Download Lecture #5

Object Identification: A Bayesian Analysis
with Application to Traffic Surveillance
By Timothy Huang and Stuart Russell
University of California at Berkeley
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• Object Identification:
A and B are the same object?
• Object Recognition:
A
?
Class 1
?
Class 2
?
….....
Class k
• Is there difference in practice?
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• k moving objects:
trajectory of object i modeled by r.v.
Prh
h   h1 , , hk 
is a prior on object trajectories
• Agent makes observations:
oi
hi
o   o1 , ,on 
is observation of some object
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• Given
 o1 , ,oa , ,ob  ,on 
• Interested in:
Pr a  b ,o 
Pr a  b o  
Pr o 
1





Pr
a

b
,
o
h
Pr
h
dh

Pr o 
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Application to traffic surveillance
Camera D
Camera U

o  o , , o
u
u
1
u
n


o  o , , o
d

o  o ,o
u
d
d
1
d
n


Find: average travel time, origin/destination counts
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• Instead of modeling trajectories:
u
1
o
d
1
o
u
2
o
d
2
o
u
n
o
d
n
o
• Matching observations is less general than
modeling trajectories
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S is the set of all possible matchings
s   1,i  ,2, j  ,,n,l   S
Pr s 
is a uniform prior on S
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
Pr a  b o , o


u
1
Pr o u , o d

1
Pr o u , o d



d


Pr a  b ,o u ,o d

u
d
Pr o , o



u
d
Pr
a

b
,
o
,
o
,s





sS


u
d
Pr
o
,
o
s Pr s 

sS : a ,b s


  
1
d u
u

Pr o o , s Pr o s Pr s  

Pr o  sS :a ,b s



d u
Pr
o
o ,s

sS : a ,b s

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Assumption:

  Pro
 
Pr o o , s 

d
u
d
j
o ,i  j
u
i
i , j s

Pr a  b o ,o d  
u

 


d u
Pr
o
 j oi ,i  j
sS : a ,b s i , j s

Computationally expensive:
(n-1)! matchings to consider
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t , time of observation

 x , lane position



w , vehicle width

l , length  height

d
oj  

h
,
mean
color
hue


c , mean color saturation

v , mean color value

C , histogram of color distribution


d
j
d
j
d
j
d
j
d
j
d
j
d
j
d
j
For each feature,

Pr x x ,i  j
d
j
u
i

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In particular:


Pr t t , x , x ,i  j 
d
j
u
i
u
i
d
j

N  xu ,x d , x u ,x d  t  t
i




 xiu ,x dj
 xiu ,x dj
j
i
j
d
j
u
i

Standard deviation and mean of
predicted link travelu times starting
upstream in lane x i
d
ending downstream in lane x j
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Appearance probability:


Pr o o ,i  j 
d
j
u
i



Pr x x ,i  j   Pr C C ,i  j
d
j
u
i
d
j
u
i
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
System learns parameters online:
Downstream
Upstream
Lane 1
Lane2
Lane 1
Lane2




2
Pr x  1 x  1,i  j 
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3
d
u
Pr x j  2 xi  2 ,i  j 
4
d
j
u
i
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Exponential forgetting:
lane x
u
i
matches
lane x
d
j
time t
 xiu , x dj     xiu , x dj   1    t

controls how fast we forget
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Matching
• Aim: find pairs (a,b) s.t. Pra  b o  1  
• Formula computed in previously
computationally intractable
• Can find most probable complete matching
in n 3 time by weighted bipartite matching
• In best matching, Pr a  b o is not
necessarily high for all (a,b)
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“Leave one out” heuristic:
a,b,c
• Upstream observations:
Downstream observations: x , y , z
• Best assignment: x  a, y  b, z  c
with probability p
• Forbid match x  a and compute new
best assignment x  b, y  a, z  c with

probability p

• If p  p  threshold accept match (a,b)
• Repeat for all matched pairs
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m upstream and n downstream observations
• n dummy upstream and m dummy
downstream observations


• odj , dummy vehicle is “new”, i.e. entered
below upstream camera
•
dummy,o 
u
i
vehicle has left before
downstream camera
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



• Pr dummy o u   , probability to exit
j
 
• Pr oid dummy   Pr oid , where  is
some coefficient and Pr oid  is prior
appearance probability
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Results
• With on-ramps and off-ramps
14% matched ---- 100% accuracy
80% matched ---- 50% accuracy
• Without on-ramps and off-ramps
37% matched ---- 100% accuracy
80% matched ---- 64% accuracy
• Link travel time --- accurate within 1% for
2 mile distance --- no bias based on speed
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