Animated Models Simplification with Local Area Distortion and Deformation Degree Control

Volume 1, Number 1, September 2014 JOURNAL OF COMPUTER SCIENCE AND SOFTWARE APPLICATION Animated Models Simplification with Local Area Distortion and Deformation Degree Control Shixue Zhang* Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, P.R. China. *Corresponding author: zhangsx@ciomp.ac.cn Abstract: In computer graphics applications, mesh simplification is an important technique to alleviate the workload of visualization processing. Compared to the extensive works on static model simplification, very little attentions have been paid to animated models. In this paper, we propose a new method to approximate animated models with local area distortion and deformation degree control. Our method uses an improved quadric error metric guided by a local area distortion measurement as a basic hierarchy. Also, we define a deformation degree parameter to be embedded into the aggregated quadric errors, so areas with large deformation during the animation can be successfully preserved. Finally, a mesh optimization process is proposed to further reduce the geometric distortion for each frame. Our approach is fast, easy to implement, and as a result good quality dynamic approximations with well-preserved sharp features can be generated at any given frame. Keywords: Animated Model; Computer Animation; Mesh Simplification; Mesh Deformation 1. INTRODUCTION In computer graphics applications, triangular mesh is most commonly used to represent threedimensional models. Especially, more and more animated models, which are also called time-varying meshes, are frequently used from scientific applications (simulation) to animation (movie, games). In many cases, some of the details of them might be unnecessary especially when viewing from a distance. Mesh simplification (or coarsening) is a process of eliminating such unnecessary or redundant details from high-resolution 3D models by repeated removing vertices, edges, or faces. Most of the existing simplification algorithms are to deal with a single static mesh, and very little work has been proposed to address the problem of accurate approximation for time-varying surfaces. In this paper, we propose a new method for generating multiresolution animated models. To simplify animated mesh sequence, one naive way is to simplify the models for each frame independently. This solution can generate the minimum geometric distortion from the original model. However, since it does not exploit the temporal coherence of the data, it can involve the unpleasant visual artifact, 1

causing the surface to vibrate and twitch. So traditional simplification methods can t be directly applied to animated models. Some previous methods focus on preserving the static connectivity, i.e. the connectivity of the timevarying surfaces remains unchanged for all frames. Such adaptations are inadequate and would cause arbitrarily bad approximations when deformation is highly non-rigid, since it does not take time-varying information into consideration. In this paper, we propose a new method to generate optimal approximations for animated meshes. The method is based on local area distortion measurement and deformation area preservation. And it is a better tradeoff between the temporal coherence and geometric distortion, i.e. we try to maximize the temporal coherence while minimizing the visual distortion during the simplification process. We propose the use of local area distortion control to preserve highly curved regions. Also we define a deformation weight, which can be added to the aggregated edge contraction cost, so areas with large deformation can be preserved. Finally, a mesh optimization method is applied on the approximations to further reduce the geometric distortion. We demonstrate that this provides an efficient means of multiresolution representation of time-varying models over all frames of an animation. The rest of this paper is organized as follows: Section 2 will review the previous related works. Section 3 will mainly introduce the procedure of our algorithm and discuss its advantage. Section 4 will show the experimental results. Finally, conclusion and some future work will be given in Section 5. 2. RELATED WORK 2.1 Mesh simplification There are now extensive papers on the approximation of dense polygonal meshes by coarser meshes that preserve surface detail. These methods can be roughly divided into five categories: vertex decimation [1], vertex clustering [2], region merging [3], subdivision meshes [4], and iterative edge contraction [5 10]. A complete review of the methods has been given in [11 13]. Among these methods, the process of iteratively edge contraction is predominantly utilized. Representative algorithms include those from Hoppe [7] and Garland [5]. In such methods, a simple multiresolution structure is generated on the surface that can be used for adaptive refinement of the mesh. Traditional mesh simplification algorithm works fine on a single static model, but it is unable to be directly applied to deforming meshes since no temporal coherence has been considered. 2.2 Mesh optimization There are many optimization techniques for vertex repositioning. Most of them are based on the idea of local optimization and require an improvement of such mesh quality parameters as aspect ratio, area, etc. Hoppe et al. [14] described an energy minimization approach to solving the mesh optimization problem. The energy function consists of three terms: a distance energy that measures the closeness of fit, a representation energy that penalizes meshes with a large number of vertices, and a regularizing term that conceptually places springs of rest length zero on the edges of the mesh. Their minimization algorithm 2

