Gradient-based Shell Generation and Deformation

Transcription

1 Gradient-based Shell Generation and Deformation Jin Huang Qing Wang Xinguo Liu Hujun Bao State Key Lab of CAD&CG, Zhejiang University Haiyang Jiang ABSTRACT Shell becomes popular in a variety of modeling techniques for representing small-scale features and increasing visual complexity. Current shell generation algorithms do not measure the volume distortion for geometric texture mapping. And when the object deforms, it is very challenging for existing algorithms to wrap the space inside the shell without large artifacts. We propose an approach to these problems by minimizing the difference between the deformation gradient of the space in the shell and the rotation component of it. Because the shell space is warped as rigid as possible, small features embedded in the shell can be preserved well. Furthermore, our algorithm can achieve shell like object deformation. We introduce a novel hierarchical dimension reduction method to solve the involved non-linear optimization problem efficiently. Finally, several examples are presented to demonstrate the usefulness of our algorithm. Keywords shell, mesh, deformation gradient 1. INTRODUCTION Shell is defined as the area between two nearly parallel surfaces of the same topology. Many recent techniques rely on a shell for modeling highly complex geometric details and surface appearances [20, 14, 21, 15, 25], which extend the concepts of conventional texturing and displacement mapping. Given a base surface S b, they first generate a shell surface S η by duplicating S b and displacing it by a distance η. Then construct the shell between S b and S η by connecting the corresponding vertices, which forms a prism tessellation of the shell space. Some techniques further tessellate the prisms into tetrahedrons [16] for creating a continuous piecewise linear map between the shell space and the texture space, called shell map. There are various choices for the displacement direction, such as normal director [6] and gradient director based on a generalized distance function [14]. There are two major limitations of the current shell generation algorithms. First, the shell distortion is not taken into account. The Correspondence authors prisms in texture space are all normal prisms (i.e. the side facets are all rectangles and perpendicular to the bottom/top triangles), while the prisms of the shell space are generally not. Consequently, some distortion errors exist in the shell map, which should be minimized for preserving the geometric features when mapped into the shell. The algorithm in [25] minimizes the stretch of the shell map for a given shell though adjusting the texture coordinates, but the shell itself is not optimized. Second, deformable objects are not addressed. The visual complexity for dynamically animated objects can be increased by extending the shell based modeling techniques, if we have deformation techniques for shells on deformable objects. In this paper, we introduce gradient-based method to address the above problems. First, we extrude some ideal prisms (normal prisms) from the base surface, and stitch them into a seamless shell. A curvature sensitive technique for creating the ideal prisms is used to improve the shell s quality. Though some surface deformation methods may be extended for shell deformation, such as [22, 24, 3], they lack a measurement for volumetric distortion and consequently cannot produce satisfactory results. In physically based deformation, Green strain which is widely used as the distortion measurement. However, due to the highly non-linear property of the Green strain, physically based method often need to update the stiffness matrix at each iteration step. Hence such methods are too expensive. Instead, We measure the distortion by shearing and stretches in the deformation gradient, and formulate a nonlinear least square equation which can be easily solved. Different to most Laplacian based mesh deformation algorithms, our distortion measurement can be used to drive thin shell effect deformation. We develop a hierarchical dimension reduction technique to improve the performance and stability of the shell effect deformation algorithm on large meshes. We solve the deformation results from the coarsest level to the finest level (original unreduced mesh), which is similar to the fast multigrid algorithm [17], but designed for a non-linear problem. 2. RELATED WORK Shell Based Modeling There are many shell based modeling and rendering techniques extending volumetric texturing and displacement mapping. Chen et al. introduce shell texture function describing irradiance fields based on pre-computed fine scale light interactions such as meso-structure shadowing and subsurface scattering [5]. Wang et al. use a shell to extend the traditional displacement map to view-dependent displacement map (VDM) [20] and generalized VDM [21] for rendering convincing small geometric details. In [15], a GPU based shell space ray tracing algorithm is proposed for real time rendering of relief textures. Zhou et al. build

