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Applied Soft Computing 72 (2018) 121–139

Contents lists available at ScienceDirect

Applied Soft Computing

journa l homepage: www.e lsev ier .com/ locate /asoc

ncertainty representation using fuzzy-entropy approach: Specialpplication in remotely sensed high-resolution satellite imagesRSHRSIs)

ritpal Singh a,∗, Gaurav Dhiman b

Smt. Chandaben Mohanbhai Patel Institute of Computer Applications, CHARUSAT Campus, Changa, Anand 388421, Gujarat, IndiaDepartment of Computer Science & Engineering, Thapar Institute of Engineering and Technology, Patiala 147004, Punjab, India

r t i c l e i n f o

rticle history:eceived 24 August 2017eceived in revised form 16 July 2018ccepted 19 July 2018vailable online 31 July 2018

eywords:uzzy sets

a b s t r a c t

Remotely sensed high-resolution satellite images contain various information in context of changes. Byanalyzing this information very minutely, changes occurred in various atmospheric phenomena can beidentified. Therefore, in this study, a novel change detection method is proposed using the fuzzy settheory. The proposed method represents the uncertain changes in the form of a fuzzy set using thecorresponding degree of membership values. By using the fuzzy set operators, such as max and minfunctions, this study derives very useful information from the images. This study also proposes a newfunction to identify the boundary of uncertain changes. Further, this study is propagated to identify the

hange recognitionntropyrobabilityranularizationonvex setemotely sensed high-resolution satellite

similarity or dissimilarity between different images of the same event that contain various uncertainchanges. To recognize the changes in a fine-grained level, this study introduces a way to represent thefuzzy information in a granular way. The utilization of the proposed method is shown by recognizingchanges and retrieving information from the remotely sensed high-resolution satellite images. Variousexperimental results exhibit the robustness of the study.

© 2018 Elsevier B.V. All rights reserved.

mages (RSHRSIs)

. Introduction

In the universe, changes always take place due to the occurrencef various uncertain events in the atmosphere. The intensity ofhese events may be moderate or severe. For example, earthquake,yclone, flood, etc., always make the changes in the earth’s sur-ace [1]. By simply monitoring, these changes cannot be observed.herefore, researchers use the spatio-temporal images to detectemporal effects in these uncertain events. Hence, change detec-ion is a technique to find out any changes in the surface of theniverse by analyzing remotely-sensed digital images, which areaptured at different time stamps. Identifying the changes are onef the most challenging tasks in the domain of pattern recognitionnd machine learning [2].

Roy et al. [3] stated that during change detection analysis, onlywo group of pixels are formed, which can be classified, as changed

lass and unchanged class. Changes in the images can be iden-ified by analyzing the corresponding pixels in a very granularevel. Therefore, during analysis, changes can be classified into

∗ Corresponding author.E-mail address: drpritpalsingh82@gmail.com (P. Singh).

ttps://doi.org/10.1016/j.asoc.2018.07.038568-4946/© 2018 Elsevier B.V. All rights reserved.

various groups of pixels, such as a low changed class, moderatechanged class, high changed class, and unchanged class. To studychange detection, three different kinds of techniques are used bythe researchers, as: (a) supervised learning [4–6], (b) unsupervisedlearning [7–14], and (c) semi-supervised learning [3,15].

Recently, researchers have focussed on fuzzy based approachesin change detection. For example, Ghosh et al. [13] have used fuzzyC-means (FCM) and Gustafson–Kessel clustering (GKC) algorithmsin change detection in multi-temporal remote sensing images.Gong et al. [16] deals with the identification of changes in syn-thetic aperture radar (SAR) images by introducing a novel imagefusion approach and modified FCM clustering algorithm. Gong et al.[17] proposed a new approach for change detection in SAR imagesby classifying changed and unchanged regions using the FCMalgorithm along with a new Markov random field (MRF) energyfunction. Li et al. [18] proposed a multi-objective fuzzy clusteringmethod for change detection in SAR images. Shi et al. [19] intro-duced a novel framework to resolve change detection problems indifference image (DI) based on fuzzy topology approach.

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22 P. Singh, G. Dhiman / Applied

.1. Problem statements and research contributions

The study shows that the change detection domain is mainlyonstrained in recognizing land cover changes by utilizing multi-emporal remote sensing images. Researchers in this domain haveot yet focused on identifying the changes in atmospheric phe-omena using remotely sensed high-resolution satellite imagesRSHRSIs), which are occurring due to dynamic events. The changesn these atmospheric phenomena (e.g., cloud density, snow cover,recipitation, etc.) can be characterized in terms of “low change”,moderate change”, “severe change”, and so on. If such changes cane detected w.r.t. their shifting from one direction to another, then

t will be easy to take a major initiative to stop the havoc causedy them. It will also be helpful to preserve the flora and fauna of aarticular region, where destruction can be caused by such changesnd their shifting. Therefore, in this study, emphasize has been putn detecting changes and their shifting, which may occur due toarious kinds of uncertain atmospheric phenomena.

The application of conventional methods (i.e., supervised, unsu-ervised and semi-supervised methods) in detecting changes inhe atmospheric phenomena is very restricted. This may be due tohe fact that these methods have not been found to be suitable forepresenting the uncertainty associated with them. Furthermore,rom the review of literature, it is also obvious that there is nony uncertainty based method proposed so far to quantify thosehanges.

