Stretch-based Tetrahedral Mesh Manipulation

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1 Stretch-based Tetrahedral Mesh Manipulation Wenhao Song Computer Graphics Department Max-Planck-Institut Informatik Saarbrücken, Germany Ligang Liu Department of Mathematics State Key Lab of CAD&CG Zhejiang University, China Figure 1: Examples of free form deformation (FFD). The object is embedded in the blue tetrahedral grids (leftmost) and is deformed by manipulating the tetrahedral grids. ABSTRACT We present a novel least scaling distortion metric to measure the deformation distortion for tetrahedral meshes. The stretch-like metric is a combination of Jacobian matrix norm and tetrahedron volume and has the properties of good shape preservation and rotation invariance. Based on our metric, we propose a uniform non-linear optimization solution to a variety of tetrahedral mesh manipulation applications including shape deformation, interpolation, deformation transfer, and deformation learning. Our approach can produce volume preserving and flip free tetrahedral mesh results, which performs much better than the previous tetrahedral manipulation approaches. We also demonstrate an efficient and practical application using free-form deformation technique. The object is embedded in a rough control tetrahedral mesh and deformed by editing the tetrahedral mesh with various constraints. Each vertex of the object can be obtained by its barycentric coordinates according to its embedding tetrahedron of the control tetrahedral mesh. CR Categories: J.6 [Computer-Aided Engineering]: Computeraided design Shape manipulation; Keywords: tetrahedral mesh, shape manipulation, volume distortion metric, free-form deformation 1 INTRODUCTION Shape manipulation plays a central role in computer modelling and animation. Good manipulation tools must be intuitive, easy to use, and robust. Most importantly, they should provide physically plausible and natural looking resultant models. As it is quite difficult to quantify the natural look of manipulated models, it is generally believed that the models should preserve as much as possible their local and global shape properties, e.g., volumes, distances, angles, etc, with minimal distortion. In some engineering applications like manipulating volumetric data, it is required to deform the shape while preserving its volume. In order to preserve shape volume, it is generally necessary to tetrahedralize the domain [9, 14]. And tetrahedral meshes are needed in some applications using finite element method [6, 2]. On the other hand, classical approaches for deforming tetrahedral mesh employ physical-based simulation such as mass spring model [16] which is linear but can not preserve the tetrahedron quality. The quality of the deformation should be measured by some geometric stretch. However, the geometric stretch for tetrahedral meshes has not well studied in the literature. Geometric stretch has been used for surface parameterization and texturing mapping community [19]. But the volume stretch metric generalized from 2D stretch metric is nonlinear and extremely difficult to be solved. In this paper, we propose a novel metric, called least scaling distortion (LSD) metric, to approximate the geometric stretch metric. The LSD metric is rotation invariant, but is dependent on scaling, which can preserve the volume constraints. Based on the new metric, we present a framework for manipulating the tetrahedral meshes including shape editing, shape interpolation, deformation transfer, and deformation learning. Furthermore, the shape manipulation can be performed in a fast way based on free-form deformation technique. The object is first embedded into a sparse control tetrahedral mesh. Then the object can be deformed by editing the control tetrahedral mesh using our approach. Our main contribution is a unified framework of tetrahedral mesh manipulation with least distortion. An efficient solver is proved to be available for solving the nonlinear optimization problem. Furthermore, all the solutions can be applied to triangular meshes, as we can measure the distortion of the deformation gradient as in [24, 25]. 2 RELATED WORK Free form deformation (FFD) is an important technique in shape modeling. Various FFD techniques have been proposed during the past two decades [20, 5]. Energy minimization based approaches [26] and multiresolution editing approaches [12] are also popular

