Contrast of a fuzzy relation

Information Sciences 180 (2010) 1326–1344

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Contrast of a fuzzy relation H. Bustince a,*, E. Barrenechea a, J. Fernandez a, M. Pagola a, J. Montero b, C. Guerra a a b

Departamento de Automática y Computación, Universidad Pública de Navarra, Campus Arrosadia s/n, P.O. Box 31006, Pamplona, Spain Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain

a r t i c l e

i n f o

Article history: Received 21 May 2009 Received in revised form 16 December 2009 Accepted 18 December 2009

Keywords: Fuzzy relation Interval-contrast Local contrast Total contrast

a b s t r a c t In this paper we address a key problem in many fields: how a structured data set can be analyzed in order to take into account the neighborhood of each individual datum. We propose representing the dataset as a fuzzy relation, associating a membership degree with each element of the relation. We then introduce the concept of interval-contrast, a means of aggregating information contained in the immediate neighborhood of each element of the fuzzy relation. The interval-contrast measures the range of membership degrees present in each neighborhood. We use interval-contrasts to define the necessary properties of a contrast measure, construct several different local contrast and total contrast measures that satisfy these properties, and compare our expressions to other definitions of contrast appearing in the literature. Our theoretical results can be applied to several different fields. In an Appendix A, we apply our contrast expressions to photographic images. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction In many information analysis and decision-making problems, the characteristics of neighboring datapoints are as important as the data itself. Indeed, in a number of fields, neglecting contextual and structural information can produce undesirable behavior in the analysis of information [43]. In order to evaluate a data element along with the information contained in its neighborhood, we consider a fuzzy set approach (see [11,24,55]). Specifically, our goal is to define and develop the concept of contrast as a measure of the variation among the membership degrees of elements in a specified region of a fuzzy relation [37,49]. In the various fields employing this concept, ‘‘contrast” is not uniquely defined (see [45]) and therefore has been studied from several points of view (see [19,20,33,44,38]). In image processing, for example, Vlachos and Sergiadis [53] state that ‘‘the very concept of image contrast lacks a precise definition.” As contrast is ill-defined, we interpret the term as qualitative rather than quantitative. Fuzzy set theory is a suitable tool for dealing with such measures ([56,57]). Peli [45] settled the difference between total contrast and local contrast. He analyzed the total contrast expressions proposed by Michelson (see [41]) and Weber (see [54]), pointing out that the latter is not normalized so must be used only in very specific cases. Peli’s work also reminds us that the root-mean-square (rms, see [17,18]) deviation of the intensities in an image can be considered a measure of the total contrast. In this paper, we try to unify the various approaches to contrast in the framework of fuzzy set theory. We present the new key concept of interval-contrast (see [4]), which will allow us to define axiomatically, characterize functionally and analyze mathematically old and new measures of local contrast and total contrast. In defining a new contrast measure, we impose several axiomatic requirements based on our analysis of contrast expressions employed in the literature

* Corresponding author. E-mail addresses: [email protected] (H. Bustince), [email protected] (J. Montero). 0020-0255/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2009.12.013

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[1,21,22,30,33,37,45,49]. We also suggest that contrast measures satisfy two new properties in addition to those usually demanded, and examine the conditions under which these properties hold. This work is organized as follows. In Section 2, we recall some preliminary definitions and results of fuzzy set theory. In Section 3, we review the concept of interval-contrast. We recover one characterization theorem by means of implication operators, and another based on automorphisms. In Sections 4 and 5 we define the local contrast of an element in a fuzzy relation, and study various methods of constructing local contrast measures based on the interval-contrast. In Sections 6–8, we define the total contrast and consider three methods of aggregating the local contrasts of a fuzzy relation. We also compare these total contrast measures to some classical measures. Section 9 concludes the theoretical analysis and offers suggestions for future work. The concepts of local contrast and total contrast have many practical applications. In an Appendix A, we use our theoretical developments to calculate the local and total contrasts of various photographic images. We have chosen this application for two reasons: it is very easy to understand an image as a fuzzy relation, and our results are very easily implemented in this context.

2. Preliminary definitions In this section we recall some definitions and results used throughout the paper. 2.1. Fuzzy relations A strong negation is a decreasing involution on [0, 1]. Any strong negation can be represented (see [50]) in terms of automorphisms (an automorphism of the interval ½a; b  R is any continuous, strictly increasing function u : ½a; b ! ½a; b such that uðaÞ ¼ a; uðbÞ ¼ b). A function N : ½0; 1 ! ½0; 1 is a strong negation if and only if there exists an automorphism u of the unit interval such that NðxÞ ¼ u1 ð1  uðxÞÞ. Let X ¼ f0; 1; . . . ; N  1g and Y ¼ f0; 1; . . . ; M  1g be two finite referential sets. A fuzzy relation on X  Y is a fuzzy set of the type

R ¼ fððx; yÞ; Rðx; yÞÞjðx; yÞ 2 X  Yg

ð1Þ

with R : X  Y ! ½0; 1. Given a fuzzy relation R, its complement is given by

NðRÞ ¼ fððx; yÞ; NðRðx; yÞÞÞjðx; yÞ 2 X  Yg;

ð2Þ

where N is a strong negation. When their support is discrete, fuzzy relations are usually described by matrices:

0 B B R¼B @

Rð0; 0Þ



Rð0; M  1Þ

Rð1; 0Þ



Rð1; M  1Þ

1 C C C: A

RðN  1; 0Þ    RðN  1; M  1Þ FRðX  YÞ represents the set of all fuzzy relations on X  Y (see [9,10]). 2.2. Interval-valued fuzzy sets or interval type 2 fuzzy sets Let us denote by Lð½0; 1Þ the set of all closed subintervals in [0, 1], that is,

Lð½0; 1Þ ¼ fx ¼ ½x; xjðx; xÞ 2 ½0; 12 and x 6 xg:

ð3Þ

Lð½0; 1Þ is a partially ordered set with respect to the relation 6L , which is defined in the following way. Given x; y 2 Lð½0; 1Þ,

: x6L y if and only x 6 y and x 6 y

ð4Þ

This relation is reflexive, transitive and antisymmetric, so ðLð½0; 1Þ; 6L Þ is a complete lattice (see [25]). The lattice operations are given by

Þ x ^ y ¼ ½minðx; yÞ; minðx; y and

Þ: x _ y ¼ ½maxðx; yÞ; maxðx; y The smallest element of this lattice is 0L ¼ ½0; 0, and the largest element is 1L ¼ ½1; 1. Notice that Lð½0; 1Þ is not a linear lattice, since it contains elements which are not comparable (for instance, [0.2, 0.7] and [0.4, 0.6]). In this paper, we are also going to use the following relationship on Lð½0; 1Þ (see [32,15]):

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: x  y if and only if y 6 x and x 6 y

ð5Þ

Definition 1. An IV negation is a function N IV : Lð½0; 1Þ ! Lð½0; 1Þ that is decreasing (with respect to 6L ), and for which N IV ð1L Þ ¼ 0L and N IV ð0L Þ ¼ 1L . If for all x 2 Lð½0; 1Þ; N IV ðN IV ðxÞÞ ¼ x; N IV is said to be involutive. The following result was proved in [25]: Theorem 1. A function N IV : Lð½0; 1Þ ! Lð½0; 1Þ is an involutive IV negation if and only if there exists a strong negation N such that

NIV ðxÞ ¼ ½NðxÞ; NðxÞ: Throughout this paper, we will restrict our analysis to these involutive IV negations N IV generated from a negation N. The origin and history of the following definition can be found in Chapter 22, pp. 491 of [12] and in [40]: Definition 2. An interval-valued fuzzy set (IVFS) (or interval type 2 fuzzy set) A on the universe U – ; is a mapping A : U ! Lð½0; 1Þ, such that the membership degree of u 2 U is given by AðuÞ ¼ ½AðuÞ; AðuÞ 2 Lð½0; 1Þ, where A : U ! ½0; 1 and A : U ! ½0; 1 are mappings defining the lower and the upper bound of the membership interval AðuÞ, respectively. Several results on representations, connective types, operations, etc. for interval-valued fuzzy sets, can be found in [15,25,35,42,51]. We denote by IVFSðUÞ the set of all IVFS on U, and by WðAðuÞÞ the length of the membership degree of element u in the interval-valued fuzzy set A:

WðAðuÞÞ ¼ AðuÞ  AðuÞ: Using W, we can construct a function that assigns a fuzzy set to each IVFS:

W : IVFSðUÞ ! FSðUÞ; where WðAÞ ¼ fðu; WðAðuÞÞÞju 2 Ug:

ð6Þ

FSðUÞ is the set of all fuzzy sets on U. 2.3. Construction of interval-valued fuzzy relations from a fuzzy relation In this paper we will use interval-valued fuzzy relations on X  Y, which can obviously be seen as IVFS defined over the referential set X  Y. Mendel [39] proved that IVFS (and hence interval-valued fuzzy relations) are a particular case of the type 2 fuzzy sets introduced by Zadeh in 1975 (see [55]). On the other hand, the interval-valued fuzzy relations (IVFR) that we shall use will always be constructed from another fuzzy relation by means of the method presented in [11,16]. In this subsection, we recall a particular instance of that method. and M1 . We define the Let R 2 FRðX  YÞ be a fuzzy relation, and consider a positive integer n less than or equal to N1 2 2 lower constructor associated with n as the mapping

Ln : FRðX  YÞ ! FRðX  YÞ given by n

Ln ðRðx; yÞÞ ¼ min ðminðRðx  i; y  jÞ; Rðx; yÞÞÞ i¼n j¼n

ð7Þ

for all x 2 X; y 2 Y. The indexes i; j only take values such that 0 6 x  i 6 N  1 and 0 6 y  j 6 M  1. Hence, by applying Ln to a fuzzy relation R, we get another fuzzy relation. For each ðx; yÞ 2 X  Y, the membership with respect to the new relation is less than or equal to the membership with respect to R. The lower constructor is based upon the t-norm minimum within a submatrix (also called a window or mask) of order ð2n þ 1Þ  ð2n þ 1Þ centered on ðx; yÞ. There is an analogous mapping based upon the t-conorm maximum, which we call the upper constructor. It is worth noting that the lower and upper constructors can be built using other t-norms and t-conorms, not just the tnorm minimum and t-conorm maximum. See [11,16] for more details. Now, recall the following basic property of fuzzy relations: Theorem 2. Take R 2 FRðX  YÞ. Then,

Ln ðRðx; yÞÞ 6 Rðx; yÞ 6 U n ðRðx; yÞÞ

ð8Þ

for all ðx; yÞ 2 X  Y. Our method of constructing interval-valued fuzzy relations is based upon this result. Construction method of IVFR: Take R 2 FRðX  YÞ. Then the mapping

Rn ðx; yÞ ¼ ½Ln ðRðx; yÞÞ; U n ðRðx; yÞÞ 2 Lð½0; 1Þ defines an interval-valued fuzzy relation on X  Y.

ð9Þ

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From now on, Rn will denote the interval-valued fuzzy relation given by Eq. (9). We represent by IVFRðX  YÞ the set of all interval-valued fuzzy relations on X  Y. Thus, we can interpret any fuzzy relation R 2 FRðX  YÞ as an interval-valued fuzzy relation such that for any ðx; yÞ 2 X  Y the membership interval is given by Rðx; yÞ ¼ ½Rðx; yÞ; Rðx; yÞ. Whenever we treat a fuzzy relation R 2 FRðX  YÞ as an interval-valued fuzzy relation, we will use the notation R 2 IVFRðX  YÞ. From the monotonicity of the minimum and maximum and from Eq. (9), we get the following theorem. M1 Theorem 3. Let n1 and n2 be two integers greater than zero and less than or equal to both N1 2 and 2 , such that n1 6 n2 . Then for each R 2 FRðX  YÞ, the following holds:

R  Rn1  Rn2 ;

ð10Þ

where  is the ordering defined in Eq. (5). Example 1. Let R 2 FRðX  XÞ, with X ¼ f0; 1; . . . ; 5g given by

0

0:01

B 0:30 B B B 0:97 R¼B B 0:10 B B @ 0:60

0:72 0:76 0:94 0:18 0:96

1

0:86 0:80 0:44 0:15 0:78 C C C 0:19 0:31 0:50 0:93 0:00 C C 0:46 0:24 0:10 0:14 0:33 C C C 0:90 0:49 0:55 1:00 0:54 A

0:47 0:02

0:69 0:44 0:69 0:31

If we apply the construction method given in Eq. (9) to this relation, taking n ¼ 1 yields the following interval-valued fuzzy relation.

0

½0:01; 0:86 ½0:01; 0:86 ½0:44; 0:94 ½0:15; 0:94 ½0:15; 0:96 ½0:15; 0:96

1

½0:10; 0:97 ½0:10; 0:86 ½0:10; 0:97 ½0:10; 0:90

½0:10; 0:93 ½0:10; 1:00

½0:02; 0:90

½0:02; 0:90

½0:10; 1:00

½0:00; 0:96 C C C ½0:00; 0:93 ½0:00; 0:93 C C ½0:00; 1:00 ½0:00; 1:00 C C C ½0:10; 1:00 ½0:14; 1:00 A

½0:02; 0:90 ½0:02; 0:90

½0:02; 0:90

½0:44; 1:00

½0:31; 1:00 ½0:31; 1:00

B ½0:01; 0:97 B B B ½0:10; 0:97 1 R ¼B B ½0:10; 0:97 B B @ ½0:02; 0:90

½0:01; 0:97 ½0:19; 0:94 ½0:15; 0:94 ½0:00; 0:96

For n ¼ 2, we have

0

½0:01; 0:97 ½0:01; 0:97 ½0:01; 0:97 ½0:00; 0:96 ½0:00; 0:96 ½0:00; 0:96

B ½0:01; 0:97 B B B ½0:01; 0:97 R2 ¼ B B ½0:02; 0:97 B B @ ½0:02; 0:97

1

½0:01; 0:97 ½0:01; 0:97 ½0:00; 0:96 ½0:00; 0:96 ½0:00; 0:96 C C C ½0:01; 0:97 ½0:01; 1:00 ½0:00; 1:00 ½0:00; 1:00 ½0:00; 1:00 C C ½0:02; 0:97 ½0:02; 1:00 ½0:00; 1:00 ½0:00; 1:00 ½0:00; 1:00 C C C ½0:02; 0:97 ½0:02; 1:00 ½0:00; 1:00 ½0:00; 1:00 ½0:00; 1:00 A

½0:02; 0:90 ½0:02; 0:90

½0:02; 1:00 ½0:02; 1:00 ½0:10; 1:00 ½0:10; 1:00

Evidently, R  R1  R2 . 3. The interval-contrast associated with a strong negation N In this section we study the concept of interval-contrast (see [4]) and derive two new properties that are of interest to our study of contrast in fuzzy relations. Definition 3. A function C I : Lð½0; 1Þ ! ½0; 1 is called an interval-contrast associated with the strong negation N if it satisfies the following conditions: (IC1) (IC2) (IC3) (IC4)

x ¼ 1; C I ðxÞ ¼ 1 if and only if x ¼ 0 and  x; C I ðxÞ ¼ 0 if and only if x ¼  If x  y, then C I ðxÞ 6 C I ðyÞ; C I ðxÞ ¼ C I ðN IV ðxÞÞ for all x 2 Lð½0; 1Þ.

The above definitions directly prove the following: Proposition 1. If u1 ; u2 are two automorphisms in the unit interval, then 1  C I ðxÞ ¼ u1 1 ðu2 ðxÞ  u2 ðxÞÞ with NðxÞ ¼ u2 ð1  u2 ðxÞÞ

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is an interval-contrast associated with the strong negation N. In particular, C I ð½0;  xÞ ¼  x if and only if u1 ðxÞ ¼ u2 ðxÞ for all x 2 ½0; 1. A mapping I : ½0; 12 ! ½0; 1 is an implication operator if and only if I satisfies the boundary conditions Ið0; 0Þ ¼ Ið0; 1Þ ¼ Ið1; 1Þ ¼ 1 and Ið1; 0Þ ¼ 0, (see pp. 21, Chapter 1 of [28] and [36]). Indeed, these two conditions are the least that we can demand for an operator to be considered an implication. Other potentially interesting properties of implication operators can be found in [8,13,14] and in Chapter 1 of [26]. Nevertheless, all fuzzy implications are obtained by generalizing the implication operator of classical logic. In this sense, Fodor and Roubens provide the following definition on p. 22 of their work [28]. Definition 4. An implication is a function I : ½0; 12 ! ½0; 1 that satisfies the following properties: (I1:) (I2:) (I3:) (I4:) (I5:)

x 6 z implies Iðx; yÞ P Iðz; yÞ for all y 2 ½0; 1; y 6 t implies Iðx; yÞ 6 Iðx; tÞ for all x 2 ½0; 1; Ið0; xÞ ¼ 1 (dominance of falsity) for all x 2 ½0; 1; Iðx; 1Þ ¼ 1 for all x 2 ½0; 1; Ið1; 0Þ ¼ 0.