A large number of studies demonstrate that the fuzzy set the-ry is an effective mathematical tool to deal with various kindsf uncertainties [20]. This theory is based on the assumption thatach element that belongs to the fuzzy set has a degree of member-hip value having the range [0,1]. But, in case of change detection,hen this theory is applied to represent the uncertainty associatedith the changes, a separate set of degree of membership is always

equired to maintain for each of the changes. This representationlso suffers from the problem that it does not have any informationbout the basis of uncertainty, which is essential to locate changesn the RSHRSIs. If anyone wants to locate the changes in a granular

ay, there is also no way in the fuzzy set to represent those changesn a fine-grained level. Recent applications of the fuzzy set theoryn change detection in land cover area motivate us to propagateur study further in the case of identifying changes in atmospherichenomena through the RSHRSIs. Hence, the main objectives alonghe contributions made in this study are presented, as follows:

1) To identify a way to represent uncertain changes using the fuzzyset theory: For this purpose, this study is persuaded to presenta new framework for the representation of uncertain changesinherited in the RSHRSIs in terms of their corresponding degreeof memberships using the fuzzy set theory. In this represen-tation, uncertain changes and their corresponding degree ofmemberships are incorporated together, which is called as afuzzy information (FI). In the FI, the basis of uncertainty for theuncertain changes is included, so that it can be easy to identifywhich basis causes which kind of changes.

2) To quantify the changes due to shifting of information: To quantifythe changes due to shifting of information, this study introducesa new function, which is termed as a fuzzy information-gain (FIG). In this function (i.e., FIG), the measure of certaininformation (i.e., entropy) is generalized [21] based on thecorresponding degree of membership of uncertain informa-tion. This FIG can quantify the uncertainty associated with theFI. Moreover, using the degree of memberships of uncertain

changes, a new function is derived, which is termed as anaverage fuzzy information-gain (AFIG). This function is used tocompute the average amount of uncertain changes associatedwith the FI. This AFIG function further helps us to determine an

mputing 72 (2018) 121–139

interesting region or space in the RSHRSIs, which is referred,as a fuzzy information convex region (FICR). This FICR is useful todetect the shifting of changes in the RSHRSIs. Various propertiesof this region are studied and presented in this article. Using theFI representation of uncertain changes, this study also providesa similarity parameter to examine distinguishes between twoatmospheric phenomena.

(3) To identify a way to represent the uncertain changes in a granularway: This study further propagates to detect the changes in agranular way. For this purpose, this study suggests the use ofgranular computing (GrC) [22]. This GrC approach is applied inthe representation of uncertain changes in a more granular wayso that changes can be characterized in a fine-grained level.

Hence, in this study, a novel approach for change detectionin the RSHRSIs is introduced, which is based on the techniques,as discussed briefly above. This approach is entitled, as a “Fuzzy-Information Retrieval and Change Detection Algorithm (FIRCDA)”.To evaluate the performance of the proposed algorithm, initialexperiments are conducted on two different land cover area imagesof the Bangong Lake and the greater Washington, D.C. Then, theexperiments are further carried out on three different satelliteimages of atmospheric phenomena, which include weather satel-lite images of India, supper Typhoon-Megi in the Philippines, andthe Mars planet. Various empirical analyses conclude that the pro-posed algorithm is more robust in change detection and shifting inthe atmospheric phenomena in comparison to the existing fuzzyset based approaches.

1.2. Organization of the article

The rest of this article is organized as follows. Section 2 estab-lishes the background required for the study. The proposed methodof information retrieval and change detection is presented in Sec-tion 3. Descriptions of data sets along with experimental set up arediscussed in Section 4. Various empirical results are discussed inSection 5. Section 6 is dedicated for the discussion and conclusions.

2. Formalization for the proposed study

In this section, we introduce the formal definitions of the fuzzyinformation (FI), measurement of FI, various definitions, theoremsand corollaries. Various examples and theorems/corollaries associ-ated with the study are presented in Appendix A.

2.1. Fuzzy information and its representation

In case of RSHRSI, if every change is considered as an individualuncertain information, then it can be represented by its corre-sponding degree of membership using the fuzzy set. For example,if U = {m1, m2, m3, . . ., mn} be a universe of discourse for n num-ber of changes, then various significant changes can be detected bycategorizing them, as “low change”, “moderate change”, “severechange”, and so on. For the representation of such changes, wecan define the fuzzy set M based on the events that belong to theuniverse of discourse U, as:

M = �(m1)/m1 + �(m2)/m2 + · · · + �(mn)/mn (1)

Here, � represents the degree of membership function used inthe fuzzy set theory. Hence, �(mk) gives the degree of membershipvalue between range [0,1] for the uncertain information mk, associ-ated with the fuzzy set M. Note that we will use � as the degree of

membership function throughout the discussion of our proposedtheories and formulas.

Assume that the universe of discourse F is composed of var-ious uncertain events E1, E2, E3, . . ., En, whose respective degree

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f memberships �(E1), �(E2), �(E3), . . ., �(En) are assumed to benown. These two sets can be represented in matrix form, as:

E] = [E1, E2, E3, . . ., En], where⋃n

k=1Ek = F (2)

[�(E)] = [�(E1), �(E2), �(E3), . . ., �(En)],

where �(Ek) ∈ [0, 1](3)

Now, Eqs. (2) and (3) contain all the information about the fuzzi-ess of F. These two equations can now be expressed in the formf fuzzy information (FI), as:

efinition 1. (Fuzzy Information (FI)). A FI is a pair set of elementsEi, �(Ei)} (i = 1, 2, . . ., n), which can be denoted by ℵ. Mathemati-ally, it can be expressed, as:

= {Ei, �(Ei)}/F, ∀ Ei ∈ F (4)

The FI {Ei, �(Ei)}/F, as defined above, often does not give com-lete information about how they are collected or evolved. In the FI,he existence of each event depends on the degree of membership,nd subsequently on the source of information, i.e., the universef discourse. Hence, to distinguish the FI in terms of its source of

nformation, it can be categorized, as: (a) undistinguished FI (UFI)nd (b) distinguished FI (DFI). Both these are defined and explainedext.

