Example Based Skeletonization Using Harmonic One-Forms

Transcription

1 Example Based Skeletonization Using Harmonic One-Forms Ying He Xian Xiao Hock-Soon Seah School of Computer Engineering Nanyang Technological University, Singapore ABSTRACT This paper presents a method to extract skeletons using examples. Our method is based on the observation that many deformations in real world applications are isometric or near isometric. By taking advantage of the intrinsic property of harmonic 1-form, i.e., it is determined by the metric and independent of the resolution and embedding, our method can easily find a consistent mapping between the reference and example poses which can be in different resolutions and triangulations. We first construct the skeleton-like Reeb graph of a harmonic function defined on the given poses. Then by examining the changes of mean curvatures, we identify the initial locations of joints. Finally we refine the joint locations by solving a constrained optimization problem. To demonstrate the efficacy of our method, we apply the extracted skeletons to pose space deformation and skeleton transfer. 1 INTRODUCTION Due to the increasing popularity of 3D laser scanners, highly detailed polygonal meshes are very common in industrial applications. Direct manipulating these models to produce faithful animation turns to be a tedious, time-consuming and error-prone procedure. A common approach to deal with this problem is to use reduced models, such as skeletons [25], free-form lattices [40], surface markers or handlers [2], to ease the animation task. Among those reduced models, skeletons are perhaps the most intuitive and popular method in video games and computer animation industry. A well defined skeleton directly represents the anatomy structure of articulated characters, meanwhile it consists of highly reduced degrees of freedom compared with detailed models, and is straightforward for artists to manipulate, pose and animate characters. Extracting skeletons from 3D objects is challenging, especially in the context of finding exact centered skeletons. There exists large number of research efforts in finding the skeletons of static models. Popular techniques include thinning [33, 37], shape decomposition [20, 26], Reeb graph [7, 32], distance transformation [12], medial axis transform [1]. However, these methods are based on a single pose and rarely discuss skinning because some dynamic parameters, such as rigid bone transformations, joint locations, vertex weights, are usually difficult to obtain from a static pose. Therefore, it is desirable to use multiple poses instead of one. This paper focuses on the problem of example based skeleton extraction raised by Schaefer and Yuksel in [39]. We are strongly motivated by the observation: many real-world deformations are isometric or near isometric from the global point of view. For example, the horse has similar metric in the rest and running poses, i.e., the deformations are caused by bending instead of stretching. From the differential geometry point of view, if the two surfaces have similar metric but different embedding (appearance in R 3 ), yhe@ntu.edu.sg xiao0003@ntu.edu.sg ashsseah@ntu.edu.sg then their Gaussian curvatures are similar but mean curvatures are NOT (see Figure 1). Usually, the deformations of articulated models, such as human and animals, occur at the joints. Therefore, the near-joint skins have similar Gaussian curvature but different mean curvature in the reference and example poses. On the other hand, the part of skins which are far from the joints can be considered as rigid, i.e., both the Gaussian curvature and mean curvature remain unchanged. Therefore, the curvatures can be used to identify and locate the joints. Although conceptually simple, the above idea is difficult to be directly applied to real-world examples. The reasons are two-fold. First, the 3D shapes are usually acquired via 3D laser scanners. It is very common that the poses have quite different resolutions and triangulations. Finding the one-to-one correspondence between poses is non-trivial. Second, the commonly-used discrete operators to compute curvature are sensitive to the resolution and connectivity. The computation results of a given model in different resolutions may look quite different. (a) Rest pose (b) Gaussian curvature (c) Mean curvature (d) Example pose (e) Gaussian curvature (f) Mean curvature Figure 1: Many deformations in real world applications are isometric or near isometric. As a result, the Gaussian curvatures do not change too much in various poses. However, mean curvature near the joints may change significantly. Please pay attention to the joints where the changes of mean curvatures are much larger than that of Gaussian curvatures. Harmonic 1-form is a differential 1-form df defined on manifold M such that f : M R is harmonic. Harmonic 1-form has many

