Generalized Rough Sets, Entropy, and Image Ambiguity Measures Debashis Sen, Member, IEEE, and Sankar K

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 1, FEBRUARY 2009 117 Generalized Rough Sets, Entropy, and Image Ambiguity Measures Debashis Sen, Member, IEEE, and Sankar K. Pal, Fellow, IEEE

Abstract—Quantifying ambiguities in images using fuzzy set theory has been of utmost interest to researchers in the field of image processing. In this paper, we present the use of rough set theory and its certain generalizations for quantifying ambiguities in images and compare it to the use of fuzzy set theory. We propose classes of entropy measures based on rough set theory and its certain generalizations, and perform rigorous theoretical analysis to provide some properties which they satisfy. Grayness and spatial ambiguities in images are then quantified using the proposed Fig. 1. Ambiguities in a grayscale image with sinusoidal gray value gra- entropy measures. We demonstrate the utility and effectiveness of dations in horizontal direction. (a) A grayscale image. (b) Fuzzy boundary. the proposed entropy measures by considering some elementary (c) Rough resemblance. image processing applications. We also propose a new measure called average image ambiguity in this context. level. This means that a gray value at a pixel represents those at its neighboring pixels to certain extents. It also means that Index Terms—Ambiguity measures, entropy, generalized rough sets, image processing, rough set theory. a gray value represents nearby gray levels to certain extents. Moreover, pixels in a neighborhood with nearby gray levels have limited discernibility due to the inadequacy of contrast. I. INTRODUCTION For example, Fig. 1(c) shows a 6 × 6 portion cut from the image EAL-LIFE images are inherently embedded with vari- in Fig. 1(a). Although this portion contains gray values sepa- R ous ambiguities. For example, imprecision of values at rated by six gray levels, it appears to be almost homogeneous. various pixels results in ambiguity, or value gradations cause Therefore, from the aforementioned analysis, we find that the vague nature of definitions such as region boundaries. Hence, ambiguities in a grayscale image are due to the following. it is natural and appropriate to use techniques that incorporate 1) Various regions have fuzzy boundaries. the ambiguities in order to perform image processing tasks. 2) Nearby gray levels roughly resemble each other, and Fuzzy set theory [1] has been extensively used in order to define values at nearby pixels have rough resemblance. various fuzziness measures of an image. The word “fuzziness” Ambiguities in a grayscale image are of two types, namely, has been, in general, related to the ambiguities arising due to grayness ambiguity and spatial ambiguity [2]. Grayness am- the vague definition of region boundaries. biguity can be quantified considering the fuzzy boundaries Let us now consider, for example, a 1001 × 1001 grayscale of regions based on global gray value distribution and the image [see Fig. 1(a)] that has sinusoidal gray value gradations rough resemblance between nearby gray levels. On the other in horizontal direction. When an attempt is made to mark the hand, spatial ambiguity can be quantified considering the fuzzy boundary of an arbitrary region in the image, an exact boundary boundaries of regions based on organization of gray values at cannot be defined as a consequence of the presence of steadily various pixels and the rough resemblance between values at changing gray values (gray value gradation). This is evident nearby pixels. from Fig. 1(b) that shows a portion of the image, where it The fuzzy set theory of Lofti Zadeh is based on the concept is known that the pixels in the “white” shaded area uniquely of vague boundaries of sets in the universe of discourse [1]. The belong to a region. However, the boundary (on the left and right rough set theory of Zdzislaw Pawlak, on the other hand, focuses sides) of this region is vague as it can lie anywhere in the gray on ambiguity in terms of limited discernibility of sets in the value gradations present in the portion. Value gradation is a domain of discourse [3]. Therefore, fuzzy sets can be used to common phenomenon in real-life images, and hence, it is wide- represent the ambiguities in images due to the vague definition ly accepted that regions in an image have fuzzy boundaries. of region boundaries (fuzzy boundaries), and rough sets can The values at various pixels in grayscale images are consid- be used to represent the ambiguities due to the indiscernibility ered to be imprecise, both in terms of the location and the gray between individual or groups of pixels or gray levels (rough resemblance). Rough set theory, which was initially developed considering Manuscript received May 7, 2008; revised August 6, 2008. First published December 2, 2008; current version published January 15, 2009. This paper was crisp equivalence approximation spaces [3], has been general- recommended by Associate Editor L. Wang. ized by considering fuzzy [4] and tolerance [5] approximation The authors are with the Center for Soft Computing Research, Indian spaces. Furthermore, rough set theory, which was initially Statistical Institute, Calcutta 700108, India (e-mail: [email protected]; [email protected]). developed to approximate crisp sets, has also been generalized Digital Object Identifier 10.1109/TSMCB.2008.2005527 to approximate fuzzy sets [4].

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In this paper, we study the rough set theory and its certain uncertainty in granulation. This is evident in a few examples generalizations to quantify ambiguities in images. Here, the given in [6], which we do not repeat here for the sake of brevity. generalizations to rough set theory based on the approximation Hence, a measure of the incompleteness of knowledge about a of crisp and fuzzy sets considering crisp equivalence, fuzzy universe with respect to only the definability of a set is required. equivalence, crisp tolerance, and fuzzy tolerance approximation The first attempt of formulating an entropy measure with spaces in different combinations are studied. All these combi- respect to the definability of a set was made by Pal et al. [11], nations give rise to different concepts for modeling vagueness, which was used for image segmentation. However, as pointed which can be quantified using the roughness measure [3]. out in [12], this measure does not satisfy the necessary property We propose classes of entropy measures which use the that the entropy value is maximum (or optimum) when the roughness measures obtained considering the aforementioned uncertainty (in this case, incompleteness of knowledge) is various concepts for modeling vagueness. We perform rigorous maximum. theoretical analysis of the proposed entropy measures and In this section, we propose classes of entropy measures, provide some properties which they satisfy. We then use the which quantify the incompleteness of knowledge about a uni- proposed entropy measures to quantify ambiguities in images, verse with respect to the definability of a set of elements giving an account of the manner in which the ambiguities are (in the universe) holding a particular property (representing a captured. We show that the aforesaid generalizations to rough category). An inexactness measure of a set, like the “roughness” set theory regarding the approximation of fuzzy sets can be used measure [3], quantifies the definability of the set. We measure to quantify ambiguities due to both fuzzy boundaries and rough the incompleteness of knowledge about a universe with respect resemblance. to the definability of a set by considering the roughness measure The utility of the proposed measures in quantifying image of the set and also that of its complement in the universe. ambiguities is demonstrated using some image processing operations like enhancement evaluation, segmentation, and edge A. Roughness of a Set in a Universe detection. A new measure called average image ambiguity (AIA) is also defined in this context. The effectiveness of Let U denote a universe of elements and X be an arbitrary some of the proposed measures is shown by qualitative and set of elements in U holding a particular property. Accord- quantitative comparisons of their use in image analysis with that ing to rough set theory [3] and its generalizations, limited of certain fuzziness measures. discernibility draws elements in U together governed by an The proposed entropy measures and their properties are indiscernibility relation R, and hence, granules of elements presented in Section II, and their utility in quantifying grayness are formed in U. An indiscernibility relation [3] in a universe and spatial ambiguities is shown in Section III. Experiments are refers to the similarities that every element in the universe presented in Section IV to demonstrate the effectiveness of the has with the other elements of the universe. The family of all proposed measures. This paper concludes with Section V. granules obtained using the relation R is represented as U/R. The indiscernibility relation among elements and sets in U results in an inexact definition of X. However, the set X can II. ENTROPY MEASURES WITH RESPECT TO THE be approximately represented by two exactly definable sets RX DEFINABILITY OF A SET OF ELEMENTS and RX in U, which are obtained as Defining entropy measures based on rough set theory has RX = {Y ∈ U/R : Y ⊆ X} (1) been considered by researchers in the past decade. Probably, the first of such work was reported in [6], where a “rough entropy” RX = {Y ∈ U/R : Y ∩ X = ∅}. (2) of a set in a universe has been proposed. This rough entropy measure is defined based on the uncertainty in granulation In the aforesaid expressions, RX and RX are called the (obtained using a relation defined over universe [3]) and the R-lower approximation and the R-upper approximation of X, definability of the set. Other entropy measures that quantify respectively. In essence, the pair of sets RX, RX is the the uncertainty in crisp or fuzzy granulation alone have been representation of any arbitrary set X ⊆ U in the approximation reported in literature [6]–[9]. An entropy measure is presented space U, R, where X cannot be defined. As given in [3], an in [10], which, although not based on rough set theory, quan- inexactness measure of the set X can be defined as tifies information with the underlying elements having limited | | − RX discernibility between them. ρR(X)=1 (3) |RX| Incompleteness of knowledge about a universe leads to granulation [3], and hence, a measure of the uncertainty in granu- where |RX| and |RX| are the cardinalities of the sets RX lation quantifies this incompleteness of knowledge. Therefore, and RX in U, respectively. The inexactness measure ρR(X) apart from the “rough entropy” in [6] which quantifies the is called the R-roughness measure of X, and it takes a value in incompleteness of knowledge about a set in a universe, the the interval [0, 1]. other aforesaid entropy measures quantify the incompleteness of knowledge about a universe. The effect of the incompleteness B. Lower and Upper Approximations of a Set of knowledge about a universe becomes evident only when an attempt is made to define a set in it. Note that the definability The expressions for the lower and upper approximations of of a set in a universe is not always affected by a change in the the set X depend on the type of relation R and whether X

