COMPUTER ANIMATION AND VIRTUAL WORLDS Comp. Anim. Virtual Worlds 2006; 17: 229–237 Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cav.126 *******************************************************************************************************************
Key-styling: learning motion style for real-time synthesis of 3D animationy By Yi Wang*, Zhi-Qiang Liu and Li-Zhu Zhou *********************************************************************************************
In this paper, we present a novel real-time motion synthesis approach that can generate 3D character animation with required style. The effectiveness of our approach comes from learning captured 3D human motion as a self-organizing mixture network (SOMN); of parametric Gaussians.The learned model describes the motion under the control of a vector variable called style variable, and acts as a probabilistic mapping from the low-dimensional style values to the high-dimensional 3D poses. We design a pose synthesis algorithm to allow the user to generate poses by specifying new style values. We also propose a novel motion synthesis method, the key-styling, which accepts a sparse sequence of key style values and interpolates a dense sequence of style values to synthesize an animation. Key-styling is able to produce animations that are more realistic and natural-looking than those synthesized with the traditional key-keyframing technique. Copyright # 2006 John Wiley & Sons, Ltd. Received: 10 April 2006; Revised: 2 May 2006; Accepted: 10 May 2006 KEY WORDS: motion synthesis; motion capture data; neural networks; self-organizing mixture network
Introduction Generally, 3D motion synthesis approaches can be categorized into three types: (1) key-framing, which requires intensive manual labor of artists to create a sequence of keyframes for each animation; (2) inverse dynamics/kinematics and physical simulation, which requires in-depth human knowledge about specific types of motions; (3) example-based synthesis, which is able to synthesize realistic animations by learning from motion capture data. Compared with the former two types, the example-based synthesis requires neither indepth human knowledge nor intensive manual labor. By designing automatic learning and convenient synthesis methods, we would be able to maximize the effectiveness of example-based synthesis. *Correspondence to: Yi Wang, Department Computer Science (Graduate School at Shenzhen), Tsinghua University at Shenzhen, Guang-Dong Province, 618055, China. E-mail:
[email protected] y Yi Wang is a Ph.D. Student, Zhi-Qiang Liu and Li-Zhu Zhou are professors. Contract/grant sponsor: Hong Kong RGC; contract/grant number: 1062/02E, CityU 1247/03E. Contract/grant sponsor: Natural Science Foundation of China; contract/grant number: 60573061.
In this paper, we present a novel and efficient example-based motion synthesis approach called keystyling. By extracting and representing characteristics of the training motion as a vector variable called style variable, we learn a novel neural network, the selforganizing mixture network (SOMN),1 from motion capture data as a probabilistic mapping from values of the style variable to 3D poses. Because a 3D pose is defined by the rotations of major body joints, it has to be represented by a high-dimensional vector. Whereas the characteristics of a certain kind of motion can usually be encoded by a few parameters, which constitute a much lower-dimensional style variable. So the learned probabilistic mapping allows the user to manipulate the lowdimensional style values instead of the complex highdimensional 3D poses. Because the learning algorithm is supervised, it ensures each dimension of the style variable has an explicit physical meaning. For example, the style variable of boxing motion may have two dimensions, where one encodes the change of body height for evading from being attacked, and the other encodes the stretch of arm for punching. Therefore, the user can give a few parameters to precisely define the style of a pose. This makes it easy and intuitive for the user to create a sparse sequence of keystyle values. The physical meaning of each dimensions of
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modeled is static poses other than dynamic motions. In Reference [4], Brand and Hertzmann proposed to learn the human motion under control of a style variable as an improved parametric hidden Markov model8 with an unsupervised learning algorithm.9 In this paper, we present a supervised learning algorithm to model the 3D poses under the control of the style variable. Compared with the works in the first column of Table 1 that model and synthesize motions, our method has the flexibility to synthesize both static poses and dynamic motions. Compared with the unsupervised pose synthesis methods as listed in the second row of Table 1, our supervised method ensures that each dimension of the style variable has explicit physical meaning, which allows the users to give precise style values to express their requirements of the synthesis results.
