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IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. X, NO. X, MONTH XXXX 1

Tiling Motion PatchesKyunglyul Hyun, Student Member, IEEE , Manmyung

Kim, Youngseok Hwang, and Jehee Lee, Member, IEEE

Abstract—Simulating multiple character interaction is challenging because character actions must be carefully coordinated to aligntheir spatial locations and synchronized with each other. We present an algorithm to create a dense crowd of virtual charactersinteracting with each other. The interaction may involve physical contacts, such as hand shaking, hugging, and carrying a heavy objectcollaboratively. We address the problem by collecting deformable motion patches, each of which describes an episode of multipleinteracting characters, and tiling them spatially and temporally. The tiling of motion patches generates a seamless simulation of virtualcharacters interacting with each other in a non-trivial manner. Our tiling algorithm uses a combination of stochastic sampling anddeterministic search to address the discrete and continuous aspects of the tiling problem. Our tiling algorithm made it possible toautomatically generate highly-complex animation of multiple interacting characters. We achieve the level of interaction complexity farbeyond the current state-of-the-art that animation techniques could generate, in terms of the diversity of human behaviors and thespatial/temporal density of interpersonal interactions.

Index Terms—Three-Dimensional Graphics and Realism, Animation

F

1 INTRODUCTION

Simulating multiple characters interacting with eachother is an emerging research topic in computer graph-ics. Modeling “interaction” between characters neces-sitates appropriate actions to occur in a carefully co-ordinated manner in both space and time. Preciselycoordinating the spatial location, timing, and actionchoices of many interacting characters is challenging andoften requires prohibitive computation and/or memoryresource.

We are particularly interested in creating a densecrowd of virtual characters that interact with each other.The interactions may occur between characters in closevicinity and often involve physical contacts, such ashand shaking, pushing, and carrying a heavy objectcollaboratively. Each character may perform an arbitrarysequence of actions, which must be spatially aligned andtemporally synchronized with other characters. Givena collection of action choices, each character can haveexponentially many action sequences and only a smallnumber of action sequences may satisfy stringent spa-tiotemporal constraints posed by interpersonal interac-tions. Finding such action sequences simultaneously formany interacting characters is extremely demanding.

We address the problem by collecting episodes ofmultiple characters. Each episode describes an interest-ing event featuring a small number of characters for ashort period of time. Tiling episodes seamlessly acrossspace and time would generate a crowd of denselyinteracting characters for an extended period of time.Each episode weaves a collection motion fragments to

• K. Hyun, M. Kim, Y. Hwang, and J. Lee are with the School of ComputerScience and Engineering, Seoul National University, Seoul, 151-744,Korea. Email: {kyunglyul, mmkim, youngseok, jehee}@mrl.snu.ac.kr

form a motion patch [1]. The motion patch has its en-tries and exits, which correspond to the beginning andend frames, respectively, of its motion fragments. Twomotion patches can be stitched together by carefullyaligning their entries and exits in both space and time.Our major challenge is tiling patches tightly such thatthe tiling does not have any dangling entries or exits,which cause sudden appearance and disappearance ofcharacters in the crowd.

The tiling is a combinatorial problem of layering acollection of motion patches. It also involves continu-ous optimization of patch locations for packing patchesdensely while avoiding collisions. Feasible solutions arerare and exhaustive search is not feasible. Our tilingalgorithm uses a combination of stochastic samplingand deterministic search to address the discrete andcontinuous aspects of the problem. Our motion patchesare deformable to add flexibility in stitching patches andthus alleviate the difficulty of tiling. The deformablepatch allows its motion fragments to warp smoothlywhile maintaining the coherence of interpersonal inter-action captured in the patch.

Our algorithm makes it possible to automatically gen-erate highly-complex animation of multiple interactingcharacters. Our algorithm takes raw motion data as in-put. The input motion data may capture a single subjector multiple subjects acting out in long clips. A collec-tion of motion patches are identified semi-automaticallyfrom the input motion data. Perfect tiling of motionpatches achieves the level of interaction complexityfar beyond the current state-of-the-art that animationtechniques could generate, in terms of the diversity ofhuman behaviors and the spatial/temporal density ofinterpersonal interactions. The spatiotemporal nature ofour tiling algorithm allows us to deal with both staticand dynamic virtual environments. We can also control

2 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. X, NO. X, MONTH XXXX

Fig. 1. Multiple characters are interacting with each other. A diversity of human behavior, such as jumping, crawling,pushing, bowing and slapping, are simulated in the automatically-generated, interpenetration-free animation.

the generation of animation by specifying when andwhich actions to occur at any specific location of theenvironment.

We further elaborate the tiling algorithm based on thereversible jump Markov Chain Monte Carlo (MCMC)framework [2] to allow the tiling to be updated incre-mentally according to the changes of the environmentand user control. The reversible jump MCMC algorithmis a flexible method for solving discrete optimizationproblems, when the number of free parameters changesthrough the optimization process. In our problem, thedimension of the free parameter vector corresponds tothe number of patch instances in the tiling. Reversiblejumps in the MCMC algorithm enable the incrementalinsertion and deletion of motion patches.

2 RELATED WORK

There exists a vast literature on crowd simulation in com-puter graphics. Crowd simulation refers to the processof simulating the movement of a number of pedestrians,which are equipped with action skills mostly for navigat-ing in virtual environments and capability of replicatinghuman collective behavior. Interaction behaviors, suchas collision avoidance [3], [4], visual perception [5], andforming groups/formations [6] to name a few recentones, have been explored. Multiple robot coordinationpursues a similar goal with much emphasis on collisionavoidance [7], [8]. Multiple robots are provided withstart and goal configurations and search for collision-free paths simultaneously. It should be noted that ourproblem is different from crowd simulation and multi-agent path planning. We assume that interpersonal inter-actions would occur frequently and thus we search fora distribution of interactions rather than moving pathsbetween start and goal configurations. Our characters areprovided with a diversity of actions (beyond the scopeof locomotion) to choose from. Every action should be

precisely coordinated to meet alignment and synchro-nization posed by frequent interactions.