Animated Models Simplification with Local Area Distortion and Deformation Degree Control partitioned the problem into two nested subproblems: an inner continuous minimization and an outer discrete minimization. Turk [15] proposed an approach for distributing a given number of points over a mesh surface evenly. These points are connected to one another to create a re-tiling of a surface that is faithful to both the geometry and the topology of the original mesh surface. Therefore, these points will eventually become the vertices of the new mesh model. In [16], a procedure was presented to improve the quality of complex polygonal surface meshes without an underlying smooth surface using numerical optimization. Recently, Liu [17] proposed a non-iterative approach using global Laplacian operator to keep the fairness, and also used the constraint condition to reserve the fidelity to the mesh. 2.3 Approximation of time-varying surfaces Shamir et al. [18, 19] designed a global multiresolution structure named Time-dependant Directed Acyclic Graph (TDAG). The TDAG is built by merging the individual multiresolution hierarchies for each frame of a mesh sequence together into a unified graph. Tags are assigned to each node specifying the time interval over which the node should be alive. The T-DAG has the advantage of being able to handle arbitrary topology changes as well as geometric deformations. The primary drawback of this approach is that it is inherently non-incremental and potentially space-inefficient. Mohr and Gleicher [20] proposed a deformation sensitive decimation (DSD) method, which directly adapts the Qslim algorithm [5] by summing the quadrics errors over each frame of the animation. The result is a single mesh that attempts to provide a good average approximation over all frames. Consequently this technique provides a pleasant result only when the original surfaces do not present strong deformation. DeCoro and Rusinkiewicz [21] introduced a method of weighing possible configuration of poses with probabilities. With articulated meshes, skeleton transformation is incorporated into standard QEM (quadric error metric) algorithm [5], and users must specify the probability distribution for each joint. This method works quite well, but is limited to a very specific class of deformations. Kircher and Garland [22] proposed a multiresolution representation with a dynamic connectivity for deforming surfaces. By their method, the simplified model for the next frame is obtained by a sequence of edge-swap operations from the simplified model at the current frame. They treat a sequence of vertex contraction and their resulting hierarchies as a reclustering process. This method seems to be particularly efficient because of its connectivity transformation. Geometry videos [23] offer a different approach, based on the geometry images of Gu et al. [24]. The mesh is first parameterized, then re-sampled over a regular grid in parameter space. The result is a 2D image with 3 channels, for x, y and z coordinates. The mesh geometry is stored or transmitted simply as an image stream (video), which can be compressed using existing video compression methods. Level of detail is obtained by sub-sampling the images, while being careful to preserve cut boundaries. While geometry images have their own advantages (particularly the implicit connectivity), they suffer the same disadvantages as all remeshing strategies i.e., it is impossible to recover the original mesh exactly. Huang et al. [25] proposed a method based on the deformation oriented decimation error metric and dynamic connectivity update. They use vertex tree to further reduce geometric distortion by allowing the connectivity to change. Their method can provide a good approximation of deforming surfaces, but 3

