Adaptive Behavioral Modeling For Crowd Simulations

Adaptive Behavioral Modeling for Crowd Simulations

Abstract Behavioral modeling for crowd simulations should consider a number of factors effecting on agents, including environment. In this study, we design an adaptive behavioral model for a dynamic virtual environment. We model the dynamic environment with behavior maps which are constructed with information theory quantities and statistical formulations. These maps are capable of capturing the dynamic nature of the environment by adaptively changing temporally and spatially. Subsequent to building this model, agents’ responses to these maps are represented with a set of formulations. In our test studies, we have observed that our model successfully produces realistic and diverse behaviors by incorporating effects of the environment.

Keywords: crowd simulation, information theory, behavioral modeling

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1 Introduction Crowd is a fundamental element in our real environment, thus it also constitutes a critical component in many virtual environment applications. In order to improve a virtual environment’s realism, crowds must be simulated believable in terms of their appearance and behavior. Recent advances in graphics hardware addresses the issue of photo-realistic rendering of crowds. However, realistic behavior of agents in crowd simulations is still a challenging problem due to the number of factors effecting agents’ behavior, hence those factors are not easy to represent mathematically. Most of the crowd simulation systems use a set of rules and scripts to model human behavior, and when used alone, this approach has limits to model the complex nature of human behavior [1]. Therefore, recent studies are generally employing agent-based behavioral modeling methods. In these methods; an agent responds to other agents and events using static and predefined behavior rules, whatever the environment conditions are. However, dynamic conditions which are inherent in the environment greatly effect an agents’ behavior and existing models are not capable of adapting themselves to these conditions.

Apart from the physical properties, a built environment also has psychological effects on people [2]. Physical properties of built environments and physical properties of agents are already successfully handled by collision avoidance algorithms, predefined rules and predefined roadmaps in most of the crowd simulations. And when dealing with humanenvironment interactions, one should have a model that can predict the environmental con2

ditions under which humans behave in a certain manner [3]. Therefore; we need an adaptive simulation which considers the psychological effects of the environment when modeling the behavior of agents.

To successfully design an adaptive crowd simulation for a dynamic virtual environment, we first build an analytical model of a crowded environment and define how agents act according to that model. • The crowd itself can be considered as one of the most prominent factor of an environment effecting agents in a simulation [4]. In order to model the effects of the crowd, we need methods to represent properties and activities of elements of a crowded environment such as built environment and agents. The resulting quantification should provide a measure to evaluate the activities of the crowd and should be utilized to represent the effects of the crowd on individual agents. A good model has to be adaptive to changes in the dynamics of the crowd both spatially and temporally. • In agent-based behavioral methods, agents are assigned some attributes to determine their behaviors; hence, their response to certain events and other agents are altered. However, as Shoda stated in [5], personality structure can be static but its behavioral output changes greatly under specific circumstances. An agent should behave differently under different environmental conditions to fulfill this statement.

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Our methods to build an analytical model of the environment borrow ideas from behavioral mapping techniques used in psychology research. These techniques involve placecentered maps, which keep track of behavior of individuals within a specific space and time. These maps display how and when a place is being used [6]. In our model, we utilize the notion of place-centered maps in simulations and propose behavior maps. A behavior map is constructed by considering both spatial and temporal dynamics of agents in a crowd simulation. It conveys information on probabilistic and statistical properties of agents’ activities. Information theory quantities, i.e. information entropy and Kullback-Leibler divergence, and statistical formulations are used to produce behavior maps. As behavior maps are updated spatially and temporally; agents adaptively respond to their environment with the contribution of these maps. This contribution is represented with numerical entities and agents’ responses are calculated with a set of formulations.

The rest of the paper is organized as follows: in Section 2, we review the related literature. In Section 3, we introduce different behavior maps. In Section 4, responses of agents to behavior maps are covered. Section 5 discusses the results obtained with the model. To conclude, final remarks are made at the conclusion section.