the style variable also makes it reasonable to interpolate the key-style sequence to generate a dense sequence of style values, which could be mapped to a realistic motion by the learned probabilistic mapping. We also develop a synthesis program, which allows the user to drag the mouse over a screen area representing the style space, which is an Euclidean space spanned by all possible style values. Because every mouse position corresponds to a style value, the trajectory of mouse movement could be mapped to a character animation in real-time. The rest of this paper is organized as follows. In Section 2, we explain our motivations by comparing various aspects of our work with previous researches. In Section 3, we explain the SOMN of parametric Gaussian model and derive its learning algorithm. Section 4 is about the synthesis of both static poses and dynamic motions. In Section 5, we show the usability and effectiveness of our prototype system by learning and synthesizing boxing motions.
Modeling Motion Style Separately from Motion Data The idea of extracting motion styles and modeling them separately from the motion data was originally proposed in the research area of gesture recognition and was well developed in Reference [8]. In Reference [4], the idea of separating style from data is used to develop a productive motion synthesis approaches that allows the users manipulate low-dimensional style values instead of the high-dimensional motion data. Because each frame of the motion data corresponds to a 3D pose, which is defined by the 3D rotations of dozens of major body joints, it has to be represented by a very high dimensional vector, usually over 60-dimensions.2 The high dimensionality makes it difficult to model and manipulate the motion data. On the contrary, the style variable is usually a low-dimensional vector (1 3dimensional in our experiments) that encodes a few important aspects of the motion. To support motion synthesis by manipulating in the low-dimensional style space and to keep maximum flexibility of synthesis, we learn a probabilistic mapping from style values to human poses, instead of motion sequences as in Reference [4], as a conditional probabilistic distribution (p.d.f.) Pðx j uÞ, where u is the style variable and x represents 3D pose.
Related Works Bayesian Learning for Example-Based Motion Synthesis Because 3D character motions are so complex, it is difficult to capture all the details with a deterministic model. In recent years, some top researches in the example-based motion synthesis area began to use the Bayesian learning framework to model the randomness feature of 3D motions. Some typical works are References [2–4], as listed in Table 1. In Reference [2], Li et al. used an unsupervised learning approach to learning possible recombinations of short motion segments as a segment hidden Markov model.5,6 In Reference [3], Grochow et al. used a non-linear principle component analysis method called Gaussian process latent variable model,7 to project 3D pose into a lowdimensional space, where each point in the space could be projected back to a 3D pose. In contrast with Reference [2], the learning approach proposed in Reference [3] is unsupervised, and the subject to be
Supervised approach Unsupervised approach
Learning (dynamic) motions
Learning (static) poses
Brand (1999)9 Li (2002);2 Brand (2000)4
This paper Grochow (2004)3
Table I. The placement of our contribution *******************************************************************************************************************
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stochastic approximation method to estimate the mixture model to achieve generally faster convergence rate and less probability of being trapped by local optima. In this paper, we derive a specific SOMN, which, unlike the traditional SOMN that models a joint distribution p(x), models a conditional probability distribution pðx j uÞ as a mixture model of,
A well-known model that represents a conditional distribution is the parametric Gaussian, whose mean vector is a function f (u). However, in order to capture the complex distribution of 3D poses caused by the complex dynamics of human motion, we model Pðx j uÞ as a mixture of parametric Gaussians. Although most mixture models are learned by the Expectation-Maximization (EM) algorithm, we derived an learning algorithm based on the (SOMN), which is in fact a stochastic approximation algorithm. Compared with the EM algorithm, which is a deterministic ascent algorithm,10 SOMN achieves faster convergence rate and is less probable to be trapped in local optima.1
pðx j u; LÞ ¼
where W j is called the style transformation matrix that together with mj and Sj form the parameter set lj of the j-th component.