requires a complex structure. Recently, Landreneau et al. [26] propose a method for simplifying a polygonal character with an associated skeletal deformation. This method works well but can be only applied to articulated meshes. Also Bruce et al. [27] propose an analytic simplification method for animated characters. They use an animation-space distance metric within a progressive mesh-based LOD scheme, and simplify individual vertices by reducing the number of bones that influence them, using a constrained least-squares optimization. Their approach is also limited to deal with meshes with skeletons, and can t be applied to non-rigid deforming surfaces. Zhang et al. [28] propose an improved method to generate progress animation models, and get elegant results. 3. OUR ALGORITHM Our algorithm consists of three parts: (1) Use the local area distortion measurement to improve QEM edge contraction cost. (2) Define a deformation degree weight during the whole animation to preserve the areas with large deformation. (3) Use the mesh optimization method to further reduce the approximation distortion. Next we will introduce the algorithm in detail. 3.1 Weighted quadric error metrics Up to now, the QEM algorithm is widely considered as one of the most efficient methods in static mesh simplification, and our method will improve QEM with local area distortion control. First of all, we should have a quick review of QEM. QEM iteratively selects an edge (v i, v j ) with the minimum contraction cost to collapse and replace this edge with a new vertex u which minimizes the contraction cost. QEM measure the edge contraction cost as the total squared distance of a vertex to the two sets of planes P(v i ) and P(v j ) adjacent to v i and v j respectively. A plane can be represented with a 4D vector p, consisting of the plane normal and the distance to the origin. Hence, the squared distance of a vertex v to a plane p equals v T (pp T )v. The QEM cost function D ij for a vertex v to replace the edge (v i, v j ) is D ij (v)=â p2p(vi ) v T (pp T )v + Â p2p(v j ) v T (pp T )v= v T Q i v + v T Q j v (1) QEM simplifies a mesh by iteratively finding the edge (v i, v j ) with the minimum D ij, performing an edge collapse operation to replace (v i, v j ) with a new vertex u ij and updating the edge contraction costs related to u ij until the desired resolution is reached. QEM can generate good results, but it neglects some basics features, such as the curvatures, etc. Next we will define the local area distortion. Let n be the surface normal at vertex v, let k 1, k 2 be its principal curvatures and e 1, e 2 are their corresponding principal directions. In the orthogonal coordinate frame with origin at v and axes e 1, e 2, n, the second-order local approximation of the surface is a patch of the form S(x,y)= x,y, k 1x 2 +k 2 y 2 2 (2) (x,y) 2 [ w 1,w 1 ] [ w 2,w 2 ] 4 By considering only the lower terms of the Taylor series approximation, we define A is a diagonal matrix

Animated Models Simplification with Local Area Distortion and Deformation Degree Control with entries The area of the surface patch is a 11 = 4 3 w3 1 w 2k 2 1 a 22 = 4 3 w 1w 3 2 k2 2 a 33 = 4w 1 w 2 a 11 +a 22 2 (3) a 11 + a 22 + a 33 = 4w 1 w 2 1 + w2 1 k2 1 + w2 2 k2 2 6 (4) Thus, the distortion factor between the area 4w 1 w 2 of the domain and the area of the surface patch is a 11 + a 22 + a 33 = 1 + w2 1 k2 1 + w2 2 k2 2 4w 1 w 2 6 (5) Observe that the area distortion depends on the term(w1 2k2 1 + w2 2 k2 2 ) 6, which we call the local area distortion measure and can be computed by a 11 + a 22 a 11 + a 22 + a 33 = trace(a) a 33 trace(a)+a 33 (6) Due to the invariance of the eigenvalues of a matrix under bijective linear transformations, the local area distortion measure can be computed in the canonical coordinate frame by trace(a) l trace(a)+l (7) where l is A s eigenvalue corresponding to the eigenvector closest to the surface normal at v. In our algorithm, we can use an additional weight to improve the quadric w i Q i + w j Q j, where the weights w i are computed by w i = trace(a i) l i (8) trace(a i )+l i Consequently, the contraction cost of every edge (v i, v j ) is computed by v T (w i Q i )v + v T (w j Q j )v (9) Moreover, to guarantee that the weight won t be overestimated, the quadric assigned to the new vertex v has to be w i Q i + w j Q j w i + w j (10) With this improved edge contraction cost, more local features can be preserved during the simplification. 5