Research on multiple character interaction has recentlyemerged. Interactive manipulation techniques for editingmultiple character motions has been studied with avarying level of control specification [9], [10], [11]. A lotof researchers have focused on interaction between twocharacters. Liu et al. [12] studied an optimization methodto synthesize two character motions with physics-basedobjectives and constraints. Kwon et al. [13] simulatedcompetitive interaction between two Taekwondo play-ers using dynamic Bayesian networks. Komura and hiscolleagues addressed a similar problem using min-maxsearch and state-space exploration [14]. Wampler et al.simulated two-player adversarial games based on game-theoretic behavior learning [15]. The state-explorationand learning-based approaches often suffer from thecurse of dimensionality. Adding extra participants causesexponential increase in either computation time or mem-ory storage usage. Thus, the techniques that worked wellwith two characters do not scale easily to deal with manycharacters.

In computer animation, metaphorical patches havebeen used to encapsulate either pre-computed or motioncapture data. Tiling patches spatially and temporallygenerates animation of larger scales. Chenney [16] dis-cussed the use of square tiles embedding flow patterns.Motion patches by Lee et al. [1] refer to a collectionof building blocks of a virtual environment, which areannotated with motion data available on the blocks.Shum et al. [17] generalized the idea further to encap-sulate short-term interactions between two charactersinto interaction patches and combined them to synthesizelarger-scale interaction among many characters. Yersinet al. [18] tiled crowd patches to generate an arbitrarilylarge crowd of pedestrians in an on-the-fly manner. Ourwork is on the extension of these patch-based synthesisapproaches. Given a collection of irregular patches, the

HYUN et al.: TILING MOTION PATCHES 3

notorious problems of tiling are gaps (temporal discon-tinuity) and overlaps (interpenetration between charac-ters). The temporal gaps between dangling entries andexits would cause characters to stay frozen over the gapor suddenly disappear and reappear at the boundaries.Our goal is tiling arbitrary irregular patches denselywithout gaps and overlaps.

The problem of packing irregular shaped pieces ina fixed area has been widely studied in geometric op-timization and computational geometry [19] [20]. Theshape packing idea has been exploited in various graph-ics applications. Kim and Pellacini [21] generated imagemosaics by filling an arbitrarily-shaped container withimage tiles of arbitrary shapes. Fu et al. [22] gener-ated a set of quad-mesh to produce a tiled surface ofcomputer-generated buildings. Merrell et al. [23] pre-sented a method for automatic generation of residentialbuilding layouts. Packing and reconfiguring geometricshapes often pose complex optimization problems thatinvolve both discrete changes to the geometric structureand continuous changes to the shape parameters. TheMCMC sampling and its variants have been explored toaddress such complex optimization problems [24] [25].Our approach were inspired by the research on geo-metric shape packing and we address further challengesthat are unique in character animation. The challengesinclude the control of complex human behavior, han-dling multi-character interaction, identifying deformablemotion patches as basic units of tiling, and the fourdimensional tiling of deformable patches.

3 DEFORMABLE MOTION PATCHESMultiple characters interacting with each other createinteresting episodes, which can serve as basic buildingblocks of complex multi-character scenes. Each episodecan be modeled as a motion patch, which describesthe actions of characters for a short period of time.Specifically, each motion patch includes a collection ofmotion fragments and their associated characters. In-teraction between characters poses constraints betweenmotion fragments. For example, handshaking indicatesthat the relative position and direction of two charactersare constrained with each other while they are shakingtheir hands together. The motion patch may also includeenvironment features and props that provide context tothe characters’ actions. The beginning and end framesof a motion fragment are called entries and exits, respec-tively, of the patch. We can stitch two motion patches ifan exit of one patch matches an entry of the other. Thecharacter’s position, direction, fullbody pose and timingshould match within user-specified thresholds such thatthe character can make a smooth transition from onemotion fragment to the other.

A tiling is a collection of motion patches that arestitched with each other. The tiling is perfect within aspatiotemporal region if it does not have any danglingentries or exits that do not have matching exits or entriesin the tiling. More specifically,

Interpersonal

interactionManipulation

handle

Fig. 2. The deformation of a motion patch, which showstwo characters coming close, facing each other, hand-shaking, and walking away. The manipulation handlesand interpersonal interaction are formulated as linear con-straints. In this example, the interpersonal interaction isthe relative body location and direction of two characterswhen their hands hold each other.

Patch Deformation Our motion patches are de-formable to allow flexibility in tiling. The deformation iscomputed in two steps: Path deformation and fullbodymotion refinement. At the first step, we consider a webof motion paths (see Figure 2). Each motion path is thetrajectory of the skeleton root in a motion fragment. Alocal coordinate system is defined at every point on themotion path such that the position and orientation ofthe character’s body can be represented with respectto the local coordinate system. A collection of motionpaths are constrained to each other using linear inter-personal constraints. Given constraints, a motion pathundergoes deformation as if it is an elastic string. The po-sition/direction/time of entries and exits operates as if itis a manipulation handle, which can also be formulatedas linear constraints. The manipulation handle is spa-tiotemporal in the sense that it specifies both body po-sition/direction and the timing of the character’s action.As a manipulation handle is dragged, a web of motionpaths undergo deformation in an as-rigid-as-possiblemanner while maintaining a collection of linear con-straints induced either from interpersonal interactions oruser manipulation of motion paths. Specifically, we usemulti-character motion editing by Kim et al. [10], whichsolves for the spatiotemporal deformation of multiplemotion paths simultaneously subject to linear constraintsbased on Laplacian equations. The character’s fullbodyposes along motion paths are refined later to maintainhand-object contacts and rectify foot sliding artifacts [26].