The Learning Algorithm In SOMN, learning of parametric Gaussians minimizes the following Kullback–Leibler divergence1 between the true distribution pðx j u; LÞ and the estimated one p^ðx j u; LÞ, Z p^ðx j u; LÞ pðx j u; LÞdx (4) Dðp^; pÞ ¼ log pðx j u; LÞ
Mixture models is able to capture complex distributions over a set of observables X ¼ fx1 ; . . . ; xN g. Denote L as the set of parameters of the model, the likelihood of an mixture model is, aj pj ðx j lj Þ
(2)
where, each component pj() is a linear parametric Gaussian distribution, pj x j u; lj ¼ N x; W j u þ mj ; Sj (3)
The SOMN of Parametric Gaussians Model
K X
ai pj ðx j u; li Þ
i¼1
Learning SOMN of Parametric Gaussians
pðx j LÞ ¼
K X
(1)
j¼1
which is always a positive number and will be zero if and only if the estimated distribution is the same as the true one. When the estimated distribution is modeled as a mixture model, taking partial derivatives of Equation 4 with respect to li and ai leads to 2 3 ^ Z ^ @ p x u; L @ 1 6 7 Dðp^; pÞ¼ 4 5pðxÞdx @li @li ^ p^ x u; L
where each pj(x) is a component of the mixture model, aj is the corresponding weight of the component, and lj denotes the parameters of the j-th component. Given the observables, X ¼ fx1 ; . . . ; xN g, learning a mixture model is in fact an adaptive clustering process, where some of the observables are used to estimate a component, whereas some other observables are used to estimate other components. A traditional approach to learning a mixture model is the EM algorithm, which, as a generalization of the K-means clustering algorithm, alternatively executes an E-step and an M-step, where in the E-step each observable xi is assigned to a component pj to the extent lij; and in the M-step each pj is estimated from those observables xi with lij > 0. It has been proven in Reference [11] that this iteration process is actually a deterministic ascent optimization algorithm. The SOMN proposed by Yin and Allinson in 20011 is a neural network that has similar properties as the wellknown clustering algorithm, the self-organizing map (SOM), but in the SOMN, each node represents a component of a mixture model. The major difference between the learning algorithm of SOMN and the EM algorithm is that the former uses the Robbins-Monro
3 ^ ^ @ p x u; L @ 1 6 7 Dðp^; pÞ¼ 4 5pðxÞdx @a @ai ^ i ^ p x u; L 2 3 K @ 4X ^ i 15 a þj @ai j ¼ 1 2 3 Z ^ ^ p a x u; l 1 i 6 i i 7 ¼ ai 5pðxÞdx 4 ^ j^ a ^i p^i x u; L Z
2
where j is a Lagrange multiplier to ensure
P
i
(5) ai ¼ 1.
1 The Kullback-Leibler is a generalized form of the likelihood. The EM algorithm learns a model by maximizing the likelihood.
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By considering p^ði j x; uÞ as the Gaussian neighborhood function,12 we can consider Equation 8 exactly as the SOM updating algorithm. Although an updating rule for DSi may be derived similarly, it is unnecessary in the learning algorithm, because the covariance of each component distribution implicitly corresponds to the neighborhood function p^ði j xÞ, or the spread range of updating a winner at each iteration. As the neighborhood function has the same form for every node, the learned mixture distribution is homoscedastic.
As in Reference [1], we choose the Robbins–Monro stochastic approximation to solve Equation 5 because the true distribution is unknown and the equation depends only on the estimated version. We obtain the following set of iterative updating rules: " # ^ 1 @p^ðx j u; LÞ ^ ^ li ðt þ 1Þ ¼ li ðtÞ þ dðtÞ ^ @li ðtÞ p^ðx j u; LÞ " # ^i Þ ai @p^ðx j u; l ^ (6) ¼ li ðtÞ þ dðtÞ ^ @li ðtÞ p^ðx j u; LÞ "
^i Þ ai ðtÞp^ðx j u; l ^ i ðtÞ þ dðtÞ ^ i ðt þ 1Þ ¼ a a ai ðtÞ ^ p^ðx j u; LÞ ^ i ðtÞ dðtÞ½p^ði j x; uÞ ai ðtÞ ¼a
#
SOMN of Parametric Gaussians for Motion Synthesis
(7)
where d(t) is the learning rate at time step t, and p^ðx j u; LÞ P is the estimated likelihood, p^ðx j u; LÞ ’ i ai p^ðx j u; li Þ. The detailed derivation of Equation 5, 6, and 7 are similar to the derivations in Reference [1]. To derive the partial derivative of the component distribution in Equation 6, we denote Zi ¼ ½W i ; mi and ^ i Þ ¼ N ðx; W i u þ mi ; Si Þ ¼ V ¼ ½u; 1T , so that p^ðx j; u; l N ðx; Zi V; Si Þ. Then, by taking derivative of Equation 6 and solving the following equation, "