2 for shape deformation. Mesh editing systems based on differential coordinates have been proposed recently [23, 27]. Some approach guides rotation deformation by harmonic scalar fields [28] or by encoding higher derivatives [15]. Some of the above approaches cannot handle large deformation while the others cannot handle distortion caused by large movements of handles. Only a few papers study the deformation of tetrahedral meshes in the literature. Volumetric constraints by sampling points inside the triangular mesh and simulating the volumetric graph Laplacian was introduced in [30]. But this approach failed to keep the tetrahedron quality and to avoid tetrahedron flips. Huang et al. use the mean value coordinate to deform the shape and introduce nonlinear hard constraints to preserve shape volume [9]. Recently, Lipman et al. [14] proposed an approach for triangular mesh deformation that can preserve local volume but tends to generate shrinkage near the bent or twisted part. To improve the quality of the tetrahedron, the approach of [8] optimize the mesh connectivity by flipping edges and faces of the tetrahedra in an iterative way. Other approaches use various distortion metrics to optimize the tetrahedral meshes [4, 7, 11, 17]. The work of [11, 17] adopt the Jacobian matrix norm, which measures the shape variation between two tetrahedra, in the distortion metric. It is still a challenge to find more efficient geometric metric for tetrahedra shape manipulation. 3 TETRAHEDRON SHAPE METRICS A tetrahedral mesh M consists of a finite set of vertices and tetrahedron elements. Throughout the paper M denotes either the set of vertices or the set of tetrahedron elements if there is no ambiguity in the context. 3.1 Jacobian matrix norm Each tetrahedron element e is defined by four vertices [v 1,v 2,v 3,v 4 ]. We define the incidence matrix A R 3 3 of e as: A(e) := [v 2 v 1,v 3 v 1,v 4 v 1 ]. That is, the incidence matrix is obtained by computing the edges of the element emanating from the first vertex in the coordinate list and concatenating them into a square matrix. The volume of the tetrahedron e is related to the determinant of A(e), which can be positive or negative depending on the labeling of the vertices. We assume throughout that all the vertices are labeled according to the right-hand rule so that the determinants of all the incidence matrices are positive for a tetrahedral mesh. A linear map f : e ē can be determined for two given tetrahedron elements e and ē with nonzero volumes. The map f has the Jacobian matrix J = A(ē)A 1 (e). This linear transformation f measures the deviation of the element ē from the element e. If two tetrahedron elements e and ē have the same shape, i.e., their edges are proportional, then for some σ > 0. A(ē) = σa(e), A(ē)A 1 (e) 2 = 3σ 2, F ( ) det A(ē)A 1 (e) = σ 3, 3.2 Volume stretch metrics The volume stretch metric is defined as the following [31]: L 2 (J) := 1 λ1 2 J F = + λ λ 3 2, (1) 3 3 where J F = tr (J T J) is the Frobenius norm, λ 1,λ 2,λ 3 are the singular values of J. This stretch was used as the texture stretch metric in its 2D form in [19, 22, 29]. It is easy to see that the volume stretch metric is invariant to the rotation of the tetrahedron but depends on the scaling of the tetrahedron. The inverse mean ratio metric was introduced by [11] as L i (J) := L 2(J) [det(j)] 2/3. However, both volume stretch metrics can not preserve the shape volume when used in deforming the tetrahedral mesh. 3.3 Least scaling distortion metric To preserve the shape volume as much as possible, we introduce a novel volume metric for tetrahedral manipulation. The volume of a tetrahedron is σ = [det(j)] 1/3. Inspired by the work of [17], we look for a metric with the form of L a,b := 1 2 ( L 2 σ + L a 2 ) where σ b 0 b < a 3 according to the positive Hessian matrix requirement. The parameters a and b should satisfy a + b = 4 if we set the volume ratio contract to one in the metric L a,b. Thus, we propose a novel volume metric, called least scaling distortion (LSD) metric, as following: L s (J) := L 2(J) 2det(J) + L 2(J) 2[det(J)] 1/3, by setting a = 3 and b = 1. When the tetrahedron element and its deformed tetrahedron element with nonzero volume have the same shape, i.e., A(ē) = σa(e),σ > 0, then L s (J) = 2 1 ( 1 σ + σ), which reaches its minimum at σ = 1. That means the LSD metric depends on the volume ratio. The volume of the tetrahedral mesh will be kept as much as possible during the manipulation. The range of the LSD metric is between two and infinity, where a value greater than two means that the tetrahedron element and its deformed tetrahedron element have different shapes. Therefore, the LSD metric for tetrahedral elements is invariant to translating and rotating the input values, but related to scaling. 