Stitching Patches. We can stitch two deformablepatches if there exist good matches at their correspond-ing entries and exits, through which characters can makesmooth transitioning from one patch to the other. Let{(pAi , tAi ) ∈ R2 × R | i = 0, · · · , N} be a motion pathin patch A, where pAi is the two-dimensional positionof the character projected onto the ground surface andtAi is its timing. Note that the frames of motion dataare usually sampled uniformly in time, though it isnot necessary. The timing of motion or the interval ofeach individual frame may change as the motion dataundergo deformation. The first frame (pA0 , θ

A0 ) is an


Exit

Entry

Matching

entry and exit

Fig. 3. Stitching deformable patches at their matchingentries and exits. Each patch is depicted as a convexpolygon that encloses its motion paths projected on theground. The convex polygons are used only for visualizingthe extent of motion patches.

entry of patch A and the last frame (pAN , θAN ) is its exit.

Assume that the dissimilarity between the configuration(the position, direction, timing, and fullbody pose) at anexit of patch A and the configuration at an entry of patchB is below user-provided thresholds. Concatenating theLaplacian equations of two patches into a single largersystem of linear equations and adding three boundaryconstraints would deform two patches to stitch themseamlessly.

pAN = pB0 (1)

pAN − pAN−1 = pB1 − pB0 , (2)

tAN = tB0 (3)

which enforce smooth transiting in position, directionand timing, respectively. If there are more than onematching entry/exit pairs, we specify three transitioningconstraints for each matching pair.

4 PATCH CONSTRUCTION

Achieving a perfect tiling is challenging even though theflexibility of patches alleviates its difficulty to a certaindegree. It is important that a collection of motion patchesshould be designed carefully for better connectivity. Themotion patch is supposed to encapsulate an interestingepisode that may entail complex spatiotemporal relation-ships among multiple characters. On the other hand, itis expected that the boundary (entries and exits) of thepatch should be as simple as possible. The simplicityof the patch boundary facilitates the process of patchstitching.

Given a collection of raw multi-character motion data,we would like to construct a collection of motion patchesto be used for tiling. It involves following steps (seeFigure 4):• Identify interesting episodes from raw motion data.

Raw Data Episode Identification Boundary Determination

Fig. 4. The procedure of patch construction

• Specify interpersonal and person-object constraintsto preserve the spatiotemporal integrity of multi-character interaction under patch deformation.

• Determine the boundary of motion patches.Though there is no limitation on the size of motionpatches, we usually design each patch to have lessthan five characters. If individual patches are too large,finding a perfect tiling could be difficult. Small patchesincluding a single character are useful because they fiteasily into narrow space. In practice, it is favorable tohave a mixture of small, single-character patches andlarge, multi-characters ones for achieving good tilings.

4.1 Episode DetectionThe episode refers to an event among multiple charactersthat is important or unusual. We use three criteria todetect such an event from raw motion data.

Contact. The physical contact between characters, suchas handshaking, pushing, high-five, and punching, indi-cates the occurrence of interesting events.

Proximity. Even though there is no physical contact, itis noteworthy if two characters come close to each otherand either of them reacts to the approach of the other.The presence of reaction is detected by measuring howmuch the character moves in a window of consecutivemotion frames. The estimated variation of motion atframe i within window [i− w, i+ w] is:

Vi =

w∑j=−w

w∑k=j+1

‖Pi+j Pi+k‖2 (4)

where Pi+jPi+k is the difference between two fullbodyposes ignoring their horizontal translation and rotationabout the vertical axis [1]. The value of Vi above a certainthreshold implies there might be a noticeable action atframe i. The threshold has been chosen empirically bytrial and error.

Synchronicity. Even though two characters do notcome close to each other, it is also noteworthy if twocharacters perform interesting actions simultaneouslyand they are aware of each other’s action. Assume thatmotion data include two characters A and B interactingwith each other. Character A has its motion data withindex i, and character B has its own data with index j.The i-th frame of A and the j-th frame of B are con-current in the global, reference time. The synchronicitybetween two characters is:

Sij = V Ai · V Bj · e−g(A,B) · e−g(B,A), (5)


where g(A,B) is the angle between the gaze directionof character A and the vector from the face of A tothe upperbody of B. This term estimates the possibilityof character A being aware of the action of B at themoment.

The detection of a noticeable event between twocharacters results in imposing interpersonal constraintsbetween relevant motion data. The constraints enforcethe relative position, direction, and their timing of ac-tions remain unchanged even though their motion dataundergo deformation. If we detect multiple events oc-curring nearby spatially and temporally, we merge theminto a single, larger event, which may include interactionamong more than two characters.

4.2 Boundary Determination

Once interesting events are detected, the next step isto determine their boundaries (entries/exits) to forma collection of tillable motion patches. There are sev-eral requirements. First, we would like to have theboundaries eventless. Any interaction/event occurringat boundaries would make the stitching process compli-cated. We want to have patches as compact as possible.Smaller patches tend to allow more flexibility in tilingand denser scenes to be created. Lastly, we want tohave as little diversity of fullbody poses at boundariesas possible. Having a wide variety of fullbody posesat entries and exits would lead to difficulty of findinggood matches in patch stitching. Note that we determinethe boundaries of all patch collection simultaneously tosatisfy the requirements.