TABLE I DIFFERENT NAMES OF RX, RX AND U, R

is a crisp [1] or a fuzzy [1] set. Here, we shall consider the where SR(u, ϕ) is a value representing the tolerance relation R upper and lower approximations of the set X when R denotes between u and ϕ. an equivalence, a fuzzy equivalence, a tolerance, or a fuzzy The pair of sets RX, RX and the approximation space tolerance relation and X is a crisp or a fuzzy set. U, R are referred to differently, depending on whether X is When X is a crisp or a fuzzy set and the relation R is a crisp a crisp or a fuzzy set; the relation R is a crisp or a fuzzy equiv- or a fuzzy equivalence relation, the expressions for the lower alence, or a crisp or a fuzzy tolerance relation. The different and upper approximations of the set X are given as names are listed in Table I.

RX = {(u, M(u)) |u ∈ U} C. Entropy Measures RX = u, M(u) |u ∈ U (4) As mentioned earlier, the lower and upper approximations of where a vaguely definable set X in a universe U can be used in the expression given in (3) in order to get an inexactness measure of the set X called the roughness measure ρR(X). The vague M(u)= mY (u) × inf max (1 − mY (ϕ),μX (ϕ)) ϕ∈U Y ∈U/R definition of X in U signifies the incompleteness of knowledge about U. M(u)= mY (u) × sup min (mY (ϕ),μX (ϕ)) (5) ϕ∈U Here, we propose two classes of entropy measures based on Y ∈U/R the roughness measures of a set and its complement in order to quantify the incompleteness of knowledge about a universe. where the membership function mY represents the belonging- One of the proposed two classes of entropy measures is ob- ness of every element (u) in the universe (U) to a granule tained by measuring the “gain in information” or, in our case, ∈ Y U/R and it takes values in the interval [0, 1] such that the “gain in incompleteness” using a logarithmic function as Y mY (u)=1, and μX , which takes values in the interval suggested in Shannon’s theory. This proposed class of entropy [0, 1], is the membership function associated with X. When X measures for quantifying the incompleteness of knowledge { } is a crisp set, μX would take values only from the set 0, 1 . about U with respect to the definability of a set X ⊆U is Similarly, when R is a crisp equivalence relation, mY would given as take values only from the set {0, 1}. The symbols (sum) and × (product) in (5) represent specific fuzzy union and in- 1 tersection operations [1], respectively, which are chosen based HL(X)=− κ(X)+κ(X ) (7) R 2 on their suitability with respect to the underlying application of measuring ambiguity. where κ(D)=ρR(D) log (ρR(D)/β) for any set D ⊆ U, β Note that, till now, we have considered the indiscernibility β denotes the base of the logarithmic function used, and X ⊆ U relation R ⊆ U × U to be an equivalence relation, i.e., R stands for the complement of the set X in the universe. The satisfies crisp or fuzzy reflexivity, symmetry, and transitivity various entropy measures of this class are obtained by calculat- properties [1]. However, if R does not satisfy any one of these ing the roughness values ρ (X) and ρ (X) considering the three properties, the expressions in (4) can no longer be used. R R different ways of obtaining the lower and upper approximations We shall consider here the case when the transitivity property is of the vaguely definable set X. Note that the “gain in incom- not satisfied. Such a relation R is said to be a tolerance relation, pleteness” term is taken as − log (ρ /β) in (7), and for β>1, and the space U, R obtained is referred to as a tolerance β R it takes a value in the interval [1, ∞]. The other class of entropy approximation space [5]. When R is a tolerance relation, the measures proposed is obtained by considering an exponential expressions for the membership values corresponding to the function [13] to measure the “gain in incompleteness.” This lower and upper approximations [see (5)] of an arbitrary set X second proposed class of entropy measures for quantifying in U are given as the incompleteness of knowledge about U with respect to the definability of a set X ⊆ U is given as M(u)=inf max (1 − SR(u, ϕ),μX (ϕ)) ϕ∈U 1 M(u)=supmin (S (u, ϕ),μ (ϕ)) E (¯ρR(X)) (ρ¯R(X )) R X (6) H (X)= ρR(X)β + ρR(X )β (8) ϕ∈U R 2

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where ρ¯R(D)=1− ρR(D) for any set D ⊆ U and β denotes From (9), (10) and (11), we deduce that the base of the exponential function used. Pal and Pal [13] had |RX| 1 |RX| considered only the case when β equaled e. Similar to the class ρR(X )=ρR(X) = . (12) L n −|RX| C n −|RX| of entropy measures HR, the various entropy measures of this class are obtained by using the different ways of obtaining the We shall now separately consider three cases of (12), where we lower and upper approximations of X in order to calculate ∞ ∞ have 1 1, it takes a value in the |RX|)=(C − 1)/C from (9). Using this relation in (12), ﬁnite interval [1,β]. Note that an analysis on the appropriate we obtain L E values that β in HR and HR can take will be given later in C Section II-E-1. 1 |RX| − ρ (X)= C 1 . (13) We shall name a proposed entropy measure using attributes R C n −|RX| that represent the class (logarithmic or exponential) it belongs to and the type of the pair of sets RX, RX considered. For After some algebraic manipulations, we deduce that example, if RX, RX represents a tolerance rough–fuzzy set and the expression of the proposed entropy in (8) is considered, 1 1 ρ (X )= . R − n − (14) then we call such an entropy as the exponential tolerance C 1 |RX| 1 rough–fuzzy entropy. Some other examples of names for the proposed entropy measures are the logarithmic rough entropy, Note that when 1