Determine the Physical Meaning of the StyleVariable A learned SOMN of parametric Gaussian model pðx j u; LÞ may be considered as a probabilistic mapping from a given style value ^u to a 3D pose x^. If the user knows the physical meaning of each dimension of the style variable u, he can give a precise style value ^u to
# @N ðx; Zi V; Si Þ Zi ðt þ 1Þ ¼ Zi ðtÞ þ dðtÞ ^ @Zi ðtÞ p^ðx j LÞ " # ai @ log N ðx; Zi V; Si Þ ¼ Zi ðtÞ þ dðtÞ N ðx; Zi V; Si Þ ^ @Zi ðtÞ p^ðx j LÞ @ log N ðx; Zi V; Si Þ ¼ Zi ðtÞ þ dðtÞ p^ði j xÞ @Zi ðtÞ 1 @ ðx Zi VÞT S1 ðx Zi VÞ ¼ Zi ðtÞ dðtÞp^ði j xÞ 2 @Zi 1 @ ðxT S1 x Zi VT S1 x xT S1 Zi V þ Zi VT S1 Zi VÞ ¼ Zi ðtÞ dðtÞp^ði j xÞ 2 @Zi 1 @ T 1 T 1 T 1 ^ ðx S x 2Zi V S x þ Zi V S Zi VÞ ¼ Zi ðtÞ dðtÞpði j xÞ 2 @Zi 1 @ @ ðZi VÞT S1 x þ ðZi VÞT S1 Zi V ¼ Zi ðtÞ dðtÞp^ði j xÞ 2 2 @Zi @Zi 1 @ @ VT ZTi S1 x þ ðVT ZTi S1 ÞðZi VÞ ¼ Zi ðtÞ dðtÞp^ði j xÞ 2 2 @Zi @Zi 1 1 T T ¼ Zi ðtÞ dðtÞp^ði j xÞS xV ZVV 2 ai
we arrived the updating rule for Zi: 1 DZi ¼ dðtÞp^ði j xÞS1 xVT ZVVT 2
specify the characteristics of the synthesized pose. The supervised learning framework presented in Section 3 allows the user to determine the physical meaning of the style variable prior to learning.
(8)
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be an Euclidean space, within which, a curve corresponds to smooth change of the style value. This is interesting for synthesizing character animations, instead of static poses, because the smooth change of motion style like body height and punch distance usually leads to smooth body movement. Experiments are shown in Section 5.
As an example, suppose that we captured boxing motion as training data, where the boxer sometimes crouches to evade from being attacked and some other times punches his fist to attack. We can use a 2dimensional style variable to describe the details of the boxing motion, where one dimension encodes the body height, which varies from crouching to standing up, and the other encodes the arm length when punching. Once the physical meaning of each dimension of the style variable is determined, the style values l ¼ fl1 ; . . . ; lN g of each of the training frames x ¼ fx1 ; . . . ; xN g can be calculated from the training motion itself. It is notable that if we carefully choose a number of dimensions of the style variable that encode visually independent characteristics of the training motion, the style space that is spanned by all possible style values, will
Generate 3D Pose from Given StyleValue Given a learned SOMN of parametric Gaussians pðx j u; LÞ with K components, mapping a given style value ^u to a 3D pose x^ can be achieved by substitute ^u into the model and draw a sample x^ from the distribution pðx j ^u; LÞ. Although the Monte Carlo sampling method
Figure 1. A prototype system for training the SOMN model from motion capture data. The graphical user interface makes it convenient for the user to browse the training 3D motion data, to discover stylistic characteristics, to specify parameters of the SOMN model, and to designate output files. *******************************************************************************************************************
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Figure 2. Two screen shots of the prototype system for real-time motion synthesis. The graphical user interface is adaptable to the change of dimensionality of the style variable. For training motions with style-dimensionality 3, the synthesis system creates a style panel including a set of slider-widgets to represent every dimensions of the style variable (as shown in the left). The user can drag the sliders to adjust the style values; For style-dimensionality < 3, users may choose to represent the style space with a screen area (as shown in the right), with the grey cross-sign denote currently select style values. If the user drags the cross-sign with mouse cursor, the system generates a motion sequences with the style of poses changes as the dragging curve.
