This presents an approach of obtainig fuzzy directed divergences and their use in binary image segmentation.

1. National Conference on Machine Intelligence Research and
Advancement (NCMIRA, 12)
Shri Mata Vaishno Devi University, Katra
Jammu & Kashmir
New Measure of Fuzzy Directed Divergence and
its Application in Image Segmentation
Surender Singh
Asstt. Prof. , School of Mathematics
Shri Mata Vaishno Devi University , Katra –182320 (J & K)
School of Mathematics, Shri Mata 21-23rd Dec., 2012
Vaishno Devi University, Katra

2.  Shannon’s Measure
 Divergence Measure
 Fuzzy set
 Fuzzy Directed Divergence
 Aggregation Operations
 Development of New measure of fuzzy directed
divergence
 Application in image segmentation
 Conclusion

3.  Shannon initially developed information theory for
quantifying the information loss in transmitting a given
message in a communication channel Shannon(1948).
 The measure of information was defined Claude E.
Shannon in his treatise paper in 1948.
n
H ( P)   pi log pi , P  n (1)
i 1
Where n
n  {P  ( p1 , p2 ,...pn ) / pi  0,  pi  1 ; n  2}
i1
is the set of all complete finite discrete probability
distributions.

4.  The relative entropy or directed divergence is a
measure of the distance between two probability
distributions. The relative entropy or Kullback-Leibler
distance Kullback and Leibler (1951) between two
probability distributions is defined as
n
D( P, Q)   pi log
pi
(2)
i 1 qi

5. A correct measure of directed divergence must satisfy
the following postulates:
a. D (P,Q) ≥ 0
b. D (P,Q) = 0 iff P = Q
c. D (P, Q) is a convex function of both
and
if in addition symmetry and triangle inequality is also
satisfied by D(P,Q) then it called a distance measure.

8.  Let a universal set X and F (X) be the set of all fuzzy
subsets .A mapping D:F (X) × F (X)→ R is called a
divergence between fuzzy subsets if and only if the
following axioms hold:
a. D (A, B)
b. D (A, B) =0 if A=B
c. max.{D( A  C, B  C), D( A  C, B  C)} 
D( A, B)
for any A, B, C ε F(X)

9.  Bhandari and Pal (1992) defined measure of fuzzy
directed divergence corresponding to (2) as follow:
n
 A ( xi )
D( A, B)    A ( xi ) log 
i 1 B ( xi )
n
(1   A ( xi ))
(1 A (xi )) log (1 B (xi ))
i 1
(3)

10. An aggregation operation is defined by the function
M : [0,1]n  [0,1]
verifying
1. M(0,0,0,...0) = 0 , M(1, 1, 1,…,1) = 1
(Boundary Conditions)
2. M is Monotonic in each argument. (Monotonicity)
If n=2 then M is called a binary aggregation operation.

11. Let U (a, b) and V (a, b) be two binary aggregation operators
then
D(P, Q)   U ( pi , qi ) V ( pi , qi ) (4)
i
Where P, Q  n
is a divergence measure.
We have A* :[0,1]2 [0,1] such that
ab
A* (a, b) 
2
and H * : [0,1]2  [0,1]
a 2  b2
such that H (a, b) 
*
a b

12. are aggregation operators.
Then following divergence measure can be defined using
the proposed method.
n
 pi2  qi2 pi  qi 
DH * A* ( P, Q)    
i 1  pi  qi 2  
n
( pi  qi )2
 (5)
i 1 2( pi  qi )

13. The measure of fuzzy directed divergence between two
fuzzy sets corresponding to (5) is defined as follow:
M H * A* ( A, B) 
F
n
( A ( xi )  B ( xi ))2  

1 1
  (x )   (x )  2   (x )   (x ) 
i 1 2  A i B i A i B i 
(6)

14. Let X  { fij , ( fij )}  fij  X , be an image of size M M
having L levels.
fij  gray level of (i, j)th pixel in theimage X. 0  ( fij )  1
( fij )  MembershipValue of (i, j )th pixel in X.
Count( f )  Numberof occurencesof the gray
level f in the image.
t  given threshold value which separates
the object and the background

15. . t
 f .count( f )
0  f 0
t
 Averagegray level of the backgroundregion
count( f )
f 0
L1
 f .count( f )
1  f t 1
L1
 Averagegray level of the object region
count( f )
f t 1

16. For bilevel thresholding
 ( fij )  exp(c. fij  0 ) if fij  t , for background
 exp(c. fij  1 ) if fij  t , for object
Where ‘t’ is chosen threshold as stated.
1
c
( f max  f min)
where fmin and fmax are the minimum and maximum gray
level in the image respectively.

17.  A ( fij ) and B ( fij ) be the membershipvaluesof the (i, j)th pixel
in the image A and B.
Then in view of equation (6) fuzzy divergence between A
and B is given by
M *F ( A, B) 
M 1 M 1 ( A ( fij )  B ( fij ))2  
 
1 1
  
i 0 j 0 2 
  A ( fij )  B ( xi ) 2   A ( fij )  B ( fij ) 

(7)

18. Chaira and Ray (2005) proposed the following
methodology for binary image thresholding.
For bi-level or multilevel thresholding a searching
methodology based on image histogram is employed
here. For each threshold, the membership values of all
the pixels in the image are found out using the above
procedure. For each threshold value, the membership
values of the thresholded image are compared with an
ideally thresholded image.

19. Thus equation (7) reduces to
M *F ( A, B) 
M 1 M 1 ( A ( fij ) 1)2 
 
1 1
  
i 0 j 0 2   A ( fij )  1 2   A ( fij ) 1
 
M 1 M 1 (  ( f )  1 2 
) 
 
1 1
A ij
  
i 0 j 0 2 
  A ( fij )  1 1   A ( fij ) 

M 1 M 1 1   A ( fij ) 
  1  A ( fij ) 
i 0 j 0 
 

(8)

20.  An ideally thresholded image is that image which is
precisely segmented so that the pixels, which are in the
object or the background region, belong totally to the
respective regions.
 From the divergence value of each pixel between the
ideally segmented image and the above chosen
thresholded image, the fuzzy divergence is found out.
 In this way, for each threshold, divergence of each
pixel is determined according to Eq. (17) and the
cumulative divergence is computed for the whole
image.

21.  The minimum divergence is selected and the
corresponding gray level is chosen as the optimum
threshold.
 After thresholding, the thresholded image leads almost
towards the ideally thresholded image.

22. In this communication an approach to develop measures
of fuzzy directed divergence using aggregation
operators is proposed.
The proposed class of fuzzy directed divergence can be
generalized in terms one , two or three parametres. The
fuzzy directed divergence is also useful to solve
problems related to decision making,pattern recognition
and so on.

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28. Thanks