We can add to this list certain properties which are not necessary, but may be required for specific applications. (I6:) (I7:) (I8:) (I9:) (I10:)

Ið1; xÞ ¼ x (neutrality of truth); Iðx; Iðy; zÞÞ ¼ Iðy; Iðx; zÞÞ (exchange property); Iðx; yÞ ¼ 1 if and only if x 6 y; Iðx; yÞ ¼ IðNðyÞ; NðxÞÞ (contraposition) with a strong negation N; I is a continuous function.

The next theorem is proven in [4]. Theorem 4. Let N be a strong negation. A function C I : Lð½0; 1Þ ! ½0; 1 is an interval-contrast associated with the strong negation N if and only if there exists a function I : ½0; 12 ! ½0; 1 satisfying I1; I8; I9, and the property Iðx; yÞ ¼ 0 if and only if x; xÞÞ. x ¼ 1 and y ¼ 0, such that C I ðxÞ ¼ NðIð Example 2. In this example, we build two interval-contrasts associated with the strong negation NðxÞ ¼ 1  x for all x 2 ½0; 1. (a) The Łukasiewicz implication ([3], pp. 3-10) is Iðx; yÞ ¼ minð1; 1  x þ yÞ. Thus,

C I ðxÞ ¼ NðIðx; xÞÞ ¼ Nðminð1; 1  x þ xÞÞ ¼ Nð1  x þ xÞ ¼ x  x ¼ Wð½x; xÞ: (b) The Fodor implication [3, pp. 3–10] is

Iðx; yÞ ¼



1 if x 6 y maxð1  x; yÞ if x > y

In this case, we have

 C I ðxÞ ¼

0 if x ¼ x Nðmaxð1  x; xÞÞ if x < x:

Any interval-contrast associated with a strong negation N obtained from Theorem 4 will be called an interval-contrast associated with the generating function I. The notion of continuity over Lð½0; 1Þ is naturally inherited from ½0; 12 . Corollary 1. Let N be a strong negation. A continuous function C I : Lð½0; 1Þ ! ½0; 1 such that C I ð½x; 1Þ ¼ NðxÞ for all x 2 ½0; 1 is an interval-contrast associated with the function I : ½0; 12 ! ½0; 1, with I satisfying I7 and I10, if and only if there exists an automorphism u of the unit interval such that

C I ðxÞ ¼ u1 ðuðxÞ  uðxÞÞ

ð11Þ

and NðxÞ ¼ u ð1  uðxÞÞ. 1

Proof. It is enough to take into account Theorem 4, the relations between the properties of I studied in [8], and the following theorem proved in 1987 by Smets and Magrez (see [48]): a function I : ½0; 12 ! ½0; 1 verifies I2; I7; I8 and I10 if and only if there exists an automorphism u of the unit interval such that Iðx; yÞ ¼ u1 ðminð1; 1  uðxÞ þ uðyÞÞÞ. h Example 3 (a) If we take uðxÞ ¼ x for all x 2 ½0; 1, then

H. Bustince et al. / Information Sciences 180 (2010) 1326–1344

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C I ð½x; xÞ ¼ x  x ¼ Wð½x; xÞ with NðxÞ ¼ 1  x for all x 2 ½0; 1. (b) if uðxÞ ¼ x2 for all x 2 ½0; 1, then

C I ð½x; xÞ ¼ ðx2  x2 Þ0:5 with NðxÞ ¼ ð1  x2 Þ0:5 for all x 2 ½0; 1.

Corollary 2. In Eq. (11) , if uðxÞ ¼ x for all x 2 ½0; 1, then the identities

C I ð½k  x; k  xÞ ¼ k  C I ð½x; xÞ

ð12Þ

with k 2 ½0; 1, and

C I ð½x þ r; x þ rÞ ¼ C I ð½x; xÞ

ð13Þ

for all x þ r;  x þ r 2 ½0; 1 hold. Conversely, if both Eqs. (12) and (13) hold, then uðxÞ ¼ x for all x 2 ½0; 1. xÞ ¼  x and C I ð½x;  xÞ ¼  x  x hold for any Proof. To see the converse, Eqs. (12) and (13) imply that the identities C I ð½0;  x;  x 2 ½0; 1 with x 6  x. This fact allows us to easily prove that uðx þ yÞ ¼ uðxÞ þ uðyÞ for any x; y 2 ½0; 1. Hence, u is linear and continuous. Since we also have uð1Þ ¼ 1, it must be the identity. The other implication is trivial. h It is interesting to notice that under the conditions given in Corollary 2, C I ðxÞ ¼  x  x. Therefore, the interval-contrast built in this setting is an interval-contrast associated with the standard (strong) negation NðxÞ ¼ 1  x, for all x 2 ½0; 1. 4. The local contrast associated with a strong negation N As pointed out in the introduction, the local contrast of an element ðx; yÞ 2 X  Y of the relation R 2 FRðX  YÞ is a measure of the variation in the membership degrees of elements in the submatrix ðx  i; y  jÞ centered on ðx; yÞ. That is, the local contrast is a property of the set of values Rðx  i; y  jÞ, with i; j ¼ n; . . . ; 0; . . . ; n, (for n fixed beforehand). In this section we introduce a formal definition of the local contrast, together with the first method for its construction. Later, we will consider two additional properties of the local contrast. We will denote by C l ðx; yÞ the local contrast of element ðx; yÞ. We require the following properties from a local contrast (see [22,52] and Chapter 2, p. 97 of [29]). Definition 5. A local contrast associated with the strong negation N is a real function on X  Y such that (LC1) 0 6 C l ðx; yÞ 6 1 for all ðx; yÞ 2 X  Y; (LC2) If the membership degrees of all the elements of the submatrix centered on ðx; yÞ are identical, then C l ðx; yÞ ¼ 0. That is, if Rðx  i; y  jÞ ¼ q0 with q0 2 ½0; 1 constant for all i; j ¼ n; . . . ; 0; . . . ; n, then C l ðx; yÞ ¼ 0. (LC3) If in the submatrix centered on ðx; yÞ there is at least one element with null membership and at least one element with a membership degree of one, then C l ðx; yÞ ¼ 1. (LC4) The local contrast of ðx; yÞ does not change if for all i; j ¼ n; . . . ; 0; . . . ; n we take NðRðx  i; y  jÞÞ instead of Rðx  i; y  jÞ. The above conditions guarantee a reasonably intuitive measurement of the local variation in membership degrees. Furthermore, ðLC1Þ; ðLC2Þ and ðLC4Þ are satisfied by all the usual expressions for contrast found in the optics literature (for instance, [45,21]). Nevertheless, the usual expressions do not verify ðLC3Þ (see [29]), because their membership degrees are not normalized. In the context of fuzzy theory, ðLC3Þ is a natural requirement. Notice that item ðLC4Þ justifies the use of involutive negations (i.e., strong) negations. Below we give some examples of local contrasts associated with an arbitrary strong negation. Example 4 (a)

C lN ðx; yÞ ¼

8 0 if the memberships of all elements in the submatrix > > > > > are the same; > > > > < 1 if in the submatrix there is at least one element with > > > > > > > > > :

membership equal to 1 and another with membership equal to 0; Pn i;j¼n

Rðxi;yjÞNðRðxi;yjÞÞ

size of the submatrix

otherwise:

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(b)

8 1 if in the submatrix there is at least one element with > > > < membership equal to 1 and another with membership C linf ðx; yÞ ¼ > equal to 0 ; > > : 0 otherwise: (c)

C lsup ðx; yÞ ¼

8 > <0 > :

if the memberships of all elements in the submatrix are the same;

1 otherwise:

Examples (b) and (c) provide bounds for any other local contrast, as the next result shows. Proposition 2. Let C l be a local contrast associated with the strong negation N. Then for any ðx; yÞ 2 X  Y, the following inequality holds:

C linf ðx; yÞ 6 C l ðx; yÞ 6 C lsup ðx; yÞ:

ð14Þ

Proof. The proposition can easily be verified by taking into account properties (LC1)–(LC4) in the definition of a local contrast. h From our point of view, it is natural to require that a local contrast satisfies the following two properties. Nevertheless, as some widely used contrasts do not satisfy them (e.g, p. 252 of [49]), we have not included them in Definition 5. (LC5) If we multiply all the membership degrees of elements in the submatrix centered on ðx; yÞ by a constant factor k 2 ½0; 1, then the local contrast of element ðx; yÞ should also be increased by a factor k; (LC6) If we increase all the membership degrees of elements in the submatrix centered on ðx; yÞ by the same quantity r, such that Rðx  i; y  jÞ þ r 2 ½0; 1 for all i; j; ¼ n; . . . ; 0; . . . ; n, then the local contrast should not change. 5. A method of constructing local contrasts from interval-contrasts and M1 . Moreover, we always Throughout this section we fix a positive integer n, which is less than or equal to both N1 2 2 speak of local contrasts and interval-contrasts associated with the same strong negation N. Theorem 5. Let C I be an interval-contrast associated with the strong negation N. Take a fuzzy relation R 2 FRðX  YÞ, and an interval-valued fuzzy relation Rn 2 IVFRðX  YÞ obtained from R using Ln and U n as specified in Eq. (9). Then

C l ðx; yÞ ¼ C I ðRn ðx; yÞÞ ¼ C I ð½Ln ðRðx; yÞÞ; U n ðRðx; yÞÞÞ

ð15Þ

is a local contrast associated with the strong negation N Proof. This result follows from the monotonicity and the idempotency of the minimum and the maximum.

h

In the following theorem we present a method to build local contrasts by means of implication operators. Theorem 6. Take R 2 FRðX  YÞ, a strong negation N, and the relation Rn 2 IVFRðX  YÞ obtained from R, Ln and U n as in Eq. (9) . Let I : ½0; 12 ! ½0; 1 be a function satisfying I1; I8 and I9 with the additional propertyIðx; yÞ ¼ 0 if and only if x ¼ 1 and y ¼ 0. Then

C l ðx; yÞ ¼ NðIðU n ðRðx; yÞÞ; Ln ðRðx; yÞÞÞÞ

ð16Þ

is a local contrast associated with the strong negation N. Proof. This result is a consequence of Theorems 2 and 4.

h

From Theorem 6 and Corollary 1, the following result follows easily. Theorem 7. Consider a fuzzy relation R 2 FRðX  YÞ and the relation Rn 2 IVFRðX  YÞ obtained from R, Ln and U n as in Eq. (9). Then C l below is a local contrast associated with the strong negation NðxÞ ¼ u1 ð1  uðxÞÞ:

C l ðx; yÞ ¼ u1 ðuðU n ðRðx; yÞÞÞ  uðLn ðRðx; yÞÞÞÞ; where u is any automorphism on the unit interval. In particular, by choosing the identity for the automorphism u in Eq. (17), we learn the following:

ð17Þ

H. Bustince et al. / Information Sciences 180 (2010) 1326–1344

1333

Corollary 3. In the setting of Theorem 7 , the local contrast C l ðx; yÞ satisfies (LC5) and (LC6) if and only if uðxÞ ¼ x for all x 2 ½0; 1. Comment 1 Note that under the hypothesis of Corollary 3, the local contrast C l1 is n

n

i¼n j¼n

i¼n j¼n

C l1 ðx; yÞ ¼ U n ðRðx; yÞÞ  Ln ðRðx; yÞÞ ¼ maxðRðx  i; y  jÞÞ  minðRðx  i; y  jÞÞ ¼ WðRn ðx; yÞÞ

ð18Þ

From R 2 FRðX  YÞ, we can build Rn 2 IVFRðX  YÞ using Ln and U n . Thanks to Corollary 3, from Rn we can construct a new fuzzy relation:

WðRn Þ ¼ fððx; yÞ; WðRn ðx; yÞÞ ¼ C l1 ðx; yÞÞjðx; yÞ 2 X  Yg

ð19Þ

Example 5. Consider the relation R from Example 1. Using the method presented in Eq. (19), and taking the local contrast C l1 given by Eq. (18) for n ¼ 1, we can construct the following fuzzy relation WðR1 Þ 2 FRðX  XÞ on X ¼ f0; . . . ; 5g:

0

0:85 B 0:96 B B B 0:87 1 WðR Þ ¼ B B 0:87 B B @ 0:88

1 0:85 0:50 0:79 0:81 0:81 0:96 0:75 0:79 0:96 0:96 C C C 0:87 0:76 0:83 0:93 0:93 C C 0:87 0:80 0:90 1:00 1:00 C C C 0:88 0:88 0:90 0:90 0:86 A

0:88 0:88 0:88 0:56 0:69 0:69 In image processing (see [16]) the membership of each element of WðRn Þ represents the variation of intensity near that element, in such a way that, if the variation is large enough, there is an edge. M1 Corollary 4. In the setting of Theorem 7 , if we take two positive integers n1 6 n2 , both less than or equal to N1 2 and 2 , then for each ðx; yÞ 2 X  Y the local contrast calculated using n1 for the lower constructor and the upper constructor is always less than or equal to the local contrast calculated in the same way but using n2 .

Proof. It follows from Eq. (10) and Theorem 7. h From Corollary 4, we conclude that the larger the submatrices used for the lower constructor and upper constructor, the larger the local contrast of the considered element. Comment 2 If C l0 and C l are two local contrasts associated with the strong negation N, then by defining

aCl0 ðx; yÞ ¼



C l0 ðx; yÞ if C l0 ðx; yÞ – 0 1 otherwise;

we can construct the local contrast

( CC l ðx; yÞ ¼

aCl0 ðx; yÞC l ðx; yÞ þ ð1  aCl0 ðx; yÞÞð1  C l ðx; yÞÞ if C l ðx; yÞ – 0 0 otherwise:

ð20Þ

Observe that the value of aC l varies from point to point. Besides, if we take C l ¼ C l0 , we recover Tizhoosh’s expression in p. 0 252 of [49]. In the Appendix A, we will compare this particular expression to some of our other local contrasts. 6. Total contrast In image processing, the classical total contrast C T of a fuzzy relation R 2 FRðX  YÞ is given by

C T 1 ðRÞ ¼

Rmax  Rmin ; Rmax þ Rmin

ð21Þ

where Rmax and Rmin are the maximum and the minimum membership value among the elements of R (see [49]). The same expression was also adopted by Michelson [41] as early as 1927, in the domain of optics. However, Eq. (21) presents two problems: (a) For a crisp relation or for any relation with at least one membership equal to zero and another equal to one, the contrast is always equal to one. (b) If Rmax ¼ Rmin ¼ 0, then C T 1 is not defined. The first problem (a) is due to the fact that Eq. (21) ignores the spatial distribution of the elements that compose the relation. (A way around this problem is to calculate the total contrast by aggregating local contrasts of individual elements of the relation). These considerations have led us to propose the following definition.

