ANI101 Lecture 10: Animation 3D Basic

by Diego
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Department
3D Animation
Course Code
3D Animation ANI101
Professor
Jason Brown

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To appear in the ACM SIGGRAPH conference proceedings Automatic Rigging and Animation of 3D Characters ∗ † Ilya Baran Jovan Popovic ´ Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology Abstract Animating an articulated 3D character currently requires manual rigging to specify its internal skeletal structure and to define how the input motion deforms its surface. We present a method for ani- mating characters automatically. Given a static character mesh and a generic skeleton, our method adapts the skeleton to the character and attaches it to the surface, allowing skeletal motion data to an- imate the character. Because a single skeleton can be used with a wide range of characters, our method, in conjunction with a library of motions for a few skeletons, enables a user-friendly animation system for novices and children. Our prototype implementation, called Pinocchio, typically takes under a minute to rig a character on a modern midrange PC. CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism—Animation Keywords: Animation, Deformations, Geometric Modeling 1 Introduction Figure 1: The automatic rigging method presented in this paper allowed us to implement an easy-to-use animation system, which Modeling in 3D is becoming much easier than before. User-friendly we called Pinocchio. In this example, the triangle mesh of a jolly systems such as Teddy [Igarashi et al. 1999] and Cosmic Blobs cartoon character is brought to life by embedding a skeleton inside (http://www.cosmicblobs.com/) have made the creation it and applying a walking motion to the initially static shape. of 3D characters accessible to novices and children. Bringing these static shapes to life, however, is still not easy. In a conventional skeletal animation package, the user must rig the character man- function. To make the optimization problem computationally feasi- ually. This requires placing the skeleton joints inside the charac- ble, we first embed the skeleton into a discretization of the charac- ter and specifying which parts of the surface are attached to which ter’s interior and then refine this embedding using continuous op- bone. The tedium of this process makes simple character animation more difficult than it could be. timization. The skin attachment is computed by assigning bone weights based on the proximity of the embedded bones smoothed We envision a system that eliminates this tedium to make an- by a diffusion equilibrium equation over the character’s surface. imation more accessible for children, educators, researchers, and Our design decisions relied on three criteria, which we also used other non-expert animators. For example, a child should be able to model a unicorn, click the “Quadruped Gallop” button, and watch to evaluate our system: the unicorn start galloping. To support this functionality, we need • Generality: A single skeleton is applicable to a wide vari- a method (as shown in Figure 1) that takes a character, a skeleton, ety of characters: for example, our method can use a generic and a motion of that skeleton as input, and outputs the moving char- biped skeleton to rig an anatomically correct human model, acter. The missing portion is the rigging: motion transfer has been an anthropomorphic robot, and even something that has very addressed in prior work [Gleicher 2001]. little resemblance to a human. Our algorithm consists of two main steps: skeleton embedding and skin attachment. Skeleton embedding computes the joint posi- • Quality: The resulting animation quality is comparable to tions of the skeleton inside the character by minimizing a penalty that of modern video games. • Performance: The automatic rigging usually takes under one ∗e-mail: [email protected] †e-mail: [email protected] minute on an everyday PC. A key design challenge is constructing a penalty function that pe- nalizes undesirable embeddings and generalizes well to new char- acters. For this, we designed a maximum-margin supervised learn- ing method to combine a set of hand-constructed penalty functions. To ensure an honest evaluation and avoid overfitting, we tested our algorithm on 16 characters that we did not see or use during devel- opment. Our algorithm computed a good rig for all but 3 of these characters. For each of the remaining cases, one joint placement hint corrected the problem. We simplify the problem by making the following assumptions. The character mesh must be the boundary of a connected volume. 1 To appear in the ACM SIGGRAPH conference proceedings The character must be given in approximately the same orientation the legs of the character in Figure 1 would be too short if a skeleton and pose as the skeleton. Lastly, the character must be proportioned extraction algorithm were used. roughly like the given skeleton. We introduce several new techniques to solve the automatic rig- Template Fitting Animating user-provided data by fitting a tem- ging problem: plate has been successful in cases when the model is fairly similar to the template. Most of the work has been focused on human mod- • A maximum-margin method for learning the weights of a lin- els, making use of human anatomy specifics, e.g. [Moccozet et al. ear combination of penalty functions based on examples, as 2004]. For segmenting and animating simple 3D models of charac- an alternative to hand-tuning (Section 3.3). ters and inanimate objects, Anderson et al. [2000] fit voxel-based ∗ volumetric templates to the data. • An A -like heuristic to accelerate the search for an optimal skeleton embedding over an exponential search space (Sec- Skinning Almost any system for mesh deformation (whether sur- tion 3.4). face based [Lipman et al. 2005; Yu et al. 2004] or volume based • Use of Laplace’s diffusion equation to generate weights for at- [Zhou et al. 2005]) can be adapted for skeleton-based deformation. taching mesh vertices to the skeleton using linear blend skin- Teichmann and Teller [1998] propose a spring-based method. Un- fortunately, at present, these methods are unsuitable for real-time ning (Section 4). This method could also be useful in existing 3D packages. animation of even moderate size meshes. Because of its simplicity and efficiency (and simple GPU implementation), and despite its Our prototype system, called Pinocchio, rigs the given charac- quality shortcomings, linear blend skinning (LBS), also known as ter using our algorithm. It then transfers a motion to the character skeleton subspace deformation, remains the most popular method using online motion retargetting [Choi and Ko 2000] to eliminate used in practice. footskate by constraining the feet trajectories of the character to the Most real-time skinning work, e.g. [Kry et al. 2002; Wang et al. feet trajectories of the given motion. 2007], has focused on improving on LBS by inferring the char- acter articulation from multiple example meshes. However, such 2 Related Work techniques are unsuitable for our problem because we only have a Character Animation Most prior research in character anima- single mesh. Instead, we must infer articulation by using the given tion, especially in 3D, has focused on professional animators; very skeleton as an encoding of the likely modes of deformation, not just little work is targeted at novice users. Recent exceptions include as an animation control structure. Motion Doodles [Thorne et al. 2004] as well as the work of Igarashi To our knowledge, the problem of finding bone weights for LBS et al. on spatial keyframing [2005b] and as-rigid-as-possible shape from a single mesh and a skeleton has not been sufficiently ad- manipulation [2005a]. These approaches focus on simplifying an- dressed in the literature. Previous methods are either mesh reso- imation control, rather than simplifying the definition of the artic- lution dependent [Katz and Tal 2003] or the weights do not vary ulation of the character. In particular, a spatial keyframing system smoothly along the surface [Wade 2000], causing artifacts on high- expects an articulated character as input, and as-rigid-as-possible resolution meshes. Some commercial packages use proprietary shape manipulation, besides being 2D, relies on the constraints to methods to assign default weights. For example, Autodesk Maya 7 provide articulation information. The Motion Doodles system has assigns weights based solely on the vertex proximity to the bone, the ability to infer the articulation of a 2D character, but their ap- ignoring the mesh structure, which results in serious artifacts when proach relies on very strong assumptions about how the character the mesh intersects the Voronoi diagram faces between logically is presented. distant bones. Skeleton Extraction Although most skeleton-based prior work 3 Skeleton Embedding on automatic rigging focused on skeleton extraction, for our prob- Skeleton embedding resizes and positions the given skeleton to fit lem, we advocate skeleton embedding. A few approaches to the inside the character. This can be formulated as an optimization skeleton extraction problem are representative. Teichmann and Teller [1998] extract a skeleton by simplifying the Voronoi skele- problem: “compute the joint positions such that the resulting skele- ton with a small amount of user assistance. Liu et al. [2003] use ton fits inside the character as nicely as possible and looks like the repulsive force fields to find a skeleton. In their paper, Katz and Tal given skeleton as much as possible.” For a skeleton with s joints (by “joints,” we mean vertices of the skeleton tree, including leaves), [2003] describe a surface partitioning algorithm and suggest skele- ton extraction as an application. The technique in Wade [2000] is this is a 3s-dimensional problem with a complicated objective func- most similar to our own: like us, they approximate the medial sur- tion. Solving such a problem directly using continuous optimiza- face by finding discontinuities in the distance field, but they use it tion is infeasible. Pinocchio therefore discretizes the problem by constructing a to construct a skeleton tree. For the purpose of automatically animating a character, however, graph whose vertices represent potential joint positions and whose skeleton embedding is much more suitable than extraction. For ex- edges are potential bone segments. This is challenging because the ample, the user may have motion data for a quadruped skeleton, graph must have few vertices and edges, and yet capture all poten- tial bone paths within the character. The graph is constructed by but for a complicated quadruped character, the extracted skeleton is likely to have a different topology. The anatomically appropriate packing spheres centered on the approximate medial surface into skeleton generation by Wade [2000] ameliorates this problem by the character and by connecting sphere centers with graph edges. techniques such as identifying appendages and fitting appendage Pinocchio then finds the optimal embedding of the skeleton into templates, but the overall topology of the resulting skeleton may this graph with respect to a discrete penalty function. It uses the still vary. For example, for the character in Figure 1, ears may discrete solution as a starting point for continuous optimization. be mistaken for arms. Another advantage of embedding over ex- To help with optimization, the given skeleton can have a lit- traction is that the given skeleton provides information about the tle extra information in the form of joint