Raw motion data include a multiplicity of fullbodyposes at frames. Our algorithm for boundary determi-nation begins by clustering fullbody poses into groups.Specifically, we use agglomerative hierarchical clusteringsuch that the dissimilarity between any poses in a groupis below a user-specified threshold. The threshold ischosen to allow smooth, seamless transitioning betweenmotion fragments if the poses selected from a singlegroup are at the matching entry and exit.

Let G1, · · · ,Gk be k largest pose groups sorted indescending order (typically, k = 10 in our experiments).We intend to choose a small number of groups thatsuggest candidate frames to be entries and exists. Then,the boundary may be determined to surround each eventtightly at frames that belong to the groups. The numberof groups used is directly related to the diversity offullbody poses, so we would like to choose as fewergroups as possible. To do so, we consider all possiblesubsets, {Gi|1 ≤ i ≤ k} of cardinality one, {(Gi,Gj)|1 ≤i < j ≤ k} of cardinality two, and so on. We examineall possible subsets and choose the one that minimizethe weighted sum of the cardinality of the subset andthe total size (number of motion frames) of motionpatches. The coefficient weighs the diversity against thecompactness of motion patches.

5 TILING ALGORITHM

The goal of tiling is packing patches in a user-specifiedspatiotemporal region without any dangling entries orexits while achieving the desired density of multi-character interactions and avoiding interpenetration be-tween them. Specifically, our tiling algorithm is basedon simulated annealing, which is known to be an ef-fective sampling method for combinatorial optimizationproblems [27]. Though simulated annealing may havethe potential to reach a global optimal solution, it takes aprohibitive computation time to remove dangling entriesand exits completely. Our algorithm is a variant ofsimulated annealing, consisting of two phases. The firstphase of the algorithm is intended to sample patchesrapidly in the spatiotemporal region, while the secondphase is a deterministic search and jittering procedurededicated to removing dangling entries/exits.

Algorithm 1 Stochastic Sampling for tiling motionpatches

P : A tiling of motion patchesR() : A random number between 0 and 1

1: T ← Tmax

2: i← 03: while i < imax do4: Pnew ← (a random patch instance)5: P′ ← P ∪Pnew

6: ∆E ← Etiling(P′)− Etiling(P)7: if e−∆E/kT > R() then8: if IsValid(P′) and NoCollision(P′) then9: P← P′

10: i← 011: end if12: end if13: T ← T −∆T14: i← i+ 115: end while

5.1 Stochastic SamplingGiven a collection of motion patches, the firstphase of our tiling algorithm is based on stochas-tic sampling of patches (see Algorithm 1). The posi-tion/direction/timing of patch instances are randomlychosen in a virtual environment and added to the tilingone by one (line 4–5). The sampling procedure minimizesan energy function

Etiling =Nd

Nc +Nd+ α(

1

Nc +Nd), (6)

where Nc and Nd are the number of connected anddangling entries/exits, respectively. The first term pe-nalizes the occurrence of dangling entries/exits, whilethe second term encourages rapid sampling of patches.As the number of patches in the tiling increases, it tendsto have more dangling entries/exits. α is a constant that


balances between the density of patches and the numberof dangling entries/exits. Each sample of a motion patchis accepted with certain probability based on simulatedannealing (line 6–7). When the simulated temperatureis high, the samples are easily accepted even thoughthey do not improve the tiling energy much. As the tem-perature cools down, the system favors steady downhilltraces to reach a stable solution. The Boltzman parameterk and the annealing schedule (Tmax and ∆T ) controls therate of convergence. The algorithm terminates when thesampling procedure fails too many times (line 3).

Stitching, Deformation and Collision. As explainedin the previous section, stitching two patches leads toa large system of linear equations that concatenatestwo systems of Laplacian equations. Adding a newpatch to the tiling of patches involves all patches inthe tiling and simply concatenating all linear systemsinto a single huge one is often infeasible (line 5). Weassume that that the patches directly adjacent to the newone is deformable and all the others are pinned down.This assumption allows a practical trade-off between thecomputation time and the quality (in terms of flexibilityand deformation) of tiling. The new patch is valid if itconnects well to the tiling without incurring interpene-tration or excessive deformation (line 8). We use a simplebounding box test on individual frames to check if thereis interpenetration between two patches. We measurethe degree of deformation on a per-frame basis. In ourexperiments, we allowed motion path deformation up topositional stretching of 3cm, bending of 7 degrees, andtime shifting of 0.018 second per frame.

5.2 Deterministic Search

Even though the energy function used in the first phasetends to minimize the number of danglings, it usuallyconverges very slowly (see Figure 6). The second phaseof the algorithm is based on combinatorial explorationand randomized jittering in a continuous domain (seeAlgorithm 2). It examines every possible patch sequencesfrom each dangling entry/exit until there remains nomore dangling (line 2). Such an exhaustive search isinfeasible in large free space. However, the patch tilingobtained from the first phase of the algorithm coversthe target domain uniformly because of the nature ofrandom sampling. The free space is fragmented intonarrow spots, in which patch connection is strictly con-strained. In the narrow, fragmented spaces, even exhaus-tive search is not computationally expensive any more.