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H. Bustince et al. / Information Sciences 180 (2010) 1326–1344

Definition 6. Let N be an strong negation. A total contrast C T associated with N is a real function on FRðX  YÞ such that: (TC1) 0 6 C T ðRÞ 6 1; (TC2) If all elements of the fuzzy relation R have the same membership degree, then C T ðRÞ ¼ 0; (TC3) If R is a crisp relation such that there exists at least one element with membership equal to one and another with membership equal to zero, then C T ðRÞ ¼ 1; (TC4) The total contrast of a fuzzy relation and that of its negation (by N) are the same; that is, C T ðRÞ ¼ C T ðNðRÞÞ. As in the case of local contrast, the most usual expressions for total contrast (see [45,21]) satisfy ðTC1Þ; ðTC2Þ and ðTC4Þ, but not ðTC3Þ (see Chapter 2, p. 97 in [29]). As before, this is due to the fact that their membership degrees are not normalized. In the context of fuzzy theory, this axiom is natural. 7. Construction of total contrasts by aggregating local contrasts In this section, all of the local and total contrasts mentioned are assumed to be associated with the same strong negation N. In the following proposition we show that aggregating local contrasts will produce a total contrast as long as we choose the aggregating function properly. Please note that usually aggregation functions are required to be increasing (see [5,17,18]). In our case, however, we drop this requirement. Proposition 3. Consider R 2 FRðX  YÞ, and let C l be a local contrast associated with the strong negation N in the sense of S ½0; 1m ! ½0; 1 be a function such that: Definition 5 . Then let F g : m2N

(1) F g ð0; . . . ; 0Þ ¼ 0 and F g ð1; . . . ; 1Þ ¼ 1. (2) If xi 2 f0; 1g for all i 2 f1; . . . ; mg, and there exist at least one component xp ¼ 1 and another component xq ¼ 0 with p; q 2 f1; . . . ; mg, then F g ðx1 ; . . . ; xm Þ ¼ 1. Under these conditions,

C T ðRÞ ¼

Fg C l ðx; yÞ x¼0;...;N1 y¼0;...;M1

ð22Þ

is a local contrast associated with the strong negation N in the sense of Definition 6. Proof. (TC1) follows from the range of F g . (TC2) is a consequence of property (LC2) in the definition of local contrasts and property (1) required of F g . (TC3) follows from (LC1) in the definition of local contrasts and property (2) required of F g . Finally, (TC4) comes from property (LC4) in the definition of local contrasts. h We point out that the domain of F g could be restricted to ½0; 1NM . Nevertheless, there is no additional difficulty in considering a more general domain, as we do here. Example 6. In this example we present two possible functions F g . The first (the maximum) is an aggregation function in the usual sense, but the second is not even monotonically increasing. (a) F g ðx1 ; . . . ; xm Þ ¼ maxðx1 ; . . . ; xm Þ. (b)

8 1 if xi 2 f0; 1g for all i 2 f1; . . . ; mg and there are > > > > > at least one xp ¼ 1 and one xq ¼ 0; < F g ðx1 ; . . . ; xm Þ ¼ p; q 2 f1; . . . ; mg; > > m > P > 1 > : xi otherwise: m i¼1

8. Three expressions for the total contrast We now present three total contrast expressions obtained by means of Eq. (22) in Proposition 3. 8.1. Maximum of the local contrasts Consider a local contrast C l associated with a strong negation N. Choosing the function in Example 6a as F g in Eq. (22), it follows that

H. Bustince et al. / Information Sciences 180 (2010) 1326–1344

C T 2 ðRÞ ¼

max C l ðx; yÞ

x¼0;...;N1 y¼0;...;M1

1335

ð23Þ

satisfies the four properties (TC1)–(TC4) required of a total contrast in Definition 6. If we consider the local contrasts constructed from Corollary 3 (that is, those associated with the strong negation NðxÞ ¼ 1  x), we find that C T 2 also satisfies the following two properties: (TC5) If we multiply all the elements of the fuzzy relation R by a constant factor k 2 ½0; 1, then the total contrast is also multiplied by k; that is, C T 2 ðk  RÞ ¼ k  C T 2 ðRÞ. (TC6) If we increase all the elements of R by some constant factor r 2 ½0; 1, such that Rðx; yÞ þ r 2 ½0; 1 for all ðx; yÞ 2 X  Y, then the total contrast C T 2 does not change. If we also adopt the conditions of Corollary 4 (by using local contrasts that are associated with the strong negation NðxÞ ¼ u1 ð1  uðxÞÞ), we obtain a total contrast C T 2 that satisfies the following two properties: (TC7) If we consider different values of n, the total contrast increases with the size of the submatrix (see Corollary 4). (TC8) If we consider the local contrast constructed in Theorem 5, and increase the maximum or decrease the minimum membership degree among the elements of the relation, then the total contrast will increase. Nevertheless, a problem arises from using Eq. (23) to construct the total contrast. In cases where the vast majority of elements have very small but non-zero local contrasts, but a single element has a local contrast equal to 1, the total contrast is also equal to one. That is, this rule permits just one element to determine the total contrast of the entire relation. This fact leads us to study other types of aggregation functions. 8.2. Arithmetic mean of the local contrasts The arithmetic mean of the local contrasts,

EðRÞ ¼

N1 M1 X 1 X C l ðx; yÞ; N  M x¼0 y¼0

ð24Þ

is not a total contrast in the sense of Definition 6 since it does not satisfy property (TC3). If we employ the local contrasts obtained from Theorem 7, then the expression

EðRÞ ¼

N1 M1 X 1 X u1 ðuðU n ðRðx; yÞÞÞ  uðLn ðRðx; yÞÞÞÞ; N  M x¼0 y¼0

ð25Þ

although not a total contrast, is an interval-valued fuzzy entropy (see [4,6,7]). Note that if we take uðxÞ ¼ x for all x 2 ½0; 1 in Eq. (25), the resulting expression is the indeterminacy index of Sambuc (see [47]).a Nevertheless, it is possible to construct a total contrast from the arithmetic mean. Based on Proposition 3, we propose a method which uses the function given in Example 6b as F g . Construction of a total contrast using the arithmetic mean: Let C l be a local contrast associated with the strong negation N. Then the expression

8 1 if C l ðx; yÞ 2 f0; 1g for all ðx; yÞ 2 X  Y; and there are at least > > > > > one ðx; yÞ such that C l ðx; yÞ ¼ 1 and another > < ðx0 ; y0 Þ such that C l ðx0 ; y0 Þ ¼ 0; C T 3 ðRÞ ¼ > > N1 > P M1 P > 1 > > C l ðx; yÞ otherwise : NM

ð26Þ

x¼0 y¼0

satisfies all four properties (TC1)–(TC4) of Definition 6. That is, C T 3 ðRÞ is a total contrast associated with the strong negation N. The most important properties of this total contrast are the following: (1) If we consider the local contrasts C l associated with the standard negation arising from Corollary 3, then the total contrast C T 3 associated with the standard negation satisfies properties (TC5)–(TC8) presented above for C T 2 . (2) If the local contrasts of some elements are small but non-zero, and there is only one element whose local contrast is equal to one, then the total contrast C T 3 is not one. That is, a single element does not determine the value of the total contrast. Comment 3. References [46] and [29] (p. 97, Chapter 2) implicitly propose the following expression for the local contrast associated with a submatrix of size ð2n þ 1Þ  ð2n þ 1Þ centered on pixel ðx; yÞ:

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H. Bustince et al. / Information Sciences 180 (2010) 1326–1344

C GP ðx; yÞ ¼

n n X X 1 ðRðx  i; y  jÞ  Rðx; yÞÞ2 ð2n þ 1Þ  ð2n þ 1Þ i¼n j¼n

ð27Þ

where Rðx; yÞ is the average membership degree over the submatrix. Nevertheless, by our definitions this expression is neither a local contrast nor the arithmetic mean. However, the expression

GP ¼

N1 M1 X 1 X C GP ðx; yÞ N  M x¼0 y¼0

ð28Þ

is a total contrast in the sense of Definition 6. 8.3. The fuzzy expected value of local contrasts In this section we use the fuzzy expected value (FEV) as function F g in the construction method for total contrasts (Proposition 3). The FEV was introduced in [34,27] as a measure that could represent the ‘‘most typical value” of the elements in a fuzzy set better than the mean or median. In this subsection, we propose a total contrast expressed in terms of the FEV of the local contrasts C l ðx; yÞ (see p. 252 of [49]). To find the total contrast associated with a strong negation N of a fuzzy relation, we are going to follow a method inspired by Van der Weken et al. [52]: Constructing a total contrast using the fuzzy expected value:Given a local contrast C l associated with the strong negation N and a fuzzy relation R 2 FRðX  YÞ, (A) Calculate all the local contrasts C l ðx; yÞ of the fuzzy relation R. (B) Select the p unique, non-zero local contrasts C l ðx; yÞ and Order them from highest to lowest: C l;1 > C l;2 . . . > C l;p . P (C) For each C l;i , record the number of appearances ni . Set ps¼1 ns ¼ V  Z. Pi n j¼1 j (D) Calculate !ðC l;i Þ ¼ VZ for each i. Observe that !ðC l;p Þ ¼ 1. (E) The total contrast associated with the strong negation N can be calculated as follows:

CT4 ¼

8 < 0 if C l ði; jÞ ¼ 0 for all ði; jÞ p

: maxðminð!ðC l;s Þ; C l;s ÞÞ otherwise

ð29Þ

:

s¼1

C T 4 represents the most typical value among all the local contrasts C l ði; jÞ. The most important qualities of the contrast C T 4 given by Eq. (29) are the following: (1) (2) (3) (4)

It satisfies properties (TC1)–(TC4) required of a total contrast in Definition 6. If we consider local contrasts C l;s ðs ¼ 1; . . . ; pÞ obtained from Corollary 3, then C T 4 does not satisfy property (TC5). If we consider local contrasts C l;s ðs ¼ 1; . . . ; pÞ obtained from Corollary 3, then C T 4 does satisfy property (TC6) With this method of construction, property (TC7) is not generally satisfied. Property (TC8) is satisfied as long as none of the non-zero contrasts vanishes (see [34]). (5) If the relation R has a single zero-valued element ðx; yÞ and a single element equal to one, and the latter is one of the eight elements adjacent to the former (that is, within the subarray ðx  i; y  jÞ for i; j 2 f1; 0; 1g centered on the zero element), then we have at most nine local contrasts C l ðx; yÞ equal to one. When the local contrasts are ordered in step 9 u 0. The other local contrasts C l;2 ; . . . ; C l;p are different from one, (A), the first will be C l;1 ¼ 1. Evidently, !ðC l;1 Þ 6 V Z

and among the ! only !ðC l;p Þ ¼ 1. Therefore, minð!ðC l;p Þ; C l;p Þ ¼ C l;p – 1; . . . ; minð!ðC l;1 Þ; C l;1 ¼ 1Þ ¼ !ðC l;1 Þ – 1, from which we can deduce that p

C T 4 ¼ maxðminð!ðC l;s Þ; C l;s ÞÞ – 1: s¼1

The existence of a single element with null intensity and a single neighbor with maximum intensity does not result in a total contrast equal to one under this measure.

Table 1 Total contrast of R with n ¼ 1. CT2

CT3

CT4

1

0.8472

0.7900

H. Bustince et al. / Information Sciences 180 (2010) 1326–1344

1337

Example 7. In Table 1 we present numerical results for the three total contrast expressions analyzed in this section. We use the fuzzy relation WðR1 Þ of Example 5, which was obtained from fuzzy relation R of Example 1. 9. Conclusions and future research This paper has studied the concept of interval-contrasts in detail. We proved a characterization theorem using automorphisms. An analysis of the properties naturally required of a contrast measure (for example, in image processing) led us to definitions of the local contrast and total contrast suitable for a fuzzy relation. From our point of view, the properties listed in each definition represent minimum requirements, since almost every classical contrast expression in the literature satisfies them (see [45]). Nevertheless, we have introduced two additional properties for the local contrast, (LC5) and (LC6), and corresponding properties for the total contrast, (TC5) and (TC6), that should also be satisfied in our opinion. Our definitions and characterization of interval-contrasts have allowed us to prove that there is only on local contrast that satisfies both (LC5) and (LC6). The measure thereby increases when the higher intensities increase or the lower intensities decrease. This contrast is obtained simply by taking the maximum and minimum intensities in the neighborhood, i.e., C l1 . These considerations suggest that C l1 is in general a good local contrast. Our theoretical results also suggest that (still in the setting mentioned above) the total contrast C T 3 built using the local contrast C l1 satisfies all required properties as well as the desirable properties (TC5)–(TC8). Hence, in general C T 3 seems to be a more appropriate measure than C T 2 . The latter still exhibits the common problem of a single element determining its value. C T 3 is also preferable to C T 4 , which does not satisfy property (TC5). Obviously, one’s choice for the local contrast expression should depend on the application. This paper has proposed a general theoretical formulation, independent of any particular application. Finally, it should be pointed out that we have used a unique aggregation function. In future research we shall explore alternative aggregation rules, taking into account the position of each information unit within the data structure. This line of research will investigate the key arguments stressed in [2,23] and [43] relative to aggregation procedures within structures. Appendix A. As we have already stated, the concept of ‘‘contrast in a fuzzy relation” makes sense in any field where it is necessary to take into account the influence of neighboring elements on the element itself. The fields of decision-making (see [43]), approximate reasoning (see [2]), and pattern recognition are examples. In the context of pattern recognition, Jahromi et al. [31] emphasized the importance of considering the influence of neighbors in the following way: The basic rationale for the Nearest Neighbors (NN) rule is both simple and intuitive: patterns close in feature space are likely to belong to the same class. The NN classifier can be represented by the following simple rule: to classify an unknown pattern, choose the class of the nearest stored training instance. In this appendix, we have chosen an image processing application to demonstrate and verify our theoretical results. An image Q of N  M pixels is interpreted as a collection of N  M elements arranged in rows and columns. A numerical value representing (grayscale) intensity, chosen from the set f0; 1; 2; . . . ; L  1g, is assigned to each element. Thus, an image is just a matrix (see [16]). Any image Q can be represented as a single fuzzy relation R such that the membership degree of each element (pixel) is its intensity divided by L  1. We will compute the following local contrasts:  C l1 , given in Eq. (18); Table 2 Lena’s images and their Histograms.