2 The number and locations of the joints and bones are totally determined by the dynamic geometry of the reference and example poses. Since symmetric Dirichlet boundary values are set for symmetric shapes, the extracted skeletons also reflect the geometric symmetry of the input model. (a) Reference pose (# of triangles = 16K) (b) Running pose (# of triangles = 30K) Figure 2: Harmonic 1-form is an intrinsic property which is determined by the metric and is independent of the embedding and resolution. Although the rest and running pose have different triangulations and appearances, their metrics are similar, thus, the harmonic 1-forms are highly consistent. Harmonic 1-forms facilitate finding the one-to-one correspondence among different poses. promising properties: Harmonic 1-form is determined by the metric and is invariant under isometric transformation. Harmonic 1-form is independent of surface representation. The surface in different resolution and triangulation has similar harmonic 1-form. Harmonic 1-form is computationally efficient and robust as we only need to solve a linear system. Symmetric harmonic 1-forms can be easily constructed on symmetric shapes. Figure 2 shows that the harmonic 1-forms of the horse in two poses (with different triangulations) are highly consistent. Therefore, harmonic 1-form can serve as a powerful tool for shape analysis, including the skeleton extraction. In this paper, we develop a robust algorithm for example-based skeleton extraction. The key idea of the proposed approach is to first compute harmonic 1-forms of the reference and example poses. Due to the small changes of the metric, the isolines of harmonic 1- forms across various poses are highly consistent. Then we use the isoline based representation of the model and extract the skeletonlike Reeb graph of the harmonic functions. Next by examining the changes of mean curvatures, we identify the initial locations of joints. Finally we refine the joint locations by solving a constrained optimization problem. Compared to the existing work, the advantages of the proposed method include: There is no restriction on the connectivity of the input meshes, i.e., the example poses could have very different triangulations and resolutions. This method is automatic except that the user needs to choose a source point on the reference pose. The remainder of this paper is organized as follows. Section 2 briefly reviews the previous work in skeletonization, Reeb graph, and harmonic 1-forms. Section 3 explains the algorithmic details for extracting skeletons from a set of example poses using harmonic 1-forms. Section 4 shows several applications of the proposed method, such as skeleton transfer and pose space deformation. Finally, we conclude the paper and briefly discuss the future research in Section 5. 2 PREVIOUS WORK 2.1 Skeleton Driven Deformation Animating characters driven by skeletons is widely applied though detailed description in academia is rare. Pioneer work can be found in [28] which introduced skeletal methods to animate hands and grasping behavior. Lewis et al. described standard method in industries thoroughly and obtained the name skeletal subspace deformation (SSD), which thereafter is also given by linear blending, soft skinning and single weight enveloping (SWE) [25]. Kavan and Žára identified that in SSD, linear blending of transformation matrix is ill defined for rotations and therefore proposes a novel method by blending associated quaternion instead [21]. Example based methods are proposed to correct underlying limitations of SSD. One type of methods is to correct collapsing parts by interpolating example shapes at run time [25, 42, 22, 23, 47]. The other category of approaches try to augment the SSD model by either introducing more parameters [49] or by adding extra joints to improve skeletal structure [31]. 2.2 Skeleton Extraction Most of the existing work focus on extracting skeletons from a static pose of a model. Mortara and Patanè defined the affine-invariant skeletal representation for 3D shape matching [32]. Katz and Tal used fuzzy clustering method to decompose a shape and then extract the skeletons [20]. Lien et al. proposed an efficient and robust approach that simultaneously generates a hierarchical shape decomposition and a corresponding set of multi-resolution skeletons [26]. Theobalt et al. took a sequences of volume data to estimate skeleton through volume decomposition of motion data [44]. Oda et al. presented a framework of extracting skeleton interactively using geodesic distances [35]. Ma et al. constructed a RBF level set for 3D model and found the maximum of gradient descent for each point and then formed the skeleton [27]. Aujay et al. presented the harmonic skeleton in which anatomical information is used to enhance the skeletons [3]. They demonstrated that very realistic skeletons can be generated using harmonic functions as well as the prior of the anatomy of the articulated models. Dellas et al. presented an automatic method to extract the animation control skeleton of virtual humans, which relies on an a-priori knowledge of the human anatomy [9]. Baran and Popovic presented an automatic method to embed a generic skeleton to a wide range of articulated 3D characters [4] Since it is usually difficult to estimate the dynamic behavior using the static pose alone, example-based techniques gain popularity in computer graphics in recent years [43, 41, 29, 39]. In [18], James and Twigg demonstrated that the conventional skinning techniques can be extended to automatically skin deformable mesh animations. Instead of specifying the hierarchical kinematic skeleton, they estimated proxy bone transformations and vertex weights for deformable shape sequences.

