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