2007,els,vlachos,if inf &app to pattern recog

8/21/2019 2007,ELS,Vlachos,If Inf &App to Pattern Recog

1/10

Intuitionistic fuzzy information Applications to pattern recognition

Ioannis K. Vlachos, George D. Sergiadis *

Aristotle University of Thessaloniki, Faculty of Technology, Department of Electrical and Computer Engineering, Telecommunications Division,

University Campus, GR54124, Thessaloniki, Greece

Received 21 July 2005; received in revised form 16 May 2006Available online 28 September 2006

Communicated by S.K. Pal

Abstract

This paper addresses the issue of information-theoretic discrimination measures for intuitionistic fuzzy sets (IFSs). Although manymeasures of distance, similarity, dissimilarity, and correlation between IFSs have been proposed, there is no reference regarding infor-mation-driven measures used for comparison between sets. In this work we introduce the concepts of discrimination information andcross-entropy in the intuitionistic fuzzy setting and we derive an extension of the De LucaTermini nonprobabilistic entropy for IFSs.Based on this entropy, we reveal an intuitive and mathematical connection between the notions of entropy for fuzzy sets (FSs) and IFSsin terms of fuzziness and intuitionism. Finally, we demonstrate the efficiency of the proposed discrimination information measure forpattern recognition, medical diagnosis, and image segmentation. 2006 Elsevier B.V. All rights reserved.

Keywords: Intuitionistic fuzzy set; Discrimination information measure; Intuitionistic fuzzy cross-entropy; Intuitionistic fuzzy entropy; Fuzziness;Intuitionism; Pattern recognition; Medical diagnosis; Image segmentation

1. Introduction

Since Zadeh introduced fuzzy sets (FSs) theory (Zadeh,1965), several new concepts of higher-order FSs have beenproposed. Among them, intuitionistic fuzzy sets (IFSs),proposed byAtanassov (1986, 1989, 1994a,b, 1999), pro-vide a flexible mathematical framework to cope, besidesthe presence of vagueness, with the hesitancy originatingfrom imperfect or imprecise information. IFSs are

described using two characteristic functions expressingthe degree of membership (belongingness) and the degreeof non-membership (non-belongingness) of elements of theuniverse to the IFS. Gau and Buehrer (1993) proposed

the notion of vague sets, which identifies with that of IFSsas pointed out byBustince and Burillo (1996).

Different aspects of IFSs have been used for pattern rec-ognition and decision making, where imperfect facts co-exist with imprecise knowledge. In the context of patternrecognition, Dengfeng and Chuntian (2002) and Mitchell(2003)applied similarity measures for IFSs, in order to per-form classification. For medical diagnosis, distance andsimilarity measures have been used in (Szmidt and Kacpr-

zyk, 2001c,b, 2004a). Another approach using intuitionisticfuzzy relations was proposed in (De et al., 2001). Correla-tion measures between IFSs and interval-valued IFSs weredefined in (Gerstenkorn and Manko, 1991; Bustince andBurillo, 1995; Hong and Hwang, 1995; Hong, 1998; Hungand Wu, 2002).

The main purpose of this paper is to introduce an infor-mation-theoretic framework for IFSs that is applicable topattern recognition problems. New concepts, such as thediscrimination information measure for IFSs and theintuitionistic fuzzy cross-entropy, are proposed and a

0167-8655/$ - see front matter 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.patrec.2006.07.004

* Corresponding author. Tel.: +30 2310 996314; fax: +30 2310 996312.E-mail address:[email protected] (G.D. Sergiadis).URL: http://mri.ee.auth.gr (G.D. Sergiadis).

www.elsevier.com/locate/patrec

Pattern Recognition Letters 28 (2007) 197206
mailto:[email protected]://mri.ee.auth.gr/http://mri.ee.auth.gr/mailto:[email protected]


2/10

generalization of the De LucaTermini nonprobabilisticentropy in the intuitionistic fuzzy setting is carried out.Moreover, a unified formulation of entropy, fuzziness,and intuitionism is thoroughly discussed. Applications ofthe proposed measures to pattern recognition, medicaldiagnosis, and image segmentation are also presented.

2. Intuitionistic fuzzy sets

In this section we present the basic elements of IFSstheory, which will be needed in the following analysis.

Definition 1. A fuzzy seteAdefined on a universeXmay begiven as (Zadeh, 1965):eA fhx; l~Axijx2 Xg; 1wherel~A :X! 0; 1is the membership function ofeA. Themembership value l ~Axdescribes the degree of belonging-ness ofx2X ineA.Definition 2. An intuitionistic fuzzy setA defined on a uni-verse Xis given byAtanassov (1986, 1989, 1994a,b, 1999):

A fhx;lAx; mAxijx2 Xg; 2

where

lA :X! 0; 1 and mA : X! 0; 1; 3

with the condition

0 6 lAx mAx 6 1; 4

for all x 2X.

The numberslA(x) andmA(x) denote thedegree of mem-bershipand thedegree of non-membershipofx to A, respec-tively. For an IFS A in Xwe call the intuitionistic indexofan element x 2X in A the following expression:

pAx 1lAx mAx: 5

We can considerpA(x) as ahesitancy degreeofx to A(Ata-nassov, 1986, 1989, 1994a,b, 1999). From(5) it is evidentthat

0 6 pAx 6 1; 6

for all x 2X.