3 Recently, Schaefer and Yuksel proposed a novel method to extract hierarchical, rigid skeletons from example poses [39]. They first defined Rigid Error Functions to find the best rigid transformation and used these error functions to estimate the transformations of the bones in the example poses. Then, they skinned the mesh by solving for vertex weights using a constrained optimization and bone influence maps. Finally, they determined the connectivity of the skeleton and the joint locations. Schaefer and Yuksel s approach is capable to estimate the complete set of parameters for skeletal animation including bone transformation, skeletal hierarchy, joint location and vertex weights. However, their method requires that the reference and example pose have the same triangulation [39]. 2.3 Reeb Graph Reeb graph is a powerful tool to analyze the topology of a shape [38]. It has wide applications in computer graphics, such as removing topological noise [48], skeleton extraction [7], shape abstraction and understanding [6, 5]. Yoshihisa and Kunii proposed an algorithm to compute the Reeb graph in time O(n 2 ), where n is the number of edges of the mesh. Cole-McLaughlin solved the problem in O(nlog n) time [8]. Recently, Pascucci et al. presented a robust online algorithm to compute Reeb graph [36] for extremely large dataset. 2.4 Harmonic One-Forms Given a scalar function f : M R defined on manifold M, harmonic 1-form df is a kind of differential 1-form such that f is harmonic, i.e., f = 0. Harmonic 1-form plays a critical role in many applications of geometry processing. Gu and Yau pioneered global conformal parameterization using holomorphic 1-form, which can be decomposed to real and imaginary part and both parts are real harmonic 1-forms [14, 19]. Holomorphic 1-form is also used to compute the affine structure of a given manifold, which is the key to construct manifold splines [13, 16, 17]. Guo et al. demonstrated that global surface conformal parameterization can be generalized to a more general setting of point based geometry by computing the holomorphic 1-form in a meshless manner [15]. Ni et al. used the harmonic Morse function to extract the topological structure of a surface [34]. Dong et al. [10] and Tong et al. [46] applied harmonic 1-forms in quadrilateral remeshing. 5. We optimize the joint locations and bone lengths by minimizing the distance between the skeletons and the Reeb graphs (Section 3.6). 3.2 Setting The Source Point Similar to [3], a source point is needed to set the boundary value for the harmonic function. This user-specified source point is one of the end points of the skeleton-like Reeb graph (see Figure 3(c)). Thus, it is desirable to locate the source point such that there is a joint nearby. Furthermore, in our algorithm, the user only needs to specify the source point on one pose, then we use the registration algorithm to map the source point and its local neighborhood to other poses. To avoid the ambiguity of the registration, we require that the source point locates at the unique salient feature of the given model [11], e.g., the nose of the Horse and Armadillo model (see Figure 3(a) and 5(a)). Note that if the input model is symmetric, the source point must locate on the symmetric plane to induce the symmetric harmonic 1-forms. 3.3 Computing Harmonic 1-Forms Given a surface M with user-specified constraint vertices, v i, i = 1,...,k, define a scalar function f : M R, such that f (v i )=a i, i = 1,...,k, where a i are the user-specified constraints. The function f is harmonic if it minimizes the harmonic energy, E( f )= f 2 (1) M Then f, the gradient of f, is a harmonic 1-form on M. The harmonic 1-form, denoted by ω, is defined for each edge [v,w], i.e., ω([v, w]) = df([v, w]) = f (w) f (v), which satisfies the following equation [14]: f (v)= k v,w ω([v,w]) = 0 (2) [v,w] M The commonly-used weights k v,w include uniform weights and cotan weights. Harmonic 1-form can be computed either by solving the Laplace equation directly or finding the minimizer of the harmonic energy. Figure 4 illustrates the harmonic 1-form on genus two surface. 3 SKELETONIZATION 3.1 Algorithm Overview Our proposed method takes several poses of a given model as input. All poses are triangle meshes and could have very different triangulations and resolutions. Our algorithm runs in five successive stages, which are illustrated in Figure 3 using the Armadillo model. (a) (b) 1. The user specifies a source point on the reference pose. We use the registration algorithm to map this source point to the other poses (Section 3.2). 2. We compute the harmonic 1-form using the user-specified source point as constraints (Section 3.3). 