4 LEAST DISTORTION OPTIMIZATION PROBLEM 4.1 Optimization problem Suppose the tetrahedral mesh M x is deformed into another mesh M y by the mapping function F : M x M y. Each tetrahedron element e T x is deformed into the corresponding element ē T y using the linear map with the Jacobian matrix J e = A(ē)A 1 (e). Some vertices on M y are usually fixed or constrained in space, while the other vertices are unrestricted. The optimization problem to minimize the average squared LSD metric over the entire mesh is then minl s (F) = 1 M y M x L s (Je 1 ) e, (2) e M x s.t. det(j e ) > 0, e M x, (3) v i = v i, i C, (4)

3 where e denotes the volume of the tetrahedron element e, M x = e Mx e is the volume of mesh M x, v i denotes the fixed vertices, and C is the index set of the fix vertex constraints during the mesh manipulation. This objective function in Eq. 2 minimizes the volume distortion of the whole tetrahedral mesh. As a consistent orientation for all the tetrahedron elements is required during the manipulation, adding the volume constraints in Eq. 3, i.e., the strict inequalities in the optimization problem, ensures a consistent orientation in the result tetrahedral mesh. Thus the tetrahedron elements will not flip during the manipulation. A similar optimization problem was also constructed for optimizing triangular and tetrahedral meshes using the inverse mean ratio metric in [17]. But the inverse mean ratio metric is dimensionless and scale invariant, which makes it unable to process mesh manipulation. 4.2 Unconstrained minimization problem The consistent orientation constraints in Eq. 3 are problematic because they involve a strict inequality. As the tetrahedral mesh is connected, it can be seen from the optimization problem in Eq. 2 that the objective function L s (F) approaches infinity if the volume of one tetrahedron element converges to zero [17]. In order for the objective function to be bounded away from infinity, all of the individual tetrahedron element functions must be bounded away from infinity. Therefore, the consistent orientation constraints can be dropped from the optimization problem as: min M y L s (F), (5) s.t. v i = v i, i C. In this case, the objective function is defined to have a value of + whenever the volume of at least one tetrahedron element is nonpositive. With this reformulation, we must provide a starting point where the orientation constraints are satisfied. 4.3 Numerical solution We now formulate the mesh manipulation as an optimization problem associated with the objective function in Eq. 5. Once the constrained vertices are fixed and removed, we have an unconstrained optimization problem with an objective function that is twice continuously differentiable on an open set. The characteristics of the objective function demand different solution techniques, and we briefly describe the method used in our system. We utilize a fast algorithm presented in [17] where a similar objective function was minimized using an inexact Newton method [10] with an Armijo line search method [3]. Denote S the objective function after removing the fixed vertices from L s (F) in Eq. 5. Given x k, a direction d k can be obtained from the symmetric linear system: 2 S(x k )d k = S(x k ), by using a conjugate gradient method with a block Jacobian preconditioner [18]. If S(x k ) T d k ρ d k 2, 2 for a positive constant ρ > 0, then, S(x k ) is used as the steepest descent direction. Using the Armijo line search method [3], a smallest nonnegative integer m is determined by satisfying S(x k + β m d k ) S(x k ) + µβ m S(x k ) T d k, where 0 < µ < 1 2 and 0 < β < 1 are constants. Therefore, the new position is iteratively updated by x k+1 = x k + β m d k. The algorithm terminates when S(x k ) 2 < Note that during the iteration, if the volume of one tetrahedron element is smaller than a given tolerance ε = 10 14, we consider the consistent orientation conditions to be violated, and the objective function is set to + in the line search. 