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
            Gradient
            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)
    – Density gradient estimation
    – Iterative hill climbing algorithm
   Benefit over the direct computation
    – Computational complexity
        Less density function evaluation
        Only local computation
Finding Mean-Shift Vector

   Gradient computation
    – 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)
   Gradient computation
    – 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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