3. We construct the skeleton-like Reeb graph of the harmonic function (Section 3.4). 4. We extract the isolines of the harmonic function and compute the average mean curvature for each isoline. By comparing the mean curvature for all poses, we can identify the joints which are closely related to the isolines whose mean curvatures are changed significantly (Section 3.5). Figure 4: Harmonic 1-form on genus two surface. The surface is cut along the red curve γ in (a). Two boundaries are denoted by γ + and γ. The Dirichlet boundary value of γ + and γ are set to be +1 and 0, respectively. Note that symmetric harmonic 1-forms can be easily constructed on symmetric shape if the boundary constraints reflect the symmetric of the input model. Specifying the boundary value is critical to the extrema of the harmonic function. Various methods are used in the previous work. Tierny et al. chose two points on the mesh which are the most distant [45]. Aujay et al. tried to find vertices v such that the distance d(v source,v) on the mesh is locally maximum [3]. Dong et al. suggested to solve a Poisson equation which mimics the curvature of the input mesh and then find the local extrema as the boundary vertices of the harmonic equation [10]. In this paper, we follow Dong

4 (a) (b) (c) (d) (e) Figure 3: Algorithm pipeline. (a) The user selects a source point on the reference pose. Usually, this point locates on the head with salient features. (b) We compute the harmonic 1-form on all poses. Note that the given rest and example poses have the same triangulation. In order to demonstrate our technique, we intentionally remesh each pose in different resolution. Two of eleven poses are shown here. Since the metric are similar in various poses, the harmonic 1-form are highly consistent. Since the Dirichlet boundary constraints are symmetric on the armadillo model, the harmonic 1-forms, as expected, also reflect the symmetric of the shape. (c) Then, we construct the skeleton like Reeb graph of the harmonic function. (d) Next, we identify the joints by finding the isolines where the mean curvature changes significantly compared to the reference and other poses. The color on the isolines illustrates the average of mean curvature. (e) Finally, we find the optimal location of the joints and bone lengths by solving a constrained optimization problem. et al. s method [10] to automatically determine the locations and values of the boundary constraints once the source point vsource is given. For each pose, there is a unique global minimal source point vsource (see Figure 5(a)) which is computed in Section 3.2. We solve the following Poisson equation as suggested in [10]: p(v) = H(v) and p(vsource ) = 0, (3) where H(v) is the mean curvature of vertex v. The goal of solving the above Poisson equation is to find a scalar field p which mimics the curvature [10]. After clustering the local extrema points which are very close to each other, the representatives of each cluster are identified as the boundary points and the function values are used as the boundary values. In order to make the harmonic 1-form of the reference and example poses as consistent as possible, we apply the same Dirichlet boundary constraints across all poses. Note that many articulated models are symmetric so that to keep the symmetry of harmonic 1-forms, we also require the boundary values of symmetric points to be identical. For example, we compute the geodesic between the source point and the local extreme points. If two or more extreme points have similar distances, we apply the same boundary value to them. This scheme guarantees the resulting harmonic 1-forms reflect the symmetric of the input shapes. Figure 5 illustrates the harmonic 1-form on one of the example pose. 3.4 Constructing The Skeleton-like Reeb Graph Reeb graph is a powerful tool when dealing with topological skeletons [38]. For a real-valued smooth function f : M R on M, the points whose derivatives of f vanishes are called critical points of f. The Reeb graph of f is a graph whose nodes corresponds to these critical points and encodes the connectivity between them. Note that the leaves of Reeb graph exactly match the local maximal and minimal of f. (a) (b) (c) (d) Figure 5: Computing the harmonic 1-forms. The global minimal point vsource is drawn in red in (a). By solving the Poisson equation, p(v) = H(v) and p(vsource ) = 0, we obtain a smooth function p which mimics the curvature of the surface shown in (b). Then we identify the local maximal points (four on feet and one on the tail tip) of p, which are shown in green in (c). Finally, the values of the extrema of the Poisson scalar field p are used as the boundary constraints of the Laplace equation. The resulting harmonic 1-form is shown in (d).