5 TETRAHEDRAL MESH MANIPULATION We illustrate our technique for tetrahedral mesh manipulation in this section. Different tetrahedral mesh models are used in our tests as shown in Fig. 2. (a) (c) Figure 2: Tetrahedral meshes used in our paper shown in cut views. (a) Bar; (b) dragon; (c) bunny; (d) gear. 5.1 Mesh deformation An editing interface like in Laplacian mesh editing [23] is used to manipulate a tetrahedral mesh in our system. The user first selects some vertices on the surface of the mesh as fixed anchors and defines the region of interest (ROI) for editing. Then, the user manipulates the handle that can either be one vertex or a group of vertices inside the ROI, and the tetrahedral mesh is reconstructed with respect to the relocation of the handle. The fixed anchors and the user handle are regarded as the constraints in the minimization problem in Eq. 5. The other vertices of the deformed tetrahedral mesh are solved by the optimization process. Fig. 3 shows various results of manipulating a bar tetrahedral mesh. The top row shows the editing results using the inverse mean ratio (IMR) metric [17]. The bottom row is the editing results using the LSD metric proposed in our paper. The original bar mesh with the fixed vertices in blue and the handle vertices in red are shown in (a). The user manipulates the handle vertices to elongate (b), squash (c), bend (d), or twist (e) the mesh respectively. It is easily seen that (b) (d)

4 Using the inverse mean ratio (IMR) metric Figure 5: Isotropic (Top row) and anisotropic (Bottom row) scaling the bunny mesh locally. (a) Using our least scaling distortion (LSD) metric (b) (c) Figure 3: Examples of manipulating a bar tetrahedral mesh. our approach has achieved much better results which are very nice and physically plausible even for large deformation manipulations. Fig. 4 shows various results of manipulating a gear tetrahedral mesh. We see that our deformation approach can preserve the sharp features of the shape. (d) (e) assume that M 0 and M 1 are compatible and each corresponding tetrahedron pair (s 0,s 1 ),s 0 T 0,s 1 T 1, has a consistent orientation, i.e., det(a(s 0 )A(s 1 )) > 0. Compatible tetrahedral meshes can be obtained by remeshing techniques [13, 1]. Alexa et al. proposed an as-rigid-as-possible morphing technique that blends the tetrahedral meshes by interpolating each individual transformation between each pair of tetrahedron [1]. But this method can not avoid the tetrahedra flips in the intermediate meshes. Let F 0t : M 0 M t and F 1t : M 1 M t be the piecewise linear maps between the tetrahedral meshes. As the LSD metric is rotation invariant, the intermediate mesh M t can be solved by the following optimization problem: min M t (1 t)l s (F 0t ) +tl s (F 1t ). (6) Figure 4: Examples of manipulating a gear tetrahedral mesh. Furthermore, the deformed meshes using our approach can almost preserve the mesh volume as the LSD metric is relevant to the scaling. If we want to control the scaling and shearing effects during the manipulation, a modified LSD metric can be used in the optimization problem as: L s (JU 1 ), where U = diag(λ 1,λ 2,λ 3 ),λ 1,λ 2,λ 3 > 0, is a positive definite diagonal matrix. Fig. 5 shows the results of isotropic and anisotropic scaling the bunny mesh locally by setting different U. The blue vertices on the surface are fixed and the other vertices are free to move. 5.2 Mesh interpolation Given two tetrahedral meshes M 0 and M 1, mesh interpolation constructs an intermediate tetrahedral mesh M t for t [0,1]. Here we Fig. 6 shows some interpolation examples between different tetrahedral meshes. Fig. 6(a) shows the morphing sequence between a bar mesh and a bent and twisted mesh with t = 0.1. The vertices in blue color are fixed as positional constraints during the optimization process in this example. Fig. 6(b) and (c) show the morphing sequences between two dragon meshes and between a bunny model and a sphere model with t = 0.2 respectively. These examples demonstrate that linear interpolation of the LSD metrics in Eq. 6 has obtained perfectly uniform transformation, e.g., uniform bending, twisting, and shearing transition from the source mesh to the target mesh. Our interpolation approach preserves the shape features and obtained high quality intermediate meshes. 5.3 Deformation transfer Deformation transfer aims to transfer the deformation exhibited by a source triangle mesh onto a different target triangle mesh [24]. We use tetrahedral meshes and represent the source deformation as a collection of Jacobian matrices that captures the linear transformations which individual tetrahedrons undergo relative to the source mesh. Given a source tetrahedral mesh X and its deformed mesh X, we want to generate a new mesh Y for the given target mesh Y such that the relationship between Y and Y is analogous to the relationship between X and X. For simplicity, we suppose X and Y are compatible tetrahedral meshes. Let G : X X and H : Y Y. We hope H = G for each corresponding element pair. Therefore, the following optimization problem should be solved min Y L s (HG 1 ), (7)