efinition 2. (Undistinguished FI (UFI)). A collection of informa-ion from a similar source, is called an undistinguished FI (UFI). It isenoted by ℵu. Mathematically, it can be expressed, as:

ℵu = [{E1, �(E1)}/F, {E2, �(E2)}/F, . . ., {En, �(En)}/F],

∀ Ei ∈ F

(5)

ere, each {Ei, �(Ei)}/F represents the individual FI w.r.t. the uni-erse of discourse F, where Ei ∈ F and �(Ei) ∈ [0, 1].

efinition 3. (Distinguished FI (DFI)). A collection of events fromifferent sources, is called a distinguished FI (DFI). It is denoted byd. Mathematically, it can be expressed, as:

d = [{Ei, �(Ei)}/F, {Hi, �(Hi)}/J], ∀Ei ∈ F, ∀Hi ∈ J (6)

n Eq. (6), {Ei, �(Ei)}/F and {Hi, �(Hi)}/J represent the individual FI.r.t. the universe of discourses F and J, respectively. Here, Ei ∈ F,here �(Ei) ∈ [0, 1]; and Hi ∈ J, where �(Hi) ∈ [0, 1].

.2. A measure of fuzzy information

The measure of a certain event is a set function M, which assign aumber M(x) to each set x in a certain class [23]. Some parametersust be imposed on the class of sets on which M is defined. Hence,

robability can be used as a type of parameter, which can be usedo measure the occurrence of any certain event.

Based on this probability parameter, Shannon [24–26] has sug-ested the following expression for the measurement of expectedmount of information, as:

(P1, P2, . . ., Pn) = −n∑i=1

Pi log2(Pi) (7)

ere, Pi (i = 1, 2, . . ., n) represents the probability of each event. Theunction H is also known as the Shannon’s entropy-function.

Suppose, � is a set of elements Ai (i = 1, 2, . . ., n), which are gen-rated with certain random experiment. Now, subsets of � can be

alled “events”, and assigned a probability. Let Ak is an event, andf the question “Does Ak ∈ � ?” has a definite answer, as yes oro, then Ak is called “certain event”. For any certain event, infor-ation can be measured utilizing Eq. (7). In the majority of the

mputing 72 (2018) 121–139 123

cases, the occurrence of events is not hundred percent sure. Addi-tionally, day by day atmospheric phenomena (e.g., cloud density,snow cover, precipitation, etc.) gradually changes, which cannot bedefined precisely based on probability measure. Therefore, to mea-sure the uncertainty involved in such kind of events, the fuzzy set isconsidered as the appropriate technique, where uncertainty asso-ciated with any kind of event can be measured based on the degreeof membership value [27]. Hence, in this study, a new function isdefined to measure the fuzziness involved in such uncertain events,which is called as a fuzzy information-gain (FIG). It can be defined,as:

Definition 4. (Fuzzy Information-Gain (FIG)). It is the measureof uncertainty regarding which event of the FI {Ei, �(Ei)}/F, wherei = 1, 2, . . ., n, has occurred or will occur in terms of degree of mem-bership. It is denoted as Gℵ. Mathematically, it can be expressed,as:

Gℵ = −n∑i=1

�(Ei) log2�(Ei) (8)

Here, Ei ∈ F, and �(Ei) ∈ [0, 1]. In Eq. (8), the function Gℵ can bereferred as a fuzzy information-gain (FIG).

The basic difference between Eq. (7) and Eq. (8) is that the formertakes probability value as an input, while another takes a degree ofmembership value as an input.

The average amount of uncertainty involved in the FI can alsomeasured using the functions of the fuzzy set theory, as:

Definition 5. (Average Fuzzy Information-Gain (AFIG)). The aver-age amount of fuzziness (expected) in a FI {Ei, �(Ei)}/F, where i = 1,2, . . ., n, can be measured in terms of degree of membership, as:

Iℵ = MAX(�(Ei)) + MIN(�(Ei))2

(9)

Here, the function Iℵ returns the AFIG value associated with any FI.In Eq. (9), each event Ei belongs to the universe of discourse F, i.e.,Ei ∈ F, and �(Ei) ∈ [0, 1].

The values of MAX(�(Ei)) and MIN(�(Ei)), defined in Eq. (9), canbe obtained, as:

MAX(�(Ei)) = − max(�(Ei)) log2[max(�(Ei))] (10)

MIN(�(Ei)) = − min(�(Ei)) log2[min(�(Ei))] (11)

In Eqs. (10) and (11), max and min represent the minimum andmaximum operations of the fuzzy sets, respectively [28]. Here, MAXand MIN can be termed as a minimum information-gain function andmaximum information-gain function, respectively. The existence ofboth these functions in Eq. (9) help to capture the average amountof uncertainty available in any FI.

Based on Example 3 (refer to Appendix A), the AFIG value for theℵd is 0.497. The MIN and MAX values for this FI can be shown withthe help of blue-cut and green-cut lines in Fig. 1, respectively. In thisfigure, red-cut line represents the outcome of the AFIG function.

2.3. Fuzzy information convexity

In this sub-section, we have discussed various definitions andproperties based on the convexity property exhibited by the AFIGfunction. These definitions and properties of the AFIG function varyfrom ones presented in [29–31]. These are discussed as follows:

Definition 6. (Convex Fuzzy Set) [29–31]. Let U be the universe ofdiscourse, R is a subset of U, is said to be convex iff for all x, y ∈ R,

we have ax + (1 − a)y ∈ R for all real a ∈ [0, 1].

Mathematically, this definition implies that x and y are twouncertain points in R, then every uncertain point p = ax + (1 − a)y,0 ≤ a ≤ 1, must also be in the subset R.

124 P. Singh, G. Dhiman / Applied Soft Computing 72 (2018) 121–139

Fig. 1. Outcomes of the MAX, MIN and AFIG functions.

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Fig. 2. FICR for the FI.