5 Building the graph. Reeb graph can be constructed by contracting the connected components of the isolines (level sets) of f to a point. Let n be the user-specified number of samples of the isovalues (200 in our experiments). We first normalize the function value of f to the unit interval [0,1] and then uniformly sample isovalues, f i = i n, i = 1,,n. Next we scan every critical point p of f, f (p) =0. If f (p) is not sampled, we insert f (p) into the isovalue set. Denote c k ( f i ) the isoline with isovalue f i on k th pose. k = 0 refers to the reference pose. (Note that each isovalue f i may have more than one isoline. We will explain eliminating the ambiguity later.) For each isoline, we compute its center as the representative. Then we use the sweep algorithm to connect these representatives and form the Reeb graph [8]. Topological filter. Once the Reeb graphs of reference and example poses are constructed, we first identify the key nodes (valence 2). Due to numerical error and/or metric changes, multiple key nodes may occur in locations where there should only be a single key node. Thus, we cluster key nodes such that the Reeb graphs of the reference and example poses have the same topological structure, i.e., the same number of key nodes (see Figure 6). Eliminating the ambiguity. Note that due to the symmetry of many articulated models, such as human and animals, two or more isolines with similar geometry may have the same isovalue, e.g., the Armadillo s elbows. Thus, we then need to distinguish these isolines with the same isovalues. In this paper, we take advantages of the articulated models which usually have some key points in the Reeb graph. Since the user picks the source point on the head, the first key point connecting to the source is the neck. So we can identify the spine easily. To further classify a branch into left and right arms (legs), we compute the sum of the signed distance from the points on the branch to the symmetry plane passing the spine. If the sum is positive, the branch is identified as a left arm or leg depending on the distance to the source point. Otherwise, the branch is identified as the right arm or leg. Note that the sign determining left and right is not important. The user can freely change the sign to swap left arm/leg and right arm/leg. Finding the correspondence among poses. Next we want to find the one-to-one correspondence between reference pose and example pose. Note that we don t need to find such a mapping between the vertices of two poses which are usually in different triangulation. Instead we will find the mapping for isolines. Isolines are the level sets of the harmonic function f, which are independent of the resolution and embedding. Therefore, even though the reference and example poses have very different resolutions, their isolines are highly consistent if the metrics are similar. However, in practice, there always exist some cases, especially the poses created by users instead of acquired by 3D scanners, where the deformations causes large changes of the metrics (although they should NOT change too much). As a result, the harmonic 1-forms and the corresponding isolines may differ a little bit. In order to find the best one-to-one correspondence of isolines between the reference and example pose, we first compute the integral of Gaussian curvature k G along each isoline i.e., c k ( f i ) k Gdl. Then for every isoline c i k in the example pose k, k 1, we find the isoline in the reference pose with the same isovalue c 0 ( f i ). If both isolines have similar integral of Gaussian curvatures, we find the ideal mapping. Otherwise, we perturb f i of the example pose in a small neighborhood and find the isovalue/isocurve such that the difference of the integral of Gaussian curvatures is minimal. As shown in Figure 8, the Reeb graphs after postprocessing are highly consistent. (a) (b) (c) Figure 6: Topological filter on the Reeb graph. (a) One key node (valence 2, drawn in red) exists in the reference pose. (b) Due to the small changes of metric in the example pose, two key nodes occur in nearby locations where there should only be a single key node. (c) We remove one key node by changing the local connectivity of the Reeb graph. (a) Figure 8: Constructing the skeleton-like Reeb graph. (a) Isolines of harmonic function f on the surfaces. (b) The red points are the critical points of the harmonic function f. The Reeb graphs of f on different poses are very consistent. 3.5 Identifying the Joints As explained above, many real-world deformations of articulated models are isometry or near isometry. It is well known that Gaussian curvature is completely determined by the metric, therefore, the Gaussian curvature of the reference and example poses remain almost unchanged. Mean curvature, however, is related to the embedding, and could be used to identify the joints. We observe that compared to the reference poses, the embedding of the joint-related skins could vary significantly in example poses. Thus, we use the average mean curvature D(i)=max{ (b) c i k Hdl k c c ik dl i k Hdl 0 c i0 dl k = 1,2, } (4) to measure the difference of the embedding of isoline c i between the reference pose and example poses. If D(i) reaches its local maximal, we mark the center of isoline c i as a joint candidate for every pose. We should also point out that for the symmetric shapes, the harmonic 1-forms are also symmetric due to the symmetric Dirichlet boundary values are specified in Section 3.3. Therefore, for symmetric parts, e.g., the Armadillo s arms, if one joint is identified, then the corresponding isolines on the symmetric parts are marked as joints too. However, in our experiments, we experienced that using the above local maximal condition alone, some extra joints may be identified. For example, due to the muscle bulge in the example

