Università di Modena e Reggio Emilia, Italy
Similarity measuress
Laboratory of Image Analysis for
Computer Vision and Multimedia
http://imagelab.ing.unimo.it
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
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
AWLG ( X |  , ) 
1
2 
e
1
 ( X   )T  1 ( X   )
2
 


X  v 
 t 
( -  0 ) mod 2 

X -   
v  v0



t  t0
Trajectory Modeling using
Expectation Maximization
•Each trajectory is encoded as a set of directions,
speeds and time value
 k , j 


T j  x1, j , x2, j , x3, j  xk , j   vk , j 
 tk , j 


•Each trajectory is modeled as a Mixture of AWLG
where number of components and parameters are
learnt trought the Expectation Maximization
• Mixture learnt components are associated to the
most similar trajectory observation using MAP

~
k
T  S i | i  1..N 
K
k  argmax AWLG(x i |  l ,  l )
l 1
“The trajectories’ are modeled as sequences of
symbols each one associated to a AWLG pdf that
better describe the associated observation
vector”
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”
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.
 KL( AWLGi | AWLG j )  KL( AWLG j | AWLGi ) 

 ( Si , T j )  

KL
(
AWLG
|
AWLG
)

KL
(
AWLG
|
AWLG
)
i
j
j
i 

1  j Ni 1
1
KL( AWLGi | AWLG j )  ln

 tr( j  i )
2 i
2 2
1
1
T
 ( i   j )  j ( i   j )
2
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
# Traj
[1]
[2]
AWLG
Direction
140
78%
73%
95%
Speed
Direction
Speed Time
Complete
108
195
80%
94%
87%
86%
99%
96%
543
90%
80%
97%
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
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