# Motivation for Visa Application by sxi11527

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```									Mean-Shift Algorithm and
Its Application

Bohyung Han
bhhan@cs.umd.edu
Introduction
   Computer vision applications and density
estimation
–   Background subtraction
–   Model representation
–   Particle filter
–   Any other statistical method
   Issues for density estimation
– How to represent density
– How to extract the important information
   Local maxima, minima
   Mode
Kernel Density Estimation

   Multivariate kernel density estimation
1    n
1 x  xi
f ( x)  d
nh
 h K( h )
i 1

   Kernels
– Gaussian                                  1 2
K N  (2 )   d / 2
exp(  x )
2
– Epanechnikov
1 / 2cd 1 (d  2)(1  x 2 ) if x  1
      
KE  

              0             otherwise
Mean-Shift Algorithm

   Basic idea
– Based on kernel density estimation
– Finding local optimum (mode)
– Iterative hill climbing algorithm
   Benefit over the direct computation
– Computational complexity
 Less density function evaluation
 Only local computation
Finding Mean-Shift Vector

– For Gaussian kernel
ˆ f ( x )  f ( x )  2
n
 x  xi 
           ˆ
nh d  2
 (x  x i ) K N  h 
i 1                      
 n            x  xi  
2          n                      h  
 x  x i   i 1
xi K N 
  x

nh d  2
  h 
KN           n
 x  xi       
   h 
i 1
KN          
 i 1                      


   Always converges to the local maximum!
Variable Bandwidth
Mean-Shift
   Motivation
– Fixed bandwidth: specification of a scale
parameter
– Difficult to find the global optimal scale
– Data-driven scale selection is required.
   Abramson’s rule
1/ 2
  
hi (x i )  h0           
 f (x i ) 
– h0 : fixed bandwidth for initial estimation
–  : geometric mean
Variable Bandwidth
Mean-Shift (cont’d)
– Also for Gaussian kernel
ˆ f ( x )  f ( x )  2  x  x i K  x  x i 
n
            ˆ                                       
d       N         
n i 1 hi               hi 
 n xi             x  xi  
                 h  
KN         
2   n
1         x  x i   i 1 hid  2     i   x
  d 2 K N         h  n 1
n i 1 hi          i                           x  xi  
  d 2 K N      h    
 i 1 hi
                  i      
Applications
   Pattern recognition
– Clustering
   Image processing
– Filtering
– Segmentation
   Density estimation
– Density approximation
– Particle filter
   Mid-level application
– Tracking
– Background subtraction
Application – Tracking (1)

   Target representation
n
qu  C  K (x* ) [b(x* )  u ]
ˆ            i        i
i 1

   Candidate representation
n
 y  xi 
ˆ u ( y )  Ch  K 
p                             [b(xi )  u ]
i 1  h 
   Bhattacharyya distance
m
 (y )  [ p(y ), q]   pu (y )qu
ˆ           ˆ      ˆ      ˆ      ˆ
u 1
Application – Tracking (2)

   Distance minimization
1 m             1 m           ˆ
qu
[p(y ), q]   pu (y 0 )qu   pu (y )
ˆ      ˆ          ˆ    ˆ          ˆ
2 u 1          2 u 1     ˆ
pu (y 0 )
 y  xi 
nh
1 m             C
  pu (y 0 )qu  h
ˆ    ˆ             h 
wi K 
2 u 1           2      i 1               

where           m
ˆ
qu
wi    [b(xi )  u ]
u 1              ˆ
pu (y 0 )

   Mean-Shift iteration
Application – Tracking (3)

   Mean-Shift tracking
algorithm

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