6 Table 1: Statistics of test examples. N p, number of poses; N t, average number of triangles in each pose (Note that we intentionally remesh the given meshes to get the different triangulation of each pose); T s, time for mapping the source point from the reference pose to example poses; T h, time for computing harmonic 1-forms; T r, time for constructing the Reeb graph; T j, time for identifying the joints; T o, time for optimizing the joint locations. Time are measured in seconds Object N p N t T s T h T r T j T o Arm 4 3k N.A Horse 16 20k Armadillo 6 60k Although encoding the topological structures nicely, the skeletonlike Reeb graphs are not the anatomical skeletons. Therefore, we need to estimate the joint locations using examples. Assume m joints are identified using the above method. Denote J j k the j-th joint, and Bl k the length of l-th bone in k th pose. We enforce that the bone lengths do not change in the deformations. For an isoline c i k in the k th pose, denote d(ci k,j1 k,,jm k ) the sum of distance between the point on the isoline and the skeleton defined by the joints {J j k }m j=1. Then, we also require that such distances are also invariant in all poses too. These constraints lead to the following constrained optimization problem: argmin J j k k 1 subject to the constraints d(c i k,j1 k,,jm k ) d(ci 0,J1 0,,Jm 0 ) 2 (5) i B l 0 = Bl k l,k (6) Figure 9: The muscle bulge may cause large change of metrics on the example poses (see Row 1 and 2). As a result, some extra joints will be identified (see Row 3). To avoid these extra joints, we use the angle constraint to test each joint. If the angles between two incident bones are greater than the user-specified threshold in ALL poses, this joint is identified as extra joint and then discarded (see Row 4). pose, the metrics in the bulged areas change a little bit larger than the rigid areas (see Figure 9). As a result, an extra joint is identified. To remove these unnecessary joints, we use the following angle constraint. For any valence-2 joint, we compute the smallest angle between the two incident bones in ALL poses. If the angle is greater than the user-specified threshold (160 degree in our implementation), this joint is considered as extra joint. As expected, this joint detection approach is highly related to the examples. The more poses involved, the more accuracy and robustness of the joints to be identified. Sixteen poses of horse model are used in our experiment. Figure 10 shows the maximal mean curvature difference max D(i) of head and leg. Obviously, the joints are closely related to the isolines whose mean curvature differs significantly to the reference pose. Figure 7 shows the detected joints on example poses. 3.6 Optimizing the Joint Locations Section 3.5 identifies the number of joints in the reference and example poses. However, the joints just locate on the Reeb graph. The objective function ensures that the relationship between the skin and bone is as rigid as possible. The constraints ensure that the bone length remains unchanged in all poses. Figure 11 shows the optimized skeletons. We tested our program on a workstation with 2.6GHz CPU and 3G memory. The statistics of the test examples are shown in Table 1. Note that no source point is specified for the Arm model. Instead, the Dirichlet boundary conditions are set as 1 and 0 on the two boundaries. 4 APPLICATIONS 4.1 Cross Parameterization and Skeleton Transfer Skeleton transfer is the direct applications of the proposed method. In this section, we demonstrate transferring the dynamic skeletons from the horse to the cat (see Figure 12). We pick a source point on the cat s nose. Then we compute the cross parameterization using harmonic 1-forms. To make the harmonic 1-forms of horse and cat as consistent as possible, we use the same boundary value conditions, i.e., the harmonic function values of the cat s nose, feet and tail are exactly the same as the corresponding values of the horse model. Once the harmonic 1-forms are ready, we can easily build the one-to-one correspondence between the isolines of horse and cat. Since each joint is associated with a unique isoline, the skeleton of horse can be easily transferred to the cat. 4.2 Pose Space Deformation Pose space deformation (PSD) provides one solution for problems induced by common underlying skinning methods such as SSD. It corrects collapsing parts of deformed models on the fly by interpolating well sculpted examples. Most literatures on PSD assume a single example as a pair of geometric mesh with associated skeleton [25][42][22][23]. Building a skeleton and rigging it to polygonal meshes are usually tedious and time-consuming to artists, not to mention posing it to align deformed models with given examples.

