Crowd Motion Model Research Articles

Defective Laplacians and paradoxical phenomena in crowd motion modeling

Defective Laplacians and paradoxical phenomena in crowd motion modeling

Prediction-correction pedestrian flow by means of minimum flow problem

We study a new variant of mathematical prediction-correction model for crowd motion. The prediction phase is handled by a transport equation where the vector field is computed via an eikonal equation [Formula: see text], with a positive continuous function [Formula: see text] connected to the speed of the spontaneous travel. The correction phase is handled by a new version of the minimum flow problem. This model is flexible and can take into account different types of interactions between the agents, from gradient flow in Wassersetin space to granular type dynamics like in sandpile. Furthermore, different boundary conditions can be used, such as non-homogeneous Dirichlet (e.g. outings with different exit-cost penalty) and Neumann boundary conditions (e.g. entrances with different rates). Combining finite volume method for the transport equation and Chambolle–Pock’s primal dual algorithm for the eikonal equation and minimum flow problem, we present numerical simulations to demonstrate the behavior in different scenarios.

ViewBirdiformer: Learning to Recover Ground-Plane Crowd Trajectories and Ego-Motion From a Single Ego-Centric View

We introduce a novel learning-based method for view birdification [1], the task of recovering ground-plane trajectories of pedestrians of a crowd and their observer in the same crowd just from the observed ego-centric video. View birdification becomes essential for mobile robot navigation and localization in dense crowds where the static background is hard to see and reliably track. It is challenging mainly for two reasons; i) absolute trajectories of pedestrians are entangled with the movement of the observer which needs to be decoupled from their observed relative movements in the ego-centric video, and ii) a crowd motion model describing the pedestrian movement interactions is specific to the scene yet unknown a priori. For this, we introduce a Transformer-based network referred to as ViewBirdiformer which implicitly models the crowd motion through self-attention and decomposes relative 2D movement observations onto the ground-plane trajectories of the crowd and the camera through cross-attention between views. Most important, ViewBirdiformer achieves view birdification in a single forward pass which opens the door to accurate real-time, always-on situational awareness. Extensive experimental results demonstrate that ViewBirdiformer achieves accuracy similar to or better than state-of-the-art with three orders of magnitude reduction in execution time.

A Fusion Crowd Simulation Method: Integrating Data with Dynamics, Personality with Common

AbstractThis paper proposes a novel crowd simulation method which integrates not only modelling ideas but also advantages from both data‐driven methods and crowd dynamics methods. To seamlessly integrate these two different modelling ideas, first, a fusion crowd motion model is developed. In this model the motion of crowd are driven dynamically by different forces. Part of the forces are modeled under a universal interaction mechanism, which describe the common parts of crowd dynamics. Others are modeled by examples from real data, which describe the personality parts of the agent motion. Second, a construction method for example dataset is proposed to support the fusion model. In the dataset, crowd trajectories captured in the real world are decomposed and re‐described under the structure of the fusion model. Thus, personality parts hidden in the real data could be locked and extracted, making the data understandable and migratable for our fusion model. A comprehensive crowd motion generation workflow using the fusion model and example dataset is also proposed. Quantitative and qualitative experiments and user studies are conducted. Results show that the proposed fusion crowd simulation method can generate crowd motion with the great motion fidelity, which not only match the macro characteristics of real data, but also has lots of micro personality showing the diversity of crowd motion.

Guiding crowds when facing limited compliance: Simulating strategies.

At traffic hubs, it is important to avoid congestion of pedestrian streams to ensure safety and a good level of service. This presents a challenge, since distributing crowds on different routes is much more difficult than opening valves to, for example, regulate fluid flow. Humans may or may not comply with re-directions suggested to them typically with the help of signage, loudspeakers, apps, or by staff. This remains true, even if they perceive and understand the suggestions. Yet, simulation studies so far have neglected the influence of compliance. In view of this, we complement a state-of-the-art model of crowd motion and crowd behavior, so that we can vary the compliance rate. We consider an abstracted scenario that is inspired by a metro station in the city of Munich, where traffic regulators wish to make some passengers abandon the obviously shortest route so that the flow evens out. We investigate the effect of compliance for two very simple guiding strategies. In the first strategy, we alternate routes. In the second strategy, we recommend the path with the lowest crowd density. We observe that, in both cases, it suffices to reroute a small fraction of the crowd to reduce travel times. But we also find that taking densities into account is much more efficient when facing low compliance rates.