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2 Related Work An overall idea of the challenges and improvements in the field can be obtained in [7]. There are several behavioral models proposed in the literature and a survey from Kasap et.al.[8] covers most of these studies. There have been many studies on agent-based crowd models to create human-like behaviors. Seminal works of Reynolds used behavioral models considering local rules [9] and create emergent flocking [10] behaviors. There is considerable work on agent-based simulation systems incorporating psychological models and sociological factors. In [11], Luo et.al. model social group and crowd related behaviors. Their main focus is a layered framework to reflect the natural pattern of human-like decision making process. Rymill et.al.[12] tried to improve the quality of agent behavior by adding theories from psychology. In their work, they tried to produce more realistic collision avoidance responses. Musse et.al. developed virtual human agents with intentions, beliefs, knowledge and perception to create a realistic crowd behavior [13]. In [1], Pelechano et.al. assigned psychological roles and communication skills to agents to produce diverse and realistic behaviors. In a more recent work, Pelechano et.al. [14] created an improved model by using psychological and geometrical rules with a social and physical forces model. In [15] Hu proposed an adaptive crowd behavior simulation, where he defines a static behavior context layer. When the behavior context is altered with a predefined event, the new context adaptively inhibits certain behavior in agents. However, this scheme is not suitable for dynamic environments. There are studies which model the virtual environment as maps to guide

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agents’ behaviors. Shao et.al.[16] modeled the environment with topological, perception and path maps to generate autonomous agents. Gayle et.al.[17] used adaptive roadmaps, which evolve with the dynamic nature of the environment. In [18], Sung et.al. assign situations and behaviors directly to environment rather than the agents themselves. The concept of behavior maps have been used in robotics and vision field. In [19], Dornhege et.al. defined behavior maps as encoding context information of the environment, and use these maps to autonomously navigate a robot on rough terrain. Berclaz used behavior maps to encode probabilities of moving in a certain direction on a specified location and used these maps to track trajectories of people and to detect anomalies in people’s behaviors [20]. In their study, they expectation maximization algorithms to detect anomalies. We integrated theories from behavioral modeling and borrowed ideas from studies representing the environment with guidance maps. To compute these maps, we employed quantities from information theory. Information theory have been introduced into computer graphics field by V´azquez et al. [21] which expresses the amount of information in a selected view.

3 Behavior Maps A behavior map spans over the virtual environment, and records all the agents’ activities. This map, B, is a 2D grid, consisting of w rows and h columns, where each cell is a square with side length l. In this work, we assume that environment consists of only horizontal

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paths.

Let A = {a1 , a2 , ..., an } be the set of agents present on the scene, where ai represents a single agent. Physical properties of an agent can be described as ai = {~u, ~v : ~u, ~v ∈ R2 } where ~u defines the position and ~v defines the velocity of agent ai . The activity of an agent is described by its position, the direction and the magnitude of its velocity. Activities of an agent is mapped to the corresponding cell in B. We compute behavior maps by using probabilistic analysis methods incorporating quantities from information theory. Figure 2 illustrates behavior map construction.

3.1 Information Theory Based Maps Information theory deals with the quantification of information. Information entropy is the key measure in information theory [22]. It provides an insight about how likely a system produces varied outcomes. Namely, it is a measure of uncertainty of a random variable.

The other concept we have utilized is Kullback-Leibler divergence (KL) [23]. Take two probability mass functions (pmf) p(x) and q(x), divergence between pmf’s p(x) and q(x) is given by: D(pkq) = −

X x∈χ

p(x) log

p(x) q(x)

(1)

which is a non-symmetric metric expressing the difference between two probability distributions.