Authorized licensed use limited to: INDIAN STATISTICAL INSTITUTE. Downloaded on May 4, 2009 at 07:56 from IEEE Xplore. Restrictions apply. SEN AND PAL: GENERALIZED ROUGH SETS, ENTROPY, AND IMAGE AMBIGUITY MEASURES 121 the proposed entropy measures do not have any unnecessary terms. However, the base parameters β’s [see (7) and (8)] of the two classes of entropy measures incur certain restrictions, so that the proposed entropies satisfy some important properties. In this section, we shall discuss the restrictions regarding the base parameters and then provide few properties of the proposed entropies. 1) Range of Values for the Base β: The proposed classes of L E entropy measures HR and HR given in (7) and (8), respectively, Fig. 2. Plots of the proposed classes of entropy measures for various rough- must be consistent with the fact that maximum information ness values A and B. (a) Logarithmic. (b) Exponential. (entropy) is available when the uncertainty is maximum and the entropy is zero when there is no uncertainty. Note that, in our case, maximum uncertainty represents maximum possible incompleteness of knowledge about the universe. Therefore, maximum uncertainty occurs when both the roughness values L E used in HR and HR equal unity, and uncertainty is zero when both of them are zero. It can be easily shown that in order to L satisfy the aforesaid condition, the base β in HR must take a ﬁnite value greater than or equal to e(≈ 2.7183), and the base β E ≥ in HR must take a value in the interval (1,e]. When β e in L ≤ E L HR and 1 <β e in HR , the values taken by both HR and E HR lie in the range [0, 1]. Note that for an appropriate β value, the proposed entropy measures attain the minimum value of zero only when ρR(X)=ρR(X )=0and the maximum value Fig. 3. Proposed entropy measures for a few different β values, when A = B. of unity only when ρR(X)=ρR(X )=1. 2) Properties: Here, we present few properties of the proposed logarithmic and exponential classes of entropy mea- sharpened versions of A and B, respectively, i.e., A∗ ≤ A L E ∗ sures expressing HR and HR as functions of two parameters and B ≤ B. L E representing roughness measures. We may rewrite the expres- 6) Symmetry: Both HR(A, B) and HR (A, B) are symmet- sions given in (7) and (8) in parametric form as follows, ric about the line A = B. L E respectively: 7) Monotonicity: Both HR(A, B) and HR (A, B) are monotonically nondecreasing functions of A and B. 1 A B 8) Concavity: Both HL(A, B) and HE(A, B) are concave HL(A, B)=− A log + B log (18) R R R 2 β β β β functions of A and B. 1 − − The plots of the proposed classes of entropies HL and HE as HE(A, B)= Aβ(1 A) + Bβ(1 B) (19) R R R 2 functions of A and B are shown in Figs. 2 and 3, respectively. L E In Fig. 2, the values of HR and HR are shown for all possible where the parameters A (∈ [0, 1]) and B (∈ [0, 1]) represent values of the roughness measures A and B considering β = e. the roughness values ρR(X) and ρR(X ), respectively. Con- Fig. 3 shows the plots of the proposed entropies for different sidering the convention 0 logβ 0=0, let us now discuss the values of the base β, when A = B. L E following properties of HR(A, B) and HR (A, B) along the lines of [15]. L ≥ E III. MEASURING AMBIGUITIES IN IMAGES 1) Nonnegativity: We have HR(A, B) 0 and HR (A, B) ≥ 0 with equality in both the cases if and only if Ambiguities in grayscale images are due to fuzzy boundaries A =0and B =0. between regions, and rough resemblance between nearby gray E L 2) Continuity: Both HR (A, B) and HR(A, B) are continu- levels and between values at nearby pixels (see Section I). In ous functions of A and B, where A, B ∈ [0, 1]. this section, we shall use the entropy measures proposed in the L E 3) Sharpness: Both HR(A, B) and HR (A, B) equal zero if previous section in order to quantify ambiguities in a grayscale and only if the roughness values A and B equal zero, i.e., image. As we shall see later, the entropy measures based on the A and B are “sharp.” generalization of rough set theory regarding the approximation L E 4) Maximality and Normality: Both HR(A, B) and HR (A, of fuzzy sets (i.e., when the set X considered in the previous B) attain their maximum value of unity if and only section is fuzzy) can be used to quantify ambiguities due to if the roughness values A and B are unity. That is, both fuzzy boundaries and rough resemblance, whereas the L ≤ L E ≤ we have HR(A, B) HR(1, 1) = 1 and HR (A, B) entropy measures based on the generalization of rough set E ∈ HR (1, 1) = 1, where A, B [0, 1]. theory regarding the approximation of crisp sets (i.e., when the L ∗ ∗ ≤ L 5) Resolution: We have HR(A ,B ) HR(A, B) and set X considered in the previous section is crisp) can be used to E ∗ ∗ ≤ E ∗ ∗ HR (A ,B ) HR (A, B), where A and B are the quantify ambiguities only due to rough resemblance.