FSs can also be represented using the notation of IFSs.An FSeA defined on Xcan be represented as the followingIFS:

A fhx;l~Ax; 1l~Axijx2 Xg; 7

with pA(x) = 0 for all x2X.The complementary set Ac ofAis defined as

Ac fhx; mAx;lAxijx2 Xg: 8

Throughout this paper by IFSX we denote the set ofall the IFSs defined on X. Similarly, FSX is the set of

all FSs on X.

3. Intuitionistic fuzzy information measures

Let p and q be two probability distributions of the dis-crete random variable X. Kullback defined the cross-entropy measure ofp from q as

Ip; q Xx2X px lnpxqx ; 9which measures the amount of discrimination ofp from q(Kullback and Leibler, 1951; Kullback, 1968). Lin (1991)pointed out that (9) is undefined ifq(x) = 0 and p(x)5 0for any x2X. To overcome this drawback, Lin proposeda modified cross-entropy measure described by

Kp; q Xx2X

px ln px

12px 1

2qx

; 10

which is well defined and independent of the values ofp(x)and q(x), x2X. Based on (10), Shang and Jiang (1997)

defined the fuzzy cross-entropy between two setseA; eB2 FSX, where hereinafter X denotes the finiteuniverse of discourse.

Definition 3 (Shang and Jiang, 1997). LeteA andeB be twoFSs defined on X. Then,

EeA; eB Xni1

l~Axi ln l~Axi

12 l~Axi l~Bxi

"

1l~Axi ln 1l~Axi

1 12 l~Axi l~Bxi

# 11

is called fuzzy cross-entropy, where n is the cardinality ofthe finite universe X.

Formula(11)is the degree of discrimination ofeA fromeB. However, EeA; eB is not symmetric with respect to itsarguments.Shang and Jiang (1997)proposed a symmetricdiscrimination information measure based on EeA; eB,given by

DFSeA; eB EeA; eB EeB; eA: 12Moreover, they showed that DFSeA; eB P 0 and

DFSeA; eB 0 if and only ifeAeB.3.1. Intuitionistic fuzzy cross-entropy and discrimination

information

Let us consider two sets A;B2 IFSX. In order toderive a cross-entropy measure for IFSs, we have to exploitthe information carried by both the membership and thenon-membership function. In an analogous manner toBhandari and Pal (1993), we consider the followingquantity:

IlA;B;xi lnlAxi

lBxi; 13

as the amount of information for discrimination oflA(xi)

from lB(xi). Therefore, the expected information for dis-

198 I.K. Vlachos, G.D. Sergiadis / Pattern Recognition Letters 28 (2007) 197206


3/10

crimination ofAagainstB, based solely on the membershipfunction, is given by

IlA;B Xni1

lAxi lnlAxi

lBxi: 14

Similarly, considering the non-membership function we

have

ImA;B Xni1

mAxi lnmAxi

mBxi: 15

Hence, the information for discrimination in favor of Aagainst Bis obtained as the sum of the quantities Il (A, B)and Im (A, B), i.e.,

I0IFSA;B Xni1

lAxi lnlAxi

lBximAxi ln

mAxi

mBxi

: 16

In order to overcome the drawback of(16)being undefined

iflB(xi) = 0 ormB(xi) = 0 for anyxi2X, a modified versionof(16)is introduced based on(10)and is given by

IIFSA;B Xni1

lAxi ln lAxi

12

lAxi lBxi

"

mAxi ln mAxi

12

mAxi mBxi

#: 17

According to Shannons inequality (Lin, 1991), one caneasily prove that IIFS (A, B) P 0 and IIFS (A, B) = 0 if andonly if A= B. A similar measure for FSs, denoted by

IFSeA; eB, was defined by Bhandari and Pal (1993). ForFSs in general IFSeA; eB 6IFSeAc; eBc. However, for IFSsthe following equality holds:

IIFSA;B IIFS Ac;Bc ; 18

where Ac and Bc are the complementary sets ofA and B,respectively. Therefore, in analogy with (11), we proceedto the following definition of the intuitionistic fuzzy cross-entropy.

Definition 4. For two sets A;B2 IFSX, IIFS (A, B) isthe intuitionistic fuzzy cross-entropy betweenAand B.

IIFS (A, B) can also be called discrimination information

for IFSs.However, one can observe thatIIFS (A, B) is not symmet-ric with respect to its arguments. Therefore, a symmetricmeasure is defined as follows:

Definition 5. For two sets A;B2 IFSX:

DIFSA;B IIFSA;B IIFSB;A 19

is called a symmetric discrimination information measurefor IFSs.

It can easily be verified that DIFS (A, B) P 0 and DIFS(A, B) = 0 if and only ifA= B. Moreover, IIFS (A, B) andDIFS (A, B) degenerate to their fuzzy counterparts when A

and Bare FSs.

3.2. Intuitionistic fuzzy entropy

Szmidt and Kacprzyk (2001a) extended the axiomsof De Luca and Termini (1972) proposing the followingdefinition for an entropy measure in the setting of IFSs.