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We need to have pmf’s regarding to the activities in the scene. The first pmf, P~vˆ, defines the velocity direction distribution. Consider an agent with normalized velocity ~vˆ, then this vector is added as a sample to one of the n bins of P~vˆ, where the value of n effects quantization resolution. The second pmf, Pk~vk , defines the velocity magnitude distribution. As agents’ speed is in a predefined range, this range is quantized into m categories and speed of an agent is added as a sample to one of these categories. We merge these two pmfs into a single pmf, P~v , with a user defined constant α, which distributes importance to direction or speed distributions, as: P~v = αP~vˆ + (1 − α)Pk~vk

(2)

As P~v is taking samples over a period of time, a Gaussian shaped filter is applied to control the importance given to temporally cumulated distributions. Let t1 and t2 be two time steps where t2 − t1 = n∆t and n ∈ N∗ , temporal filter is applied as; P~v t1 →t2 = λ0 P~vt2 + λ1 P~v t2 −∆t + . . . + λn P~v t2 −n∆t −(x−µ)2 1 λn = √ e c2 , µ = 0 σ 2π

(3) (4)

where, n is defined as historical depth which defines the maximum age to consider, where age means the time passed from the moment the distribution have occurred. With ∆t defining the time interval between two adjacent frames,

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3.1.1

Entropy Map

Entropy measures the uncertainty of a random variable. If we interpret activities of an agent as the random variable, entropy values reflect behavioral patterns of the crowd. Entropy values denote where agents move either more independently or where they move in a pattern. Locations with smaller entropy values mark where agents move with similar velocities. Conversely, locations with higher entropy values represent greater disorder in agents’ behavior. To build the entropy map, E, we begin by considering a random variable, Xi,j (i,j indicating location on E), drawn according to pmf (P~v

(t−n∆t)→t

)i,j . Then, E can be defined

as; E t = {H(Xi,j ) ; 0 ≤ i < w, 0 ≤ j < h}

(5)

, where H(Xi,j ) is the entropy of Xi,j [22].

3.1.2

Expectance Map

Probability distribution of activities on a scene gives an idea on the type of activities that is likely to occur at specific locations. This distribution is defined by pmf P~v in our calculations. In order to quantize the expectance of the current activities on the scene, we compare the current probability distribution on the scene with P~v . We employ Kullback-Leibler divergence (D, Equation 1) to compute the difference between two probability distributions. Itti et.al. [24] used KL divergence to discover surprising events in video. They employed a principled approach to prove that KL is a powerful measure to represent surprise. We postulate

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that KL calculations can be represented as behavior maps, so called expectance maps. Let P~v t define the current (at time t) probability distribution of the scene, and P~v

(t−n∆t)→(t−∆t)

define the cumulative (between time steps (t − n∆t),(t − ∆t)) distribution of activities, expectance map KL is defined as; KLt = {(D(P~v

(t−n∆t)→(t−∆t)

kP~v t ))i,j ; 0 ≤ i < w, 0 ≤ j < h}

(6)

A high KL value represents that current activities taking place at that location can be regarded as surprising, while in areas with lower KL values, the current status of the crowd is as expected.

3.2 Density Map In addition to the proposed information theory based maps, a map called density map, F , is also integrated into our behavioral simulation model. This map indicates how crowded a specific location is. In order to produce a measure that is less prune to noise, a temporal filter is applied on F . (t−n∆t)→(t−∆t)

F t = {fi,j

; 0 ≤ i < w, 0 ≤ j < h}

(7)

where f is a function giving the number of agents on location i, j between time steps (t − n∆t) and (t − ∆t) with temporal filter applied on count values in the same manner as Equation 3.

