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Similarity measuress
Laboratory of Image Analysis for Computer Vision and Multimedia
http://imagelab.ing.unimo.it
Università di Modena e Reggio Emilia, Italy
Simone Calderara, Rita Cucchiara
Motivations• People Trajectories are rich descriptor of human
activity• Long Trajectories can be acquired using automatic
Video Surveillance Systems
• Trajectories are time series of low-dimensional feature points
“Data automatically extracted are subject to noise and must be robustly modeled”
“People Trajectories have different lengths and point numbers”
A possible solution could be:
“ Use Robust Statistics to learn the principal trajectory components and an elastic measure for the comparison ”
Time Series Modeling• Point to Point vs Statistical: use a point-to-point
comparison or exploit statistical data representation and a correspondent pattern recognition approach
• Original vs Transformed: use the original feature space or provide a feature extraction step after a space transformation
• Complete vs Selected:use all the temporal data or select a subset of them
Trajectory Modeling using Expectation Maximization• Each trajectory is encoded as a set of directions,
speeds and time value
• Each trajectory is modeled as a Mixture of AWLG where number of components and parameters are learnt trought the Expectation Maximization
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Semi-directional Approximated Wrapped and Linear Gaussian pdf• Gaussian distributions are not suitable for periodic
angular variable such as the trajectory directions because its dependence on the data origin
• Multivariate distribution that jointly model scalar and periodic variables must account for the different nature of the data.
• The Approximated Wrapped and Linear Gaussian is:• circularly defined along specific dimensions thus
independent from the value set as data origin• periodic every 2𝜋 interval on angular dimensions and not periodic along scalar ones
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Elastic Comparison between Symbols Sequences “We transform comparison between two sequences of features in the comparison between two sequences of symbols, with every symbol corresponding to a single AWLG distribution”
• Due to uncertainty and spatial/temporal shifts, exact matching between sequences is unsuitable for computing similarities
• We use Global Alignment between two sequences, basing the distance as a cost of the best alignment of the symbols
• Dynamic Programming reduce computational time to O (N · M)
“Using global alignment instead of local one is preferable because the former preserves both global and local shape characteristics”
• Mixture learnt components are associated to the most similar trajectory observation using MAP
“The trajectories’ are modeled as sequences of symbols each one associated to a AWLG pdf that better describe the associated observation vector”
Symbol to Symbol similarity measure:“Since the symbols we are comparing correspond to pdf, match/mismatch should be proportional to the distance between the two corresponding pdfs”
• AWLG pdf is a single wrap of a wrapped Gaussian
• KL Divergence can be used to compare AWLG distributions
• The Alignment Cost between is proportional to the Average Resitor difference of KL Divergence.
Andrea Prati is with Dipartimento di Scienze e Metodi dell’Ingegneria, University of Modena and Reggio Emilia, Italy. Simone Calderara and Rita Cucchiara are with Dipartimento di Ingegneria dell’Informazione, Università di Modena e Reggio Emilia, Italy. Email: {andrea.prati, rita.cucchiara,simone.calderara}@unimore.it
Experimental resultsOur model has been tested on >500 Trajectories as distance measure for the K-Medoids Clustering Algorithm• Clusters have been compared against a manual
Ground Truth• The method has been compared with two state of the
art approaches [1] and [2] that use different representations
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# Traj [1] [2] AWLG
Direction 140 78% 73% 95%
Speed 108 80% 87% 99%
Direction Speed Time
195 94% 86% 96%
Complete 543 90% 80% 97%