Definition 6 (Szmidt and Kacprzyk,2001a). An entropy on

IFSX is a real-valued functional E: IFSX !0; 1, satisfying the following axiomatic requirements:

IFS1: E(A) = 0 iff A is a crisp set, i.e., lA(xi) = 0 orlA(xi) = 1 for all xi2X.

IFS2: E(A) = 1 ifflA(xi) = mA(xi) for all xi2X. IFS3: E(A) 6 E(B) if A 6 B, i.e., lA(xi) 6 lB(xi) andmA(xi) P mB(xi), for lB(xi) 6 mB(xi) or lA(xi) P lB(xi)and mA(xi) 6 mB(xi), for lB(xi) P mB(xi) for any xi2X.

IFS4: E(A) = E(Ac).

A different definition and interpretation of entropy for

IFSs, which will be discussed in Section 4, has been pre-sented byBurillo and Bustince (1996).

Shang and Jiang (1997) proved that the De LucaTermini entropy for FSs described by:

EFSLTeA 1

n ln 2

Xni1

l~Axi ln l~Axi

1l~Axi ln1l~Axi; 20

can be directly derived from(12)according to the followingtheorem.

Theorem 7 (Shang and Jiang,1997). IfeA 2 FSX, thenEFSLT

eA 2n ln 21DFSeA;fAc 1; 21wherefAc is the complementary set ofeA.

Since we have showed that the symmetric discriminationinformation measure DIFS (A, B) reduces to its fuzzy coun-terpartDFS(A, B) whenA and Bare FSs, it is expected thatan extension ofTheorem 7to the intuitionistic fuzzy settingwill still hold.

Theorem 8. For a set A2 IFSX:

EIFS

LTA 2n ln 21

DIFSA;Ac

1 22

is an intuitionistic fuzzy entropy measure.

Proof. In order for(22)to be qualified as a sensible mea-sure of intuitionistic fuzzy entropy, it must satisfy the setIFS1IFS4 of axiomatic requirements. Substituting DIFS(A, Ac) in(22)with its expression from(19)and taking intoaccount(17), yields

EIFSLTA 1

n ln 2

Xni1

lAxi ln lAxi mAxi ln mAxi

1pAxi ln1pAxi pAxi ln 2: 23

I.K. Vlachos, G.D. Sergiadis / Pattern Recognition Letters 28 (2007) 197206 199


4/10

IFS1: LetA be a crisp set with membership values beingeither 0 or 1 for all xi2X. Taking into account thatpA(xi) = 0 for all xi2 X, from (23) we obtain that

EIFSLTA 0. Suppose now that EIFSLTA 0. However,

1 pA(xi) = lA(xi) + mA(xi) for all xi2X and thus (23)can be rewritten as

EIFSLTA 1

n ln 2Xni1

lAxi ln lAxi

lAxi mAxi

mAxi ln mAxi

lAxi mAxipAxi ln 2

: 24

Since every term in the summation is non-positive, wededuce that every term should equal zero, i.e.,

lAxi ln lAxi

lAxi mAxi 0 and

mAxi ln mAxi

lAxi mAxi 0 25

and also pA(xi) = 0, i.e., mA(xi) = 1 lA(xi), for all xi2X.This set of equations implies that Ais a crisp set.

IFS2: Let lA(xi) = mA(xi) for all xi2X. Applying thiscondition to(23) yields EIFSLTA 1. Let us now supposethat EIFSLTA 1. Then, from(23)we obtain that:Xn

i1


lAxi mAxi lnlAxi mAxi

2

0: 26

Consider now the following function:

fx x lnx; 27wherex2(0,1]. From(27)we have thatf0(x) = 1 + lnxand

f00x 1x>0, since x2(0,1]. Consequently, fis a convex

function. Therefore, for any two points x1 and x2 in (0,1]the following inequality holds:

fx1 fx2

2 P f

x1x22

: 28

Substituting(27)into(28)yields

x1lnx1 x2lnx2 x1x2 lnx1x2

2 P 0; 29

with the equality holding only forx1=x2. One can observe

that (29) has the same form as (26). Moreover, ifx1= x2= 0 the equality in (29) will still hold since0ln0 = 0. Therefore, from (29) follows immediately that(26)is satisfied if and only iflA(xi) = mA(xi) for all xi2X.

IFS3: In order to show that (23) fulfills requirementIFS3, it suffices to prove that the function:

gx;y 1

ln 2x lnxylny xy lnxy

1xy ln 2; 30

wherex,y2[0, 1], is increasing with respect to its first argu-ment xand decreasing for y. Taking the partial derivative

ofgwith respect to x yields

ogx;y

ox

ln 2x

xyln 2

: 31

In order to find the critical point ofgwe set ogx;yox

0. Solv-ing for the critical point xcp, we obtain

xcp y: 32

From(31) and (32) we have that:

ogx;y

ox P 0; when x 6 y 33

and

ogx;y

ox 6 0; when x P y 34

for any x, y 2[0,1]. Thus,fis increasing with respect to xfor x 6 y and decreasing when x P y.