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As mentioned in Section I, ambiguities in a grayscale image The fuzzy sets ΥT and ΥT previously considered capture the are of two types, namely, grayness and spatial ambiguities. fuzzy boundary aspect of the ambiguities. Furthermore, we Grayness ambiguity measure can be obtained by considering consider limited discernibility among the elements in G that the fuzzy boundaries of regions based on global gray value results in vague definitions of the fuzzy sets ΥT and ΥT , and distribution and the rough resemblance between nearby gray hence, the rough resemblance aspect of the ambiguities is also levels. In this case, the image should be considered as an captured. array of gray values, and the measure of consequence of the Granules, with crisp or fuzzy boundaries, are induced in G as incompleteness of knowledge about the universe of gray levels its elements are drawn together due to the presence of limited in the array would quantify the ambiguities. Spatial ambiguity discernibility (or indiscernibility relation) among them, and this measure can be obtained by considering the fuzzy boundaries process is referred to as the gray-level granulation. We assume of regions based on organization of gray values at various pixels that the indiscernibility relation is uniform in G, and hence, the and the rough resemblance between values at nearby pixels. granules formed have a constant support cardinality (size) ω. In this case, the image should be considered as a universe of Now, using (4)–(6), we get general expressions for the differ- pixels, and the measure of the incompleteness of knowledge ent lower and upper approximations of ΥT and ΥT obtained about the universe of pixels would quantify the ambiguities. In considering the different indiscernibility relations discussed in the aforesaid discussion, the measure of the incompleteness of Section II-B as follows: knowledge about a universe with respect to the definability of a set should be used, as the set would be employed to capture the Υ = l, M (l) , Υ = l, M (l) T ΥT T ΥT vagueness in region boundaries. Note that although the discussion in this section will be on grayscale images, it is also applicable to images obtained ΥT = l, MΥ (l) , ΥT = l, M (l) (22) T ΥT by carrying out operations on grayscale images, for example, images of edge strengths. where l ∈ G.Wehave γ MΥ (l)= mω (l) × inf max m¯ ω (ϕ),μT (ϕ) A. Grayness Ambiguity Measure: Ambiguities in an Image T i ϕ∈G i Represented as an Array of Gray Values i=1 γ Let G be the universe of gray levels and ΥT beasetinG, i.e., ω × ω MΥ (l)= m (l) sup min m (ϕ),μT (ϕ) ΥT ⊆ G, whose elements hold a particular property to extents T i ∈ i i=1 ϕ G given by a membership function μT defined on G. Let us now take up the problem of quantifying ambiguities in an image I γ × considering it as an array of gray values. Let OI be the gray- MΥ (l)= mω (l) inf max m¯ ω (ϕ), μ¯T (ϕ) T i ϕ∈G i level histogram of the image I. The fuzzy boundaries and rough i=1 resemblance in I causing the ambiguities are related to the γ incompleteness of knowledge about G, which can be quantified M (l)= mω (l) × sup min mω (ϕ), μ¯ (ϕ) (23) i i T ΥT ∈ using the proposed entropy measures in Section II-C. i=1 ϕ G We shall consider ΥT such that it represents the category “dark areas” in the image I, and the associated property when equivalence indiscernibility relation is considered, with m¯ ω (ϕ)=1− mω (ϕ) and μ¯ (ϕ)=1− μ (ϕ), and we “darkness” given by the membership function μT shall be i i T T modeled as have ⎧ ¯ ≤ − MΥ (l)=inf max Sω(l, ϕ),μT (ϕ) ⎪ 1,l T Δ T ϕ∈G ⎪ 2 ⎨ − (l−(T −Δ)) − ≤ ≤ 1 2 2Δ ,T Δ l T M (l)=supmin (Sω(l, ϕ),μT (ϕ)) μT (l)= 2 (20) ΥT ⎪ (l−(T +Δ)) ϕ∈G ⎪ 2 ,T≤ l ≤ T +Δ ⎩ 2Δ 0,l≥ T +Δ ¯ MΥ (l)=inf max Sω(l, ϕ), μ¯T (ϕ) T ϕ∈G where l ∈ G and T and Δ are called the crossover point and the M (l)=supmin (Sω(l, ϕ), μ¯T (ϕ)) (24) Υ bandwidth, respectively. We shall consider that Δ is a constant T ϕ∈G and that different definitions of the property “darkness” can be obtained by changing the value of T , where T ∈ G. when tolerance indiscernibility relation is considered, with ¯ − In order to quantify the ambiguities in the image I using Sω(l, ϕ)=1 Sω(l, ϕ). In (23), γ denotes the number of granules formed in the universe G, and mω (l) gives the mem- the proposed classes of entropy measures, we consider the i ω following sets: bership grade of l in the ith granule i . These membership grades may be calculated using any concave, symmetric, and normal membership function (with support cardinality ω), such ΥT = {(l, μT (l)) |l ∈ G} as the one having a triangular, trapezoidal, or bell (for example, { − | ∈ } ΥT = (l, 1 μT (l)) l G . (21) the π function) shape. Note that the sum of these membership

The ambiguity measure Λ of I is obtained as a function of T , which characterizes the underlying set ΥT , as follows: 1 ΛL(T )=− κ(Υ )+κ Υ (26) ω 2 T T κ ⊆ where (D)= ω(D) logβ( ω(D)/β) for any set D G. Note that the aforementioned expression is obtained by using ω(ΥT ) and ω(ΥT ) in the proposed logarithmic class of entropy functions given in (8), instead of roughness measures. When the proposed exponential class of entropy functions is used, we get (¯ω (ΥT )) (¯ω (ΥT )) ω(ΥT )β + ω ΥT β ΛE(T )= (27) ω 2

where ¯ω(D)=1− ω(D) for any set D ⊆ G. It should be noted that the values ω(ΥT ) and ω(ΥT ) in (25) are obtained by considering “weighted cardinality” measures instead of car- Fig. 4. Different forms that the lower and upper approximations of ΥT can dinality measures, which are used for calculating roughness take when used to get the grayness ambiguity measure. (a) Crisp ΥT and ω ω ω values [see (3)]. The weights considered are the number of Crisp i . (b) Fuzzy ΥT and Crisp i . (c) Crisp ΥT and Fuzzy i . ω × →{ } (d) Fuzzy ΥT and Fuzzy i . (e) Crisp ΥT and Sω : G G 0, 1 . occurrences of gray values given by the gray-level histogram (f) Fuzzy Υ and S : G × G →{0, 1}.(g)CrispΥ and S : G × G → T ω T ω OI of the image I. [0, 1]. (h) Fuzzy Υ and S : G × G → [0, 1]. T ω The ambiguity measures obtained using (26) and (27) are referred to as the grayness ambiguity measures, and they lie in the range [0, 1], where a larger value means higher ambiguity. grades over all the granules must be unity for a particular value of l. In (24), Sω : G × G → [0, 1], which can be any B. Spatial Ambiguity Measure: Ambiguities in an Image concave and symmetric function, gives the relation between Represented as a Universe of Pixels any two gray levels in G.ThevalueofSω(l, ϕ) is zero when the difference between l and ϕ is greater than ω, and S (l, ϕ) Let us now take up the problem of quantifying the ambi- ω guities in an image I considering it as a universe of pixels equals unity when l equals ϕ. (associated with gray values). Let P be the universe of pixels The lower and upper approximations of the sets ΥT and and Υ beasetinP , i.e., Υ ⊆ P , whose elements which Υ take different forms, depending on the nature of rough T T T are associated with gray values hold a particular property to resemblance considered and whether the need is to capture am- extents given by the membership function μ [see (20)] defined biguities due to both fuzzy boundaries and rough resemblance T on G. Now, the fuzzy boundaries and rough resemblance in or only those due to rough resemblance. The nature of rough I causing the ambiguities are related to the incompleteness of resemblance may be considered such that an equivalence rela- knowledge about P , which can be quantified using the proposed tion between gray levels induces granules having crisp (crisp classes of entropy measures in Section II-C. In order to quantify ω) or fuzzy (fuzzy ω) boundaries, or there exists a tolerance i i the ambiguities in the image I using the proposed classes of relation between gray levels that may be crisp (S : G × G → ω entropy measures, we consider the following sets: {0, 1}) or fuzzy (Sω : G × G → [0, 1]). When the sets ΥT and Υ considered are fuzzy sets, ambiguities due to both fuzzy T Υ = (p ,p ,μ l boundaries and rough resemblance would be captured, whereas T 1 2 T p1,p2 when the sets ΥT and Υ considered are crisp sets, only the T Υ = (p ,p , 1 − μ l (28) ambiguities due to rough resemblance would be captured. The T 1 2 T p1,p2 different forms of the lower and upper approximations of Υ T where p1,p2∈P and lp ,p is the gray value at the pixel are shown graphically in Fig. 4. 1 2 p1,p2. Hereafter in this paper, we shall use p12 ≡p1,p2. We shall now quantify the ambiguities in the image I by mea- The fuzzy sets ΥT and ΥT previously considered capture the suring the consequence of the incompleteness of knowledge fuzzy boundary aspect of the ambiguities. Limited discernibil- about the universe of gray levels G in I. This measurement is ity is considered among the elements in P that results in vague done by calculating the following values: definitions of the fuzzy sets ΥT and ΥT in (28), and hence the rough resemblance aspect of ambiguities is also captured. M (l)O (l) Granules, with crisp or fuzzy boundaries, are induced in l∈G ΥT I ω(ΥT )=1− P as its elements are drawn together due to the presence of ∈ M (l)OI (l) l G ΥT an indiscernibility relation among them, and this process is referred to as the spatial granulation. The indiscernibility re- l∈G MΥ (l)OI (l) Υ =1− T . (25) lation is assumed uniform in P , and hence, the granules formed ω T M (l)O (l) l∈G I have a constant support cardinality (size) ω1 × ω2 denoted as ΥT