2 a low-distortion shell map for synthesizing high quality geometric texture over a base surface [25]. Mesh Deformation Our work is closely related to many mesh deformation techniques which preserve the fine scale surface details. Gradient domain techniques preserve surface details by maintaining the length of the differential coordinates or curvatures [22, 18, 12]. Such techniques deform an object by manipulating the orientation of the differential coordinates and adding some positional constraints, followed by mesh reconstruction via a Poisson equation. Various techniques have been proposed for manipulating the orientation, including distance based propagation [22, 24], harmonic guidance based interpolation [23], and rotation-invariant representation [13]. In [10], the orientation is automatically optimized, which eliminates some manual interactions. Our work is also related to the as-rigid-as-possible shape interpolation [1] and deformation transfer techniques [19], which solve for the deformed shapes based on triangle-to-triangle transformatiosn, known as deformation gradients. They treat the target transformations for the deformed triangles as known as a prior. Consequently they can obtain the deformation results by solving linear systems. Recently a surface deformation technique based on shape matching is proposed in [3], which augments the surface meshes with rigid prisms and minimizes the elastic energy between neighboring prisms. Though it uses some prisms, it is not suitable for shell deformation, since it dose not measure the volumetric distortions at all. Shell deformation has been studied in [9], which only considers thin shell effect surface deformation. We deal with the space inside shells in this paper. 3. VOLUME DEFORMATION Our algorithm is an energy-based deformation method[4]. We first formulate an energy function for deforming volume objects including shells. Let Ω be an object in R 3, the deformation of Ω can be represented as a function x(u) : Ω R 3. Given an infinitesimal volume dτ centered at a point u Ω, and let x(u) = UσV t be the singular value decomposition (SVD) of the deformation gradient x(u), where σ is a diagonal matrix formed by the singular values σ 1,2,3, and U and V are respectively the left and right eigen matrices. Then, the stretch distortion of the infinitesimal volume can be measured by Σ 3 i=1 (σ i 1) 2 = σ I 2 F = U(σ I))V t 2 F = UσV t UV t 2 F = x(u) ρ( x(u)) 2 F, (1) where F denotes the Frobenius norm, ρ( x(u)) denotes UV t that is the rotation part of x(u) in polar decomposition. Integrating the stretch distortion error over the object, we have the following energy function for deformation: E = x(u) ρ( x(u)) 2 F dτ (2) Ω Since ρ( x(u)) are pure rotations, minimizing the above energy will constrain the deformation gradients to be rotational transformations as closely as possible, which in turn gives as rigid as possible deformation results. Botsch et al. also proposed a distortion measurement in terms of volume change [2], which is however not sufficient and requires additional regularization constraints to eliminate extra degree of freedoms. Our distortion measurement overcomes such a problem, and as rigid as possible results are often better than volume invariant results. For example, using our measurement, the change of shell height will be greatly less than the using volume invariant one where the base surface is highly bended. In practice, we tessellate Ω into a set of tetrahedrons. Let { T 1,...,T m } and { u 1,...,u n } respectively denote the tetrahedron set and vertex set. Then the deformation function x(u) can be approximated by a piecewise linear function: x(u) = n i=1 x iφ i (u), where x i x(u i ) is the displaced position of u i, and φ i (u) are some piecewise linear basis functions valued 1 at u i and 0 elsewhere. Let T i be a tetrahedron made by vertex u i1,...,u i4, then deformation gradient in T i is x Ti = 4 j=1 x ( ) i j φi j Ti. It is well known that φi j (u) are actually the barycentric coordinates of u in T i and have constant gradients in T i. As a result, x Ti is constant too. Assemble the constant gradients φ i into a 3m 3n matrix G, we can rewrite the deformation energy in Eq. (2) as follows xt 1 r t 2 1 E(x) = G..... Gx h 2 F, (3) x t n r t m F where r i denotes ρ( x Ti ) for simplicity, and we omit the volume weights for clarity without loss of generality. We also transpose the deformation gradient to follow the convention that x is at the right