attributes: for example, expected structure of the character, which may be difficult to ob- joints that should be approximately symmetric should be marked as tain from just the geometry. So although we could use an existing such; also some joints can be marked as “feet,” indicating that they skeleton extraction algorithm and embed our skeleton into the ex- should be placed near the bottom of the character. We describe the tracted one, the results would likely be undesirable. For example, attributes Pinocchio uses in a supplemental document[Baran and 2 To appear in the ACM SIGGRAPH conference proceedings Figure 2: Approximate Medial Sur- Figure 3: Packed Spheres Figure 4: Constructed Graph Figure 5: The original and face reduced quadruped skeleton Popovic 2007a]. These attributes are specific to the skeleton but are spheres. In fact, this step typically takes less than 1% of the time of independent of the character shape and do not reduce the generality the entire algorithm. of the skeletons. Graph Construction The final discretization step constructs the 3.1 Discretization edges of the graph by connecting some pairs of sphere centers (Fig- ure 4). Pinocchio adds an edge between two sphere centers if the Before any other computation, Pinocchio rescales the character to spheres intersect. We would also like to add edges between spheres fit inside an axis-aligned unit cube. As a result, all of the tolerances are relative to the size of the character. that do not intersect if that edge is well inside the surface and if that edge is “essential.” For example, the neck and left shoulder Distance Field To approximate the medial surface and to facili- spheres of the character in Figure 3 are disjoint, but there should still be an edge between them. The precise condition Pinocchio tate other computations, Pinocchio computes a trilinearly interpo- lated adaptively sampled signed distance field on an octree [Frisken uses is that the distance from any point of the edge to the surface et al. 2000]. It constructs a kd-tree to evaluate the exact signed dis- must be at least half of the radius of the smaller sphere, and the closest sphere centers to the midpoint of the edge must be the edge tance to the surface from an arbitrary point. It then constructs the endpoints. The latter condition is equivalent to the requirement that distance field from the top down, starting with a single octree cell and splitting a cell until the exact distance is within a tolerance τ of additional edges must be in the Gabriel graph of the sphere centers the interpolated distance. We found that τ = 0.003 provides a good (see e.g. [Jaromczyk and Toussaint 1992]). While other conditions can be formulated, we found that the Gabriel graph provides a good compromise between accuracy and efficiency for our purposes. Be- balance between sparsity and connectedness. cause only negative distances (i.e. from points inside the character) are important, Pinocchio does not split cells that are guaranteed not Pinocchio precomputes the shortest paths between all pairs of to intersect the character’s interior. vertices in this graph to speed up penalty function evaluation. 3.2 Reduced Skeleton Approximate Medial Surface Pinocchio uses the adaptive dis- tance field to compute a sample of points approximately on the The discretization stage constructs a geometric graph G = (V,E) medial surface (Figure 2). The medial surface is the set of C - 1 into which Pinocchio needs to embed the given skeleton in an op- timal way. The skeleton is given as a rooted tree on s joints. To discontinuities of the distance field. Within a single cell1of our oc- tree, the interpolated distance field is guaranteed to be C , so it is reduce the degrees of freedom, for the discrete embedding, Pinoc- necessary to look at only the cell boundaries. Pinocchio therefore chio works with a reduced skeleton, in which all bone chains have traverses the octree and for each cell, looks at a grid (of spacing been merged (all degree two joints, such as knees, eliminated), as shown in Figure 5. The reduced skeleton thus has only r joints. τ) of points on each face of the cell. It then computes the gradient vectors for the cells adjacent to each grid point—if the angle be- This works because once Pinocchio knows where the endpoints of tween two of them is 120 or greater, it adds the point to the medial a bone chain are in V , it can compute the intermediate joints by surface sample. We impose the 120 condition because we do not taking the shortest path between the endpoints and splitting it in ac- cordance with the proportions of the unreduced skeleton. For the want the “noisy” parts of the medial surface—we want the points where skeleton joints are likely to lie. For the same reason, Pinoc- humanoid skeleton we use, for example, s = 18, but r = 7; with- chio filters out the sampled points that are too close to the character out a reduced skeleton, the optimization problem would typically surface (within 2τ). Wade discusses a similar condition in Chap- be intractable. Therefore, the discrete skeleton embedding problem is to find ter 4 of his thesis [2000]. the embedding of the reduced skeleton into G, represented by an r- Sphere Packing To pick out the graph vertices from the medial tuple v = (v 1...,v )rof vertices in V , which minimizes a penalty surface, Pinocchio packs spheres into the character as follows: it function f(v) that is designed to penalize differences in the embed- sorts the medial surface points by their distance to the surface (those ded skeleton from the given skeleton. that are farthest from the surface are first). Then it processes these points in order and if a point is outside all previously added spheres, 3.3 Discrete Penalty Function adds the sphere centered at that point whose radius is the distance The discrete penalty function