3.2 Deformation degree measurement The deformation sensitive decimation (DSD) [20] algorithm addresses this problem by summing the edge contraction costs across all frames. Based on the DSD method, we add an additional weight to the DSD cost which measures the deformation degree. The deformation is measured by the change of frame-to-frame edge collapse cost. For areas with large deformation, the change of the collapsing cost must be prominent, while collapsing cost may change slightly in areas with small deformation. So the contraction cost for edge (v i, v j ) can be represented by cost ij = DSD ij + k 1 Â f t=1 Dt ij D ij = Â f t=1 Dt ij + k 1 Â f t=1 Dt ij D ij (11) where D t ij is the collapse cost of edge (v i, v j ) in frame t, and D ij is the average collapse cost of edge (v i, v j ) over all of the frames. And k 1 is a user-specified coefficient to adjust the influence the deformation degree. In our experiment, we set k 1 to around 1. 3.3 Mesh optimization For the simplified models, we finally apply a mesh optimization process to further reduce the visual distortion. The basic operations in our algorithm are edge split and edge contraction. We define the shape factor function for an edge as follows: S(e)= 6 l(e) Â ei 2Neighbour(e) l(e i ) (12) where l(e) is the length of the edge, and Neighbour(e) is the set of edges that are incident to e. Normally we should guarantee that the shape factor value for each edge satisfy the condition 0.5 apple S(e) apple p 2. If this value is less than 0.5, we should perform an edge contraction operation on this edge; on the other hand, if this value is bigger than p 2, we should perform a corresponding edge split operation. For an edge split operation, we should decide the position of the new vertex. In our method, we perform the following operations. Suppose the two incident vertices of edge e are P 1 (x 1,y 1,z 1 ) and P 2 (x 2,y 2,z 2 ), and the opposite angles are P 3 (x 3,y 3,z 3 ) and P 4 (x 4,y 4,z 4 ) as shown in Figure 1. The corresponding sphere can be decided as the following equation: x 2 + y 2 + z 2 x y z 1 x1 2 + y2 1 + z2 1 x 1 y 1 z 1 1 x2 2 + y2 2 + z2 2 x 2 y 2 z 3 1 x3 2 + y2 3 + z2 3 x 3 y 3 z 3 1 x4 2 + y2 4 + z2 4 x 4 y 4 z 4 1 = 0 (13) 6 The midpoint of e is P m =(P 1 +P 2 )/2. Suppose the normal vectors for P 1 and P 2 are n 1 and n 2, and we estimate the normal vector of P m as n m =(n 1 +n 2 )/2. So the line pass through P m and with direction of n m

Animated Models Simplification with Local Area Distortion and Deformation Degree Control Figure 1. The sphere decided by P 1, P 2, P 3 and P 4. can be represented as: P(t)=P m +t n m (14) Equation (13) and Equation (14) can decide two points of intersection, and we select the nearer one to P m as our new vertex. 3.4 Algorithm outline Our algorithm can be summarized into the following steps: 1. For each vertex v i on the original models, compute the quadric matrix Q i and the corresponding weight parameter w i. 2. Compute the edge contraction cost for each edge and get the unified cost by adding the quadric errors of each frame. 3. Measure the deformation degree weight, and add it to the aggregated edge contraction cost for the whole animation. 4. Iteratively perform the edge collapse operation and update the connection information until the desired resolution is reached. 5. For each frame model, use the mesh optimization method to further reduce the visual distortion. 4. EXPERIMENTAL RESULT We test the result of our algorithm on a computer with Pentium4 3.2G CPU and 2G memory. We use OpenGL to render the models. The simplification of a horse- gallop animation is shown in the Figure 7

Figure 2. Horse-gallop animation with 48 frames. Top row: original sequence with 8431 vertices. Bottom row: simplified models with 800 vertices. Figure 3. An elephant-to-horse morphing sequence with 200 frames. Top: The original sequence with 42875 vertices and 85746 faces. Middle: Reduce 80% of the original components using our method, 8576-vertices and 17148-faces approximation. Bottom: Reduce 95% of the original components, 2187-vertices and 4370-faces approximation. 2. This animation contains 48 frames. Compared to original models on the upper row, the bottom shows our simplified results when 90% of its components are reduced. We could see that most of the features in the original models are preserved. Figure 3 shows another more complicated example on an elephant-to-horse animation. The second row and third row show the simplification results when 80% and 95% of its components are reduced respectively. We can see that even very little component is left, the simplified models can still be rendered faithfully to the original ones. Next, we should make a comparison of our method with previous ones. In Figure 4, we show the simplification result of Kircher s method(left) and ours(right) on a horse-to-man morphing sequence. By enlarging the detail of some particular features, we could see that our result preserves the hands and feet better, moreover the triangulation in this area is more reasonable than Kircher s result. In the area of the human s feet, our result has more triangles than the left one. That is because the local sharp featrues in this area are relative greater, and also the deformation degree is larger, so more edge collapse operation is postponed in this area. Figure 5 shows another example of the elephant-to-horse morphing sequence. In most of the areas, our result preserves more local sharp features than Kircher s result. Especially in the area of the horse tail, our result obviously consists of more triangles, and preserves more details. 8 Figure 6 and Figure 7 show the RMS error statistics of horse-gallop and horse-to-man animation results of different methods. The RMS error line of our result(red) lies between DSD and independent