side of the matrix G. Constraints can be formulated as extra energy terms after being appropriately weighted. In this paper we only consider positional constraints, and handle them by appending one row to G and h for each position constraint. The deformation energy is nonlinear, since h depends on x via a non-linear polar decomposition procedure. Similar to [10], by virtue of the quasi-linear property of our deformation energy, we can take the following inexact Gauss-Newton iterative algorithm to solve it: 1. Fix h and solve for x by (G t G)x = G t h; 2. If not converge, update h using the latest x and goto step 1. The first step involves solving x with fixed G t G. 4. SHELL GENERATION AND DEFORMA- TION In this section we will introduce an optimized shell generation technique formulated as a deformation problem, and then introduce a shell deformation algorithm for deformable objects. 4.1 Shell Generation In the shell generation application, we first create no distortion prisms (i.e. ideal prisms) for each triangle of the base surface, then stitch them to generate a seamless shell by deforming the prisms. Let f be a triangle on the base surface S b. The prism corresponding to f in the shell space has no distortion if and only if it is a normal prism. Such normal prism can be constructed by offsetting f along its normal by distance η. Obviously, these ideal prisms cannot form a seamless shell It is necessary to glue together the corresponding vertices, which requires deforming the normal prisms. Based upon the above discussion, we develop a shell generation method as follows:

3 We find that this is a result of trading-off between the stretches in the normal direction and tangent plane by our energy function. Consider an intuitive case: the optimal shell surface of a sphere should be another concentric sphere, but the prisms are obviously distorted a lot, because the top triangle of each prism is stretched by a factor of (1+η/γ), where γ is the radius of the base sphere and η is the shell thickness. When we minimize the stretch distortion of Eq. (2), the height of the prisms will be compressed to reduce the area stretch of the top triangles until a balance between the height compression and top triangle stretch is reached. As a result the prisms will be a little shorter than expected thickness η. Since such thickness defect is more noticeable near sharp features than in relative flat regions, there occurs a groove-like defect on the Bunny ear as shown in Figure 1. Based on this observation, we can reduce such artifacts near the sharp features by appropriately stretching the top triangles of the normal prisms before using them as the ideal prisms for shell generation. Figure 1: Groove-like defects on Bunny model. Top: defects on the ear and a zoomed view; Bottom: improved result by the curvature sensitive method. 1. First construct some normal prisms; 2. Then tessellate the ideal prisms into tetrahedrons; 3. At last, deform the ideal prisms so that the corresponding vertices coincide, producing a seamless shell. In step (2) the shell map algorithm [16] is adopted to triangulate the prisms, such that neighboring prisms in the output shell have consistent triangulation. In step (3) we use the distortion energy function of Eq. (3) to deform the ideal prisms. We d like to point out that Eq. (3) is applicable for cracked tetrahedrons by assigning each set of corresponding cracked vertices with a common displaced position. Let x b and x η respectively denote the vertex of S b and S η. Then the deformation energy in Eq. (3) can be rewritten as: E(x b, x η ) = ( Gb G ) ( x ) 2 b η h x η. F Now that x b is known and fixed for shell generation, the above energy function can be further simplified as E(x η ) = Gη x η (h G b x b ) 2 F. (4) We apply the iterative solver to minimize the energy function in Eq. (4). The initial guess is provided by the simple offset method which offsets the base surface along the vertex normal. We adopt the Gauss-Seidel method to solve the linear system in each iteration, and we found that it converges very fast in our experiments, since the vertices on the base surface S b are known and fixed. For the bunny model of vertices, it takes about 5 iterations within 10 seconds to generate a good shell on average. Curvature Sensitive Method Note that the above basic shell algorithm may produce undesired results near high curvature regions when the shell thickness is large. As shown in Figure 1 (Top), there is a noticeable groove-like defect along the sharp edge of the ear of the Bunny when we increase the shell thickness. Recall that the top triangle of each prism is stretched by a factor of (1 + η/γ), or (1 + ηκ) using curvature κ. This motivates us to construct curvature sensitive prisms as follows. Let f be a