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ω1,ω2. Now, using (4)–(6), we get general expressions for of pixels P . This measurement is done by calculating the the differentde lower and upper approximations of Υ and Υ following roughness values: T T given in (28) as follows: | | Υ ΥT T ρω(ΥT )=1− ρω Υ =1− (32) T |ΥT | ΥT ΥT = p12,MΥ (p12) ΥT = p12,M (p12) T ΥT where ω ≡ ω12. Now, the ambiguity measure Λ of I is obtained Υ = p12,M (p12) Υ = p12,M (p12) (29) as a function of T as follows: T Υ T Υ T T 1 ΛL(T )=− κ(Υ )) + κ Υ (33) ω 2 T T ∈ where p12 P .WehaveMΥT (p12), M (p12), M (p12), ΥT ΥT where κ(D)=ρω(D) log (ρω(D)/β) for any set D ⊆ P . and M (p12), respectively, as β Υ T Note that, in the aforementioned discussion, the ambiguity γ measure is obtained by using the roughness values associated ω ω m 12 (p12)infmax m¯ 12 (ϕ12),μT (lϕ12 ) with ΥT and ΥT in order to calculate the proposed logarithmic i ϕ12∈P i i=1 class of entropy measures that quantifies the incompleteness of γ knowledge about P . When the proposed exponential class of ω ω m 12 (p12)supmin m 12 (ϕ12),μT (lϕ12 ) i ∈ i entropy measures is used, we get i=1 ϕ12 P γ 1 E (¯ρω (ΥT )) (ρ¯ω (Υ )) Λ (T )= ρω(ΥT )β +ρω Υ β T (34) ω ω ω T m 12 (p12)infmax m¯ 12 (ϕ12), μ¯T (lϕ12 ) 2 i ϕ12∈P i i=1 − ⊆ γ where ρ¯ω(D)=1 ρω(D) for any set D P . ω ω The ambiguity measures obtained using (33) and (34) are m 12 (p12)supmin m 12 (ϕ12), μ¯T (lϕ12 ) (30) i ∈ i i=1 ϕ12 P referred to as the spatial ambiguity measures, and they lie in the range [0, 1], where a larger value means higher ambiguity. when equivalence indiscernibility relation is considered, with ω − ω − m¯ 12 (ϕ12)=1 m 12 (ϕ12) and μ¯T (lϕ12 )=1 μT (lϕ12 ), i i C. Average Image Ambiguity (AIA) and we have As mentioned earlier, the elements in the set considered for ¯ quantifying ambiguities in a grayscale image hold a particular MΥT (p12)= inf max Sω12 (p12,ϕ12),μT (lϕ12 ) ϕ12∈P property, which is given by a membership function. This membership function is characterized by certain parameters, which, MΥ (p12)= sup min (Sω12 (p12,ϕ12),μT (lϕ12 )) T ∈ ϕ12 P in turn, characterize the set under consideration. The property ¯ M (p12)= inf max Sω12 (p12,ϕ12), μ¯T (lϕ12 ) can be defined in different ways by changing some of these ΥT ∈ ϕ12 P parameters. Different ambiguity (Λ) measures are obtained for M (p12)= sup min (Sω (p12,ϕ12), μ¯T (lϕ )) (31) Υ 12 12 different definitions of the property, and the average of all these T ϕ12∈P measures gives us a characteristic measure of the image under when tolerance indiscernibility relation is considered, with consideration. Therefore, we obtain a class of characteristic S¯ (p ,ϕ )=1− S (p ,ϕ ). In the aforementioned measures of an image based on rough set theory and its certain ω12 12 12 ω12 12 12 generalizations as follows: expressions, we use ϕ12 ≡ϕ1,ϕ2 and ω12 ≡ω1,ω2, γ denotes the number of granules formed in the universe P , 1 Λ= Λ(i) (35) and mω12 (p12) gives the membership grade of p12 in the ith | | i Θ ∈ ω12 i Θ granule i . These membership grades may be calculated using any concave, symmetric, and normal 2-D membership where Θ is a set of all possible combinations of values of the function (with support cardinality ω1 × ω2). Note that the sum parameters that are used to define the property in different ways. of these membership grades over all the granules must be We shall refer Λ as AIA, and its value lies in the range [0, 1], × → unity for a particular value of p12. In (31), Sω12 : P P where a smaller value means that various parts of the image are [0, 1], which can be any concave and symmetric 2-D function, better distinguishable from each other, in a holistic sense. gives the relation between any two pixels in P .Thevalueof In our case, as mentioned earlier, we use the “darkness” prop- Sω12 (p12,ϕ12) is zero when the spatial separations between p1 erty, and different definitions of this property can be obtained by and ϕ1, and p2 and ϕ2 are greater than ω1 and ω2, respectively, changing a parameter T ∈ G, where G is the universe of gray and Sω12 (p12,ϕ12) equals unity when p12 equals ϕ12.The levels. Therefore, we get discussion in the case of grayness ambiguity measure, on the 1 different forms that the lower and upper approximations of ΛL = ΛL(T ) the sets Υ and Υ take, is also applicable here, when we con- ω |G| ω T T T ∈G ω12 ω sider i , Sω12 , and P instead of i , Sω, and G, respectively. 1 We shall now quantify the ambiguities in the image I by ΛE = ΛE(T ) (36) ω |G| ω measuring the incompleteness of knowledge about the universe T ∈G

L E where Λω and Λω are the AIA measures obtained using the logarithmic and exponential classes of entropies, respectively. Note that ω in (36) is a constant with respect to the “darkness” property.