Similarly, we obtain that:

ogx;y

oy 6 0; when x 6 y 35

and

ogx;y

oy P 0; when x P y: 36

Let us now consider two sets A;B2 IFSXwithA 6 B.Assume, that the finite universe X is partitioned into twodisjoint sets X1 and X2, with X1[X2= X. Let us furthersuppose that all xi2X1 are dominated by the conditionlA(xi) 6 lB(xi) 6 mB(xi) 6 mA(xi), while for all xi2X2lA(xi) P lB(xi) P mB(xi) P mA(xi) holds. Then, from themonotonicity of f and (23) we obtain that EIFSLTA 6

EIFSLTB when A 6 B.IFS4: Evaluating (23) for Ac = {hxi,mA(xi), lA(xi)ij

xi2X} yields EIFSLTA E

IFSLTA

c. h

Proposition 9. IfeA2 FSX the intuitionistic fuzzyentropy EIFSLT

eA reduces to the De LucaTermini entropyEFSLT

eA for FSs.Proof. The proof is trivial. h

From Theorem 8 it is evident that 0 6 DIFS (A, Ac) 6

2n ln2. It can also be proved that the same holds for

DIFS (A, B), i.e.,0 6 DIFSA;B 6 2n ln 2; 37

which means that DIFS (A, B) is finite when the IFSs A andBare finite.

4. Entropy, fuzziness, and intuitionism a unified

formulation

Burillo and Bustince (1996), prior toSzmidt and Kacpr-zyk (2001a), proposed a different set of axioms for intui-tionistic fuzzy entropy measures. Instead of the axiomaticrequirements IFS1IFS4, they argued that an entropy mea-

sure should possess the following properties:



5/10

IFS1 0: E(A) = 0 iffAis an FS, that is A 2 FSX. IFS2 0:E(A) = Cardinal(X) = nifflA(xi) = mA(xi) = 0 for

all xi2X. IFS3 0: E(A) P E(B) if A^ B, i.e., lA(xi) 6 lB(xi) andmA(xi) 6 mB(xi), for all x i2X.

IFS4 0: E(A) = E(Ac).

Moreover, they introduced an entropy measure satisfy-ing their set of requirements, given by

EBBA Xni1

pAxi: 38

It is well known that for FSs entropy is a measure offuzziness. Since the notion of IFSs is a generalization ofthat of FSs, it is expected that a direct connection betweenthe concepts of entropy and fuzziness among the two set-tings should exist. Using the proposed generalization ofthe De LucaTermini entropy, we will demonstrate that

such a connection is present.Let us rewrite(23)as

EIFSLTA EfuzzA EintuitA; 39

where the terms Efuzz(A) and Eintuit(A) are described by

EfuzzA 1

n ln 2

Xni1


1pAxi ln1pAxi 40

and

EintuitA Xni1pAxin

; 41

respectively. One can observe thatEintuit(A) is the normal-ized entropy EBB(A) of(38), which measures the degree ofintuitionismof the IFS A (Burillo and Bustince, 1996).

At first sight, an intuitive interpretation of the termEfuzz(A) may seem to be a little more complicated. How-ever, if(40)is closely examined, we can notice that the termEfuzz(A) describes the intrinsic fuzziness of set A. Thatis, Efuzz(A) measures how fuzzy the IFS A is, in termsof its elements being dominated by the condition

lAxi mAxi 12for allxi2X. Therefore, we callEfuzz(A)

the degree of fuzziness of the IFS A.The above analysis leads us to draw an interesting con-

clusion regarding the concept of entropy in the intuitionis-tic fuzzy setting. According to (39), intuitionistic fuzzyentropy consists of two intuitively and mathematically dis-

tinct components: one expressing the degree of fuzzinessand the other the degree of intuitionism of an IFS A. Ifthe set A is an FS, the fuzziness component reduces tothe De LucaTermini definition of entropy for FSs, whilethe intuitionistic component becomes zero. On thecontrary, when A is totally intuitionistic, i.e., lA(xi) =mA(xi) = 0 for all xi2X, the entropy of the IFS dependssolely on the intuitionistic component, since the degree offuzziness is zero. Therefore, we can conclude that there isa conceptual and qualitative difference among the defini-tions of entropy in the fuzzy and the intuitionistic fuzzysetting. In FSs theory entropy is indeed a measure of fuzz-iness, while for IFSs entropy measures both fuzziness and

intuitionism.Atanassov (1999)proposed a geometrical representation

of IFSs in an Euclidean plane with Cartesian coordinates.Szmidt and Kacprzyk (2000) extended this approach byconsidering all three parameters of IFSs and proposedthe geometric interpretation of IFSs as a mapping into asimplex in the unit intuitionistic fuzzy cube. Let us nowconsider an IFS A in X= {x}. Using the latter representa-tion,Figs. 1a and b illustrate the two components Efuzz(A)

Fig. 1. Components of intuitionistic fuzzy entropyEIFSLT. Degrees of (a) fuzziness Efuzz and (b) intuitionism Eintuit for IFSs defined in X= {x}.

Fig. 2. Intuitionistic fuzzy entropy EIFSLT for IFSs defined in X= {x}.



6/10

and Eintuit(A), respectively, which comprise the intuitionis-tic fuzzy entropy measureEIFSLTA. The color of each point(lA(x), mA(x), pA(x)) on the simplex denotes the entropyvalue of the set A= {hx, lA(x), mA(x)ijx} correspondingto that point. Finally, Fig. 2depicts the entropy EIFSLTA.