In the following, the convexity is discussed based on outcomef the AFIG function, which can be applied on any FI. Hence, thisefinition is termed as a FI Convex (FIC), and can be defined, as:

efinition 7. (FI Convex (FIC)). Since, the universe of discourse is composed of various uncertain events, E1, E2, E3, . . ., En, andhe Iℵ is the representation of AFIG function, where each Ei ∈ F.hen, the outcome of Iℵ is said to be FIC iff satisfying the conditionIN(�(Ei)) ≤ Iℵ ≤ MAX(�(Ei)), for all Iℵ ∈ [0, 1].

Mathematically, this definition implies that outcome of Iℵ is anncertain point in a degree of membership function, and it satisfieshe convexity if its value lies between MIN(�(Ei)) and MAX(�(Ei)),.e., MIN(�(Ei)) ≤ Iℵ ≤ MAX(�(Ei)). The convex property shown byhe AFIG function, is illustrated in Fig. 1.

Based on above discussion, we can now define FI Convex RegionFICR), as:

efinition 8. (FI Convex Region (FICR)). Since, the universe of dis-ourse F is composed of various uncertain events E1, E2, E3, . . ., En.or each Ei, we can define a FICR based on three functions, viz., ˛min,ℵ, and ˛max. The FICR of F is determined by each event Ei ∈ F, hencet can be denoted by Cr(Ei). Mathematically, it can be expressed, as:

r(Ei) =⋃

Ei ∈ F

{Cr(Ei) : {˛min, Iℵ, ˛max} ⊆ F} (12)

Mathematically, this definition implies that FICR is the combina-ion of these three uncertain points, viz., ˛min, Iℵ, and ˛max, wheremin ≤ Iℵ ≤ ˛max.

In Fig. 2, a FICR is shown. Here, outcome of Iℵ is shown, whichs bounded between ˛min and ˛max. Therefore, this outcome is alsoncluded in the representation of FICR.

The FICR for I can be represented, as: {˛ , I , ˛max} ∈ F.

ℵ min ℵThe unbounded property of Iℵ (i.e., AFIG) function leads to devel-

pment of a new definition, which is called as a FI Concave RegionFICOR), which is presented, as follows:

Fig. 3. FI with unbounded region (i.e., FICOR).

Definition 9. (FI Concave Region (FICOR)). Since, the uni-verse of discourse F is composed of various uncertain eventsE1, E2, E3, . . ., En. For each Ei, we can define a FICOR for outcomeof the function Iℵ, which is not bounded by MIN(�(Ei)) (or, ˛min)and MAX(�(Ei)) (or, ˛max). The FICOR of F can be denoted by Vr(Ei).Mathematically, it can be expressed, as:

Vr(Ei) =⋃

Ei ∈ F

{Vr(Ei) : {Iℵ} ⊆ F} (13)

The necessary and sufficient conditions for outcome of the func-tion Iℵ, to be concave, are:

(1) For each Ei ∈ F, iff �(E1) = �(E2) = · · · = �(En), and(2) For each Ei ∈ F, iff ˛min = ˛max.

The FICOR is shown in Fig. 3, where outcome of the function Iℵ,which is �1 (say), is completely unbounded, i.e., Iℵ : [0, 1] →)�1(.This concave region can be expressed as the combination of {Iℵ},i.e., {Iℵ} ⊆ Vr(Ei).

2.4. Measurable fuzzy information convex region

In this sub-section, we discuss the measurable fuzzy informationconvex region (FICR) in terms of basic integral process.

Let H is a real-valued convex function, which is able to measurethe region bounded between interval [˛min, ˛min]. Here, we developmore general integration process by applying Riemann integral [32]on H, by considering that H is continuous. This integration process isdifferent from one presented in [33]. This integration can be appliedto the function, provided that certain “measurability” is possible.

It is shown that collection of information can be measured interms of two functions, viz., FIG and AFIG. Here, the AFIG functionrelies upon two principle measures, as ˛min and ˛max. Here, theAFIG function generates a region, which is bounded between ˛minand ˛max. This region is referred as the FICR (refer to Definition 8),which is a measurable quantity. Hence, this integration process isapplied here to measure this FICR, which can be defined, as:

Definition 10. (Measurable function for FICR). Since, F is com-posed of various uncertain events E1, E2, E3, . . ., En. For each Ei,Cr(Ei) is the convex region, if it satisfies the Definition 8. Therefore,Cr(Ei) is a fuzzy-space, and H is a function that can be measurable.

If outcome of the experiment is �, then we may capture the valuefor H(�). Here, H(�) represents the function of measurable regionor space covered by the Cr(Ei). Mathematically, we wish to com-pute �{� : H(�) ∈ L}, where L = [˛min, ˛max]. For this to be possible,L must belongs to Cr(Ei). Since, ˛min and ˛max are both measurable

can be computed.If Iℵ (i.e., AFIG) be a measurable function for the F, and H(�) be

a function that measures the space covered by each information

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i, where Ei ∈ F; then we can define this fuzzy-space in terms ofntegral of ˛min and ˛max functions w.r.t. �, as:

efinition 11. (Integral of FICR). Integral of FICR boundedetween ˛min and ˛max can be defined from a to b, as:

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ere, a to b can be defined, as:

˜ = max[�(E1), �(E2), . . ., �(En)] (15)

˜ = min[�(E1), �(E2), . . ., �(En)] (16)

ere, max and min represent the minimum and maximum opera-ions of the fuzzy sets [28], respectively; and a, b ∈ [0, 1].

emark 1. The larger is the value of H(�), more is the uncertaintynvolved in the FICR.