5 (a) (b) (c) Figure 6: Interpolation sequence between different tetrahedral meshes. to obtain the deformed target mesh Y. Fig. 7 demonstrates our method used to transfer different deformations of a bar mesh onto other bar meshes with various thickness. Top row shows the results of transferring bending effects: bending effect of the example (a) is transferred to a thicker bar (b) and to a thinner bar (c) respectively. Bottom row shows the results of transferring twisting effects: twisting effect of the example (d) is transferred to a thicker bar (e) and to a thinner bar (f) respectively. Note that the thicker mesh is more rigid in the bending direction as it is bent much less by transferring while the thinner mesh is softer in the bending direction as is bent more by transferring, see top row of Fig. 7. Similar effects occur in twist examples in the bottom row of Fig. 7. Our approach deformation transfer has obtained physically plausible results. (a) (b) (c) (d) (e) (f) Figure 7: Examples of transferring different deformations of a bar mesh onto other bar meshes with various thickness. 5.4 Deformation learning The goal of deformation learning is to position a small number of handle vertices and produce a meaningful overall deformation from a set of given example meshes to indicate the class of meaningful deformations [25]. We also use tetrahedral representation for meshes. Each example is represented with a a collection of Jacobian matrices that captures the linear transformations which individual tetrahedrons undergo relative to a reference pose. Our approach searches among all meshes with specified vertex positions to find the one that has the least LSD metric difference with the given example meshes. Given a deformation example of the source tetrahedral mesh X as G : X X. We want to edit the target tetrahedral mesh Y so that the style of the deformation H : Y Y is analogous to the style of the deformation G. If n deformation examples G i : X i X,i = 1,2,,n, are given, i.e., X i are meaningful deformed shapes of X, we could edit Y by learning from all the examples. For each tetrahedron element X and its corresponding element Y, the Jacobian matrices are respectively represented by G i and H. We decompose the Jacobian matrix G i into scale/shear and rotational components using polar factorization [21]: [ G i] 1 = S i R i (8) Like in the paper [25], we blend the example shapes by using the matrix exponential and logarithm functions: [ [G(ϖ)] 1 n n ] = ϖ i S i exp ϖ i log(r i ), i=1 i=1 where ϖ = (ϖ 1,ϖ 2,,ϖ n ) is the weight vector. As LSD metric is rotation invariant, the rotation part could actually be removed from the metric. Thus the following optimization problem could be solved to obtain the edited shape Y : ( [ n ]) i S i=1ϖ i minl s Y H. (9)

6 It can be proved that the Hessian matrix 2 ϖ i ϖ j L s (Y ) is positive definite. Thus we solve the above optimization problem by using the following Hessian matrix [ ] 2 yi y j L s (Y ) deformed result using our learning approach by twisting the top side with a larger angle is shown in (e). 2 ϖ i ϖ j L s (Y ) which is used to simultaneously search the steepest descent direction of the geometric coordinates Y and the blending weights ϖ. Fig. 8 demonstrates deformation learning results from a bent deformation example shown in brown color in (a). Two example poses of bar are shown in (a). The bottom of the unbent bar is constrained to stay in place while the vertices on the top side are moved. Fig. 8(b) shows the deformed result using the deformation technique in Eq. 5, which is not reasonable. The deformed results using MeshIK learning approach [25] and our learning approach are shown in (c) and (d) respectively. Our approach obtains the better result as the shape near the handle vertices in (c) is much distorted. ( a ) ( b ) ( c ) ( d) Figure 8: Deformation learning from a bent deformation example. Fig. 9 shows the result of deformation learning from one bent and twist deformation example shown in brown color in (a). The bottom of the unbent bar is constrained to stay in place. The vertices on the top side are manipulated with a rotation movement. The deformed result using MeshIK learning approach [25] shown in (b) is rather bad as the bent result is affected by the twist information in the example. Our approach produces a