Proximity Functions. Application to Fuzzy Thresholding - EUSFLAT

EUSFLAT - LFA 2005


       
  

  
    
 


       
   
 
    

  




         
  
  
 
                 
 
               
  
   
    
            
     
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     → [0, 1]    



I :

 
     I1   x ≤ z 
I(x, y) ≥ I(z, y)   y ∈ [0, 1] I2   y ≤ t 
I(x, y) ≤ I(x, t)   x ∈ [0, 1] I3  I(0, x) = 1   x ∈ [0, 1] I4  I(x, 1) = 1   x ∈ [0, 1] I5  I(1, 0) = 0  [0, 1]2

1 

    
 I %      %   # I6 I(1, x) = x  I7 I(x, I(y, z)) = I(y, I(x, z))  I8 x ≤ y     I(x, y) = 1  I9 I(x, 0) = n(x)     I10 I(x, y) ≥ y  I11 I(x, x) = 1  I12 I(x, y) = I(n(y), n(x))      n I13 I  

 


EUSFLAT - LFA 2005

             
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M : [0, 1] → [0, 1]  
n ≥ 2           
 

 A1. M (x1 , · · · , xn ) = 0  

  xi = 0   i ∈ {1, · · · , n} A2. M (x1 , · · · , xn ) = 1  

  xi = 1   i ∈ {1, · · · , n} A3.   
  (x1 , · · · , xn ) 
 (y1 , · · · , yn )
   xi, yi ∈ [0, 1]   i ∈ {1, · · · , n}  xi ≤ yi   i ∈ {1, · · · , n} 
M (x1 , · · · , xn ) ≤ M (y1 , · · · , yn ); M  

 
  
 


    
  A4. M      


  
 



M (x1 , · · · , xn ) = M (xp(1) , · · · , xp(n) ) 
  
p 
{1, · · · , n}

       

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  A, B ∈ F(X)     #

   
  

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SM : F(X) × F(X) → R     F(X)  SM  

  
   



 


(SM 1) SM (A, B) = SM (B, A)   A, B ∈ F(X) (SM 2) SM (A, Ac ) = 0  

  A (SM 3) SM (C, C) = MaxA,B,∈F (X) SM (A, B)    C ∈ F(X) (SM 4)    A, B, C, D ∈ F(X) 

A ≤ B ≤ C ≤ D


SM (A, D) ≤ SM (B, C)

 Ac       A    Ac = {(x, µAc (x) = c(µA (x)))|x ∈ X}        (SM 4)  6
   #   A, B, C ∈ F(X)%  A ≤ B ≤ C %  SM (A, B) ≥ SM (A, C)  SM (B, C) ≥ SM (A, C) !   %       
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χ : [0, 1]2 → [0, 1]      

     

 
 

 1) χ(x, y) = χ(y, x)   x, y ∈ [0, 1] 2) χ(x, y) = 1  

  x = y  3) χ(x, y) = 0  

  x = 1 
 y = 0  x = 0 
 y = 1 4) χ(x, y) = χ(c(x), c(y))   x, y ∈ [0, 1] c  
  


 5)    x, y, z ∈ [0, 1]  x ≤ y ≤ z  
χ(x, y) ≥ χ(x, z) 
 χ(y, z) ≥ χ(x, z)        8$  6
  #   x, y, z, t ∈ [0, 1]%  x ≤ y ≤ z ≤ t,  χ(y, z) ≥ χ(x, t)

    
 ϕ1, ϕ2        
 
   


χ(x, y) = ϕ−1 1 (1 − |ϕ2 (x) − ϕ2 (y)|)

 c(x) = ϕ2 (1 − ϕ2 (x))      



  

 

   


 

  x ∈ [0, 1]  

  ϕ1 (x) = ϕ2 (x)   x ∈ [0, 1] χ(1, x) = x

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EUSFLAT - LFA 2005

    
 χ     


 ϕ 
     
 
  F1 = ϕ ◦ χ     



 


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          I : [0, 1]2 → [0, 1]%   

    I 1     
  
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χ : [0, 1]2 →

  χ(1, x) = x   x ∈ [0, 1]     

     

I : [0, 1]2 → [0, 1]  I  
 I7 
 I13  

      
    ϕ  
 
    

[0, 1]

⎧ χ(x, y) = ϕ−1 {∧(1, ∧(1 − ϕ(x) + ϕ(y), ⎪ ⎪ ⎪ ⎪ 1 − ϕ(y) + ϕ(x)))} = ⎨ ϕ−1 (1 − |ϕ(x) − ϕ(y)|) ⎪ ⎪  ⎪ ⎪ ⎩ c(x) = ϕ−1 (1 − ϕ(x)).

n

i=1

%" 
& ϕ(x) = x4 %    

 1 χ(x, y) = ∧ 1, ∧(1 − x4 + y 4 , 1 − y 4 + x4 ) 4 % 1  c(x) = (1 − x4 ) 4

    
!  M : [0, 1]n → [0, 1]  

   

! 

     
    i) SM (A, B) = SM (B, A)   A, B ∈ F(X) ii) SM (A, Ac ) = 0  

  A 

 iii) SM (A, B) = 1  

  A = B  iv)  A ≤ B ≤ C  
SM (A, B) ≥ SM (A, C) 
 SM (C, B) ≥ SM (C, A) v) SM (Ac , Bc ) = SM (A, B)

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1 1 1 4 χ(x, y) = ∧ 1, ∧(1 − x + y 4 , 1 − y 4 + x 4 ) % 1  c(x) = (1 − x 4 )4 &
 )     χ   1 4




SM (A, B) = M χ(µA (xi ), µB (xi ))

χ(x, y) = ∧ 1, ∧(1−x+y, 1−y +x) %  c(x) = 1 − x

1 4

 

SM : F(X) × F(X) → [0, 1],

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 & ϕ(x) = x   %

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& ϕ(x) = x

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M (x1 , · · · , xn ) = 0  

  x1 = · · · = xn = 0 M (x1 , · · · , xn ) = 1  

  x1 = · · · = xn = 1 M  
  
 χ : [0, 1]2 → [0, 1]     






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EUSFLAT - LFA 2005

     
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