1338

Table 3 Contrast CT2; CT3 and CT4. C l1 n ¼ 1; 10; 20; 50

CC l1 n ¼ 1; 10; 20; 50

C lN n ¼ 1; 10; 20; 50

CC lN n ¼ 1; 10; 20; 50

IM01

CT 2 CT 3 CT 4

0.1647 0.0231 0.0693

0.1725 0.0970 0.1451

0.1765 0.1288 0.1569

0.1765 0.1601 0.1725

0.9922 0.9495 0.8942

0.9767 0.8284 0.7689

0.9691 0.7776 0.7355

0.8359 0.7315 0.7248

0.2500 0.2460 0.2494

0.2500 0.2476 0.2490

0.2498 0.2476 0.2487

0.2491 0.2476 0.2485

0.6383 0.6233 0.6259

0.6353 0.6274 0.6263

0.6346 0.6274 0.6267

0.6294 0.6274 0.6268

IM02

CT 2 CT 3 CT 4

0.4510 0.0628 0.1333

0.4745 0.2636 0.3294

0.4863 0.3509 0.4039

0.4863 0.4376 0.4510

0.9922 0.8909 0.8163

0.9322 0.6375 0.5816

0.9246 0.5604 0.5250

0.6320 0.5117 0.5050

0.2500 0.2366 0.2467

0.2499 0.2367 0.2435

0.2490 0.2366 0.2420

0.2453 0.2367 0.2408

0.7133 0.6388 0.6312

0.6886 0.6389 0.6342

0.6833 0.6389 0.6348

0.6499 0.6387 0.6364

IM03

CT 2 CT 3 CT 4

0.7922 0.1113 0.1843

0.8353 0.4668 0.4940

0.8549 0.6206 0.6043

0.8549 0.7720 0.7608

0.9922 0.8304 0.7572

0.8962 0.5827 0.5410

0.8621 0.5785 0.5582

0.7519 0.6597 0.6529

0.2500 0.2143 0.2441

0.2499 0.2144 0.2373

0.2498 0.2143 0.2328

0.2430 0.2138 0.2224

0.8209 0.6658 0.6364

0.7508 0.6645 0.6447

0.7377 0.6640 0.6502

0.6913 0.6641 0.6611

IM04

CT 2 CT 3 CT 4

1 0.1908 0.2588

1 0.7346 0.6667

1 0.9050 0.8081

1 0.9914 0.9271

1 0.6706 0.6881

1 0.7538 0.6360

1 0.8792 0.7542

1 0.9847 0.9157

1 0.1480 0.2302

1 0.4050 0.2966

1 0.6704 0.6018

1 0.9234 0.9039

1 0.6558 0.6613

1 0.8155 0.7014

1 0.8885 0.7285

1 0.9690 0.9039

IM05

CT 2 CT 3 CT 4

1 0.2085 0.3384

1 0.8253 0.7481

1 0.9587 0.8904

1 1 1

1 0.4597 0.6512

1 0.8557 0.7355

1 0.9544 0.8731

1 1 1

1 0.1157 0.2223

1 0.6582 0.6300

1 0.8807 0.8643

1 1 1

1 0.5146 0.7075

1 0.9241 0.8252

1 0.9725 0.8932

1 1 1

H. Bustince et al. / Information Sciences 180 (2010) 1326–1344

IM

1339

H. Bustince et al. / Information Sciences 180 (2010) 1326–1344

 C lN , given in Example 4;  CC l1 , constructed from Eq. (20) taking C l0 ¼ C l ¼ C l1 ; and  CC lN , constructed from Eq. (20) taking C l0 ¼ C l ¼ C lN . Each one is associated with the standard negation. In Table 2 we show a sequence of images for Lena (IM01-IM05) and their associated intensity histograms. Recall that the histogram of a digital image with intensity levels in the range f0; . . . ; L  1g is a discrete function hðr k Þ ¼ nk , where r k is the kth intensity value and nk is the number of pixels in the image with intensity rk . In our sequence, the width of the intensity histograms increases from one column to the next. In Table 3 we present the numerical values of C T 2 ; C T 3 and C T 4 obtained using local contrasts C l1 ; CC l1 ; C lN and CC lN , for submatrix sizes n ¼ 1; 10; 20 and 50. In Table 4, we consider an arbitrary group of images (IM06-IM12) with their corresponding histograms. Table 5 presents the same total contrast calculations as Table 3 for images IM06–IM12. From an analysis of Table 3 and 5, we can draw several conclusions. (1) For each image, the greatest value of the total contrast is always obtained with C T 2 . This result comes from the fact that for any given local contrast function, the total contrast based on the aggregation maximum provides a larger result than those based on the arithmetic mean and the fuzzy expected value. Nevertheless, the last two functions are not comparable (see [27]). (2) Within the first column of Table 3, we see that C T 2 grows progressively larger at each row. This property follows from the fact that C l1 satisfies Property (TC8), as the maximum is an increasing aggregation function. Similar considerations hold for C T 3 . However, we can say nothing about the expected behavior of C T 4 , since the fuzzy expected value is not an increasing function (see [34]). In the other three columns, the values are not easily comparable. (See, for instance, the sequence of C T 3 values for local contrast C lN in Table 3.) The construction methods we have used to get local contrasts different from C l1 do not permit a simple relationship between the intensity distribution and the contrast. (3) In the last two rows of Table 3, all the values of C T 2 are 1. This means that there is at least one submatrix with at least one pixel intensity equal to one and another pixel intensity equal to zero. (4) C T 3 and C T 4 satisfy the following two rules:  If the value in the first column (total contrasts using C l1 ) is less than or equal to 0.5, then the corresponding value in the second column (total contrasts using CC l1 ) is always greater than the value in the first column (see Table 3). Where the values in the first column are greater than 0.5, the corresponding values in the second common follow no particular pattern. (See, for instance, IM10 in Table 5.)  If the value in the third column (total contrasts using C lN ) is less than or equal to 0.5, then the corresponding value in the fourth column (total contrasts using CC lN ) is always greater than the value in the third column. Where the values in the first column are greater than 0.5, the corresponding values in the second common follow no particular pattern. (See, for instance, IM04 in Table 3.) These two statements can be easily proven theoretically by considering the construction of CC l1 and CC lN , and the fact that C T 3 is the arithmetic mean of the local contrasts. (5) In general, there is no relationship between the corresponding values of C l1 and C lN . By construction, these local contrasts are not comparable. (6) Corollary 4 states that as the size of the submatrix increases, the amplitudes of the intervals constructed with our methods also increase. This rule can be checked by comparing the numerical values corresponding to n ¼ 1; 10; 20 and n ¼ 50 in Tables 3 and 5. For example, the values of C l1 are greater for n ¼ 20 than for n ¼ 1, as seen in the first column of each table.

Table 4 Original images and their histograms.

4

2

4

2000

x 10

3.5

6000

x 10

12000

10000

10000

8000

8000

6000

6000

4000

4000

3500

1400

1.4

12000

4000 5000

3

1600

1.6

2.5

4000

3000

1200

1.2

2500

2

1000

1 0.8

800

0.6

600

0.4

400

0.2

200

3000 2000

1.5

2000

0

50

100

150

200

250

300

0

1500

1

1000 1000

0.5

0

4500

1800

1.8

0

50

100

150

200

250

300

0

2000

2000

500

0

50

100

150

200

250

300

0

0

50

100

150

200

250

300

0

0

50

100

150

200

250

300

0

0

50

100

150

200

250

300

0

0

50

100

150

200

250

300

1340

Table 5 Contrast CT2; CT3 and CT4. IM

CC l1 n ¼ 1; 10; 20; 50

C lN n ¼ 1; 10; 20; 50

CC lN n ¼ 1; 10; 20; 50

IM06

CT 2 CT 3 CT 4

0.8314 0.0790 0.1729

0.9725 0.3995 0.5020

0.9725 0.6164 0.6314

0.9725 0.8940 0.8500

0.9922 0.6264 0.7747

0.9922 0.5656 0.5936

0.9922 0.6377 0.6163

0.9466 0.8183 0.7689

0.2499 0.1562 0.2432

0.2461 0.1742 0.2381

0.2449 0.1875 0.2269

0.2431 0.1981 0.2151

0.9677 0.4986 0.6340

0.7816 0.5809 0.6481

0.7417 0.6420 0.6613

0.7372 0.6863 0.6699

IM07

CT 2 CT 3 CT 4

0.7294 0.1377 0.2108

0.7882 0.5073 0.5211

0.7882 0.6238 0.6078

0.7882 0.7308 0.7137

0.9922 0.7908 0.7193

0.9246 0.5438 0.5216

0.6847 0.5520 0.5345

0.6662 0.6106 0.6017

0.2500 0.2112 0.2420

0.2491 0.2111 0.2347

0.2482 0.2111 0.2318

0.2376 0.2109 0.2287

0.7989 0.6692 0.6397

0.7705 0.6685 0.6467

0.7581 0.6680 0.6497

0.7266 0.6678 0.6559

IM08

CT 2 CT 3 CT 4

0.9804 0.6459 0.9804

0.9804 0.9804 0.9804

0.9804 0.9804 0.9804

0.9804 0.9804 0.9804

0.9616 0.6335 0.9616

0.9616 0.9616 0.9616

0.9616 0.9616 0.9616

0.9616 0.9616 0.9616

0.0128 0.0063 0.0128

0.0110 0.0095 0.0110

0.0098 0.0095 0.0098

0.0097 0.0095 0.0097

0.9873 0.6463 0.9747

0.9847 0.9811 0.9783

0.9837 0.9811 0.9805

0.9821 0.9811 0.9808

IM09

CT 2 CT 3 CT 4

0.8588 0.1887 0.2378

1 0.6727 0.6353

1 0.7615 0.7056

1 0.8584 0.8078

0.9922 0.8045 0.7070

1 0.5897 0.5556

1 6525 0.6043

1 0.7626 0.7248

0.2500 0.0934 0.2197

1 0.1018 0.1367

1 0.1207 0.1161

1 0.2429 0.1649

0.9921 0.8291 0.7051

1 0.8337 0.7828

1 0.8352 0.8075

1 0.8587 0.8235

IM10

CT 2 CT 3 CT 4

0.9294 0.0775 0.1578

0.9451 0.4070 0.4874

0.9608 0.5368 0.5531

0.9804 0.7736 0.6863

0.9922 0.8753 0.7805

0.9767 0.6581 0.5583

0.9691 0.6521 0.5609

0.9616 0.7166 0.6360

0.2500 0.1279 0.2203

0.2419 0.1283 0.2003

0.2270 0.1289 0.1939

0.2106 0.1311 0.1846

0.9775 0.7894 0.6917

0.9505 0.7854 0.7018

0.9463 0.7829 0.7111

0.9313 0.7766 0.7176

IM11

CT 2 CT 3 CT 4

0.8157 0.0426 0.1305

0.8745 0.2669 0.3882

0.8745 0.4058 0.5175

0.8745 0.6295 0.6784

0.9922 0.6368 0.8295

0.9922 0.6491 0.5880

0.9922 0.6273 0.5665

0.9844 0.6614 0.6017

0.2500 0.1293 0.2455

0.2500 0.1760 0.2469

0.2500 0.1839 0.2463

0.2500 0.1980 0.2371

0.9991 0.4995 0.6403

0.9990 0.6354 0.6429

0.9969 0.6545 0.6467

0.8346 0.6848 0.6558

IM12

CT 2 CT 3 CT 4

0.8196 0.0344 0.1205

0.8627 0.1965 0.2821

0.8706 0.3142 0.3779

0.8784 0.5495 0.6118

0.9922 0.5688 0.8428

0.9922 0.6662 0.6662

0.9922 0.6680 0.6201

0.9844 0.7014 0.6486

0.2500 0.1314 0.2478

0.2500 0.1910 0.2480

0.2500 0.2052 0.2477

0.2500 0.2228 0.2456

0.9957 0.4190 0.6301

0.9860 0.5756 0.6298

0.9755 0.6125 0.6302

0.7903 0.6539 0.6353

H. Bustince et al. / Information Sciences 180 (2010) 1326–1344

C l1 n ¼ 1; 10; 20; 50

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H. Bustince et al. / Information Sciences 180 (2010) 1326–1344