For any dangling, the algorithm iterates over all pos-sible patch choices that have an entry/exit matchingthe dangling (line 3). We first check if the numberof danglings is increased by the new patch (line 5).Otherwise, we check further if the new patch incursinterpenetration or excessive deformation (line 8). If thenew patch is not valid, we explore a continuous searchdomain by randomly jittering its spatial location, direc-tion and timing (line 12). In our experiments, the jittering

Algorithm 2 Deterministic search for removing dan-glings

P : A tiling of motion patchesD : A set of dangling entries and exitsN(P) : the number of dangling entries and exits in P

1: while D 6= ∅ do2: for d ∈ D do3: for (Pnew that can resolve d) do4: P′ ← P ∪Pnew

5: if N(P′) ≤ N(P) then6: i← 07: while i < imax do8: if IsValid(P′) and NoCollision(P′) then9: P← P′

10: break11: end if12: P′ ← P ∪ Jitter(Pnew)13: i← i+ 114: end while15: end if16: if d is not dangling then17: break18: end if19: end for20: if d is dangling then21: P← P \P(d)22: end if23: end for24: end while

range is initially set as a circle of radius 0.6m in XZ-plane, an angle interval of -15 to 15 degree, and a timeinterval of -0.3 to 0.3 seconds. As the iteration continues,the jittering range expands gradually to explore a widerrange. The maximum range is a circle of radius 1.0m, anangle interval of -25 to 25 degree, and a time interval of -0.5 to 0.5 seconds. If the dangling entry/exit has no patchto connect with it, the patch with the dangling entry/exitis removed from the tiling (line 20–21). The removal ofthe patch makes yet another dangling entries/exits. Asthe algorithm iterates, it will examine other possibilitiesthat has not been explored before to eventually removedanglings completely (line 3). We keep track of samplinghistory so that patches removed in line 21 is not chosenagain in line 3.

5.3 User Control

Though our tiling algorithm is capable of automaticallygenerating a crowd of interacting characters, it can alsoallow the user to have direct and indirect control over thegeneration of animation. The user control may influencethe motion of each character indirectly by specifying avariety of environment features or the ratio/preferencesof actions. The user control can be more direct and imme-diate to specify which action to occur when and where


Fig. 5. A small example for illustrating the tiling procedure. The spatiotemporal volume consists of a square spatialregion of 10 m × 10 m and a time interval of 600 seconds. The time interval is depicted as dashed lines. We croppedthe animation at dashed lines. The screenshots on the right side were taken at the frame the red lines indicate. (Up)The first phase of our algorithm generated 67 patches featuring 29 characters. (Down) The second phase of thealgorithm added new patches to fill in gaps. The final result included 197 patches featuring 23 characters. Note thatthe number of characters decreased, even though it has more patches. It is because the new patches filled in the gapsbetween the trajectories of characters 5 and 7, 8 and 20, 9 and 25, 16 and 29, and 21 and 23 to merge them.

Fig. 6. The plots of the tiling energy and the numberof dangling entries/exits with respect to the computationtime. The red line shows the time when the second phaseof the algorithm begins.

in the environment. We annotate each patch with labels,such as slow/fast and friendly/hostile, and allows theuser to decide how frequently each class of actions tooccur in the animation. More specifically, we define apreference function

F = e−f1(x,y)/σ1 · e−f2(θ)/σ2 · e−f3(t)/σ3 · e−f4(l)/σ4 , (7)

where (x, y) is the position of a patch instance, θ is itsorientation, t is the timing in the reference time, and lis its label. fi controls the tendency of patch distributionand σi weighs the significance of the terms. For example,f1(x, y) = (xd − x)2 + (yd − y)2 if we want to control thecharacters to move to their target location (xd, yd). If wedo not have any preference over labels, then f4(l) = 0for any l. fi can be defined in various ways to providethe user with flexibility and versatility in control spec-ification. The preference function applies differently tothe aforementioned algorithms. In the random samplingprocess (line 4, Algorithm 1), the normalized preferencefunction is used as a probability distribution of samplingPnew. In the deterministic search (line 3, Algorithm 2),the preference function prioritizes the candidates to de-


termine which patches to examine first.Interactive Control. The second phase of the algo-

rithm can be easily modified to deal with interactivecontrol of a small number (less than a dozen) of char-acters. Two minor modifications are required. One isthe way patch instances are sampled in time. Whilethe original algorithm has a fixed time horizon, theinteractive version of the algorithm advances the timehorizon incrementally and patch instances are sampledcontinuously at runtime to fill up the horizon. The otheris the way the tiling is deformed when a new patchinstance is added. The original algorithm allows both thenew patch and existing patches in the tiling to undergodeformation, while the interactive version allows onlythe new patch to deform. The rationale is that patchinstances in the tiling is already in the past and shouldnot change.

6 LOCAL UPDATE ALGORITHM

Given a perfect tiling of motion patches, we may wantto update the tiling locally according to the changesof the environment and user control. This involves theinsertion and deletion of motion patches incrementallyfrom the tiling. The first phase of the algorithm describedin the previous section only allows the insertion of patchinstances. We employ a variant of the Reversible JumpMarkov Chain Monte Carlo (MCMC) framework to de-sign a local update algorithm. The simulated annealing isa classical MCMC method. The reversible jump MCMCprovides better flexibility and generalization capabilityover simulated annealing.

6.1 Reversible Jump MCMCLet T be the space spanned by given patches suchthat an element x ∈ T is a tiling of those patches.The dimension of x depends on the number of patchinstances in the tiling. If we define probability distri-bution P on T , we can sample random tilings fromthat distribution and it belongs to the class of MCMCsampling. Metropolis-Hastings algorithm is well-knowntechnique for MCMC when direct sampling is difficult.The algorithm generates a sequence of random samplesusing probability distribution P and proposal density Q.Intuitively speaking, P (x) explains how good the tilingis. The goodness is evaluated based on the target density,the number of danglings, and the number of collisions.Q elucidates our preference of which patches to samplewhere/when in the tiling. If we do not have any pref-erence, Q is a uniform distribution (constant value). Ittakes new sample x′ based on the current sample x andtheir proposal density. Then it either accepts or rejectsthe new sample with acceptance ratio

αx→x′ = min

{1,P (x′)Q(x|x′)P (x)Q(x′|x)

}.