.5. Fuzzy information similarity matrix

This similarity matrix is used to study the similarity or theissimilarity between different RSHRSIs that contain various uncer-ain information. This uncertain information is represented as a FI.ased on this FI contained in different RSHRSIs, the similarity orhe dissimilarity between them can be studied. In this study, thisarameter is called as a fuzzy information similarity parameter (FISP),hose representation is given below, as:

efinition 12. (Fuzzy Information Similarity Parameter (FISP)).et two universe of discourses M and N are defined for the setf events {m1, m2, . . ., mm} and {n1, n2, . . ., nn}, respectively. Theegree of memberships of occurrence of each of the events, say mind ni are �(mi) (i = 1, 2, . . ., m) and �(ni) (i = 1, 2, . . ., n), respec-ively. The similarity between each of the events in terms of mi and

˜ j associated with the M and N, respectively, can be given, as:

(mi, nj) = MIN[�(mi), �(nj)]MAX[�(mi), �(nj)]

(17)

Here, the function S is called as the FISP. The values ofIN[�(mi), �(nj)] and MAX[�(mi), �(nj)], defined in Eq. (17), can

e obtained, as:

AX[�(mi), �(nj)] = − max[�(mi), �(nj)] log2[max(�(mi), �(nj))]

(18)

IN[�(mi), �(nj)] = − min[�(mi), �(nj)] log2[min(�(mi), �(nj))]

(19)

In Eqs. (18) and (19), max and min represent the minimum andaximum operations of the fuzzy sets [28], respectively.

In Definition 12, the FISP is introduced for the recognition ofverall similarity or dissimilarity between different RSHRSIs. In thisomputation, rather than only focusing on the identified changesthat are marked with the dark circle, for example as in the casef Figs. 9–13), overall changes in the RSHRSIs have been consid-red. In some cases, detection of minimum or insignificant changess not possible. In such situation, it is not possible to provide aegion for that changes. Therefore, this study provides a bound-ry for that changes using the MAX and MIN functions. Hence, inhe computation of similarity or dissimilarity (refer to Eq. (17)),

nly the degrees of memberships returned by the MAX and MINunctions are utilized.

In this matrix, S(mi, nj) gives the strength of the relationshipetween two uncertain events in terms of corresponding degree

mputing 72 (2018) 121–139 125

of memberships. Hence, the value of S(mi, nj) lies between interval[0,1].

2.6. Granular representation of fuzzy information

The objective of this study is to represent the FI in a gran-ular way. This representation is only possible if the degree ofmembership based on a secondary domain (granular FI) coincideswith the degree of membership based on the primary domain(original FI).

Granular computing (GrC) is an emerging technique for informa-tion processing [22]. GrC is used to extract fine-grained informationfrom the complex information or data. It is applied in variousdomains, such as data discretization [34], data classification [35],partition of universe to solve complex problems [36], analysis ofnon-geometric patterns [37], and decision-making [38]. GrC tech-niques can be regarded as an indispensable part of FI granulation,rough sets and interval computations [39]. GrC is also used in datamining for rule representation, rule mining, and soft computing(especially in fuzzy and rough sets) [40].

In fuzzy sets, information granulation implies discretization onthe basis of information or source of information into a fine-grainedlevel. Such information can be ordered, as primary domain basedinformation, secondary domain based information, and so on. Fol-lowing definitions illustrate the concept of primary and secondarydomains based information.

Definition 13. (Primary Domain Based FI (PDBFI)). Each informa-tion Ei, which is initially depends on the universe of discourse F, iscalled as a primary domain based FI (PDBFI). Here, the F is referred asthe primary domain, and this representation is called as the PDBFI.This representation of FI can be denoted as Gp. Mathematically, itcan be expressed, as:

Gp =⋃m

i=1{Ei, �(Ei)}/F |∀Ei ∈ F (20)

Here, each Ei is satisfying Eqs. (2) and (3). This representation canalso be considered as a zero-order information granularization (0-OIG).

Definition 14. (Secondary Domain Based FI (SDBFI)). Let F ={F1, F2, . . ., Fn}, where each Fi, i = 1, 2, . . ., m, is a subset of F, insuch a way that

⋃mi=1Fi = F. Then, the elements of F, which are

the subsets of F is called a granularization of F, and intervals F1,F2, . . ., Fn are called the granules (or, blocks). Similarly, assume thatan information Ek = {e1, e2, . . ., en}, where each ei, i = 1, 2, . . ., m, isa subset of Ek. Then, the elements of Ek, which are subsets of the Ek,is called a granularization of Ek, and information e1, e2, . . ., en arecalled as a granule-information. Now, if any granule-information ei,can be able to defined on Fi, where Fi ∈ F, and ei ∈ Ek, then each Fiis called a secondary-domain. This representation of information iscalled a secondary domain based FI (SDBFI), and can be denoted asGs. Mathematically, it can be expressed, as:

Gs =⋃m

i=1{ {ei, �(ei)}/Fi{Ek, �(Ek)}/F

}|∀ei ∈ Fi, Ek ∈ F (21)

Here, each Ek is satisfying Eqs. (2) and (3). This representationcan also be considered as a first-order information granularization(1-OIG). By applying discretization techniques [41–43], one canachieve various levels of granularization.

Remark 2. (i) More granularization reduces the fuzziness of theinformation that belongs to the universe of discourse [44], therebyincreasing the FIG and AFIG values of any uncertain event. (ii) More

thinner the limit of an interval, less is the fuzziness available in it.Therefore, for discovering more changes, one can make the limitthinner at a specific level; otherwise, it will convert the fuzzy setinto the crisp set [20].

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26 P. Singh, G. Dhiman / Applied

Following are the challenges and issues in FI granularization, as:

1) The properties of granulation provide a dynamic approach toprocess the FI into small sub-modules to solve the particularproblem. In this situation, finding the optimal number of user-defined granules are a noteworthy issue for the researchers tosolve the particular problem.

