figure 4 are some deformation results of 3D objects. We embed the 3D object with coarse tetrahedron mesh, and run physically-based simulation method [Mu ller and Gross 2004; Huang et al. 2006a] to manipulate the control mesh. The target object embedded in the control mesh is deformed by using our modified-bi. In Figure 6 are some deformation results of the Dragon model. For this example, the coarse tetrahedron mesh is manipulated by regarding the tetrahedron control mesh as a volumetric graph and using the volumetric graph Laplacian [Zhou et al. 2005] representation. To relax the user from specifying the rotation Figure 1 is a typical twisting example. We computed the discon- 244

5 Model Bunny Dragon Bugman Phonograph Vertices 35 K 249 K 264 K 74 K Tetrahedrons Precompute the Cholesky factorization Control Vertices Solve for the local transformations 1 Cholesky Time 9.8ms 21.1ms 45.3ms 20.2ms 3 Interpolation in CPU 2 Solve Time 4.0ms 2.0ms 14.0ms 1.92ms 3 CPU Time 17.0ms 141.1ms 120.0ms 35.3ms 4 GPU Time 3.3ms 18.3ms 16.3ms 3.9ms 4 Interpolation in GPU Table 1: Performance list with model sizes. The performance is tested on a desktop PC with a 2.8GHZ Intel Pentium IV CPU, 1GB RAM and a GeForce 7900 graphics card. Figure 5: More deformation results of Phonograph model. information for the deformation handles, we use the iterative method [Huang et al. 2006c] to solve for the deformation. The target object is also deformed by using our modified-bi. Figure 7: Deformation by dragging a few control vertices. On top left is the original 3D model with the control mesh, and the others are deformation results. The small balls represent the positions of the dragged control vertices. (a) (b) (c) Figure 8: Image object warping with boundary constraints. (a) Original image object, (b) Warping result by our modified-bi, (c) Warping result with boundary constraints. M1,..., Mn = arg min Etotal (x) + γ E pos (x), x1,..., xn m,x Figure 6: Deformation results of the Dragon model. The original model is shown with the wireframe of control mesh. The others are two views of a deformation result. where γ is weighting coefficient. We automatically determined the value of γ, such that the Hessian matrices of γ E pos (x) and Etotal (x) have the same Frobenius norm. We have implemented both methods, which produce almost undistinguished results. Interactive Coarse Mesh Control It is often desired to enable the user to interactively control the coarse mesh vertices. Though the number is very small, it is still too much for users to manipulate all of them. And some internal vertices are very difficult to select and manipulate. Fortunately, we can automatically optimize the positions of those free vertices using the same energy function in Equation (8). Because both Etotal (x) and E pos (x) are quadric functions of the x, the above minimization problem can be also solved in the same way as solving the minimization problem in Equation (9). Figure 7 shows some deformation results achieved by dragging only a very few control vertices and minimizing the deformation energy in Equation (11). This technique gets rid of the requirement of an external coarse mesh deformation technique and enables the user to interactively manipulate individual control vertices. Let {x ci }ki=1 be the manipulated vertices, {x f j }rj=1 be the free vertices, and E pos (x) = ki=1 xci x ci 2. Then the free vertices can be obtained by solving one of the following two energy minimization problems: M1,..., Mn = arg x f1,..., x fr min r m, {x fi }i=1 2D Image Object Deformation The energy minimization framework in Equation (9) and Equation (11) also works for 2D image objects covered by planar triangular control meshes. Some examples are shown in Figure 3, Figure 8, and Figure 9. In these examples, we manually draw the control meshes to cover the 2D objects. Etotal (x), s.t. xci = x ci, 1 i k. (11) (10) 245

