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)

National Conference on Machine Intelligence Research and

Advancement (NCMIRA, 12)

Shri Mata Vaishno Devi University, Katra

Jammu & Kashmir

21-23rd Dec., 2012 School of Mathematics, Shri Mata Vaishno Devi University, Katra

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

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.

(1)

Where

is the set of all complete finite discrete probability

distributions.

nii PppPH

,log)(n

1i

}2;1,0/),...,({n

1i21

npppppP iinn

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

)2(log),(1

n

i i

ii

q

ppQPD

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.

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.

for any A, B, C ε F(X)

),(

)},(),,(.{max

BAD

CBCADCBCAD

Bhandari and Pal (1992) defined measure of fuzzy

directed divergence corresponding to (2) as follow:

)3())(1(

))(1(log))(1(

)(

)(log)(),(

1

1

n

i iB

iAiA

n

i iB

iAiA

x

xx

x

xxBAD

An aggregation operation is defined by the function

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.

]1,0[]1,0[: nM

Let and be two binary aggregation operators then

(4)

Where

is a divergence measure.

We have such that

and

such that

),( baU ),( baV

i

iiii qpVqpUQPD ),(),(),(

nQP ,

]1,0[]1,0[: 2* A

2),(* ba

baA

]1,0[]1,0[: 2* H

ba

babaH

22* ),(

are aggregation operators.

Then following divergence measure can be defined using

the proposed method.

)5()(2

)(

2),(

1

2

1

22

**

n

i ii

ii

n

i

ii

ii

ii

AH

qp

qp

qp

qp

qpQPD

The measure of fuzzy directed divergence between two

fuzzy sets corresponding to (5) is defined as follow:

)6(

)()(2

1

)()(

1

2

))()((

),(

1

2

**

n

i iBiAiBiA

iBiA

F

AH

xxxx

xx

BAM

Let be an image of size

having L levels.

,)}(,{ XfffX ijijij MM

X.in pixel)th j(i, of Value Membership )( ijf

image. in the f level

gray theof occurences ofNumber )( fCount

background theandobject the

separates which valuesholdgiven thre t

X. image in the pixelj)th (i, of levelgray ijf 1)(0 ijf

.

regionobject theof levelgray Average

)(

)(.

region background theof levelgray Average

)(

)(.

1

1

1

1

1

0

0

0

L

tf

L

tf

t

f

t

f

fcount

fcountf

fcount

fcountf

For bilevel thresholding

Where ‘t’ is chosen threshold as stated.

where fmin and fmax are the minimum and maximum gray level in the image respectively.

objectfor ,).exp(

backgroundfor ,).exp()(

1

0

tfiffc

tfiffcf

ijij

ijijij

)(

1

minmax ffc

Then in view of equation (6) fuzzy divergence between A

and B is given by

B. andA image in the

pixelj)th (i, theof valuesmembership thebe )( and )( ijBijA ff

)7(

)()(2

1

)()(

1

2

))()((

),(

1

0

21

0

*

M

j ijBijAiBijA

ijBijAM

i

F

ffxf

ff

BAM

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.

Thus equation (7) reduces to

)8(

)(1

)(1

)(1

1

1)(

1

2

)1)((

1)(2

1

1)(

1

2

)1)((

),(

1

0

1

0

1

0

21

0

1

0

21

0

*

M

j ijA

ijAM

i

M

j ijAijA

ijAM

i

M

j ijAijA

ijAM

i

F

f

f

ff

f

ff

f

BAM

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.

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.

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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Thanks