5. Applications

IFSs are a suitable tool to cope with imperfectly definedfacts and data, as well as with imprecise knowledge. In thissection we present applications of the symmetric discrimi-nation information measure for IFSs, in the context ofpattern recognition, medical diagnosis, and image segmen-tation.

5.1. Pattern recognition

In order to demonstrate the application of the intro-duced symmetric discrimination information measure for

IFSs to pattern recognition, we considered the problem dis-cussed in Dengfeng and Chuntian (2002) and Mitchell(2003). We are given three known patterns P1, P2 and P3,which have classifications C1, C2 and C3 respectively.The patterns are represented by the following IFSs inX= {x1, x2, x3}:

P1 fhx1; 1:0; 0:0jx1i; hx2; 0:8; 0:0jx2i; hx3; 0:7; 0:1jx3ig;

42

P2 fhx1; 0:8; 0:1jx1i; hx2; 1:0; 0:0jx2i; hx3; 0:9; 0:0jx3ig;

43

P3 fhx1; 0:6; 0:2jx1i; hx2; 0:8; 0:0jx2i; hx3; 1:0; 0:0jx3ig:44

Given an unknown pattern Q, represented by the IFS

Q fhx1; 0:5; 0:3jx1i; hx2; 0:6; 0:2jx2i; hx3; 0:8; 0:1jx3ig;

45

our aim is to classifyQ to one of the classes C1,C2and C3.According to the principle of minimum discriminationinformation between IFSs, the process of assigning Q toCk is described by

k arg mink

DIFSPk; Qf g: 46

Table 1presentsDIFS(Pk,Q), k2{1,2,3}. One can observethatQ has correctly being classified to C3, a result which isin agreement with the ones obtained in (Dengfeng andChuntian, 2002; Mitchell, 2003).

5.2. Medical diagnosis

IFSs have drawn the attention of many researchers inorder to perform medical diagnosis. Different elements ofthe IFSs theory have been utilized to carry out this task,such as the normalized Hamming and Euclidean distances

(Szmidt and Kacprzyk, 2001b,c), a similarity measureinvolving both similarity and dissimilarity between sets(Szmidt and Kacprzyk, 2004b,a), and intuitionistic fuzzyrelations (De et al., 2001). In this paper, we propose analternative approach to medical diagnosis using the newlydefined concept of the symmetric discrimination informa-tion measure.

Let us consider the same data as in (De et al., 2001;Szmidt and Kacprzyk, 2001c,b, 2004a), consisting of a setof patients P = {Al, Bob, Joe, Ted}, a set of diagnosesD= {Viral fever, Malaria, Typhoid, Stomach problem,Chest pain}, and a set of symptoms S= {Temperature,Headache, Stomach pain, Cough, Chest pain}.Table 2pre-

sents the characteristic symptoms for the diagnoses consid-ered. The symptoms for each patient are given inTable 3.Each element of the tables, is given in the form of a pair ofnumbers corresponding to the membership and non-mem-bership values, respectively, e.g., inTable 2the temperaturefor viral fever is described by (l,m) = (0.4,0.0).

In order to find a proper diagnosis, we calculate foreach patientpi2P, where i2{1, . . .,4}, the symmetric dis-crimination information measure for IFSs DIFS(s(pi), dk)between patient symptoms and the set of symptoms thatare characteristic for each diagnosis dk2D, with k2{1, . . .,5}. Similarly to (46), the proper diagnosis dk for

the ith patient p iis derived according to:

k arg mink

fDIFSspi; dkg: 47

Therefore, we assign to the ith patient the diagnosis whosesymptoms have the lowest symmetric discrimination infor-

Table 1Symmetric discrimination information measure DIFS(Pk,Q), with

k2{1,2,3}

P1 P2 P3

Q 0.4492 0.3487 0.2480

Table 2Symptoms characteristic for the diagnoses considered

Viralfever

Malaria Typhoid Stomachproblem

Chestproblem

Temperature (0.4, 0.0) (0.7, 0.0) (0.3, 0.3) (0.1, 0.7) (0.1, 0.8)

Headache (0.3, 0.5) (0.2, 0.6) (0.6, 0.1) (0.2, 0.4) (0.0, 0.8)Stomach

pain(0.1, 0.7) (0.0, 0.9) (0.2, 0.7) (0.8, 0.0) (0.2, 0.8)

Cough (0.4, 0.3) (0.7, 0.0) (0.2, 0.6) (0.2, 0.7) (0.2, 0.8)Chest pain (0.1, 0.7) (0.1, 0.8) (0.1, 0.9) (0.2, 0.7) (0.8, 0.1)

Table 3Symptoms characteristic for the patients considered

Temperature Headache Stomach pain Cough Chest pain

Al (0.8, 0.1) (0.6, 0.1) (0.2, 0.8) (0.6, 0.1) (0.1, 0.6)Bob (0.0, 0.8) (0.4, 0.4) (0.6, 0.1) (0.1, 0.7) (0.1, 0.8)Joe (0.8, 0.1) (0.8, 0.1) (0.0, 0.6) (0.2, 0.7) (0.0, 0.5)

Ted (0.6, 0.1) (0.5, 0.4) (0.3, 0.4) (0.7, 0.2) (0.3, 0.4)



7/10

mation measure from patients symptoms. The results forthe considered patients are given inTable 4.