If the new sample x′ is rejected, the previous sample x isconsidered as next sample. Note that we only need the

ratio of probabilities P (x′)/P (x), but exact probabilityP (x) is not required. In the context of patch tiling, anew tiling x′ is chosen by inserting a new patch instanceto tiling x, removing a patch instance from tiling x, orjittering the location of a patch instance. Therefore, thedimension of x′ may not match the dimension of x. Thediscrepancy in dimensionality could make the transitionirreversible. The presence of irreversible transitions canbe problematic, because the Markov chain cannot samplea certain portion of the space anymore and may lead tolocal optima.

Reversible jump MCMC is a variant of MCMC meth-ods, which allows dimension jumping [2]. The dimen-sion of tilings in T is not uniform, because the dimensionof a tiling depends on the number of patch instances.The dimension matching function fn,m(xm, un,m) of re-versible jump MCMC enables transitions between dif-ferent dimensions. un,m is a random variable whichsupplement generating a new state x′m ∈ Rm fromprevious state xn ∈ Rn. The reversibility in dimensiontransition is guaranteed with acceptance ratio

αxn→x′m = min

{1,P (x′m)Q(xn|x′m)

P (xn)Q(x′m|xn)

∣∣∣∣∂fn,m(xn, un,m)

∂(xn, un,m)

∣∣∣∣} .The proposal function Q(x′m|xn) = J(n,m)Q(un,m|n,m)where J(n,m) is the probability of jumping from dimen-sion n to dimension m and Q(un,m|n,m) is the proposaldensity of the random variable un,m. The Jacobian ofthe dimension matching function prevents irreversiblejumps.

6.2 Tiling with MCMCWhile incrementally updating a given tiling, we wouldlike to minimize the energy function

E(P) = we|ρo − ρ|+ wdNd + wcN2o ,

where Nd is the number of danglings, No is the numberof collisions, and ρo and ρ are the target and currentdensity of tiles. we, wd and wc weighs three terms withrespect to each other. In our experiments, weight valuesare 3, 0.02, and 0.01, respectively. The target probabilityis

P (P) ∝ e−E(P).

Note that we do not need to normalize the probabilityfunction, since only the ratio of probabilities P (x′)/P (x)is required in the sampling algorithm. We set jumpingprobability equally for insertion and removal.

J(n, n+ 1) = J(n, n− 1) =1

2.

For n ≤ m, function fn→m appends random variablesto match the dimension of the source and target statessuch that xm = fn→m(xn, un,m) = (xn, un,m) where un,mis a uniform random variable that consists of the type,position, orientation, and timing of a new patch intance.We define the proposal density Q such that

Q(un,m) = U(C)× U(x, y)× U(θ)× U(t),


Algorithm 3 Delayed Rejection Reversible Jump MCMCP : A tiling of motion patchesR() : A random number between 0 and 1

1: accept← 12: reject← 03: while reject

accept+reject < threshold do4: action ←INSERT or REMOVE5: if action is INSERT then6: Pnew ← (a random patch instance)7: P′ ← P ∪Pnew

8: else9: Premove ← (a random patch in P)

10: P′ ← P \Premove

11: end if12: if R() < αP→P′ then13: P← P′14: accept← accept+ 115: continue from line 316: end if17: P′0 ← P18: j ← 119: for j ≤ jmax do20: d← a random dangling entry or exit.21: for Pnew that can resolve d do22: P′j ← P′j−1 ∪Pnew

23: if not IsValid(P′j) then24: continue25: end if26: end for27: if R() < αP→P′j then28: P← P′j29: accept← accept+ 130: continue from line 331: end if32: end for33: reject← reject+ 134: end while

where U(C), U(x, y), U(θ), U(t) are the uniform distri-bution on types, locations, orientations, and timings ofpatch instances, respectively. Alternatively, we can usethe preference function in Equation (7) to design theproposal density.

Ideally, the algorithm terminates if the tiling is perfectwithout any collision. In practice, achieving a perfecttiling based solely on MCMC sampling takes too muchcomputation, if patches are packed tightly. Our MCMCsampling algorithm terminates if the rate of rejectionis above a certain threshold. Then, we move on to thesecond phase of the algorithm presented in Section 5.2to remove any residual danglings.

6.3 Delayed Rejection

Introducing a new patch into a tightly packed tilingoften lead to a high rejection rate in MCMC sampling.

Persistent rejection, often in particular parts of the tiling,is a computational bottleneck. The tiling is often locallystable and inserting/deleting a single patch instancecannot make a significant change to escape from thelocally stable state. We employ a delayed rejection policyto alleviate persistent rejection [28]. On rejection of aproposal, a second subsequent attempt to move is madein hope that the subsequent move may resurrect theinitial proposal. Specifically, when a jump from xn tox′m is rejected, we perform a repair step to generate anew sample x′′l . The composite move xn → x′m → x′′l iseither accepted or rejected with acceptance ratio

αxn→x′′l = min

{1,P (x′′l )Q(xn|x′′l )[1− αx′′l →x∗ ]P (xn)Q(x′′l |xn)[1− αxn→x′m ]

}where x∗ is x′′l with dimension matching such thatfn,m(x∗, un,m) = x′′l . We usually generate multiple sec-ondary attempts for every proposal. The proposal isfinally rejected if none of the composite moves areaccepted. The reversible jump MCMC algorithm withdelayed rejection is described in Algorithm 3.

7 EXPERIMENTAL RESULTS

The timing data in this section was measured on a2.93GH Intel Core i7 computer with 8Gbyte main mem-ory and an ATI Radeon HD 4550 graphics accelerator. Inour experiments, parameter α, k, Tmax and ∆T are 0.2,0.03, 200 and 1, respectively, for all examples. We do notneed to tune parameters for each individual example.