IV. APPLICATIONS AND EXPERIMENTAL RESULTS Fig. 5. Visual quality of the original and enhanced images of a tire. A plethora of image processing techniques based on measur- (a) Original image. (b) Enhanced by histogram equalization. (c) Enhanced by ing ambiguities in images using fuzzy set theory are available unsharp masking. in literature, with some of them representing the state of the art. In this section, we shall demonstrate the utility of the A. Enhancement Evaluation proposed entropies in measuring ambiguities in images by Quantitative evaluation of image enhancement operations is considering a few elementary image processing applications, an important task in image processing. Among quite a few such as enhancement evaluation, segmentation, and edge de- works of enhancement evaluation reported in literature, the one tection, where ambiguity-measure-based techniques have been in [16] employs a fuzziness-based image quality measure. We previously used. shall now consider this image quality measure for enhancement As mentioned in Section I, ambiguities in images are due to evaluation and compare it to the use of the proposed logarithmic fuzzy boundaries and rough resemblance. In Sections II and III, tolerance fuzzy rough–fuzzy entropy. If it is considered that we have shown that the proposed classes of entropy measures the quality of an image is a term that describes how well its based on rough set theory and its certain generalizations have different parts are distinguishable, then the proposed AIA mea- the following advantages over most fuzzy-set-theory-based un- sures in (36) can readily be used as image quality measures for certainty measures. quantitative evaluation of image enhancement with a smaller 1) There are no terms in the expressions of the proposed value signifying better quality. Note that, on the contrary, a classes of entropy measure that “theoretically” convey the larger value of the measure used in [16] means better image same. quality. 2) Some of the proposed entropy measures can be used to Consider the images in Fig. 5. The image in Fig. 5(a) is quantify ambiguities due to both fuzzy boundaries and the original image, and the images in Fig. 5(b) and (c) are rough resemblance. the enhanced ones using the histogram equalization [17] and unsharp masking [17] techniques, respectively. The image qual- In this section, we shall also compare the use of the proposed ity measure used in [16] orders these images as (b), (a), and entropies in measuring ambiguities with certain existing use (c) with the measures 0.49309, 0.3073, and 0.27942, respec- L of fuzziness measures in order to observe whether the afore- tively, whereas the AIA measure Λω orders these images as (c), mentioned advantages translate into better performance. Thus, (a), and (b) with the AIA values 0.20095, 0.22729, and 0.25046, the effectiveness of some of the proposed entropy measures in respectively. The AIA values suggest that Fig. 5(c) has the best quantifying ambiguities in images shall be demonstrated. quality; as in Fig. 5(c), several details have cropped up due to Throughout this section, we shall consider the proposed the enhancement without compromising much on the overall ambiguity measure given in (26), which signiﬁes measuring contrast of the image, unlike Fig. 5(b). the grayness ambiguity using the proposed logarithmic class of entropy functions. The measures in (25) which are used in B. Segmentation (26) are calculated considering that the pairs of lower and upper approximations of the sets ΥT and ΥT represent a tolerance Segmentation is one of the core tasks in image processing, fuzzy rough–fuzzy set. The aforesaid statement, according to and a vast number of simple to complex techniques have the terminology given in Section II-C, signiﬁes that logarithmic been reported in literature. In order to compare the use of tolerance fuzzy rough–fuzzy entropy is used in this section to the proposed logarithmic tolerance fuzzy rough–fuzzy entropy get the grayness ambiguity measure. We consider the values of in segmentation with a fuzzy-set-theory-based method, we the parameters Δ and ω as eight and six gray levels, respec- shall consider the fuzzy-entropy-based segmentation technique tively, and the base β as e, without loss of generality. proposed in [18]. The technique in [18] uses a membership Note that the logarithmic tolerance fuzzy rough–fuzzy en- function like the one given in (20) and determines the fuzzy tropy is a representative of the proposed entropies which can be entropy measure of the underlying image for all values of T . used to capture ambiguities due to both fuzzy boundaries and The appropriate number of minima in the fuzzy entropy (as rough resemblance. The expression of this entropy measure, a function of T ) curve is then chosen as the thresholds for like those of all the other proposed entropy measures, does segmentation. In order to have a fair comparison, we use the not have terms that “theoretically” convey the same. Hence, proposed ambiguity measure in (26) (based on the aforesaid en- the utility of all the proposed entropy measures and the effec- tropy) instead of the fuzzy entropy in the same technique given tiveness of some of them can be demonstrated by considering in [18]. We apply the aforementioned segmentation algorithms the proposed logarithmic tolerance fuzzy rough–fuzzy entropy to the grayscale images corresponding to Ohta’s color features alone. I1, I2, and I3 [19] of a color image and use a technique similar

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Fig. 6. Object extraction using algorithms employing the proposed grayness ambiguity measure and fuzzy entropy, and the mean shift method. (a) An image Fig. 8. Comparison of performance of the algorithms employing the proposed of islands. (b) Using proposed. (c) Using fuzzy entropy. (d) By mean shift ambiguity measure and fuzzy entropy, and the mean shift method using the β method. -index measure.

numbers of regions. The β-index values for the I1, I2, and I3 features of the image in Fig. 7(a) corresponding to the results in Fig. 7(b) and (c) are 8.1027, 9.8338, and 28.399 and 7.4253, 9.9947, and 28.726, respectively. From visual inspection (the areas marked in circles) and the β-index values, we may say that the algorithm using the proposed measure outperforms the Fig. 7. Segmentation using algorithms employing the proposed grayness one using fuzzy entropy. Considering the comparison between ambiguity measure and fuzzy entropy, and the mean shift method. (a) The the proposed ambiguity-measure-based algorithm and the mean pepper image. (b) Using proposed. (c) Using fuzzy entropy. (d) By mean shift shift method in Fig. 7, we find that there are considerable dif- method. ferences in the segmentation results obtained. These differences to the one used in [20] to obtain the segments in the color are due to the fact that the mean shift method forms regions in image. Secondary to visual inspection, we shall also use the an image by considering a compromising combination of gray β-index [21], where a larger value signifies better segmentation, value/color similarity and spatial proximity, whereas the other to compare the performance of the algorithms. two segmentation techniques mentioned in this section consider We shall also consider a state-of-the-art segmentation only the gray value/color similarity. It is visually evident in technique proposed in [22], which uses the mean shift pro- Fig. 7 that the algorithm using the proposed ambiguity measure cedure, for comparison with the aforementioned segmentation gives better results than the mean shift method in terms of color technique based on the proposed ambiguity measure. The afore- uniformity within regions. On the other hand, the mean shift said comparison would let us know whether the segmentation method outperforms the proposed ambiguity-measure-based results obtained by the proposed ambiguity-measure-based algorithm when the compactness of a given region in terms of technique are comparable to that of the state-of-the-art tech- spatial proximity is considered. nique, even when the technique using the proposed ambi- In order to carry out a rigorous analysis, we first consider guity measure considered here is not a sophisticated one. 45 grayscale images from the Universidad de Granada im- When color images are considered, the β-index cannot be age database (http://decsai.ugr.es/cvg/dbimagenes/index.php), used for the aforesaid comparison, as the mean shift segmen- where images 1–18 are that of galaxies, images 19–34 are tation method in [22] works on a color image as a whole, Brain MRIs, and images 35–45 are that of nematodes. We unlike the technique using the proposed ambiguity measure then perform object/background separation in the images of which works on the grayscale images corresponding to the galaxies and nematodes and segment the Brain MRIs into three underlying color features. It should be noted that the parame- regions employing the algorithms using the proposed ambiguity ters required in the mean shift method (see [22]) are man- measure and fuzzy entropy, and the mean shift segmentation ually adjusted in accordance to the underlying segmentation method. The corresponding β-index values are put against the problem. image numbers in the bar chart shown in Fig. 8. It is evident In Fig. 6, the aforementioned algorithms are applied to from the figure that the algorithm using the proposed measure, separate the objects in the color image (intensity feature shown) in general, produces results which correspond to the larger from the backgrounds. The β-index values for the I1, I2, and I3 β-index value signifying better segmentation performance than features of the image in Fig. 6(a) corresponding to the results the algorithm using fuzzy entropy. Note that when a grayscale in Fig. 6(b) and (c) are 7.162, 1.1985, and 1.9872 and 7.162, image is under consideration, the β-index evaluates segmen- 2.6443, and 1.8787, respectively. It is visually evident that the tation performance in terms of gray value uniformity within algorithm using the proposed ambiguity measure outperforms regions. Therefore, as evident from Fig. 8, we find that the the one using fuzzy entropy, and in this case, the larger β- β-index suggests that the algorithm using the proposed ambigu- index value for I2 corresponding to the result in Fig. 6(c) ity measure gives better results than the mean shift method, as proves insignificant compared to the larger β-index value for the mean shift method compromises on the gray value similarity I3 corresponding to the result in Fig. 6(b). As can be seen during segmentation. from Fig. 6(b) and (d), the object extraction results obtained by the algorithm using the proposed ambiguity measure are C. Edge Detection comparable to that of the mean shift method. In Fig. 7, the aforementioned algorithms are applied to An important process in most of the edge detection sys- segment the color image (intensity feature shown) into specified tems existing in literature is to determine the edges through a