more reasonable result shown in (c). This example demonstrates that our learning approach can extract bending information from the example with complex hybrid deformation. ( a ) ( b ) ( c) Figure 9: Deformation learning from a bent and twisted deformation example. Fig. 10 demonstrates that our learning approach can extract twisting deformation from complex twisting example. Two example poses of bar are shown in brown color in (a) where the two side of the unbent bar are twisted in two different directions. The deformed result using the deformation technique in Eq. 5 shown in (b), where the twist occurs uniformly along the bar. The approach of MeshIK learning [25] does not wok for this case, see (c), while our learning approach obtain a reasonable twist result shown in (d). Note that the bottom part of the bar is not twisted as much as the upper part as the bottom of the bar is constrained to be fixed. The ( a ) ( b ) ( c ) ( d ) ( e) Figure 10: Deformation learning from a bent and twisted deformation example. 5.5 tetrahedral free-form deformation A tetrahedral mesh with a large number of tetrahedra will cause much computation cost. We use the technique of free form deformation(ffd) to deform the shape effectively. We can first embed the shape in a sparse control tetrahedral mesh and then get its deformation by editing the control tetrahedral mesh. Each point inside the control tetrahedral mesh can be computed by the barycentric coordinates of its embedded tetrahedron. Fig 1 demonstrates various deformation examples. The original model is embedded by a sparse control tetrahedral mesh that is shown in blue lines. Then the object is deformed to different shapes by editing the control tetrahedral mesh in different ways. It is shown that the deformed objects are smoothly changed and have no shrinkage even in large bending and twisting. 6 DISCUSSION All the techniques presented in this paper are implemented and tested on a 1.8G Intel computer with 1G memory. The main computational core of our system is solving the optimization problem. We apply an efficient implementation of an inexact Newton algorithm [10] using the conjugate gradient method with an Armijo line search method [3] to solve the unconstrained optimization problem. We could also linearize the optimization process as in [25] to speed up the computation for large meshes. However, it is still a far way to manipulate the tetrahedral meshes using our approach in real time. In each iteration of the inexact Newton method, it involves solving the linear system by the conjugate gradient. Thus, one iteration time for the inexact Newton method varies according to the iteration inside the conjugate gradient. We record the range of the one iteration time related to the number of the iterations inside the conjugate gradient in Table 1. Usually, our algorithms converge in 4 8 steps according to the step size of the manipulated points. As to deformation learning, it involves extra weight Hessian matrix evaluation, where the time is multiplied with the increase of the examples meshes. Table. 2 shows the geometric stretch of the examples shown in the paper. The stretch metric of the deformed tetrahedral mesh is measured as e L 2 (J)det(J)/ e det(j). From Table. 2, it is shown that our approach preserves the tetrahedron quality and is better than both the IMR metric and MeshIK method. 7 CONCLUSIONS We introduce a novel least scaling distortion metric for tetrahedral mesh manipulation and achieve good results. The mesh manipulation is formularized as a nonlinear optimization problem based

7 Example #Vertex #Tet #IN(sec) #CG Bar(Fig. 3) Bunny(Fig. 5) Dragon(Fig. 6(b)) Fig. 6(c) Fig Fig Table 1: Timing statistics of the examples shown in the paper. #Vertex: the number of vertices; #Tet: the number of tetrahedra; #IN: the computation time in second of each iteration of inexact Newton method; #CG: the iteration time of the conjugate gradient method. Example (b) (c) (d) (e) Fig 3(IMR) Fig 3(LSD) Fig Fig Fig Table 2: Geometric stretch for tetrahedral mesh deformations in our paper. on the LSD metric which can be solved using an inexact Newton method with an Armijo line search method. All deformation techniques of triangular meshes are implemented on tetrahedral meshes in a unified framework. We demonstrate the superiority of our approach over other approaches by many experimental results. The main weakness of our approach is the speed of the solver as the optimization problem is nonlinear. 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