has great impact on the generality and to the surface. In other words, the largest spheres are added first, quality of the results. A good embedding should have the propor- and no sphere contains the center of another sphere (Figure 3). tions, bone orientations, and size similar to the given skeleton. The Although the procedure described above takes O(nb) time in the paths representing the bone chains should be disjoint, if possible. worst case (where n is the number of points, and b is the final num- Joints of the skeleton may be marked as “feet,” in which case they ber of spheres inserted), worst case behavior is rarely seen because should be close to the bottom of the character. Designing a penalty most points are processed while there is a small number of large function that satisfies all of these requirements simultaneously is 3 To appear in the ACM SIGGRAPH conference proceedings Good embeddings (p ’s): difficult. Instead we found it easier to design penalties indepen- i dently and then rely on learning a proper weighting for a global Bad embeddings (q ’i): penalty that combines each term. b2 The Setup We represent the penalty function f as a linear com- P k bination of k “basis” penalty functions: f(v) = i=1 γi i(v). Pinocchio uses k = 9 basis penalty functions constructed by hand. Best Γ They penalize short bones, improper orientation between joints, length differences in bones marked symmetric, bone chains shar- ing vertices, feet away from the bottom, zero-length bone chains, Margin improper orientation of bones, degree-one joints not embedded at extreme vertices, and joints far along bone-chains but close in the graph [Baran and Popovic 2007a]. We determine the weights 0 b1 Γ = (γ 1...,γ )ksemi-automatically via a new maximum margin approach inspired by support vector machines. Figure 6: Illustration of optimization margin: marked skeleton em- Suppose that for a single character, we have several example em- beddings in the space of their penaltiesi(b ’s) beddings, each marked “good” or “bad”. The basis penalty func- tions assign a feature vector b(v) = (b 1v),...,b (k)) to each example embedding v. Let p ,.1.,p m be the k-dimensional fea- Learning Procedure The problem of finding the optimal Γ does ture vectors of the good embeddings and let 1 ,...,q ne the fea- ture vectors of the bad embeddings. not appear to be convex. However, an approximately optimal Γ is acceptable, and the search space dimension is sufficiently low Maximum Margin To provide context for our approach, we re- (9 in our case) that it is feasible to use a continuous optimization view the relevant ideas from the theory of support vector ma- method. We use the Nelder-Mead method [Nelder and Mead 1965] chines. See Burges [1998] for a much more complete tuto- starting from random Γ’s. We start with a cube [0,1] , pick random rial. If our goal were to automatically classify new embeddings normalized Γ’s, and run Nelder-Mead from each of them. We then into “good” and “bad” ones, we could use a support vector ma- take the best Γ, use a slightly smaller cube around it, and repeat. chine to learn a maximum margin linear classifier. In its sim- To create our training set of embeddings, we pick a training set plest form, a support vector machine finds the hyperplane that of characters, manually choose Γ, and use it to construct skeleton separates the p is from the q is and is as far away from them embeddings of the characters. For every character with a bad em- as possible. More precisely, if Γ is a k-dimensional vector with bedding, we manually tweak Γ until a good embedding is produced. ▯Γ▯ = 1` the classification margin of th´ best hyperplane normal to We then find the maximum margin Γ as described above and use Γ is 1 min i=1 Γ q −imax i=1 Γ p i . Recalling that the total this new Γ to construct new skeleton embeddings. We manually 2 T classify the embeddings that we have not previously seen, augment penalty of an embedding v is Γ b(v), we can think of the maxi- mum margin Γ as the one that best distinguishes between the best our training set with them, and repeat the process. If Γ eventually “bad” embedding and the worst “good” embedding in the training stops changing, as happened on our training set, we use the found set. Γ. It is also possible that a positive margin Γ cannot be found, in- In our case, however, we do not need to classify embeddings, dicating that the chosen basis functions are probably inadequate for finding good embeddings for all characters in the training set. but rather find a Γ such that the embedding with the lowest penalty f(v) = Γ b(v) is likely to be good. To this end, we want Γ to For training, we used 62 different characters (Cosmic Blobs distinguish between the best “bad” embedding and the best “good” models, free models from the web, scanned models, and Teddy models), and Γ was stable with about 400 embeddings. The weights embedding, as illustrated in Figure 6. We therefore wish to max- imize the optimization margin (subject to ▯Γ▯ = 1), which we we learned resulted in good embeddings for all of the characters in define as: our training set; we could not accomplish this by manually tuning the weights. Examining the optimization results and the extremal n T m T i=1 Γ q − iin i=1 . i embeddings also helped us design better basis penalty functions. Although this process of finding the weights is labor-intensive, Because we have different characters in our training set, and be- it only needs to be done once. According to our tests, if the basis cause the embedding quality is not necessarily comparable between functions are carefully chosen, the overall penalty function gener- alizes well to both new characters and new skeletons. Therefore, different characters, we find the Γ that maximizes the minimum margin over
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