2) If there is a static event, the granulation level is already char-acterized. For instance, the University examination systemincludes the well-defined universe of marks for each subject togenerate the particular grade of a student. However, if there isan occurrence of a dynamic event, the given universe differs ina given phase. For example, the universe for an every day tem-perature, universe for closing price of a stock index, and so on.In this situation, choosing a specific granulation is significantlya complex task.

3) Representation of the FI into various level of granulation is com-putationally very complex and expensive. Therefore, to get thefinal decision from such granular FI, various levels of defuzzifi-cation are required.

. The proposed algorithm for information retrieval andhange detection

In this section, the proposed algorithm “Fuzzy-Informationetrieval and Change Detection Algorithm (FIRCDA)” is presented.his algorithm has the following advantages, as:

It is applicable in retrieving various information from the RSHRSIsby employing the functions, such as FIG, MAX, MIN, AFIG, andFISP.This algorithm is useful to locate the regions of uncertain changesin RSHRSIs, which is called as the FICR (as defined in Definition 8).This algorithm is also useful to visualize the shifting of changesw.r.t. time and date.

.1. Fuzzy-Information Retrieval and Change Detection AlgorithmFIRCDA)

Each phase of the FIRCDA is explained as follows.Step 1. Read the RSHRSI, and convert it into gray form. Expla-

ation: Initially, RSHRSI is uploaded into the system. Then, it isonverted into the gray form.

Step 2. Find the color intensities in each pixel of RSHRSI. Expla-ation: For RSHRSI, color intensity values corresponding to eachixel is found. We consider this set of color intensity values, as aector C. This vector C, can be represented, as:

=

⎡⎢⎢⎢⎢⎣

ci,j ci,j+1 ci,j+2 . . . ci,j+n

ci+1,j ci+1,j+1 ci+1,j+2 . . . ci+1,j+n

......

.... . .

...

ci+m,j ci+m,j+1 ci+m,j+2 . . . ci+m,j+n

⎤⎥⎥⎥⎥⎦ (22)

ere, m and n represent the maximum number of rows and columnsn any RSHRSI. Here i, j = 1, 2, 3, . . ., n

Step 3. Define the universe of discourse for the color intensityalues. Explanation: The universe of discourse CU for the matrix, ashown in Eq. (22), can be defined, as CU = [cmin, cmax]. Here, cmin andmax represent the minimum and maximum color intensity valuesrom the matrix, as shown in Eq. (22).

Step 4. Discretize the universe of discourse. Explanation: Weiscretize the universe of discourse CU = [cmin, cmax], into severalqual lengths of intervals, as a1 = [lb1, ub1], a2 = [lb2, ub2], a3 = [lb3,b3], . . ., an = [lbn, ubn]. Here, each lbi and ubi represents the lower and

mputing 72 (2018) 121–139

upper bounds of an interval ai, where i = 1, 2, . . ., n. Here, ubn ≤ cmax,and ai ∈ CU .

Step 5. Define fuzzy set for each of the intervals, and fuzzify eachof the color intensity values. Explanation: For n number of inter-vals, we can define total n number of fuzzy sets, as A1, A2, . . ., An,on the universe of discourse CU . For all these fuzzy sets, asdefined above, we adopt the linguistic values, as A1 = (very low) ∈a1A2 = (not very low) ∈ a2, . . ., An = (very very high) ∈ an. Now,to fuzzify the set of color intensity values, find the interval ai, whereeach color intensity value ch,k (as shown in Eq. (22)), belongs to. Forexample, a color intensity value c1,1 belongs to the interval a1 = [lb1,ub1]. For this interval, we have defined the fuzzy set, as A1. Hence,this color intensity value is fuzzified, as A1. In this way, all the colorintensity values are fuzzified.

Step 6. Define the FI for the color intensity values. Explanation:For the color intensity values, as shown in Eq. (22), where each ch,k(i = 1, 2, . . ., n) initially belongs to the universe of discourse CU , canbe represented based on the definition of FI (refer to Definition (1)),as:

ℵu = {ch,k, �(ch,k)}/CU | ch,k ∈ CU (23)

Here, each {ch,k, �(ch,k)}/CU represents the individual FI w.r.t.CU .Step 7. Calculate the degree of membership value for each of

the color intensity values. Explanation: For each color intensityvalue ch,k, its corresponding degree of membership value w.r.t. theuniverse of discourse CU = [cmin, cmax], can be computed, as:

�(ch,k) = (ch,k − cmin)/(cmax − cmin) (24)

Here, each �(ch,k) represents the degree of membership value forthe color intensity value ch,k.

The nature of membership function depends on the context ofthe application [45]. In the proposed method, degree of member-ship of color intensity value ch,k is determined using Eq. (24), whosecomputation depends on the limit of the universe of discourseCU = [cmin, cmax]. However, one can use the other fuzzy member-ship functions, for example, triangular function, Gaussian function,and so on [46].

Step 8. Compute the FIG value for each of the color intensityvalues. Explanation: For each color intensity value ch,k, its corre-sponding FIG value can be computed based on Eq. (8), as:

Gℵ = −�(ch,k) log2�(ch,k) (25)

Here, the function Gℵ is called the “FIG”.Step 9. Calculate the MAX and MIN values from the set of color

intensity values. Explanation: Based on Eqs. (10) and (11), we cancompute the MAX and MIN values from the set of color intensityvalues C, as:

MAX(�(C)) = − max(�(C)) log2[max(�(C))] (26)

MIN(�(C)) = − min(�(C)) log2[min(�(C))] (27)

In Eqs. (26) and (27), the max and min represent the minimum andmaximum operations of the fuzzy sets, respectively [28].

Step 10. Obtain the AFIG value for the set of color intensity val-ues. Explanation: Based on Eqs. (26) and (27), the AFIG value canbe obtained, as:

MAX(�(C)) + MIN(�(C))

Iℵ =

2(28)

Here, the function Iℵ returns the AFIG value associated with the setof color intensity values C.