Motion Patch Construction. We captured two sub-jects performing a variety of interactive actions, such asdancing, tagging, and bumping each other. Some actionsinvolved a lot of physical contacts, while our subjectssometimes exhibited interaction without touching eachother. We also designed some patches manually usingour motion editing system we developed based on theidea of Kim et al. [10]. We constructed multi-characterpatches by putting together several pieces of single-person motion capture data and specifying interpersonalconstraints. In our experiments, 43 kinds of motionpatches are used and 27 of them are single-characterpatches. A big patch is used to demonstrate Keyframingexample and 16 box-related patches are additionallyused in Box-movers example.

Static Environments. Our first example was con-structed in a static environment of 12m × 12m square.We generated an animation of 600 frames (20 seconds). Inthe first phase of the algorithm, we assigned preferencesvalues to the patches such that larger (more characters,larger spatial extent, extended period of time) patchesare more likely to be sampled. The second phase of thealgorithm added mostly small, single-character patchesthat can easily squeeze into narrow spots (see Figure 5).The computation time increases pseudo-exponentiallywith the density of patches. Constructing dangling-freetiling with 173, 374, 490, 657, and 907 patches took 0.3,


Example # of patches # of patch types # of characters Size Time of Scene Computation time (minutes)(# of unary patches) (m2) (seconds) Phase1 Phase2

Static 173 43(27) 16 12 × 12 20 0.1 0.2Static 374 43(27) 34 12 × 12 20 0.2 1.8Static 490 43(27) 48 12 × 12 20 1 11Static 657 43(27) 60 12 × 12 20 3 14Static 907 43(27) 70 12 × 12 20 58 45

Dynamic 528 43(27) 66 12 × 12 20 53 44Control 628 44(27) 55 12 × 12 23 17 14

Repeating 813 43(27) 55 10 × 10 27 83 43Object 526 59(39) 36 10 × 10 27 1.5 5Edit(*) 1463 43(27) 77 12 × 12 20 7 8

Fig. 7. Examples and Statistics. (*) denotes that the scene is generated using our incremental update algorithm.

2, 12, 17, and 103 minutes, respectively (see top row inFigure 7).

Dynamic Environment. Our spatiotemporal tilingalgorithm is inherently capable of dealing withdynamically-changing environments. Whenever a newpatch instance is sampled, the interpenetration with(either static or dynamically-moving) obstacles and otherpatch instances are detected and avoided. The envi-ronment may have sources and sinks where charactersare newly created or disappear from the scene. Thealignment of sources and sinks generates the flow ofcharacter’s movements. The environment in the secondexample has two dynamically-moving obstacles (see leftimage, middle row in Figure 7). One of the obstaclesis tall and the other is shorter. The characters can jumpover the shorter obstacle, but cannot jump over the other.It took about 90 minutes to construct a tiling with 528patches featuring 66 characters that playbacks for 20seconds.

Chicken Hopping. We captured two subjects chicken-hopping, which is a play between human players. Theplayer must stand straight, lift up one of his/her leg,hold the ankle with hands, and hop on the other leg.The players bump each other to make the opponentlose balance. We recorded 50 seconds of the play andour patch construction algorithm identified 6 two-player

patches from the data. Many of the patches exhibit twoplayers bumping and some patches shows one playerdashes and the other evades. We manually picked 15single-player patches that allow each individual charac-ter to navigate and steer. We made two demos using thedata. The first demo shows two teams of players (redand blue) aligning at opposite sides, rushing towardseach other, and bumping. Each team has 15 players.The player bumps the opponent, but avoids the fellow.The animation is 22 seconds long. The generation of theanimation took 7 minutes. The second demo shows in-teractive control of small groups of characters. Each teamhas four players and the user selects target locationsinteractively to control them. Our interactive algorithmrun on the average at the rate of 45 frames per second.

Keyframing. The user can directly control the posi-tion/direction/timing of specific patch instances in thetiling. To do so, the user selects several patch instancesand arranges them in the spatiotemporal domain ashe/she wants. We use the arrangement as the initialconfiguration of the tiling in Algorithm 1, which will addnew patch instances around the user-specified instancesto make them fully connected. The resultant tilingthus obtained will include the user-specified patchesat desired spatiotemporal location. Specifically, we con-structed a large patch with ten characters aligning in two


rows to bow to each other. We placed three instancesof this large patch at 1.5, 11.5, and 21.5 seconds in theinitial configuration of the tiling. The result shows thetwo-row formation is formed and dispersed repeatedlyin complex multi-character animation (see the secondimage, middle row in Figure 7).

Spatially Repeating Tile. We can construct a seamlesstile that repeats either spatially or temporal. Given asymmetric spatiotemporal region, the construction ofrepeating tiles allows patch instances to wrap aroundthe symmetric region. Spatial repetition of tiles cancover an arbitrarily large spatial region seamlessly, whiletemporal repetition of tiles generates an infinitely longanimation. Our 10m× 10m square tile repeats seamlesslyon regular grids (see the third image, middle row inFigure 7). The repetition of square tiles is visible atlarge scale. It might be possible to construct aperiodictilings, such as Wang tiles [29] and Penrose tiles [30], bysampling motion patches on triangular or rectangularregions with proper boundary conditions.

Box movers. In our box mover example, we havefour patches that involve props. The character can pickup a box from a stack of boxes, carry it around, passit to another character, put it down on the ground,and sit down on the box(see right image, middle rowin Figure 7). The characters are precisely aligned andsynchronized with each other when they pass a box fromone character to the other.

Incremental Update. We generated a tiling for a staticenvironment and updated the tiling to cope with a newobstacle at the middle of the environment (see Figure 8).Most parts of the initial tiling remains unchanged. Ourlocal update algorithm efficiently removes interpenetrat-ing patches and repairs the tiling to make it perfectwithout danglings. The generation of the initial tilingfrom scratch took 97 minutes. The local update is muchfaster, so it took only 15 minutes.

