Similarity measuress

1. 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 • 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 vsSelected:use all the temporal data or select a subset of them • 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 definedalong specific dimensionsthus independent from the value set as data origin • periodic every 2𝜋 interval on angular dimensions and not periodic along scalar ones • Trajectory Modeling using Expectation Maximization • Each trajectory is encoded as a setof 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

2. 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. • 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 Alignmentbetween 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” • Experimental results • Our 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 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