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4 Response to Behavior Maps In our crowd simulation engine, agents have internal properties to determine their behavior. Internal properties consist of two elements; behavioral constants and behavioral state. Behavioral constants can be regarded as personality attributes of an agent and behavioral state determines at what level behavioral constants effect agent’s behavior. Throughout the simulation, behavioral state is altered adaptively by behavior maps. In Figure 1, the overall structure of our model can be seen. Stages in the figure are listed as: 1)Activities of agents are analyzed between time t0 and tn to produce behavior maps. 2)Each agent is mapped to a specific behavior map cell. 3)Each agent has predefined roadmap, internal properties and goals. 4)Considering all these factors, an agent modifies its internal properties and orientation. In this step, ai ’s properties are modified in order to respond to the behavior map. 5)Agent’s modified properties and preferred velocity is fed into the navigation system. 6)Navigation system considers predefined rules and agent’s properties to produce a collision free path.

To cover diverse agent behaviors we incorporate the following methods into our simulation model: 1) Composite agents: A composite agent is a special agent equipped with a proxy agent, ri , to model a number of emergent behaviors realistically [25]. A proxy agent is a virtual agent, which is visible to all agents in the simulation except its parent ai . ri moves according to ai ’s choices and ai displays particular behaviors by controlling ri . 2)Preferred velocity: In crowd simulations, agents have an initial state and a desired final 11

state. Each agent seek for a collision-free path to its destination [7]. In order to reach their goal as quick as possible, agents prefer to move in a pattern that brings them directly to their goal or subgoals. A preferred velocity, vp , of an agent ai is the optimal velocity that would bring the agent to its goal. 3) Safety factor: Safety factor is the range an agent considers while calculating possible future collisions. A high value means, the agent takes more of the possible collisions into account and behaves more careful by avoiding them. On the other hand, with a lower value the agent becomes reckless and constitutes a higher possibility of making collisions. In Figure 3-a, these methods are illustrated briefly.

Definition of an agent has to include internal properties, in addition to the physical properties. Definition of an agent ai is extended as: ai = {type, ~u, ~v , ~vp , d, s, ri , δ, hf1 , .., fn i , β : ~vp ∈ R2 ; fn , β, d, s ∈ [0, 1]}

(8)

These parameters are: 1) type : Indicates whether an agent is composite agent, proxy agent or a normal agent. A composite agent both has a proxy agent ri and personal attributes. A proxy agent doesn’t carry any property, its behavior is dependent solely on its parent. A normal agent only carries personal properties. 2) ~vp : Indicates the velocity an agent prefers to move. 3) ri : Indicates the proxy agent associated with the current agent (available for composite agents). 4) d : Distance to set between ri [~u] and ~u (available for composite agents). The longer the distance, the further ai can proceed with suffering less collisions. 5) s : Size of the area ri occupies (available for composite agents). The larger the area, the easier ai can move. 6) δ : Indicates the range in which an agent considers possible 12

collisions, modifies safety factor. 7) fn : Indicates a behavior constant. Each constant can be utilized to mimic certain personality attributes. They are static properties assigned to ai at the beginning of the simulation. fn values are randomly distributed over agents and this distribution is a method for creating behavioral variety. 8) β : Indicates the behavioral state of an agent.

Response to entropy map Entropy map displays how predictable crowd moves in a particular location. The notion of predictability can be utilized to modify behaviors of agents. An agent, ai ’s, response to entropy maps can be formulated as; ′

′

ai [β] = kEi,j ; f0 = β(1 − f0 ) ; f1 = βf1 ′ ′ ai [~vp ] = (~vpo\ + f0~vb )f0 m0

(9)

′

ai [δ] = f1 δ0 where k is a constant to normalize β values, δ0 is the default value for δ and ~vpo is the optimal velocity leading to agent’s goal. The modified behavioral constants are utilized to modify ′

preferred velocity or modify δ value. ~vp ’s direction is calculated with f0 and ~vb , and its ′

magnitude is calculated with f0 and the default speed m0 . ~vb can be calculated as a vector leading to lower entropy zones found as a result of a local search in an agent’s location or can be a random vector. In the former case, agents tend to move to locations with lower entropy values. Figure 3-b displays how β values modify agents’ internal properties.