Note that, as mentioned earlier, we have taken specific values of the parameters Δ, ω, and β in this section to calculate grayness ambiguity measure. In order to calculate spatial ambiguity measure, apart from Δ and β, we need to assign values of the parameters ω1 and ω2 instead of ω. Assigning appropriate values to the aforesaid parameters is a matter of subjective analysis. As analyzed in [25], in order to get a suitable value of Δ, the global gray value distribution of the underlying image can be considered. The width of base regions corresponding to all the peaks in the distribution may then be found, and as mentioned in [25], certain base regions with widths below a certain Fig. 9. Edge extraction using thresholds determined by algorithms employing threshold may be marked as “unnecessary”. The minimum of the proposed ambiguity measure and fuzzy entropy. (a) The intensity features half of the width of the remaining base regions can be chosen as of the color images considered. (b) Extraction by algorithm using the proposed ambiguity measure. (c) Extraction by algorithm using fuzzy entropy. the value of Δ.Thevalueofω can be based on Weber’s law [26] and the related concept of just noticeable difference. Following Weber’s law, the difference of any gray value from another that decision making method after the edge strength at each pixel makes them just distinguishable can be considered as the value in the image has been obtained. A well-known example of of ω. Choosing appropriate values of ω and ω is analogous such a decision making method is the hysteresis thresholding 1 2 to the problem of choosing a suitable window size, which is [23] which uses two predefined thresholds, where one of them encountered in many image processing tasks. In most of such is usually obtained by multiplying the other with a constant. image processing tasks, the window size is chosen arbitrarily The process of determining thresholds in histograms of edge as 3 × 3or5× 5, and in similar manner, one can consider strength, which are generally unimodal and positively (right) that both ω and ω equal three or five. A difference in the skewed, is considered here in order to compare the use of the 1 2 choice of the base β amounts only to a change in the unit of proposed logarithmic tolerance fuzzy rough–fuzzy entropy with measuring ambiguity. Therefore, any suitable value of β can be a fuzzy-set-theory-based method. We use the Canny operator considered, and the value β must not be changed through an for color images [24] and the nonmaximum suppression [23] experiment. to determine the edge strength at each pixel in a color image and then apply the previously mentioned algorithms, which use the proposed ambiguity measure and fuzzy entropy, in order V. C ONCLUSION to determine a threshold corresponding to each algorithm from the histogram of edge strength. Note that we have not used the Ambiguities in grayscale images are due to fuzzy boundaries threshold determined in the hysteresis process but instead used between regions, and rough resemblance between nearby gray only the single threshold to extract the edges because our prime levels and between values at nearby pixels. The use of rough aim here is to compare the use of the proposed entropy with that set theory and its certain generalizations for quantifying ambi- of fuzzy entropy. guities in images has been proposed in this paper. New classes In Fig. 9, we consider a few color images such that the of entropy measures based on rough set theory and its certain amount of edges present in them varies significantly. It is generalizations have been proposed, and rigorous theoretical visually evident from the figure that the algorithm using the analysis of the proposed entropies has been carried out. The proposed measure satisfactorily extracts the edges in the images proposed entropies have then been used to quantify ambiguities considered, whereas the algorithm using fuzzy entropy fails in images, and it has been shown that some of the proposed miserably in some. This shows that the ambiguities in images of entropies can be used to quantify ambiguities due to both fuzzy edge strength are better represented by the proposed measure. boundaries and rough resemblance. The utility and effective- From the different image processing applications considered ness of the proposed entropy measures have been demonstrated in this section, we see that quantifying ambiguities in images by considering some elementary image processing applications using the logarithmic tolerance fuzzy rough–fuzzy entropy, and comparisons with the use of certain fuzziness measures. in general, results in better performance than the use of cer- A new measure called average image ambiguity has also been tain fuzziness measures like the fuzzy entropy. Note that, as defined in this context. mentioned earlier, the aforesaid proposed entropy used in this The proposed classes of entropy measures based on rough set section is a representative of the proposed entropies which can theory and its certain generalizations are not restricted to the be used to capture ambiguities due to both fuzzy boundaries few applications discussed in this paper. They are, in general, and rough resemblance. Thus, the effectiveness of some of the applicable to all tasks where ambiguity-measure-based tech- proposed entropies, which can be used to capture ambiguities niques have been found suitable, provided that the rough resem- due to both fuzzy boundaries and rough resemblance in images, blance aspect of ambiguities exists. It would be interesting to is demonstrated by the performance improvement observed. We carry out such investigations as the proposed measures possess have also carried out analyses considering the quantification of certain advantages over most fuzzy-set-theory-based uncer- spatial ambiguity, where similar performance improvement has tainty measures, which have been the prime tool for measuring been observed. ambiguities.