8 DISCUSSIONWe explored a patch-based approach to create adense crowd of characters interacting with each other.The formulation was reduced to tiling spatiotemporalpatches without temporal gaps and spatial overlaps. Ourstochastic sampling algorithm demonstrated the powerof patch-based approaches in creating very complexanimation by tiling a small collection of precomputedmotion patches.

Our new perspective allows complex interactions andphysical contacts between multiple characters to be for-mulated as a sampling problem. Comparing to agent-based approaches, most of agent-based methods steereach individual character independently. Interpersonalinteractions are formulated as heuristic rules or potentialfields, but may not be enforced precisely. Our patch-based method would be reduced to an agent-basedmethod if only single-character patches are used fortiling, because the string of single-character patches cor-responds to the motion of each individual character. In

that sense, the patch-based approach can be consideredas a generalization of an agent-based approach.

There is no guarantee that the stochastic samplingconverges while achieving all desired properties, suchas the density of the crowd, collision avoidance, obeyinguser control, and coping with environment features. Inpractice, the convergence of the algorithm is closelyrelated to the diversity and flexibility of motion patches.Including various single-character patches facilitates theconvergence because it allows each individual characterto be steered independently.

The collision detection and avoidance is a bottleneckof computation. More than half of the computationtime is spent for bounding box tests and jittering-basedcollision avoidance. Collision avoidance can be moreefficient if the penetration depth between two motionpatches can be computed. Choi et al. [31] demonstratedan interactively-controllable character that is equippedwith path planning capability and can navigate throughhighly-constrained environments in real-time. It waspossible because all the obstacles had simple geome-try and the interpenetration between the character andobstacles could be easily resolved. Unfortunately, wehave no efficient algorithm to compute the penetra-tion depth, which directly indicates the resolution ofinterpenetration, between four-dimensional spatiotem-poral objects. Efficient computation of penetration depthwould achieve significant performance gain for our tilingalgorithm.

Patch-based motion synthesis has been explored byseveral research groups [1], [17], [18]. Specifically, wemake two major improvements over the previous work.The first is the perfect tiling of motion patches. Shum etal. [17] has a lengthy discussion on why the perfect tilingis difficult and their observation motivated our work.We have found a solution to the problem that remainunsolved in the previous work. The two-phase algorithmand deformable patches played a key role in constructingthe perfect tiling. The second improvement is the auto-matic algorithm for identifying motion patches from acollection of raw multi-character motion data. The over-all contribution is a system that generates arbitrarily-large/long animations from a collection of raw motiondata. The baseline algorithms are fully automatic. Theanimator may intervene to change and control the resultby tuning parameters, defining control functions, andadding special features.

We presented two stochastic sampling algorithms. Oneallows only the insertion of patches to rapidly growtilings. The other based on MCMC sampling allowsboth the insertion and deletion of patches to providemore flexibility and versatile. Theoretically, the latteralgorithm subsumes the functionality of the former al-gorithm. In practice, there exists a trade-off between per-formance and flexibility. The former algorithm withoutpatch deletion is usually faster than MCMC samplingwhen we want to generate a tiling from scratch. On theother hand, the flexibility of MCMC sampling allows us


Fig. 8. Incremental update. (Left) The tiling is initially generated in an empty space. (Middle and Right) Our reversiblejump MCMC algorithm updates the tiling incrementally to introduce a new obstacle and enlarge the obstacle at themiddle of the space.

to locally update existing tilings.Patch-based motion synthesis has demonstrated its

promise in controlling animated characters in large scalewhile coping with the complexity of environments, thediversity of human behavior, and the computation oflong-term planning. The data-driven animation commu-nity have considered fragments of motion capture dataas basic building blocks of human motion synthesis. Weadvocate motion patches as building blocks of large-scalemotion synthesis.

ACKNOWLEDGMENTSThis research was supported by Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Science, ICT &Future Planning (MSIP) (No.2013-003303 and No. 2007-0056094).

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Kyunglyul Hyun received the B.S. degree inComputer Science and Engineering from SeoulNational University, Korea, in 2004. He is cur-rently working toward the Ph.D. degree in theDepartment of Computer Science and Engineer-ing, Seoul National University, Korea. His re-search interests include character animation andcrowd simulation.

Manmyung Kim received the B.S. degree inMechanical and Aerospace Engineering and theM.S. and Ph.D. degrees in Computer Scienceand Engineering from Seoul National University.He is currently a manager at Lineage EternalTeam, NCSOFT Corp. His research interestsinclude synthesizing character animation andartificial intelligence.

Youngseok Hwang received the B.S. degreein Electrical and Computer Engineering and theM.S. degree in Computer Science and Engineer-ing from Seoul National University. He is workingin X3 Team, XLgames Corp. His research inter-ests include computer graphics and animation.

Jehee Lee is a professor of Computer Scienceand Engineering at Seoul National University. Heis leading the SNU Movement Research Labo-ratory. His research interests are in the areasof computer graphics, clinical gait analysis, androbotics. More specifically, he is interested indeveloping new ways of understanding, repre-senting, and animating human movements. Thisinvolves full-body motion analysis and synthesis,biped control and simulation, animal locomotion,motion capture, motion planning, data-driven

and physically based techniques, interactive avatar control, and crowdsimulation. He served as a member of international program committeesof top-tier conferences including ACM Siggraph, ACM Siggraph Asia,ACM/Eurographics Symposium on Computer Animation, and PacificGraphics. He also served as a program co-chair of ACM/EurographicsSymposium on Computer Animation 2012.

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