Brian E. Moore

Chaotic invariants of Lagrangian particle trajectories for anomaly detection in crowded scenes

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 work-flow 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.

Identifying Behaviors in Crowd Scenes Using Stability Analysis for Dynamical Systems

A method is proposed for identifying five crowd behaviors (bottlenecks, fountainheads, lanes, arches, and blocking) in visual scenes. In the algorithm, a scene is overlaid by a grid of particles initializing a dynamical system defined by the optical flow. Time integration of the dynamical system provides particle trajectories that represent the motion in the scene; these trajectories are used to locate regions of interest in the scene. Linear approximation of the dynamical system provides behavior classification through the Jacobian matrix; the eigenvalues determine the dynamic stability of points in the flow and each type of stability corresponds to one of the five crowd behaviors. The eigenvalues are only considered in the regions of interest, consistent with the linear approximation and the implicated behaviors. The algorithm is repeated over sequential clips of a video in order to record changes in eigenvalues, which may imply changes in behavior. The method was tested on over 60 crowd and traffic videos.

A streakline representation of flow in crowded scenes

Based on the Lagrangian framework for fluid dynamics, a streakline representation of flowis presented to solve computer vision problems involving crowd and traffic flow. Streaklines are traced in a fluid flow by injecting color material, such as smoke or dye, which is transported with the flow and used for visualization. In the context of computer vision, streaklines may be used in a similar way to transport information about a scene, and they are obtained by repeatedly initializing a fixed grid of particles at each frame, then moving both current and past particles using optical flow. Streaklines are the locus of points that connect particles which originated from the same initial position. In this paper, a streakline technique is developed to compute several important aspects of a scene, such as flow and potential functions using the Helmholtz decomposition theorem. This leads to a representation of the flow that more accurately recognizes spatial and temporal changes in the scene, compared with other commonly used flow representations. Applications of the technique to segmentation and behavior analysis provide comparison to previously employed techniques, showing that the streakline method outperforms the state-of-theart in segmentation, and opening a new domain of application for crowd analysis based on potentials.

Backward error analysis for multi-symplectic integration methods

Summary.A useful method for understanding discretization error in the numerical solution of ODEs is to compare the system of ODEs with the modified equations obtained through backward error analysis, and using symplectic integration for Hamiltonian ODEs provides more incite into the modified equations. In this paper, the ideas of symplectic integration are extended to Hamiltonian PDEs, and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations. In particular, local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize, but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution. These results are also applied to the nonlinear wave equation.

Multi-symplectic integration methods for Hamiltonian PDEs

Recent results on numerical integration methods that exactly preserve the symplectic structure in both space and time for Hamiltonian PDEs are discussed. The Preissman box scheme is considered as an example, and it is shown that the method exactly preserves a multi-symplectic conservation law and any conservation law related to linear symmetries of the PDE. Local energy and momentum are not, in general, conserved exactly, but semi-discrete versions of these conservation laws are. Then, using Taylor series expansions, one obtains a modified multi-symplectic PDE and modified conservation laws that are preserved to higher order. These results are applied to the nonlinear Schrodinger (NLS) equation and the sine-Gordon equation in relation to the numerical approximation of solitary wave solutions.

Linear PDEs and Numerical Methods That Preserve a Multisymplectic Conservation Law

Multisymplectic methods have recently been proposed as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. Their excellent long time behavior for a variety of Hamiltonian wave equations has been demonstrated in a number of numerical studies. A theoretical investigation and justification of multisymplectic methods is still largely missing. In this paper, we study linear multisymplectic PDEs and their discretization by means of numerical dispersion relations. It is found that multisymplectic methods in the sense of Bridges and Reich Phys. Lett. A, 284 (2001), pp. 184-193] and Reich J. Comput. Phys., 157 (2000), pp. 473-499], such as Gauss-Legendre Runge-Kutta methods, possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity. A certain CFL-type restriction on

Visual crowd surveillance through a hydrodynamics lens

\Delta t/\Delta x

Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations

might be required for methods higher than second order in time. It is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of

Conformal conservation laws and geometric integration for damped Hamiltonian PDEs

\Delta t/\Delta x

Second Order Conformal Symplectic Schemes for Damped Hamiltonian Systems

despite being multisymplectic in the sense of discrete variational mechanics [J. E. Marsden, G. P. Patrick, and S. Shkoller, Commun. Math. Phys., 199 (1999), pp. 351-395].