Similarity measuress
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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 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 j k j k j k j k j j j j t v x x x x T , , , , , 3 , 2 , 1 , , 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 2 interval on angular dimensions and not periodic along scalar ones 0 0 0 ) ( ) ( 2 1 2 mod ) - ( - 2 1 ) , | ( 1 t t v v X t v X e X AWLG X X T
description
Università di Modena e Reggio Emilia , Italy. Laboratory of Image Analysis for Computer Vision and Multimedia http://imagelab.ing.unimo.it. Similarity measuress. Simone Calderara, Rita Cucchiara. Motivations People Trajectories are rich descriptor of human activity - PowerPoint PPT Presentation
Transcript of Similarity measuress
Similarity measuressLaboratory 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%