In (Szmidt and Kacprzyk, 2001b, 2004a) a comparisonbetween different methods for medical diagnosis based onIFSs theory was carried out. The authors stated that usingthe similaritydissimilarity measure proposed in (Szmidtand Kacprzyk (2004a,b)) results in more proper diagnosesthan using distance-based measures, since it avoids draw-ing conclusions about strong similarity based on small

distances between IFSs. Moreover, they showed that theirmeasure performs better compared to the intuitionisticfuzzy relations method ofDe et al. (2001). The similaritymeasure used was defined as

SimA;B lIFSA;B

lIFSA;Bc

; 48

wherelIFS(A, B) denotes the normalized Hamming distancebetween the IFSs Aand Band is given by

lIFSA;B 1

2n

Xni1

jlAxi lBxij jmAxi mBxij

jpAxi pBxij: 49

The results obtained using the proposed symmetric dis-crimination information measure are an aggregation of thediagnoses obtained using the distance-based methods andthe similaritydissimilarity measure. For example, theresults for Al, Bob, and Joe are in agreement with the onesobtained using (48), while for Ted we derived the samediagnosis as when using the normalized Hamming distancemeasure. Therefore, the proposed measure is characterizedby both the relevant criterion of the similaritydissimilaritymeasure and the absolute behavior of distance-based meth-ods. Moreover, the symmetric discrimination informationmeasure has the advantage of being finite according to(37), while the similaritydissimilarity measure of (48) isnot. Furthermore, let us consider the case of two IFSs Aand A0, with each one being equidistant from the IFSsB and Bc. Thus, the similarity measure of (48) yieldsSim(A, B) = Sim(A0, B) = 1, despite the fact that maybeone of the sets A and A 0 is closer to B than the other.However, this is not the case with the proposed measure.

5.3. Image segmentation

Let us consider an imageAof sizeM Npixels, havingLgray levelsgranging between 0 andL1. According toPal

and King (1980, 1981, 1982), the image can be considered as

an array of fuzzy singletons. Each element of the arraydenotes the membership value of the gray level gij, corre-sponding to the (i,j)th pixel, with respect to an image prop-erty. Therefore, the image can be represented as the FS:

eA fhgij;l~Agijijgij2 f0;. . .;L1gg; 50

wherei2{1, . . ., M} and j2{1, . . ., N}. For the task of im-age segmentation we consider as the property the distanceof gray levels from the means of their respective classes.

Chaira and Ray (2003)proposed an image segmentationmethod based on the minimization of fuzzy divergence.Motivated by their work, we extend the aforementionedapproach to the intuitionistic fuzzy setting. Given a certainthresholdTthat separates the foreground (object) from thebackground, the average gray level of the background isgiven by

mB XT

g0ghAg

XTg0hAg ; 51while for the foreground as

mF

XL1gT1

ghAgXL1gT1

hAg; 52

where hA is the histogram of image A. Chaira and Ray(2003) defined the membership function of each pixel inthe image to be determined using the gamma distribution as

l~A

g; T

exp 1

gmaxgminjgmBj

; if g6 T

exp 1gmaxgmin

jgmFj

; if g>T;

8>>>>>:

53

where gmin and gmax are the minimum and maximum graylevels of the image, respectively.

Based on the FS described by (53), we construct corre-sponding membership and non-membership functionsgiven by

lAg; T kl~Ag; T 54

and

mAg; T 1l~Ag; Tk; 55

where k2 [0,1]. Therefore, the image in the intuitionisticfuzzy domain is represented as the following IFS:

A fhg;lAg; mAgijg2 f0;. . .;L1gg: 56

It is easy to verify that A is a proper IFS, since0 6 lA(g; T) + mA(g; T) 6 1 for any k2[0, 1] and for allg2{0, . . ., L1}. If k = 1, then the IFS reduces to thecorresponding FS described by(53).

Let us denote by T an ideally segmented image, forwhich foreground and background regions have been pre-cisely segmented. The underlying idea of the described

approach of Chaira and Ray (2003) is the minimization

Table 4Symmetric discrimination information measure DIFS(s(pi), dk) betweeneach patients symptoms and the considered set of possible diagnoses

Viral fever Malaria Typhoid Stomachproblem

Chestproblem

Al 0.4304 0.6045 0.5065 1.8899 2.2681Bob 1.4777 2.6161 0.8239 0.2276 1.4692

Joe 0.6762 0.7365 0.4582 1.9779 2.3405Ted 0.3593 0.7252 0.6585 1.2585 1.5547



8/10

of the fuzzy divergence between the actual and the ideallythresholded image, which leads to the maximum belonging-ness of each foreground pixel to the foreground region andthat of each background pixel to background. Conse-quently, after thresholding, it is expected that the threshol-ded image will approximate to the ideally segmented image.

Therefore, from(19)and taking into account that lTg 1 for all g2{0, . . ., L1}, we obtain that

DIFSA;T; T XL1g0

hAg lAg; T ln 2lAg; T

1lAg; T

mAg; T ln 2ln 2

1lAg; T

: 57

The optimization criterion can be formulated as

Topt arg minT

fDIFSA;T; Tg; 58

where T2{gmin, . . .,gmax} and Topt is the optimal thresh-

old. It should be mentioned that ifk = 1, thenDIFSreduces

to DFS and the algorithm degenerates to its fuzzycounterpart.