is generally applicable for most complex distributions, to avoid the intensive computation and achieve realtime performance, we designed the following two-step algorithm as shown in Algorithm 1 to calculate the pose x^ with the highest probability. The first step of the algorithm calculates the poses f^ xj gKj¼1 that are most probable for each component pj of the learned model; and then the algorithm selects and returns the most probable one x^ among all the f^ xj gKj¼1 .
input: The given new style ^u output: The synthesized pose x^ calculate the most porbable pose from each component; foeach j 2 ½1; K do j x^j W j ^u þ mj ; end select the most probable one among the calculation result; j argmaxj aj pJ ð^ xj j^u; LÞ; x^ x^j ;
Synthesis with the Key-Styling Approach
Algorithm 1. synthesize pose from given style value
We developed an interactive graphical user interface (GUI) system as shown in Figure 2 to ease the pose and motion synthesis. With the parameter adjustment panel (to the left of the main window), the user is able to specify a style value by adjusting every dimension of the style variable. The changed style value is instantly input to Algorithm 1, and the synthesized pose x^ is displayed in real-time. With this GUI program, users can also create animations by (1) select a sparse sequence of key-styles to define the basic movement of a motion segment, (2)
produce a dense sequence of style values interpolating the key-styles, and (3) map each style value into a frame to synthesize the motion sequence. As the traditional method of producing character animations is called keyframing, which interpolate a sparse sequence of keyframes, we fittingly name our method key-styling. A known problem of keyframing is that the synthesized animation appears rigid and robotic. This is because the keyframes is represented by a highdimensional vector consisting of 3D joint rotations.
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use a 2-dimensional style variable to encode the body height and the distance of punching. Once the dimensionality of the style variable is determined, labeling the training frames with style values is not a difficult problem. For the application of automatic motion synthesis, we must have the skeleton (the connections of joints) for rendering the synthesized motion and must have the rotations of joints as training data. With these two kinds of information, it is easy to compute the style value ui for each training frame xi . ‘We wrote a motion browser program, as shown in Figure 1, to help the user browsing the captured motions, discovering motion styles, and writing embedded Perl scripts to derive style values from motion data.’ After estimating an SOMN of parametric Gaussians from the labeled training frames, we can give new style value by dragging the slide bars of our prototype motion
Evenly interpolating the rotations cannot ensure evenly interpolated dynamics. On the contrary, interpolating the key-styles results in smooth changes of the major dynamics and style-to-pose mapping adds kinematics details to the motion. So the change of kinematics details does not need to be evenly.
Experiments To demonstrate the usability our synthesis approach, we captured a segment of boxing motion of about 3 minutes under the frame-rate of 66 frame-per-second as the training data. Some typical poses in the motion is shown in Figure 3 (a), (b), and (c). Because the boxer sometimes crouches to evade and some other times punches out, we
Figure 3. (a)(c): Some typical poses in our training boxing motion, where, (a) a small body height value and small punch distance, (b) a large body height value and small punch distance, (c) large body height value and small large punch distance. (d) A short segment of synthesis motion — punching while crouching, which is generated by simply dragging the slide bar to change the punch distance. The starting pose is similar to the one shown in (a), while the ending pose has never appeared in the training motion. *******************************************************************************************************************
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synthesis system (as shown in Figure 2). The short animation shown in Figure 3(d) is generated by dragging the slide bar that represents the punch distance synthesizes. The accompanied demo video shows real-time synthesis of boxing animation by dragging mouse over a screen area representing the style space of boxing motion.