Splitting scheme for a macroscopic crowd motion model with congestion for a two-typed population

<p style='text-indent:20px;'>We study the extension of the macroscopic crowd motion model with congestion to a population divided into two types. As the set of pairs of density whose sum is bounded is not geodesically convex in the product of Wasserstein spaces, the generic splitting scheme may be ill-posed. We thus analyze precisely the projection operator on the set of admissible densities, and link it to the projection on the set of measures of bounded density in the mono-type case. We then derive a numerical scheme to adapt the one-typed population splitting scheme.</p>

Parameter calibration with stochastic gradient descent for interacting particle systems driven by neural networks

We propose a neural network approach to model general interaction dynamics and an adjoint-based stochastic gradient descent algorithm to calibrate its parameters. The parameter calibration problem is considered as optimal control problem that is investigated from a theoretical and numerical point of view. We prove the existence of optimal controls, derive the corresponding first-order optimality system and formulate a stochastic gradient descent algorithm to identify parameters for given data sets. To validate the approach, we use real data sets from traffic and crowd dynamics to fit the parameters. The results are compared to forces corresponding to well-known interaction models such as the Lighthill–Whitham–Richards model for traffic and the social force model for crowd motion.

A Perceptually-Validated Metric for Crowd Trajectory Quality Evaluation

Simulating crowds requires controlling a very large number of trajectories and is usually performed using crowd motion algorithms for which appropriate parameter values need to be found. The study of the relation between parametric values for simulation techniques and the quality of the resulting trajectories has been studied either through perceptual experiments or by comparison with real crowd trajectories. In this paper, we integrate both strategies. A quality metric, QF, is proposed to abstract from reference data while capturing the most salient features that affect the perception of trajectory realism. QF weights and combines cost functions that are based on several individual, local and global properties of trajectories. These trajectory features are selected from the literature and from interviews with experts. To validate the capacity of QF to capture perceived trajectory quality, we conduct an online experiment that demonstrates the high agreement between the automatic quality score and non-expert users. To further demonstrate the usefulness of QF, we use it in a data-free parameter tuning application able to tune any parametric microscopic crowd simulation model that outputs independent trajectories for characters. The learnt parameters for the tuned crowd motion model maintain the influence of the reference data which was used to weight the terms of QF.

Modeling of crowds in regions with moving obstacles

<p style='text-indent:20px;'>We present a model of crowd motion in regions with moving obstacles, which is based on the notion of measure sweeping process. The obstacle is modeled by a set-valued map, whose values are complements to <inline-formula><tex-math id="M1">\begin{document}$ r $\end{document}</tex-math></inline-formula>-prox-regular sets. The crowd motion obeys a nonlinear transport equation outside the obstacle and a normal cone condition (similar to that of the classical sweeping processes theory) on the boundary. We prove the well-posedness of the model, give an application to environment optimization problems, and provide some results of numerical computations.</p>

Weighted-averaging model of crowd motion generalizes to different turn angles and crowd sizes

From schools of fish to pedestrians on a busy street, it appears that collective motion emerges from local interactions between individuals. The crux of the problem is to understand these local interactions, including the rules of engagement and the neighborhood of interaction over which they operate. In Rio, Dachner & Warren’s (2018) behavioral model, a pedestrian aligns their heading with a weighted average of the heading directions of all neighbors in the field of view (the rule), whose weights decay exponentially with distance (the neighborhood). The model was based on and fit to data with neighbors turning only ±10°. Here we test how the model generalizes to moderate turn angles of ±0-20° with fixed parameters (for large angles, see Wirth & Warren, VSS, 2017, 2019). In the first experiment, a participant (N=12) was instructed to “walk with” a virtual crowd of 12 virtual humans presented in a Samsung Odyssey HMD. On each trial, the crowd began walking forward; after 2-3s, a subset of neighbors (0, 3, 6, 9, or 12) turned left or right by 5°, 10°, 15°, or 20°. The participant’s heading and speed were recorded. Mean final heading increased with subset turn angle (p < .001), as well as subset size (p < .001), confirming a proportional influence over this turn range. Mean final speed showed no main effects of turn angle or subset size, although there was a significant interaction (p = 0.05), indicating a slight slowing with a ±20˚ turn as subset size increases to maintain stability (Hicheur, et al. 2005). Next we plan to manipulate the size of the crowd to test whether neighbor influence is averaged or additive. The results indicate that the weighted-averaging model generalizes over a moderate range of turn angles.