triangle on the base surface. We first compute the averaged principle curvatures κ 1, κ 2 and the principle directions e 1, e 2, as shown in Figure 2. Then construct the idea prisms by scaling the top triangle of the normal prism along the principle direction e i by a factor of (1 + ηκ i ) for i = 1,2. To avoid triangle flipping and degeneration, we set two values s min and s max to clamp the scale factors. To compute the averaged curvatures, we adopt the normal cycle algorithm [7] using a 2-ring neighboring triangles. After constructing the ideal prisms, we stitch them by minimizing Eq. (4) to generate a seamless shell. In Figure 1 (Bottom) is result generate by the improved algorithm. Compared with Figure 1 (Top), we can see that the groove-like artifact disappears. In Figure 3 there are more shell examples produced by our method. Table 1 gives the stretch distortion errors (defined in Eq. (2)) of the shells we have generated in this paper. Comparing the stretch distortion errors between our method and the simple offsetting method shows that our method can generate less distorted shells than the simple offsetting method. Figure 4 is a comparison of the shells generated by our method and by the simple offset method. We measure the stretch distortion 3 i=1 (σ i 1) 2 for all tetrahedrons and visualize them on the base surface by mapping large distortion to red color and small distortion to green color. The zoomed view shows that our method can avoid almost all local self-intersections for reasonable large thickness value. This advantage is achieved by the gradient based energy which prevents the tetrahedrons from being inverted. Figure 5 is an example of geometric texture mapping based on our shell generation result, which shows the meso-scale of geometry details are well preserved. 4.2 Shell Deformation In this section, we consider shell deformation for deformable objects. When the object is deformed, one could simply re-generate the shell on the deformed base surface. However, automatically generated shell could be further edited by customized applications, which make the re-generation method not applicable. Therefore, it is better to deform the shell surface S η. We deform the shell surface by minimizing the volume deformation energy Eq. (3). Note that the deformation energy for shell deforma-

4 Torus Bunny Bunny Venus Head Horse Armadillo 32.7/ / / / / / / / / / / /1.52 Table 1: Model size and stretch distortion errors. The format of each cell is vertex number/triangle number ( 10 3 ) in top and mean error/max error in bottom. Shells marked with are generated by using the simple offset method and shown here for comparison. f ' f ' f ' e 2 e 2 e 2 f e 1 f e 1 f e 1 Figure 2: Construction of an ideal prism for a triangle f. Left: normal prism; Middle: stretch the top triangle along principle direction e 1 ; Right: stretch along principle direction e 2. tion measures the stretches with respect to the original shell, which should be distinguished from the deformation energy for shell generation that measures the stretches with respect to the ideal prisms. Shell deformation can be divided into the following two types. When base surface is pre-deformed to a shape, we only optimize the shell surface using the base surface as the position constraints. After decorating a model with a layer of geometric details, such a deformation method can coat these details to a deformed shape of this model. In Figure 6, we embed a textured rectangle in the shell of a plane, then deform the plane. The effect of shell space warping is demonstrated by the warped rectangle. Our optimized method generates less stretch than simple offset method and can avoid local self intersection. Such a deformation method can be also used in animation decoration. Figure 8 shows such an example on the Bunny model. We first deform the base surface by simulation, then use the deformed mesh sequence to drive the shell deformation. In the first frame, flowers are embedded in the tessellation of the shell space by barycentric coordinates interpolation to increase visual effects. During the deformation, the relative distortion to the original shell is minimized, so that the geometric details can be naturally stretched along the surface without artifacts. This shell model has 69K vertices, and the deformed shell can be solved in about 1.0 seconds for each base surface deformation. Another type of shell deformation is simultaneously solving the deformations of both the base surface and the shell surface. In this case, the boundary conditions are user specified positional constraints on the shell. Unlike the Laplacian based mesh deformation method, our deformation energy is much closer to the physical model of thin shell. Hence the object tends to collapse instead of keeping its mean curvature when deformed, which is just like the behavior of a real-world shell object, for example, a basketwork(figure 9). In such shell deformation, both the base surface and the shell surface are unknown variables to be optimized, whose large degree of freedoms cause the inexact Gauss-Newton iterative solver very slow, particularly when the object is densely tessellated. Next, we will develop a hierarchical dimension reduction algorithm using a hierarchical mesh structure to improve the stability and converge speed. Figure 3: Shell generation results. The numbers below the shells are the ratios of the shell thickness over the model s bounding sphere radius. 