For evaluating the performance of the proposedsymmetric discrimination information measure DIFS forimage segmentation, we have considered various syntheticand real-world images of size 256 256 pixels having

8 bits-per-pixel gray-tone resolution. Figs. 35 illustratethe results obtained using the proposed intuitionisticfuzzy measure, as well as its fuzzy counterpart. Simula-tions have showed that setting parameter k= 0.2 yieldsthe overall best result with respect to the minimizationof the empirical discrepancy measures described in theend of this section. From the images of Figs. 35 onemay observe that the results obtained using the proposedsymmetric discrimination information measure DIFSoutperform the ones derived using its fuzzy counterpart.Additionally, geometrical properties of the objects,such as the area and the shape, were successfully pre-served, even for the low-contrasted images of Figs. 3

and 5.

Fig. 3. (a) Test image (adopted from Tizhoosh (2005)) and (b) ideal manually segmented image. Images obtained using (c) fuzzy discrimination

information measure (Topt= 35) and (d) its proposed intuitionistic counterpart (Topt= 25).

Fig. 4. (a) Test image and (b) ideal manually segmented image. Images obtained using (c) fuzzy discrimination information measure (Topt= 103) and (d)

its proposed intuitionistic counterpart (Topt= 123).

Fig. 5. (a) Test image (adopted from Tizhoosh (2005)) and (b) ideal manually segmented image. Images obtained using (c) fuzzy discrimination

information measure (Topt= 101) and (d) its proposed intuitionistic counterpart (Topt= 1).



9/10

In order to further assess the performance of the DIFSmeasure in a quantitative way, we have considered the fol-lowing empirical discrepancy criteria, proposed byYasnoffet al. (1977) and Zhang (1965), which measure the disparitybetween the segmented and a reference binary image in thecase of bi-level thresholding. These criteria are character-ized by a large dynamic range and by their capability to

distinguish small segmentation degradation. It should bementioned that the reference image required for calculatingthe empirical discrepancy measures is obtained by manualsegmentation.

Misclassification error (ME): The misclassificationerror is given as

ME 1jBR\BSj jFR\FSj

jBRj jBSj ; 59

whereBRandFRdenote the background and foregroundpixels of the reference image, respectively. BS and FS

refer to corresponding pixels of the ideal manually seg-mented image, while j jdenotes the cardinality of a set. Probability error (PE): The probability error is

expressed as

PE PF PBjF PB PFjB; 60

whereP(BjF) (P(FjB)) is the probability of error in clas-sifying foreground (background) as background (fore-ground) and P(F), P(B) are the a priori probabilities offoreground and background, respectively.

Relative ultimate measurement accuracy (RUMA): TheRUMAcriterion measures the discrepancy of object fea-tures between the reference and the actual segmented

image. If we consider the area property, the criterionis formulated as

RUMAjAR ASj

AR; 61

where AR and AS are the area features of the referenceand the segmented images, respectively.

FromFig. 6one may observe that the algorithm basedon the proposedDIFSmeasure produces segmented imagesthat are closer to the ideally segmented ones, than theapproach using the DFS. This is justified in terms of signif-

icantly smaller values characterizing the corresponding

empirical discrepancy measures for the intuitionistic fuzzyapproach.

6. Conclusions

In this paper we presented an information-theoreticapproach to discrimination measures for IFSs. A symmetric

discrimination information measure was proposed and thenotion of intuitionistic fuzzy cross-entropy was introduced.Furthermore, a generalized version of the De LucaTermininonprobabilistic entropy was derived for IFSs. Based onthis generalization, a connection between the concepts offuzziness and entropy in the fuzzy and the intuitionisticfuzzy setting was established. Finally, we demonstratedthe efficiency of the proposed symmetric discriminationinformation measure in the context of pattern recognition,medical diagnosis, and image segmentation.

References

Atanassov, K.T., 1986. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 8796.

Atanassov, K.T., 1989. More on intuitionistic fuzzy sets. Fuzzy Sets Syst.33, 3746.

Atanassov, K.T., 1994a. New operations defined over the intuitionisticfuzzy sets. Fuzzy Sets Syst. 61, 137142.

Atanassov, K.T., 1994b. Operators over interval-valued intuitionisticfuzzy sets. Fuzzy Sets Syst. 64, 159174.

Atanassov, K.T., 1999. Intuitionistic Fuzzy Sets: Theory and Applica-tions. Studies in Fuzziness and Soft Computing, vol. 35. Physica-Verlag, Heidelberg.

Bhandari, D., Pal, N.R., 1993. Some new information measures for fuzzysets. Inform. Sci. 67, 209228.

Burillo, P., Bustince, H., 1996. Entropy on intuitionistic fuzzy sets and on

interval-valued fuzzy sets. Fuzzy Sets Syst. 78, 305316.Bustince, H., Burillo, P., 1995. Correlation of interval-valued intuitionistic

fuzzy sets. Fuzzy Sets Syst. 74, 237244.Bustince, H., Burillo, P., 1996. Vague sets are intuitionistic fuzzy sets.