7 Figure 10: The column graphs show the maximal mean curvature difference D(i) of the head and one leg. The red bars are the local maximums which correspond to joints (green points). In previous sections, the extracted skeletons only compose of joint positions in world space. To employ the skeletons in real productions, it is desirable to compute joint transformations as parameters of pose space. Although an example mesh can be expressed as a reference mesh with a group of affine transformations [18], animators prefer more meaningful parameters to control such as joint rotations or translations. In this paper, instead of computing pure rotations [24] or rotations and translations [30], we parameterize example pose space as rotations and scaling values based on the observation that scaling joints can capture more accurate deformations than translations. This strategy can also be justified by the fact that articulate deformation is more likely resulted from muscle flexion while bone length should be fixed. We first align the source joint of reference mesh to one example mesh. And then starting from root joint, rotation angles and scaling values for each joint will be recursively computed in a manner of width first search. Given joint positions of a pair of parent and child joint in reference pose and one example pose, two bones in respective reference and example pose form a quaternion that rotates reference bone into example bone. Converting this quaternion to euler rotations in the local coordinate system of one parent joint is trivial. The offset between rotated child joint in reference pose and the child in example pose can serve as the amount of joint scaling. Once pose space parameters of each example mesh are obtained, we can perform PSD with the method described in [49]. Figure 13 shows that additional poses can be easily created using PSD. 5 CONCLUSIONS In this paper, we presented an approach to example based skeleton extraction. The proposed method is automatic except that the user needs to specify a source point on the reference pose. Due to many promising properties of harmonic 1-forms, such as resolution and embedding independence, we constructed a highly consistent one-to-one correspondence between reference and example poses. Therefore, even though the input models are in different resolutions, the extracted skeletons are highly consistent. The number and locations of the joints are totally determined by the dynamic geometry of the reference and example poses. Thus, the more poses are involved, the more accurate and robust skeletons we will obtain. Furthermore, for symmetric shapes, the extracted skeletons can also reflect the geometric symmetry of the input model. We applied the extracted skeletons in several applications such as skeleton transfer and pose space deformation. The limitation of the proposed method exists and demand further improvement in the future. Our current method only applies to the model which we can eliminate the ambiguity using the distance from critical points to the source point (usually on the head). However, this method does not work for octopus which has equal sized arms. In the near future, we plan to improve our method to apply to a wide range of objects. ACKNOWLEDGEMENTS This work was supported in part by the NTU SUG19/06 and AcRF RG69/07. The models are courtesy of Robert Sumner and Jovan Popovic (Horse) and Shin Yoshizawa (Armadillo). Figure 13: Constructing new poses using the extracted skeletons. REFERENCES [1] N. Amenta, S. Choi, and R. K. Kolluri. The power crust. In Proceedings of the sixth ACM symposium on Solid modeling and applications, pages , [2] O. K.-C. Au, H. Fu, C.-L. Tai, and D. Cohen-Or. Handle-aware isolines for scalable shape editing. ACM Trans. Graph., 26(3):83, [3] G. Aujay, F. Hétroy, F. Lazarus, and C. Depraz. Harmonic skeleton for realistic character animation. In Symposium on Computer Animation, pages , [4] I. Baran and J. Popovic. Automatic rigging and animation of 3d characters. ACM Trans. Graph., 26(3):72, 2007.

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9 high low Figure 7: Identifying the joints. 4 of 16 poses are shown here. Row 1 The color on each isoline illustrates the changes of the mean curvature between the reference and running poses. Row 2 By finding the local maximal of mean curvature changes, we can identify the joint candidates (green points) on the Reeb Graph. Note that the joints are closely related to the isolines whose mean curvatures change significantly. Figure 11: Optimizing the joint locations by solving a constrained optimization problem. (Please see the accompanying video demonstration). Figure 12: Cross parameterization and skeleton transfer using harmonic 1-form.