Graph-based topic models for trajectory clustering in crowd videos

Probabilistic topic modelings, such as latent Dirichlet allocation (LDA) and correlated topic models (CTM), have recently emerged as powerful statistical tools for processing video content. They share an important property, i.e., using a common set of topics to model all data. However, such property can be too restrictive for modeling complex visual data such as crowd scenes where multiple fields of heterogeneous data jointly provide rich information about objects and events. This paper proposes graph-based extensions of LDA and CTM, referred to as GLDA and GCTM, to learn and analyze motion patterns by trajectory clustering in a highly cluttered and crowded environment. Unlike previous works that relied on a scene prior, we apply a spatio-temporal graph to uncover the spatial and temporal coherence between the trajectories of crowd motion during the learning process. The presented models advance the conventional approaches by integrating a manifold-based clustering as initialization and iterative statistical inference as optimization. The output of GLDA and GCTM are mid-level features that represent the motion patterns used later to generate trajectory clusters. Experiments on three different datasets show the effectiveness of the approaches in trajectory clustering and crowd motion modeling.

Learning crowd behavior from real data: A residual network method for crowd simulation

Traditional methods of crowd evacuation simulation reduce the visual realism of crowd motion modeling due to hypothetical scenarios and rules. Data-driven methods are an effective way to enhance the visual realism of crowd simulation. However, the existing work mainly focuses on training models for specific scenarios and applies them to the same scenario, so it lacks consideration of scene adaptability. To address this problem, we present a residual network based scene-independent crowd simulation (ResNet-SICS) framework to simulate crowd motion. First, a data-driven crowd properties quantization (DCPQ) method is proposed. This method divides crowd properties into physical properties and social properties: the physical properties (such as the position and the velocity) are extracted from a large number of real videos, and the social properties (such as local cohesiveness and global collectiveness) are then quantified according to the physical properties of crowds. Second, a residual network for crowd behavior properties learning (ResNet-CBPL) motion prediction model is established. This model takes the crowd properties as parameters to construct the residual network, and uses real data to learn the movement rules of the crowd. The ResNet-CBPL motion prediction model can fit the movement behavior of the crowd more accurately than can other models. Finally, we implement a crowd simulation system based on ResNet-CBPL to visualize the results of theoretical analysis in a graphical way. The experimental results show that the proposed method can simulate the crowd motion more realistically, and the trained crowd simulation framework can be applied to various scenarios.

Lagrangian Discretization of Crowd Motion and Linear Diffusion

We study a model of crowd motion following a gradient vector field, with possibly additional interaction terms such as attraction/repulsion, and we present a numerical scheme for its solution through a Lagrangian discretization. The density constraint of the resulting particles is enforced by means of a partial optimal transport problem at each time step. We prove the convergence of the discrete measures to a solution of the continuous PDE describing the crowd motion in dimension one. In a second part, we show how a similar approach can be used to construct a Lagrangian discretization of a linear advection-diffusion equation. Both discretizations rely on the interpretation of the two equations (crowd motion and linear diffusion) as gradient flows in Wasserstein space. We provide also a numerical implementation in 2 dimensions to demonstrate the feasibility of the computations.

Global asymptotic stability of nonconvex sweeping processes

Building upon the technique that we developed earlier for perturbed sweeping processes with convex moving constraints and monotone vector fields (Kamenskii et al, Nonlinear Anal. Hybrid Syst. 30, 2018), the present paper establishes the conditions for global asymptotic stability of global and periodic solutions to perturbed sweeping processes with prox-regular moving constraints. Our conclusion can be formulated as follows: closer the constraint to a convex one, weaker monotonicity is required to keep the sweeping process globally asymptotically stable. We explain why the proposed technique is not capable to prove global asymptotic stability of a periodic regime in a crowd motion model (Cao-Mordukhovich, DCDS-B 22, 2017). We introduce and analyze a toy model which clarifies the extent of applicability of our result.

Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model

This paper concerns optimal control of a nonconvex perturbed sweeping process and its applications to optimization of the planar crowd motion model of traffic equilibria. The obtained theoretical results allow us to investigate a dynamic optimization problem for the microscopic planar crown motion model with finitely many participants and completely solve it analytically in the case of two participants.

Optimal control of a nonconvex perturbed sweeping process

The paper concerns optimal control of discontinuous differential inclusions of the normal cone type governed by a generalized version of the Moreau sweeping process with control functions acting in both nonconvex moving sets and additive perturbations. This is a new class of optimal control problems in comparison with previously considered counterparts where the controlled sweeping sets are described by convex polyhedra. Besides a theoretical interest, a major motivation for our study of such challenging optimal control problems with intrinsic state constraints comes from the application to the crowd motion model in a practically adequate planar setting with nonconvex but prox-regular sweeping sets. Based on a constructive discrete approximation approach and advanced tools of first-order and second-order variational analysis and generalized differentiation, we establish the strong convergence of discrete optimal solutions and derive a complete set of necessary optimality conditions for discrete-time and continuous-time sweeping control systems that are expressed entirely via the problem data.

Local interactions underlying collective motion in human crowds

It is commonly believed that global patterns of motion in flocks, schools and crowds emerge from local interactions between individuals, through a process of self-organization. The key to explaining such collective behaviour thus lies in deciphering these local interactions. We take an experiment-driven approach to modelling collective motion in human crowds. Previously, we observed that a pedestrian aligns their velocity vector (speed and heading direction) with that of a neighbour. Here we investigate the neighbourhood of interaction in a crowd: which neighbours influence a pedestrian's behaviour, how this depends on neighbour position, and how the influences of multiple neighbours are combined. In three experiments, a participant walked in a virtual crowd whose speed and heading were manipulated. We find that neighbour influence is linearly combined and decreases with distance, but not with lateral position (eccentricity). We model the neighbourhood as (i) a circularly symmetric region with (ii) a weighted average of neighbours, (iii) a uni-directional influence, and (iv) weights that decay exponentially to zero by 5 m. The model reproduces the experimental data and predicts individual trajectories in observational data on a human 'swarm'. The results yield the first bottom-up model of collective crowd motion.

Crowd motion and evolution PDEs under density constraints

This is a survey about the theory of density-constrained evolutions in theWasserstein space developed by B. Maury, the author, and their collaborators as a model for crowd motion. Connections with microscopic models and other PDEs are presented, as well as several time-discretization schemes based on variational techniques, together with the main theorems guaranteeing their convergence as a tool to prove existence results. Then, a section is devoted to the uniqueness question, and a last one to different numerical methods inspired by optimal transport.

Optimality conditions for a controlled sweeping process with applications to the crowd motion model

The paper concerns the study and applications of a new class of optimal control problems governed by a perturbed sweeping process of the hysteresis type with control functions acting in both play-and-stop operator and additive perturbations. Such control problems can be reduced to optimization of discontinuous and unbounded differential inclusions with pointwise state constraints, which are immensely challenging in control theory and prevent employing conventional variation techniques to derive necessary optimality conditions. We develop the method of discrete approximations married with appropriate generalized differential tools of modern variational analysis to overcome principal difficulties in passing to the limit from optimality conditions for finite-difference systems. This approach leads us to nondegenerate necessary conditions for local minimizers of the controlled sweeping process expressed entirely via the problem data. Besides illustrative examples, we apply the obtained results to an optimal control problem associated with of the crowd motion model of traffic flow in a corridor, which is formulated in this paper. The derived optimality conditions allow us to develop an effective procedure to solve this problem in a general setting and completely calculate optimal solutions in particular situations.

Projection Process with Definable Right-Hand Side

We study the relation between sweeping processes with the cone of limiting normals and projection processes. We prove the existence of a solution of a perturbed sweeping process with the cone of limiting normals and of nonstationary projection processes, provided the sets involved are definable (and are moving in a definable way) in some o-minimal structure. An application to a crowd motion model is presented.