similarity measuress laboratory of image analysis for computer vision and multimedia università di...

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%