5. HIERARCHICAL DIMENSION REDUC- TION We build our hierarchical dimension reduction algorithm on the traditional skeleton mesh skinning method to handle general deformable objects. The basic idea is to create a hierarchical skeleton structure for the object and apply the skeleton-based subspace method at each level to aggressively reduce the dimensionality, so that we can solve the deformation from the coarsest level to the finest level following the idea of multi-resolution algorithm [26] and multigrid algorithm [17]. 5.1 Skeleton Hierarchy Since the shell surface has the same topology, we only need to create a hierarchical skeleton structure for the base surface and then apply it to the shell surface. There are several techniques that can be adopted for building the hierarchy for our purpose, such as hierarchical fuzzy mesh decomposition [11], successive neighborhood clustering [3]. In order to generate nearly equal size of clusters at each level, we take the following top-down subdivision approach. First, a single cluster containing the whole mesh is created as the coarsest level at the top. Then, the cluster is recursively subdivided into two sub clusters until a desired subdivision depth is reached or subdivision cannot be continued. In our experience, using a few hierarchical levels (about 5) can successfully accelerate the conver-

5 Figure 4: Shell comparison using Stanford Bunny model. Top: offset method; Bottom: our method. Left: shell surface; Middle: translucent rendering of the shell and a zoom view; Right: false-color visualization of the stretch distortion distribution. Figure 6: Shell space warping comparison. The base surface (in green) is deformed from a planar rectangle. The chessboard texture is a vertical slice of the volumetric texture to show the space warping. On the bottom are the zoomed views of the top. On the left is the result of the simple offset method, which suffers from large stretches and some local self intersections. On the right is an optimized result by our method. Figure 5: Geometric texture mapping comparison. Our method (a) can achieve high quality geometric texture mapping due to the optimized shell. But simple offset method (b) generates many artifacts, such as large distortion and local self intersections. gence. See Figure 7 for an example of clusters at different levels. Let C l {c 1 l,...,cm l l } be the clusters of level l. For each cluster c j l C l, we assign it with a transformation matrix M j l and a translation vector t j l, called the skeleton parameters of c j l, to represent its deformation as x i = [M j l, t j l ]ū i, u i c j l, where ū i denotes the homogeneous coordinate. By assembling ū i into a sparse matrix B l and packing [M j l, t j l ] into a vector z l, we have xt. 1 x.. (M j. = B l l )t x t (t j B l )t l z l. (5) n. Then we have the following dimension reduced energy function for shell deformation by substituting x = B l z l in Eq. (3): E(z l ) = (GB l )z l h 2 F. (6) Applying the inexact Gauss-Newton iterative solver, we have A l z l = b l for each iteration, where A l = (GB l ) t (GB l ) and b l = (GB l ) t h. (7) Because there are a few neighbors for a cluster, the matrix GB l is still very sparse, and the equation can be solved efficiently. Inspired by [8], we can compute h at level l without perform- Figure 7: Clusters in different levels. ing the polar decompositions for every tetrahedron. Note that in the reduced deformation space, tetrahedrons whose nodes are in a same vertex cluster c j l C l share identical skeleton parameters, and the corresponding rigid parts of the deformation gradients are also same. So we treat them as a cluster. For tetrahedrons whose nodes are in multiple vertex clusters, each of them makes a single element cluster. So based on the vertex clusters C l, we can create tetrahedron clusters to fast update the rigid transformation h. Then we can compute polar decomposition only once for one tetrahedron cluster. 