Chaotic Invariants of Lagrangian Particle Trajectories
for Anomaly Detection in Crowded Scenes
Shandong Wu,
Brian E. Moore,
Mubarak Shah
Computer Vision Lab, University of Central Florida, USA
sdwu@eecs.ucf.edu,
Crowd Scene Analysis
Video
clip
• Challenges
– Very high density of objects
– Diverse level of coherency of motions
• Traditional methods
– Only suitable for sparse scenes
– Suffer from severe occlusions, small
object sizes, similar appearance
bmoore@math.ucf.edu,
Optical
flow
Particle
trajectory
Experiment 1: Anomaly Detection
Particle advection
Calculating optical flow
Clustering particle trajectory
Representative
trajectory
Chaotic
description
Scene
model
Anomaly
shah@eecs.ucf.edu
Learning model
Detection &Localization
Calculating chaotic invariants
Query clip
Particle trajectory
Chaotic Description
Particle Advection
Video clip
–
The Algorithm
X time series of a representative trajectory
• To identify the dynamics of representative trajectories
Embedding (m: embedding dimension; J: time delay)
Chaotic dynamics by measurable chaotic invariants
• Chaotic feature set
Optical flow
–
–
–
Sub-pixel level 2D optical
flow interpolation
L
D
M
Clustering
Orbit
: Largest Lyapunov exponent
: Correlation dimension
: Mean of representative trajectories
• Only necessary for position-caused anomalies
Normality model
For
, find
its nearest neighbor
Representative trajectory
p: mean period of orbit
Pairs of nearest neighbors
Euler’s method
• 2D trajectory embedding: in terms of two 1D time series
d(t): average divergence
Particle’s position
Calculating L
Calculate mean rate of
Separation of the nearest
neighbors
Use the distances of nearest neighbors
L
D
Correlation sum
Calculating D
Particle trajectory
A set of approximately parallel lines each with
a slope roughly proportional to L
A bunch of adjacent particle trajectories may
belong to a single sub-object
L
–
–
Step 1: Remove relatively motionless particles and
trajectories that carry minor information
Step 2: Cluster by k-means according to position
information
Output: Representative trajectories
Likelihood of representative trajectories w.r.t normality model
D
Clip 29
Clip 30 (Anomaly source GT)
Advantages of the Algorithms
Method: clustering
–
: Heaviside step function
: threshold distance
: number of points
(crucial for small and noisy data set)
Observation:
•
Experiment 2: Anomaly Localization
Fit the average line
Cluster Particle Trajectories
–
Dataset: Unusual crowd activity dataset from University of Minnesota
for jth pair of nearest neighbors
: distance between the jth pair of nearest neighbors
•
ROC curve and comparison with previous work
Detection example
•
•
(a) All representative
trajectories
(b) Clustering of trajectories
with low probability
(c) Localized major anomaly
sources
The above algorithms proven to be insensitive to the changes in
– time delay
– embedding dimension
– size of data set and
– to some extent noise
More importantly, ensure L>0 for the condition of chaotic analysis
A more efficient and manageable crowd scene representation
**
Anomaly Detection
•
•
•
Definition of anomaly
– Spatiotemporal change of system dynamics (chaotic or/and positions)
• Global anomaly: entire change of dynamics
• Local anomaly: dynamic changes near particular spatial points
Normality model
– Multi-variate GMM: 4D ( [Lx , Ly , Dx , Dy ] ) or 6D ( plus [M x , M y ] )
– Learning by: EM + IPRA algorithm
Experiment 3: Position-caused Anomaly Localization
Chaotic features
Position feature
Synthesized anomaly
Detected anomaly
Anomaly Localization
Novelties / Contributions
Localize the anomaly in terms of position & size
– 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
• A novel combination of Lagrangian particle dynamics approach together with
chaotic modeling
• Unique utilization of clustering of particle trajectories for modeling crowded
scenes (novel modeling element: representative trajectory)
• Chaotic dynamics are introduced into the crowd context with effective
algorithms for calculating chaotic invariants
• Our methods are able to deal with both coherent and incoherent flows