In the figure of Table 6, there are thirty-six pixels with maximum intensity (255, or 1.0) distributed in a 6  6 square with, and one pixel adjacent to that square with intensity zero. All the others intensities are between 0.078 and 0.157 (grayscales 20 through 40). We use C l1 as the local contrast. Exactly six pixels have local contrasts equal to 1, while all others have very small local contrasts (the mean of the non-zero local contrasts, provided by C T 3 , is equal to 0.0175). In this situation C T 2 ¼ 1, hence only 37 pixels have fixed the total contrast of the image. In our opinion, this behavior invalidates C T 2 as a total contrast measure. In Table 7, we experimentally verify our theoretical developments regarding local contrast properties (LC5) and (LC6) (see Corollary 3 for an example), as well as total contrast properties (TC5) and (TC6). 74 150 and 255 ). In the second The first row of the table shows an image with intensities between 74 and 150 (for us, between 255 row, we have the same image with 100 added to the grayscale intensity of each pixel. Finally, the third row divides each intensity of the original image by two. Examining the three total contrasts (all using local contrast C l1 ), we notice that the numerical values of C T 2 in the first and second rows are equal. The same is true of C T 3 and C T 4 . This behavior follows directly from the fact that C l1 satisfies (LC6). The third image in Table 7 demonstrates the fact that since C T 2 and C T 3 satisfy (TC5), their numerical values in the third row are half of those calculated for the first row. Again, if the local contrast satisfies (LC5), then so will the total contrast based upon it. C T 4 does not follow this rule, because it is constructed using the fuzzy expected value instead. Hence, (TC5) does not hold.

Table 6 A special case: total contrasts CT2; CT3; CT4 with C l1 . Image

Histogram

CT2

CT3

CT4

7000

1

0.0163

0.0784

6000

5000

4000

3000

2000

1000

0

0

50

100

150

200

250

300

Table 7 Contrast CT2; CT3 and CT4 for n ¼ 1. Image

Histogram 4000

3500

C l1

CC l1

C lN

CC lN

CT 2

0.2667

0.9922

0.2500

0.6692

CT 3 CT 4

0.0336 0.0784

0.9215 0.8824

0.2339 0.2483

0.6281 0.6296

CT 2

0.2667

0.9922

0.2138

0.9615

CT 3 CT 4

0.0336 0.0784

0.9215 0.8824

0.1375 0.1966

0.7541 0.7008

CT 2

0.1333

0.9922

0.2074

0.7799

CT 3 CT 4

0.0168 0.0471

0.9384 0.9246

0.1615 0.1922

0.7028 0.6969

3000

2500

2000

1500

1000

500

0

0

50

100

150

200

250

300

4000

3500

3000

2500

2000

1500

1000

500

0 0

50

100

150

200

250

300

7000

6000

5000

4000

3000

2000

1000

0 0

50

100

150

200

250

300

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Analysis of the other columns makes it clear that total contrast properties (TC5) and (TC6) hold whenever the corresponding local contrasts satisfy (LC5) and (LC6). A.1. Comparison with other expressions of contrast In Table 8 we show 24 images ðIM13  IM36Þ grouped into threes. In each group the central image is the original; the images on the left and right have been manipulated using Photoshop to minimize and maximize contrast, respectively. In Table 9, we show the numerical results from applying the following expressions from Table 8 to each image.    

C T 1 , Michelson’s Eq. (21); C T 4 , constructed using Tizhoosh’s expression as the local contrast (that is, C l ¼ C l0 ¼ C l1 in Eq. (20)); GP, constructed as indicated in 3; C T 3 , constructed with Eq. (26) and local contrast C l1 ;

The last measure is our preferred definition of the total contrast. Comments 1–6 above, although referring specifically to Tables 3 and 5, are further validated by the results in Tables 8 and 9. Finally, Table 10 presents two more special cases. Image IM37 is black and white only. In image IM38, nearly all of the . Two adjacent pixels are set to zero and one instead. An analysis of this table leads us to the following pixels have intensity 125 255 conclusions. (1) Michelson’s expression takes the maximum contrast in both IM37 and IM38. This result is inacceptable in an image such as IM38, where nearly all pixels have the same intensity, so the visual contrast is zero with the exception of an isolated pair of pixels. Our best proposal, C T 3 with C l ¼ C l1 , does not share this behavior.

Table 8 Original images.

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H. Bustince et al. / Information Sciences 180 (2010) 1326–1344 Table 9 Total contrast comparison. Image

Michelson

Tizhoosh

GP

Our best proposal

IM13 IM14 IM15 IM16 IM17 IM18 IM19 IM20 IM21 IM22 IM23 IM24 IM25 IM26 IM27 IM28 IM29 IM30 IM31 IM32 IM33 IM34 IM35 IM36

0.3516 0.6953 1 0.5 0.9922 1 0.25 0.5063 0.96 0.3773 1 1 0.3723 0.9767 1 0.4150 0.8387 1 0.4620 0.5502 0.6776 0.4859 1 1

0.8824 0.8423 0.7519 0.8688 0.807 0.7196 0.9540 0.9246 0.8688 0.8962 0.8107 0.6993 0.7632 0.6399 0.5947 0.7292 0.6268 0.6280 0.7355 0.6923 0.6478 0.8295 0.7573 0.6401

0.0016 0.0061 0.0219 0.0049 0.0190 0.0605 0.0001 0.0005 0.0022 0.0015 0.0094 0.0338 0.0152 0.0933 0.2920 0.0184 0.0718 0.2597 0.0194 0.0300 0.0497 0.0048 0.0187 0.0439

0.02 0.0387 0.0459 0.0443 0.0874 0.1459 0.0083 0.0165 0.0326 0.0299 0.0741 0.1337 0.0898 0.2222 0.3247 0.0930 0.1834 0.3375 0.1145 0.1424 0.1558 0.0424 0.0841 0.1006

Table 10 Borderline cases. Image

Michelson

Tizhoosh

GP

Our best proposal

1

1

0.0756

1

1

0.5002

0.0031

0.025

(2) Tizhoosh’s expression is a total contrast in our sense of the term. For IM37, the value is one as expected. Nevertheless, we consider its result for IM38 too high, bearing in mind that most of the image has the same intensity. For this reason, we consider our best proposal more appropriate than that of Tizhoosh. (3) The numerical results provided by GP make clear that this measure is not a total contrast in our sense of the term; it does not assign the value one to images in black and white. In conclusion, from our point of view, at both the theoretical and the applied levels, the best expression for measuring local contrast is C l1 , and the best expression for total contrast is C T 3 . References [1] M. Adel, D. Zuwala, M. Rasigni, S. Bourennane, Filtering noise on mammographic phantom images using local contrast modification functions, Image Vis. Comput. 26 (2008) 1219–1229. [2] A. Amo, J. Montero, E. Molina, Representation of consistent recursive rules, Euro. J. Oper. Res. 130 (2001) 29–53. [3] M. Baczyn´ski, B. Jayaram, Fuzzy Implications (Studies in Fuzziness and Soft Computing), Springer, Berlin, 2008. [4] E. Barrenechea, H. Bustince, M. Pagola, J. Fernandez, Construction of interval-valued fuzzy entropy invariant by translations and scalings, Soft Comput., in press. doi:10.1007/s00500-009-0480-7. [5] G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners (Studies In Fuzziness and Soft Computing), vol. 221, Springer, 2007, pp. 1–37. [6] P. Burillo, H. Bustince, Entropy on intutionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets Syst. 78 (1996) 305–316.

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