6 References ALEXA, M Differential coordinates for local mesh morphing and deformation. The Visual Computer 19, 2, BARR, A. H Global and local deformations of solid primitives. In SIGGRAPH, BEIER, T., AND NEELY, S Feature-based image metamorphosis. Computer Graphics 26, 2, BOTSCH, M., AND KOBBELT, L An intuitive framework for real-time freeform modeling. ACM Trans. Graph. 23, 3, Figure 9: Examples of 2D image object deformation. On top left is the original image with coarse control mesh, and the others are deformation results. The small red balls represent the deformation handles. In Figure 8, the user intends to elongate the human face by dragging several control points. As shown in Figure 8(b), the straight line of the top boundary is bent. Such bending is caused by the local transformation for minimizing the gradient discontinuity and vibrations. To void such undesired bending, we constrain the local transformations of the outmost control vertices to be zero when minimizing E total (x) in Equation (8). The result of such constrained method is shown in Figure 8(c), where the bending is effectively avoided. Figure 9 shows several 2D deformation results, obtained by dragging only a few (4 6) control vertices, and solving the energy minimization problem described in Equation (11). 6 Conclusion and Future Work We have presented an efficient deformation method based on a novel interpolation technique modified barycentric interpolation (modified-bi). The experimental results have demonstrated that modified-bi is very successful in 2D image and 3D shape deformation, Using the modified-bi, extremely coarse mesh can be used to greatly reduce the simulation/deformation cost. The major limitation of modified-bi is that it requires solving a linear system for transformation optimization, but this is not critical since it is performed on the coarse control mesh. An avenue for future work is to explore more efficient solvers for the local transformations. Another limitation is that the continuity of the modified- BI is not theoretically proved though numerically validated in experiments. In future, we will intensively study the continuity problem of the modified-bi. One possibility is to take advantage of the tetrahedron subdivision method[schaefer et al. 2004]. We will also explore more deformation methods by taking advantage of the modified-bi technique, and develop more control techniques and support more user constraints. 7 Acknowledgments We would like to thank the reviewers for their valuable comments. This work is supported in partial by the 973 Program of China (No.2002CB and No.2006CB303102), NSFC (No ), the Program for New Century Excellent Talents in University of China (No. NCET ) and the National High Technology Research and Development Program of China (No. 2007AA01Z336). BOTSCH, M., AND KOBBELT, L Real-time shape editing using radial basis functions. Computer Graphics Forum 24, 3, COHEN-OR, D., SOLOMOVIC, A., AND LEVIN, D Threedimensional distance field metamorphosis. ACM Trans. Graph. 17, 2, COQUILLART, S Extended free-form deformation: a sculpturing tool for 3d geometric modeling. In SIGGRAPH, DER, K. G., SUMNER, R. W., AND POPOVIĆ, J Inverse kinematics for reduced deformable models. In ACM SIG- GRAPH, FENG, J.,MA, L.,AND PENG, Q A new free-form deformation through the control of parametric surfaces. Computers & Graphics 20, 4, FLOATER, M. S Mean value coordinates. Comput. Aided Geom. Des. 20, 1, HSU, W. M., HUGHES, J. F., AND KAUFMAN, H Direct manipulation of free-form deformations. In SIGGRAPH, HUANG, J.,LIU, X.,BAO, H.,GUO, B.,AND SHUM, H.-Y An efficient large deformation method using domain decomposition. Computers & Graphics 30, 6, HUANG, J.,SHI, X.,LIU, X.,ZHOU, K.,GUO, B.,AND BAO, H Geometrically based potential energy for simulating deformable objects. The Visual Computer 22, 9, HUANG, J.,SHI, X.,LIU, X.,ZHOU, K.,WEI, L.-Y.,TENG, S., BAO, H.,GUO, B.,AND SHUM, H.-Y Subspace gradient domain mesh deformation. ACM Trans. Graph. 25,3. IGARASHI, T.,MOSCOVICH, T.,AND HUGHES, J. F Asrigid-as-possible shape manipulation. ACM Trans. Graph. 24, 3, JOSHI, P., MEYER, M., DEROSE, T., GREEN, B., AND SANOCKI, T Harmonic coordinates for character articulation. ACM Trans. Graph. 26, 3, 71. JU, T., SCHAEFER, S., AND WARREN, J Mean value coordinates for closed triangular meshes. ACM Trans. Graph. 24, 3, KHO, Y., AND GARLAND, M Sketching mesh deformations. In Proceedings of the symposium on Interactive 3D graphics and games, LIPMAN, Y.,SORKINE, O.,LEVIN, D., AND COHEN-OR, D Linear rotation-invariant coordinates for meshes. ACM Trans. Graph. 24, 3,

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