3. Grochow K, Martin SL, Hertzmann A, Popovic´ Z. Stylebased inverse kinematics. In Proceedings of ACM SIGGRAPH, 2004; pp. 522–531. 4. Brand M, Hertzmann A. Style machines. In Proceedings of ACM SIGGRAPH, 2000; pp. 183–192. 5. Ostendorf M, Digalakis VV, Kimball OA. From HMM’s to segment models: a unified view of stochastic modeling for speech recognition. IEEE Transactions on Speech and Audio Processing 1996; 4(5): 360–378. 6. Gales MJF, Young SJ. The theory of segmental hidden Markov models. Technical report, Cambridge Univ. Eng. Dept., 1993. 7. Lawrence ND. Gaussian process latent variable models for visualisation of high dimensional data. In Proc. 16th NIPS, 2004. 8. Wilson AD, Bobick A. Parametric hidden markov models for gesture recognition. IEEE Transactions on Pattern Analysis Machine Intelligence 1999; 21(9): 884–900. 9. Brand M. Pattern discovery via entropy minimization. In Artificial Intelligence and Statistics, vol. 7. Heckerman D, Whittaker C (eds). Morgan Kaufmann, Los Altos, 1999. 10. Yin H-J, Allinson NM. Comparison of a bayesian som with the em algorithm for gaussian mixtures. In Proceedings of Workshop on Self-Organizing Maps, 1997; pp. 304– 305. 11. Ormoneit D, Tresp V. Averaging, maximum penalised likelihood and bayesian estimation for improving gaussian mixture probability density estimates. IEEE Transactions on Neural Networks 1998; 9: 639–650. 12. Teuvo Kohonen. Self-Organizing Maps. Springer, Berlin, 2001.
Conclusion and Discussion In this paper, we presented a novel approach to synthesizing 3D character animations automatically and conveniently. The first step of our approach is to learn a probabilistic mapping from a low-dimensional style variable to high-dimensional 3D poses. By modeling the probabilistic mapping with an SOMN of parametric Gaussians, we designed a learning algorithm which is less prone to being trapped in local optima and converges faster than previous EM-based algorithms for learning mixture models. The supervised learning framework gives the user the flexibility to specify the physical meaning of each dimension of the style variable. As a result, given a learned model and using our prototype motion synthesis system, the user is able to create 3D poses by simply dragging slide-bar widgets and/or to produce character animations by our keystyling technique.
Authors’ biographies:
ACKNOWLEDGEMENTS
We sincerely appreciate Dr. Hu-Jun Yin of University of Manchester for his fruitful suggestions and detailed explanation on the SOMN model. We sincerely appreciate Professor Wen-Ping Wang of Hong Kong University for the constructive discussion about the motion synthesis approach. We gratefully acknowledge Microsoft Research Asia for providing the motion capture data. This work is supported in part by Hong Kong RGC Project No. 1062/02E, CityU 1247/03E, and Natural Science Foundation of China No. 60573061.
Yi Wang received the B.Sc. degree with first-class honors in computer science and engineering from the Changsha Institute of Technology, China, and is currently a Ph.D. student at the Tsinghua University, Beijing, China. He has been working at Microsoft Research Asia on photorealistic rendering as a visiting student, and working at the City University of Hong Kong on the theory of Bayesian learning. His major interests are designing programming languages and writing parsers/compilers/interepters for fun.
References 1. Yin H-J, Allinson NM. Self-organizing mixture networks for probability density estimation. IEEE Transactions on Neural Networks 2001; 12(2): 405–411. 2. Li Y, Wang T-S, Shum H-Y. Motion texture: a two-level statistical model for character motion synthesis. Proceedings of ACM SIGGRAPH, pages 465–472, 2002.
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Zhi-Qiang Liu (S -82-M -86-SM -91) received the M.A.Sc. degree in Aerospace Engineering from the Institute for Aerospace Studies, The University of Toronto, and the Ph.D. degree in Electrical Engineering from The University of Alberta, Canada. He is currently with School of Creative Media, City University of Hong Kong. He has taught computer architecture, computer networks, artificial intelligence, programming languages, machine learning, pattern recognition, computer graphics, and art & technology. His interests are scuba diving, neural-fuzzy systems, painting, gardening, machine learning, mountain/beach trekking, smart media systems, computer vision, serving the community and fishing. !
Li-Zhu Zhou is a full Professor of Department of Computer Science and Technology at Tsinghua University, Beijing, China. He received his Master of Science degree in Computer Science from University of Toronto in 1983. His major research interests include database systems, digital resource management, web data processing, and information systems.
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