Animated Models Simplification with Local Area Distortion and Deformation Degree Control Figure 4. Comparison of Kircher s result (left) [22] and our result (right) on a horse-to-man morphing sequence, the approximated versions contains 3200 vertices and 6392 triangles. Figure 5. Comparison of Kircher s result (left) [22] and our result (right) on elephant-to-horse morphing sequence, the approximated versions contains 800 vertices and 1592 triangles. QEM on the whole, and is obvious lower than other methods. Especially in the statistics of horse-gallop animation, the RMS error of our result is very close to the independent QEM error. So our method can be demonstrated to generate better result both in visual effect and error statistic. To demonstrate the various applicability of our algorithm, we finally show a skinning animation example in Figure 8. Our method can still produce faithful models for different level of details. 5. CONCLUTION AND FUTURE WORK In this paper, we propose a simplification method for animation models with local area distortion and deformation degree control. Given a sequence of meshes representing time-varying 3D data, our method produces a sequence of simplified versions that are good approximations of the original ones for 9

Figure 6. RMS error metric statistics of the horse-gallop animation (48 frames). Vertical axis indicates the error value, and the horizontal axis indicates the animation frame. Figure 7. RMS error metric statistics of the horse-to-man animation (200 frames). Vertical axis indicates the error value, and the horizontal axis indicates the animation frame. each frame. Our method uses a weighted quadric error metric to measure the edge contraction cost. A deformation degree weight is added to the edge collapse cost to preserve the areas with large deformation. Finally, a mesh optimization method is applied on the approximations to further reduce the geometric distortion. Our method is easy to implement and can produce better approximations than other previous methods. There are certainly further improvements that could be made to our algorithm. For example, we believe that there must be a way to extend our algorithm to be view-dependent. 6. ACKNOWLEDGMENTS We wish to thank Scott Kircher for providing the elephant-to-horse sequence, Robert W. Sumner and Jovan Popovic for providing the horse-gallop data. This work was supported by the National High-Tech 10

Animated Models Simplification with Local Area Distortion and Deformation Degree Control Figure 8. A head-expression animation sequence with 192 frames. Top: The original sequence with 15941 vertices and 31620 faces. Middle: Reduce 80% of the original components using our method, 3200-vertices and 6320-faces approximation. Bottom: Reduce 95% of the original components, 826-vertices and 1608-faces approximation. Research & Development Program of China (No. 2008AA8080202). References [1] W. J. Schroeder, J. A. Zarge, and W. E. Lorensen, Decimation of triangle meshes, in ACM Siggraph Computer Graphics, vol. 26, pp. 65 70, ACM, 1992. [2] K.-L. Low and T.-S. Tan, Model simplification using vertex-clustering, in Proceedings of the 1997 symposium on Interactive 3D graphics, pp. 75 82, ACM, 1997. [3] M. Garland, A. Willmott, and P. S. Heckbert, Hierarchical face clustering on polygonal surfaces, in Proceedings of the 2001 symposium on Interactive 3D graphics, pp. 49 58, ACM, 2001. [4] A. Lee, H. Moreton, and H. Hoppe, Displaced subdivision surfaces, in Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pp. 85 94, ACM Press/Addison- Wesley Publishing Co., 2000. [5] M. Garland and P. S. Heckbert, Surface simplification using quadric error metrics, in Proceedings of the 24th annual conference on Computer graphics and interactive techniques, pp. 209 216, ACM Press/Addison-Wesley Publishing Co., 1997. [6] A. Guéziec, Locally toleranced surface simplification, Visualization and Computer Graphics, IEEE Transactions on, vol. 5, no. 2, pp. 168 189, 1999. [7] H. Hoppe, Progressive meshes, in Proceedings of the 23rd annual conference on Computer graphics and interactive techniques, pp. 99 108, ACM, 1996. [8] H. Hoppe, View-dependent Refinement of Progressive Meshes, in ACM SIGGRAPH 1997 Conference Proceedings, pp. 189 198, ACM, 1997. [9] H. Hoppe, New quadric metric for simplifiying meshes with appearance attributes, in Proceedings of the conference on Visualization 99: celebrating ten years, pp. 59 66, IEEE Computer Society Press, 1999. [10] J. Van, P. Shi, and D. Zhang, Mesh simplification with hierarchical shape analysis and iterative edge contraction, Visualization and Computer Graphics, IEEE Transactions on, vol. 10, no. 2, 11

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