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P. Singh, G. Dhiman / Applied

Step 11. Compute the similarity among RSHRSIs. Explanation:sing FISP (as defined in Definition 12), similarity among RSHRSIsan be computed, as:

(Ii, Ij) = MIN[�(ci), �(cj)]MAX[�(ci), �(cj)]

(29)

Here, Ii and Ij represent two different RSHRSIs, where indices ind j represent their corresponding capturing information (such asime, date, year, etc.), respectively. In Eq. (29), MAX and MIN valuesor the Ii and Ij can be obtained from Eqs. (26) and (27).

Step 12. Find the columns’ index values (jth) from the vector Cas shown in Eq. (22)), whose ci,j value corresponds to MIN value.ocate all these MIN values in RSHRSIs, which occurs maximumumber of times among all these values. Store all these values in

vector, which is represented by a vector Rcol , and can be defined,s:

col = MAXoccur(j1, j2, j3, . . ., jh) (30)

ere, h defines the maximum number of index values, whose ci,jalue corresponds to MIN value.

Step 13. Find the rows’ index values (ith) from the vector C (ashown in Eq. (22)), whose ci,j value corresponds to MIN value, andts columns’ index values (jth) are same as previously obtained jthndex value from the vector Rcol (refer to Step 1). Calculate the aver-ge absolute value of these rows’ index values (ith), and store in aector, which is represented by Rrow . This vector can be defined, as:

row =| AVG(i1, i2, i3, . . ., iq) | (31)

ere, q defines the maximum number of index values, whose ci,jalue corresponds to MIN value.

Step 14. Locate Rrow (ith) index and Rcol (jth) index of the vector , which are the co-ordinate points of the MIN regions.

Step 15. Repeat Steps 12–14 for the co-ordinate points of theAX and AFIG regions.

Step 16. Make a region by connecting all these co-ordinateoints of MIN, MAX and AFIG regions. This region is called as theICR, as defined in Definition (8).

.2. Computational complexity

In this subsection, the computational complexity of the pro-osed algorithm is discussed. Both the time and space complexitiesf the proposed algorithm are computed and given below.

Time complexity for the proposed FIRCDA:(1) It requires O(N) time to read the image and convert it into

gray form.(2) To find the color intensities in each pixel, the algorithm uses

O(N) time.(3) It requires O(M × N) time to define the fuzzy set for each of

the intervals, where M defines the number of fuzzy sets andN defines to discretize the universe of discourse.

(4) The algorithm requires O(N) time to compute the degree ofmembership and FIG value for each of the color intensityvalue.

(5) The computational cost to calculate the MAX, MIN, AFIG valuesfrom the set of color intensity values, and similarity amongRSHRSIs (using Eq. (29)), is O(N) time.

(6) It requires O(M × N) time to compute the MIN, MAX, and AFIGregions, where M and N defines the row and column pixelvalues of a given RSHRSI, respectively.

(7) The algorithm uses O(N) time to make a region by connectingall the co-ordinates points.

Therefore, the total computational complexity of all abovesteps for maximum number of iterations is O(M × N).

mputing 72 (2018) 121–139 127

• Space complexity of the proposed FIRCDA:The space complexity of the proposed algorithm is the maxi-

mum amount of space, which is considered at any one time duringits initialization process. Thus, the total space complexity of theproposed FIRCDA is O(M × N).

4. Descriptions of data sets

To demonstrate the effectiveness of the proposed algorithm (i.e.,FIRCDA), the experiment is carried out on two different kinds ofdata sets, as land cover area data set and atmospheric phenomenadata set. A detailed description of data sets is provided next.

4.1. Data set corresponding to the land cover area

In this category of the data set, two different categories of LAND-SAT satellite images have been selected, as:

(a) Images of Bangong Lake, Himalayas: Two LANDSAT satelliteimages of Bangong Lake in the Tibet Autonomous Region ofChina are acquired on 24/08/1998 and 02/09/2013. This dataset is acquired from USGS (Source: https://remotesensing.usgs.gov/gallery/index.php). During this time interval, changes areobserved in this lake, which have expanded the area of lakealong the marshy southwestern and northern shorelines. Theseshoreline changes adversely affect the levels of local salinity,whose affect can be observed in the vegetation as well as in theliving organisms. Fig. 4(a) and (b) show the images of BangongLake for 24/08/1998 and 02/09/2013, respectively. The refer-ence map is generated using these images, which is depicted inFig. 4 (c). This reference map consists of 56,335 changed pixelsand 4,62,065 unchanged pixels. This reference map is furtherused for detecting the changes in the Bangong lake.

(b) Images of the greater Washington, D.C.: Two LANDSAT satel-lite images of the greater Washington, D.C. area are acquiredon 23/12/1972 and 08/05/2012. This data set is acquired fromUSGS (Source: https://remotesensing.usgs.gov/gallery/index.php). These two images are shown in Fig. 5(a) and (b). Inthese images, highly red colors indicate forests and large grassyregion, while light colors indicate fields and the highly urbandevelopment areas. A comparison between these two imageshas been clearly demonstrating the significant growth in thegreater Washington, D.C. from the last 40 years. For furtheranalysis, the reference map is generated using these images,which is depicted in Fig. 5 (c). This reference map consists of1,25,623 changed pixels and 3,92,777 unchanged pixels.