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Response to expectance map An expectance map indicates the level of surprise in a specific location. Agents give different responses to unexpected events, which can be formulated as; ′

′

ai [β] = kKLi,j ; f0 = βf0 ; f1 = β(1 − f0 ) ′ ′ ai [~vp ] = (~vpo\ + f1~vb )f0 m0

′

(10)

′

ai [d] = f0 d0 ; ai [s] = f0 s0 ; ai [δ] = 1/βδ0 ′

′

In this formulation, f0 value modifies the proxy agent ri , as f0 gets higher, d and s values ′

are amplified. f0 value also increases the speed of an agent in areas with high KL value. ′

f1 value, on the other hand, amplifies the deviation from optimal velocity ~vpo . The final response is the change in δ value that is inversely proportional to β values.

Response to density map While some agents tend to avoid crowded areas, others don’t. This behavioral difference is related to the behavioral constant f0 . An agents’ responses can be described with the same formulation in Equation 10 but with different f0 values.

5 Results & Test Cases We tested our methods through a number of scenarios. Our test environment is a modified version of multi-agent simulation system called RVO proposed in [26]. We implemented our adaptive behavioral model on top of this system. Our approach produces realistic and diverse crowd behavior. Figure 4 displays snapshots from test scenarios. Accompanying 14

videos presents comparative results of our simulation system performing in real-time on a system with Intel QuadCore 2.8 GHz and Nvidia GeForce GTX-280.

In order to test our behavioral model, we create a simulation containing 200 agents. We initialize the crowd by creating three groups of agents with specific behavioral constants. First of these groups constitutes of 20 agents with high f0 and low f1 values which can be considered either as aggressive agents. Second of these groups contain 20 agents with low f0 and high f1 values which can be considered as calm agents. The last group consist of 160 standard agents which do not display any adaptive behavior. As we run our simulation with these agents, their activities are cumulated and analyzed either in one of the behavior maps or in a combination of them, i.e. entropy, expectance or density maps. These maps are accessed by all the non-standard agents and they modify their behavior adaptively by using the associated behavior map value. To test this adaptive process, we perform three scenarios, where responses to each behavior map is observed distinctively. In order to observe each map’s effects separately, in each scenario, we use a single behavior map.

To observe responses given to entropy maps, we adopt a scenario where specific agents pass through other agents in a piazza and a pavement to reach their goal. As implied by the formulation of entropy maps, a piazza produces higher entropy values and as agents behave in an ordered manner, pavements produce lower entropy values. Confident agents move directly to their goal as their preferred velocity is not effected from higher β values.

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However, in areas with lower entropy all the agents behave identical, as the effect of f0 value is reduced by β values. Besides, agents do not search for a more “safe” (collisionfree) velocity in areas with low entropy, where δ values are lowered by β.

To analyze responses given to expectance maps, we created a scenario where a number of agents exit a building into a crowded street. A congestion occurs at the entrance of the building and it creates an area of high KL value. On these locations, aggressive agents clear their way through the congestion and move directly to their goal. On the other hand, agents in the second group behave unexpectedly in areas with high KL value. This response mimics panicking behavior when an unexpected event happens. Figure 3-c illustrates how, at micro level, agents respond to expectance map. In this figure, a1 belongs to the first group and while a2 is in the second. In time interval t1 , a1 and a2 behave identical. In t2 , they enter a high KL zone. a1 responds by enlarging s and d values to keep its ~vp as close as possible to optimal. However, a2 mimics a panicking behavior and behaves in an unexpected manner. At t3 , agents return to their initial state. Notice that at the end of t3 , a1 proceeded further.

In response to density maps, agents display similar behaviors in expectance map case. To test these responses, a concert scenario is prepared where all the agents’ destination is the stage. Agents in the first group do not avoid crowded areas and they try to clear their way more aggressively as it gets more crowded. On the other hand, agents in the second group avoid crowded zones and stay away from the stage.