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ACKNOWLEDGMENT Debashis Sen (S’05–M’06) received the M.A.Sc. degree in electrical engineering from Concordia Uni- The authors would like to thank the anonymous referees for versity, Montreal, QC, Canada, in 2005 and the their valuable suggestions. S. K. Pal would also like to thank the B.Eng. (Hons.) degree in electronics and communi- cation engineering from the University of Madras, Government of India for the J. C. Bose National Fellowship. Chennai, India, in 2002. He is currently working toward the Ph.D. degree in computer engineering at the Center for Soft Computing Research, Indian REFERENCES Statistical Institute, Calcutta, India. He was with the Center for Signal Processing and [1] G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Communications and with the Video Processing and Applications. New Delhi, India: Prentice-Hall, 2005. Communications Group, Concordia University, in 2003–2005. His research [2] S. K. Pal, “Fuzzy models for image processing and applications,” Proc. interests include image and video processing, probability theory, and soft Indian Nat. Sci. Acad., vol. 65, no. 1, pp. 73–90, 1999. computing. [3] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data. Dordrecht, The Netherlands: Kluwer, 1991. [4] D. Dubois and H. Prade, “Rough fuzzy sets and fuzzy rough sets,” Int. J. Gen. Syst., vol. 17, no. 2/3, pp. 191–209, 1990. [5] A. Skowron and J. Stepaniuk, “Tolerance approximation spaces,” Fundamenta Informaticae, vol. 27, no. 2/3, pp. 245–253, Aug. 1996. [6] T. Beaubouef, F. E. Petry, and G. Arora, “Information-theoretic measures of uncertainty for rough sets and rough relational data bases,” Inf. Sci., vol. 109, no. 1–4, pp. 185–195, Aug. 1998. [7] M. J. Wierman, “Measuring uncertainty in rough set theory,” Int. J. Gen. Syst., vol. 28, no. 4, pp. 283–297, 1999. [8] J. Liang, K. S. Chin, C. Dang, and R. C. M. Yam, “A new method for Sankar K. Pal (M’81–SM’84–F’93) received the measuring uncertainty and fuzziness in rough set theory,” Int. J. Gen. Ph.D. degree in radio physics and electronics from Syst., vol. 31, no. 4, pp. 331–342, 2002. the University of Calcutta, West Bengal, India, in [9] J.-S. Mi, X.-M. Li, H.-Y. Zhao, and T. Feng, “Information-theoretic 1979 and another Ph.D. degree in electrical engineer- measure of uncertainty in generalized fuzzy rough sets,” in Rough Sets, ing along with DIC from Imperial College, Univer- Fuzzy Sets, Data Mining and Granular Computing. Berlin, Germany: sity of London, London, U.K., in 1982. Springer-Verlag, 2007, pp. 63–70. He is the Director and a Distinguished Scientist of [10] R. R. Yager, “Entropy measures under similarity relations,” Int. J. Gen. the Indian Statistical Institute. Currently, he is also Syst., vol. 20, no. 4, pp. 341–358, 1992. a J. C. Bose Fellow of the Government of India. [11] S. K. Pal, B. U. Shankar, and P. Mitra, “Granular computing, rough He founded the Machine Intelligence Unit and the entropy and object extraction,” Pattern Recognit. Lett., vol. 26, no. 16, Center for Soft Computing Research: A National pp. 2509–2517, Dec. 2005. Facility in the Institute in Calcutta. He was with the University of California, [12] D. Sen and S. K. Pal, “Histogram thresholding using beam theory and Berkeley, and the University of Maryland, College Park, in 1986–1987; the ambiguity measures,” Fundamenta Informaticae, vol. 75, no. 1–4, NASA Johnson Space Center, Houston, TX, in 1990–1992 and 1994; and the pp. 483–504, Jan. 2007. U.S. Naval Research Laboratory, Washington, DC, in 2004. [13] N. R. Pal and S. K. Pal, “Entropy: A new definition and its applica- Dr. Pal is a Fellow of the Academy of Sciences for the Developing World tions,” IEEE Trans. Syst., Man, Cybern., vol. 21, no. 5, pp. 1260–1270, (TWAS), Italy, the International Association for Pattern Recognition, U.S., the Sep./Oct. 1991. International Association of Fuzzy Systems, U.S., and all the four National [14] N. R. Pal and J. C. Bezdek, “Measuring fuzzy uncertainty,” IEEE Trans. Academies for Science/Engineering in India. Since 1997, he has been a Fuzzy Syst., vol. 2, no. 2, pp. 107–118, May 1994. Distinguished Visitor of the IEEE Computer Society (U.S.) for the Asia–Pacific [15] B. R. Ebanks, “On measures of fuzziness and their representations,” Region and has been holding several visiting positions in Hong Kong and J. Math. Anal. Appl., vol. 94, no. 1, pp. 24–37, Aug. 1983. Australian universities. He is a coauthor of 14 books and more than 300 [16] H. R. Tizhoosh, G. Krell, and B. Michaelis, “λ enhancement: Contrast research publications in the areas of pattern recognition and machine learning, adaptation based on optimization of image fuzziness,” in Proc. IEEE Int. image processing, data mining and Web intelligence, soft computing, neural Conf. Fuzzy Syst., 1998, vol. 2, pp. 1548–1553. nets, genetic algorithms, fuzzy sets, rough sets, and bioinformatics. He has [17] R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. received the 1990 S.S. Bhatnagar Prize (which is the most coveted award New Delhi, India: Pearson Educ., 2002. for a scientist in India) and many prestigious awards in India and abroad, [18] S. K. Pal, R. A. King, and A. A. Hashim, “Automatic grey level including the 1999 G.D. Birla Award, the 1998 Om Bhasin Award, the 1993 thresholding through index of fuzziness and entropy,” Pattern Recognit. Jawaharlal Nehru Fellowship, the 2000 Khwarizmi International Award from Lett., vol. 1, no. 3, pp. 141–146, 1983. the Islamic Republic of Iran, the 2000–2001 FICCI Award, the 1993 Vikram [19] Y. I. Ohta, T. Kanade, and T. Sakai, “Color information for region Sarabhai Research Award, the 1993 NASA Tech Brief Award (U.S.), the 1994 segmentation,” Comput. Graphics Image Process., vol. 13, no. 3, pp. 222– IEEE TRANSACTIONS ON NEURAL NETWORKS Outstanding Paper Award 241, Jul. 1980. (U.S.), the 1995 NASA Patent Application Award (U.S.), the 1997 IETE-R.L. [20] Y. W. Lim and S. U. Lee, “On the color image segmentation algorithm Wadhwa Gold Medal, the 2001 INSA-S.H. Zaheer Medal, the 2005–2006 based on the thresholding and the fuzzy C-means techniques,” Pattern ISC-P.C. Mahalanobis Birth Centenary Award (Gold Medal) for Lifetime Recognit., vol. 23, no. 9, pp. 935–952, 1990. Achievement, the 2007 J. C. Bose Fellowship of the Government of India, and [21] S. K. Pal, A. Ghosh, and B. U. Shankar, “Segmentation of remotely the 2008 Vigyan Ratna Award from Science and Culture Organization, West sensed images with fuzzy thresholding, and quantitative evaluation,” Int. Bengal. Prof. Pal is/was an Associate Editor for the IEEE TRANSACTIONS J. Remote Sens., vol. 21, no. 11, pp. 2269–2300, Jul. 2000. ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE (2002–2006), the [22] D. Comaniciu and P. Meer, “Mean shift: A robust approach toward feature IEEE TRANSACTIONS ON NEURAL NETWORKS [1994–1998 and 2003–2006], space analysis,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 24, no. 5, Neurocomputing (1995–2005), Pattern Recognition Letters,theInternational pp. 603–619, May 2002. Journal of Pattern Recognition and Artificial Intelligence, Applied Intelligence, [23] J. Canny, “A computational approach to edge detection,” IEEE Information Sciences, Fuzzy Sets and Systems, Fundamenta Informaticae,the Trans. Pattern Anal. Mach. Intell., vol. PAMI-8, no. 6, pp. 679–698, LNCS Transactions on Rough Sets,theInternational Journal of Computational Nov. 1986. Intelligence and Applications, IET Image Processing,theJournal of Intelligent [24] A. Koschan and M. Abidi, “Detection and classification of edges in Information Systems,andtheProceedings of INSA-A. He is also a Book color images,” IEEE Signal Process. Mag., vol. 22, no. 1, pp. 64–73, Series Editor of Frontiers in Artificial Intelligence and Applications (IOS Jan. 2005. Press) and Statistical Science and Interdisciplinary Research (World Scientific, [25] C. A. Murthy and S. K. Pal, “Histogram thresholding by minimizing Singapore); a member of the Executive Advisory Editorial Board of the IEEE graylevel fuzziness,” Inf. Sci., vol. 60, no. 1/2, pp. 107–135, Mar. 1992. TRANSACTIONS ON FUZZY SYSTEMS,theInternational Journal of Image [26] A. K. Jain, Fundamentals of Digital Image Processing. New Delhi, and Graphics,andtheInternational Journal of Approximate Reasoning;and India: Prentice-Hall, 2001. a Guest Editor for the IEEE Computer.

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