Fuzzy Sets Syst. 79, 403405.Chaira, T., Ray, A.K., 2003. Segmentation using fuzzy divergence. Pattern

Recognition Lett. 24, 18371844.De, S.K., Biswas, R., Roy, A.R., 2001. An application of intuitionistic

fuzzy sets in medical diagnosis. Fuzzy Sets Syst. 117, 209213.De Luca, A., Termini, S., 1972. A definition of a nonprobabilistic entropy

in the setting of fuzzy sets theory. Inform. Control 20, 301312.Dengfeng, L., Chuntian, C., 2002. New similarity measures of intuition-

istic fuzzy sets and application to pattern recognitions. PatternRecognition Lett. 23, 221225.

Gau, W.L., Buehrer, D.J., 1993. Vague sets. IEEE Trans. Systems Man

Cybernet. 23 (2), 610614.

MEMEME PEPEPE RUMARUMARUMA

DFSDFSDFS

DIFSDIFSDIFS

0.20.2

0.40.4

0.60.6

0.80.8

000

11

0.01

0.02

0.03

0.04

Fig. 6. Comparison of the empirical discrepancy measures for the fuzzy and intuitionistic fuzzy approaches, for the images of (a)Fig. 3, (b)Fig. 4, and (c)Fig. 5.



10/10

Gerstenkorn, T., Manko, J., 1991. Correlation of intuitionistic fuzzy sets.Fuzzy Sets Syst. 44, 3943.

Hong, D.H., 1998. A note on correlation of interval-valued intuitionisticfuzzy sets. Fuzzy Sets Syst. 95, 113117.

Hong, D.H., Hwang, S.Y., 1995. Correlation of intuitionistic fuzzy sets inprobability spaces. Fuzzy Sets Syst. 75, 7781.

Hung, W.-L., Wu, J.-W., 2002. Correlation of intuitionistic fuzzy sets bycentroid method. Inform. Sci. 144, 219225.

Kullback, S., 1968. Information Theory and Statistics. Dover Publica-tions, New York.

Kullback, S., Leibler, R.A., 1951. On information and sufficiency. Ann.Math. Statist. 22, 7986.

Lin, J., 1991. Divergence measures based on the Shannon entropy. IEEETrans. Inform. Theory 37, 145151.

Mitchell, H.B., 2003. On the DengfengChuntian similarity measure andits application to pattern recognition. Pattern Recognition Lett. 24,31013104.

Pal, S.K., King, R.A., 1980. Image enhancement using fuzzy set. Electron.Lett. 16, 376378.

Pal, S.K., King, R.A., 1981. Image enhancement using smoothing withfuzzy sets. IEEE Trans. Systems Man Cybernet. 11, 495501.

Pal, S.K., King, R.A., 1982. A note on the quantitative measure of imageenhancement through fuzziness. IEEE Trans. Pattern Anal. Mach.Intell. 4, 204208.

Shang, X.-G., Jiang, W.-S., 1997. A note on fuzzy information measures.Pattern Recognition Lett. 18, 425432.

Szmidt, E., Kacprzyk, J., 2000. Distances between intuitionistc fuzzy sets.Fuzzy Sets Syst. 114, 505518.

Szmidt, E., Kacprzyk, J., 2001a. Entropy of intuitionistic fuzzy sets. FuzzySets Syst. 118, 467477.

Szmidt, E., Kacprzyk, J., 2001b. Intuitionistic fuzzy sets in intelligent dataanalysis for medical diagnosis. In: Alexandrov, V.N., Dongarra, J.J.,Juliano, B.A., Renner, R.S., Tan, C.J.K. (Eds.), ICCS 2001, vol.LNCS 2074. Springer-Verlag, Berlin Heidelberg, pp. 263271.

Szmidt, E., Kacprzyk, J., 2001c. Intuitionistic fuzzy sets in some medicalapplications. In: Reusch, B. (Ed.), Fuzzy Days 2001, vol. LNCS 2206.Springer-Verlag, Berlin, Heidelberg, pp. 148151.

Szmidt, E., Kacprzyk, J., 2004a. A similarity measure for intuitionisticfuzzy sets and its application in supporting medical diagnosticreasoning. In: Rutkowski, L., Siekmann, J.H., Tadeusiewicz, R.,Zadeh, L.A. (Eds.), ICAISC 2004, vol. LNAI 3070. Springer-Verlag,Berlin, Heidelberg, pp. 388393.

Szmidt, E., Kacprzyk, J., 2004b. Similarity of intuitionistic fuzzy sets andthe Jaccard coefficient. In: Proc. of the 10th Int. Conf. IPMU 2004.pp. 14051412.

Tizhoosh, H.R., 2005. Image thresholding using type II fuzzy sets. PatternRecognit. 38, 23632372.

Yasnoff, A.A., Mui, J.K., Bacus, J.W., 1977. Error measures for scenesegmentation. Pattern Recognit. 9, 217231.

Zadeh, L.A., 1965. Fuzzy sets. Inform. Control 8, 338353.Zhang, Y.J., 1965. A survey on evaluation methods for image segmen-

tation. Pattern Recognit. 29, 13351346.


2007,els,vlachos,if inf &app to pattern recog

Documents