Chaotic Invariants of Lagrangian Particle Trajectories for Anomaly Detection in Crowded Scenes
 Introduction
 Significance of Crowd Scene Analysis
 Challenges
 The Idea
 The Novelties
 Particle Advection
 Cluster Particle Trajectories
 Chaotic Invariants
 Feature Set
 Advantages of the Algorithm
 Anomaly Detection
 Modeling Learning
 Anomaly Localization
 Experiment Results
 Conclusions
 Related Publication
Introduction
A novel method for crowd flow modeling and anomaly detection is proposed for both coherent and incoherent scenes. The novelty is revealed in three aspects. First, it is a unique utilization of particle trajectories for modeling crowded scenes, in which we propose new and efficient representative trajectories for modeling arbitrarily complicated crowd flows. Second, chaotic dynamics are introduced into the crowd context to characterize complicated crowd motions by regulating a set of chaotic invariant features, which are reliably computed and used for detecting anomalies. Third, a probabilistic framework for anomaly detection and localization is formulated.The overall workflow begins with particle advection based on optical flow. Then particle trajectories are clustered to obtain representative trajectories for a crowd flow. Next, the chaotic dynamics of all representative trajectories are extracted and quantified using chaotic invariants known as maximal Lyapunov exponent and correlation dimension. Probabilistic model is learned from these chaotic feature set, and finally, a maximum likelihood estimation criterion is adopted to identify a query video of a scene as normal or abnormal. Furthermore, an effective anomaly localization algorithm is designed to locate the position and size of an anomaly. Experiments are conducted on known crowd data set, and results show that our method achieves higher accuracy in anomaly detection and can effectively localize anomalies.
Significance of Crowd Scene Analysis
 Management of large gatherings of people at events or in confined spaces
 Anomaly detection, localization, and alarm
 Crowd surveillance, public place monitoring, security control, etc.
Figure 1. Crowd scenarios with different levels of cherency.
Challenges
 Very high density of objects
 Diverse level of coerency of motions

Traditional methods
 Only suitable for sparse scenes
 Suffer from the problems due to severe occlusions, small object sizes, similar appearance
The Idea
 Langragian particle dynamic + chaotic invariants
Figure 2. Framework for anomaly detection and localization.
The Novelties
 Unique utilization of clustering of particle trajectories for modeling crowded scenes
 Chaotic dynamics are introduced into the crowd context
 Being able to deal with both coherent and incoherent flows
Particle Advection
Figure 3. Particle trajectories overlayed on three crowd scenes. Top row shows zoomin view of parts of each scene.
Cluster Particle Trajectories
 Principle: A bunch of adjacent particle trajectories may belong to a single subobject

Method: clustering
 Step 1: Remove relatively motionless particles and trajectories that carry minor information
 Step 2: Cluster by kmeans according to position information
 Output: Representative trajectories
Figure 4. Trajectories after low variance particles are removed. Top row shows zoomin view of parts of each scene.
Figure 5. Trajectories clustered according to position information, (left) and representative trajectories for two clusters (right).
Figure 6. Representative trajectories for three scenes. Top row shows zoomin view of parts of each scene.
Chaotic Invariants
 Representation of scenes: Representative trajectories
 To identify the scene's dynamics in terms of the dynamics of representative trajectories: lChaotic dynamics by measurable chaotic invariants
Feature Set
F = { L, D, M } L: Largest Lyapuno exponent
 D: Correlation dimension
 M: Mean of representative trajectories (Only necessary for positioncaused anomalies)
Figure. The algorithm for computing L and D.
Advantages of the Algorithm
 Proven to be insensitive to the changes in time delay, embedding dimension, size of data set and to some extent noise
 Ensure L>0 for condition of chaotic analysis
Figure 7. Largest Lyapunov exponents for representative trajectories using our method (left) and the method of [7] (right).
Anomaly Detection

Definition of anomaly: Spatiotemporal change of scene/system dynamics (chaotic or/and positions)
 Global anomaly: entire change of dynamics
 Local anomaly: dynamics changes near particular spatial points
 Approach: Probabilistic model
Modeling Learning
 Normality model: Multivariate GMM
 Learning by: EM + IPRA algorithm
 Principle for judging a query as normal or abnormal: Probability of the query belonging to the normality model + ML criterion
Anomaly Localization
 Localize the anomaly in terms of position & size

Steps
 Calculate the likelihoods contributed by each representative trajectory
 Localize those trajectories with low likelihoods
 Cluster them according to position information
 Filter out the clusters with fewer number of trajectories
 The remaining clusters reveal the major abnormal regions
Experiment Results

Dataset
Unusual crowd activity dataset from University of Minnesota
Other coherent and incoherent crowd motions
10frames clips and interpolate to 500 points 
Three experiments
Global anomaly detection (Exp. 1)
Figure 8. Sample frames from three crowd scenes. The first two frames in each row show normal behavior, and the third frame shows abnormal escape panic
Figure 9. Representative trajectories for three clips in a sequence, the first one shows normal behavior and the last two are abnormal.
Figure 10. Marginal PDF of two chaotic features of x (left) and y (right) of learned 4D mixture of Gaussian model.
Figure 11. Likelihood profile for testing clips and corresponding ground truth.
Figure 12. ROC curves for (a) our method, and (b) method of [9].
Due to change of chaotic dynamics (Exp. 2)
Figure 13. (a) Normal clapping behavior, and (b) introduction of abnormal dancing behavior.
Figure 14. For clip 30 correctly detected anomalies, red points below threshold correspond to abnormal representative trajectories, while blue points above threshold correspond to normal.
Figure 15. A frame from a clip with abnormal behavior, (a) representative trajectories, (b) candidates for local anomalies, (c) correct localization of anomalies.
Figure 16. Positioncaused anomaly localization
Conclusions
 A novel combination of Lagrangian particle dynamics approach together with chaotic modeling.
 Representative trajectory: serve as a compact, yet informative, modeling element in crowd flows.
 A representative feature set to reliably capture the system dynamics.
 An effective anomaly detection & localization algorithm.