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Each one of the three behaviour map methods is generating satisfactory results in different setting. To address the general case for a dynamic virtual environment we combined all three models into a single one. We calculate a weighted average response by adjusting the contribution of each behavior map to agent’s responses. The resulting simulation outperformed s’ngle behavior map simulations in terms of variability and realism. For a visual comparison between each method, please refer to the accompanying video.

6 Conclusion In this paper, we proposed an adaptive behavioral model for crowd simulations. Our model incorporates the dynamics of a virtual environment through building an analytical model of crowd’s activities and formulates agents’ responses.

We ran our model over a number of scenarios and observed that agents’ behaviors are adaptively altered under certain environmental conditions. Results show that our methods add complexity and diversity in agents’ behaviors, thus improve realism. These methods can be integrated into either scripted behavioral models to increase their behavioral variation or autonomous agent systems to improve their realism. Presented methods will provide lifelike and evolving environments for games, industrial crowd simulations,i.e. emergency, urban planning and can reduce production times for movie pre-visualizations.

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Our model performed in real-time with several hundreds of agents. As a future work, we will improve the scalibilty of our model with parallel computing techniques. Another future work is introducing static preset maps into our model which will enable a designer to encode environmental factors.

References [1] N. Pelechano, K. OBrien, B. Silverman, and N. Badler. Crowd simulation incorporating agent psychological models, roles and communication. In First International Workshop on Crowd Simulation, 2005. [2] Tommy Grling and Reginald G.Golledge, editors. Behavior and environment : psychological and geographical approaches. North-Holland. [3] R. De Young. Environmental psychology. Encyclopedia of environmental science, Kluwer Academic Publishers, Hingham, MA, USA, 1999. [4] P.A. Bell. Environmental Psychology. Wadsworth Pub Co, 2001. [5] Yuichi Shoda. Computational modeling of personality as a dynamical system. Handbook of Research Methods in Personality Psychology, pages 633 – 652, 2007. [6] R. Sommer and B. Sommer. A Practical Guide To Behavioral Research: Tools and Technique. Oxford University Press, 5 edition, 2002.

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[7] Daniel Thalmann and Soraia Raupp Musse. Crowd Simulation. Springer, 2007. [8] Z. Kasap and N. Magnenat-Thalmann. Intelligent Virtual Humans with Autonomy and Personality : State-of-the-Art, pages 43–84. Studies in Computational Intelligence, Springer, 2008. [9] C.W. Reynolds. Steering behaviors for autonomous characters. In Game Developers Conference, 1999. [10] C.W. Reynolds. Flocks, herds and schools: A distributed behavioral model. In Proceedings of the 14th annual conference on Computer graphics and interactive techniques, pages 25–34, 1987. [11] Linbo Luo, Suiping Zhou, Wentong Cai, Malcolm Yoke Hean Low, Feng Tian, Yongwei Wang, Xian Xiao, and Dan Chen. Agent-based human behavior modeling for crowd simulation. Comput. Animat. Virtual Worlds, 19(3-4):271–281, 2008. [12] S.J. Rymill and N.A. Dodgson. A Psychologically-Based Simulation of Human Behaviour. In Theory and Practice of Computer Graphics, pages 35–42, 2005. [13] S. R. Musse and D. Thalmann. Hierarchical model for real time simulation of virtual human crowds. IEEE Transactions on Visualization and Computer Graphics, 7:152– 164, 2001.