5.2 Hierarchical solver Though we can simply apply the fast multigrid algorithm [17] for each linearized step, there are many places that can be improved. For example, when the coarse level is solved, not only vector z but also vector b can be updated for the fine level to accelerate the convergence. Similar to the top-down scheme in [3], to solve the deformation A l z l = b l at level l, we first solve the deformation at level l 1 using the reduced vectors z l 1 and b l 1. Then correct vectors z l and b l at the current level, and perform a series of Gauss-Seidel iterations to smooth the skeleton parameters z l. Please be reminded that we cannot replace a sub cluster s skeleton parameters by its parent cluster s during consecutive manipulations. Because it will lose the high frequency deformation component obtained in previous iterations. But we can propagate the parameter changes from the parent cluster to the sub clusters. First we compute the rotation

6 change R l 1 and centroid translation t l 1 of the parent cluster as soon as the coarse level l 1 is solved. Then accumulate them to fine level parameters for better initial value of Gauss-Seidel iterations. In our experience, about 5 Gauss-Seidel iterations for each level is enough. The hierarchical solver can speedup the deformation of basketwork over 10 times compared to a single level solver. Figure 9 shows a sequence of shell deformation results on a basketwork. about 8K vertices, which takes about 0.12 seconds for each deformation on average using the hierarchical solver. An optimized shell generation and shell deformation algorithm based on stretch distortion minimization; A hierarchical dimension reduction technique designed for solving nonlinear optimization problems on arbitrary mesh. Though stretch distortion is inevitable in shell generation and deformation, our algorithm can minimize it to produce optimal results. To the best of our knowledge, it is the first shell generation and deformation algorithm which takes into account and minimizes the volume distortion. Examples have shown that it can produce highquality shells in terms of the fairness of the shell surface and the number of local self-intersections. And our volume deformation technique can be naturally applied to animating shells. Our shell deformation method can deform the base surface and the shell surface consistently and decorate animation with geometrical details without artifacts, which is difficult to achieve by extending existing surface deformation methods. The major limitation of our algorithm is that there is no hard constraints to guarantee that there is no self-intersection. Currently we do not take the embedded details in the shell into account when deforming the shell. It should be interesting and useful to develop a content sensitive shell deformation algorithm in future work. At last, the hierarchical dimension reduction algorithm we have proposed is very useful in solving nonlinear optimization problem on arbitrary mesh. We believe that it can find applications in many digital geometry processing problems. Figure 8: Animation sequence of a bunny model decorated with flowers. Figure 9: Direct shell deformation of a basketwork. 6. CONCLUSION We have introduced a shell generation and deformation algorithm which is very useful for shell based modeling, rendering and animation. Our main contributions are: A quasi-linear stretch distortion energy function formulated for volume deformation; Acknowledgement This work is partly supported by the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No ), the National Basic Research Program of China (Project No. 2006CB303102) and National Natural Science Foundation of China(No ). 7. REFERENCES [1] M. Alexa, D. Cohen-Or, and D. Levin. As-rigid-as-possible shape interpolation. In SIGGRAPH 00, pages , [2] M. Botsch and L. Kobbelt. Multiresolution surface representation based on displacement volumes. Computer Graphics Forum, 22(3): , Sep [3] M. Botsch, M. Pauly, M. Gross, and L. Kobbelt. Primo: coupled prisms for intuitive surface modeling. In Eurographics Symposium on Geometry Processing, pages 11 20, [4] G. Celniker and D. Gossard. Deformable curve and surface finite-elements for free-form shape design. In SIGGRAPH 91: Proceedings of the 18th annual conference on Computer graphics and interactive techniques, pages , New York, NY, USA, ACM Press. [5] Y. Chen, X. Tong, J. Wang, S. Lin, B. Guo, and H.-Y. Shum. Shell texture functions. ACM Trans. Graph., 23(3): , [6] J. Cohen, A. Varshney, D. Manocha, G. Turk, H. Weber, P. Agarwal, F. Brooks, and W. Wright. Simplification envelopes. In SIGGRAPH 96, pages , [7] D. Cohen-Steiner and J.-M. Morvan. Restricted delaunay triangulations and normal cycle. In SCG 03: Proceedings of the nineteenth annual symposium on Computational geometry, pages , [8] K. G. Der, R. W. Sumner, and J. Popović. Inverse kinematics

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