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[14] N. Pelechano, J. M. Allbeck, and N. I. Badler. Controlling individual agents in highdensity crowd simulation. In Proceedings of the 2007 ACM SIGGRAPH/Eurographics symposium on Computer animation, pages 99–108. Eurographics Association, 2007. [15] X. Hu. Context-Dependent Adaptability in Crowd Behavior Simulation. In Information Reuse and Integration, 2006 IEEE International Conference on, pages 214–219, 2006. [16] W. Shao and D. Terzopoulos. Autonomous pedestrians. Graphical Models, 69(56):246–274, 2007. [17] R. Gayle, A. Sud, E. Andersen, S.J. Guy, M.C. Lin, and D. Manocha. Interactive Navigation of Heterogeneous Agents Using Adaptive Roadmaps. Visualization and Computer Graphics, IEEE Transactions on, 15(1):34–48, 2009. [18] M. Sung, M. Gleicher, and S. Chenney. Scalable behaviors for crowd simulation. In Computer Graphics Forum, volume 23, pages 519–528, 2004. [19] C. Dornhege and A. Kleiner. Behavior maps for online planning of obstacle negotiation and climbing on rough terrain. In Intelligent Robots and Systems, 2007. IROS 2007. IEEE/RSJ International Conference on, pages 3005–3011, 2007. [20] J. Berclaz, F. Fleuret, and P. Fua. Multi-Camera Tracking and Atypical Motion Detection with Behavioral Maps. In proceedings of the European Conference on Computer Vision. Springer. 20

[21] Pere-Pau V´azquez, Miquel Feixas, Mateu Sbert, and Wolfgang Heidrich. Viewpoint selection using viewpoint entropy. In Proceedings of the Vision Modeling and Visualization Conference 2001, pages 273–280. Aka GmbH, 2001. [22] C. E. Shannon. A mathematical theory of communication. Bell Systems Technical Journal, 27:623–656, 1948. [23] Solomon Kullback. Information Theory and Statistics (Dover Books on Mathematics). Dover Publications, 1997. [24] Laurent Itti and Pierre Baldi. A principled approach to detecting surprising events in video. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 631–637. IEEE Computer Society, 2005. [25] Yeh Hengchin, Curtis Sean, Patil Sachin, van den Berg Jur, Manocha Dinesh, and Lin Ming. Composite agents. In Symposium on Computer Animation - SCA’08, 2008. [26] J. van den Berg, M. Lin, and D. Manocha. Reciprocal Velocity Obstacles for real-time multi-agent navigation. In Robotics and Automation, 2008., pages 1928–1935, 2008.

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Agents, A

Behavior Map, B 1 t0

tn

Agent, ai

2

ai :(u, v)current

3 Behavior Map Values

Goals & Roadmaps Internal Properties

Modify Internal Properties & PrefferedVelocity with Behavior Map Values

Navigation System

ai :(u, v)new

4

5

6

Figure 1: Overall structure of our model. (Details given in Section 4) 22

P

P 3

+ ..... + 0

n

Cumulative Probabilistic Model

4

Entropy Map, E

H(x)

1

Current Velocity Distribution

Agents, A

Expectance Map, KL

2 D(x)

5

Density Map, F Density Distribution

6

Figure 2: Behavior map construction. 1) List of agents is fed to the system. 2) Activities of the crowd are mapped to the underlying grid to form the current distribution function of the activities of the crowd 3) Older distributions are merged with a temporal filter. 4) Entropy map of the scene is built by calculations on merged pmf’s from (t − ∆n to t). 5) Expectance map is formed by calculating KL divergence between the probabilistic model and the current distribution. 6) Density map is formed by calculating the current densities on a specific cell.

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(a)

ai

ri

(f0, .., fn), β

s vp d δ

(b)

δ = f0* β* δ0 vp o

f0 * β * vb vp

(c)

t1

t2

a1

a1

a2

a2

t3

a1

a2

Low KL

High KL (surprise zone)

Low KL

Figure 3: a) A composite agent ai and its associated proxy agent ri b) Effect of β values on agents’ internal properties c) Responses of agents to expectance map (Details in Section 5) 24

Figure 4: Screenshots from our test environment. Each column represents successive time steps. Each row displays responses to entropy, expectance and density map consecutively. Agents behave distinctively in response to each of these maps. Note that, aggressive agents (red) and calm agents (blue) behave accordingly to our model.

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