4.2. Data set corresponding to the atmospheric phenomena

In this category of data set, three different kinds of weathersatellite images have been selected, as:

a) Images of India: In this category of data set, four weather satel-lite images of India are included. These images are collectedfrom the site of National Satellite Meteorological Centre, Min-istry of Earth Sciences, Govt. of India [47], and depicted inFig. 6(a)–(d). The size of these four images are 720 × 720 pix-els, which were captured on date 01/08/2016 at four differenttime intervals, as 05:30 am, 06:00 am, 06:30 am, and 07:00 am.In these images, changes are observed continuously in termsof cloud density. In Fig. 6(e), a reference map is given, which

showing the changes between 05:30 am and 07:00 am in theatmospheric phenomena. This reference map is obtained fromFig. 6(a)–(d). This reference map consists of 47,680 changed and4,70,720 unchanged pixels.

128 P. Singh, G. Dhiman / Applied Soft Computing 72 (2018) 121–139

Fig. 4. Images of the Bangong Lake, Himalayas: (a) Image is acquired on 24/08/1998, (b) Image is acquired on 02/09/2013, and (c) the reference map for showing the changesin between 24/08/1998 and 02/09/2013.

Fig. 5. Images of the greater Washington, D.C.:(a) Image is acquired on 23/12/1972, (b) Image is acquired on 08/05/2012, and (c) the reference map for showing the changesin between 23/12/1972 and 08/05/2012.

F stam0

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ig. 6. Images of India: (a)–(d) captured on date 01/08/2016 at four different time7:00 am.

b) Images of supper Typhoon-Megi, Philippines: This data set consistsof three images of typhoons with the size of 720 × 720 pixels,which were observed in the Philippines on dates 20/10/2010,21/10/2010 and 22/10/2010. These images are available onthe website of the Earth Observatory Center, NASA [48].These images are depicted in Fig. 7(a)–(c). A typhoon is amature tropical cyclone, which mostly occurs in the north-

western Pacific Ocean. In this part of the world, mostly Japan,the Philippines and Hong Kong is drastically suffered by thetyphoon. In Fig. 7(d), a reference map is given, which show-

ps, and (e) the reference map for showing the changes in between 05:30 am and

ing the changes between 20/10/2010 and 22/10/2010 in thesupper Typhoon-Megi. This reference map is obtained fromFig. 7(a)–(c). This reference map consists of 38,966 changed and4,79,434 unchanged pixels.

(c) Images of the Mars planet: Finally, two weather satelliteimages of the Mars planet captured on dates 09/07/1997 and10/07/1997 with the size of 720 × 720 pixels are used to ana-

lyze the behaviors and variations of dust on the Mars planet.These images are available on the website of the Hubble Euro-pean Space Agency, Germany [49]. These images are depicted

P. Singh, G. Dhiman / Applied Soft Computing 72 (2018) 121–139 129

Fig. 7. Images of supper Typhoon-Megi: (a)–(c) captured on dates 20/10/2010, 21/10/2010 and 22/10/2010, respectively, and (d) the reference map for showing the changesin between 20/10/2010 and 22/10/2010.

F 0/07/10

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ig. 8. Images of the Mars planet: (a) and (b) captured on dates 09/07/1997 and 19/07/1997 and 10/07/1997.

in Fig. 8(a) and (b). In Fig. 8(c), a reference map is given, whichshows the changes between 09/07/1997 and 10/07/1997 in theMars planet. This reference map is obtained from Fig. 8(a) and(b). This reference map consists of 36,623 changed and 4,81,777unchanged pixels.

.3. Evaluation indexes

The visualization approach is used to show the change detectionn images. Apart from this, there are some other measurementso calculate the values of indexes for evaluation. In this work,ercentage of correct classification (PCC) and Kappa coefficientre considered as evaluation indexes for experimentation [18].he mathematical formulation of PCC and Kappa coefficients areescribed as follows:

TP + TN

CC =N

(32)

appa = PCC − PV1 − PV

(33)

997, respectively, and (c) the reference map for showing the changes in between

where,

PV = (TP + FP) × RC + (TN + FN) × RUN2

(34)

In Eq. (34), FP represents the number of pixels unchanged originallybut falsely detected as changed, FN represents the number of pix-els is changed but falsely detected as unchanged, TP represents thenumber of changed pixels, TN represents the number of unchangedpixels, RC represents the number of actual changed pixels, RU rep-resents the number of actual unchanged pixels, and N is the totalnumber of pixels.

4.4. Experimental setup

The performance of the proposed algorithm is compared withvarious well-known approaches to demonstrate its applicability.The maximum number of iterations for the proposed algorithmis set according to the pixel values of an image. In this paper, we

have utilized the images with size of 720 × 720 pixels. Therefore,the maximum number of iterations for the row-wise computationis 720, and for the column-wise computation is 720. The wholesimulation process is carried out in Matlab R2017a environment

130 P. Singh, G. Dhiman / Applied Soft Computing 72 (2018) 121–139

F propoa CDA, (t

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ig. 9. Changes showing in the Bangong Lake, Himalayas based on: (a) and (b) the

nd 02/09/2013 in the Bangong Lake, Himalayas obtained by: (c) the proposed FIRhe MOFCM approach.

perating on Microsoft Windows 8.1 with 64 bits on Core i-5 pro-essor, and 2.40 GHz processor with 4 GB main memory.

. Empirical results

In this section, the effectiveness of the proposed algorithm (i.e.,IRCDA) is evaluated by conducting experiments on two differ-nt kinds of data sets, as discussed in Section 4. As our proposedlgorithm is based on the fuzzy set theory, therefore to assess theerformance of the proposed algorithm, only this theory basedethods have been adopted, which are FCM S1 [50], MRFFCM [17],

CM [51], and MOFCM [18].

.1. Change detection analysis

1) In terms of land cover area data set: In this category of data set, wehave analyzed the changes from the images of the Bangong lakeand greater Washington, D.C. Initially, various information hasbeen retrieved from these images using the proposed functions,viz., FIG, MAX, MIN, and AFIG. In Fig. 9(a) and (b), the regions ofMAX, MIN, and AFIG are identified in the images of BangongLake, using the green, red and blue color spots, respectively.These spots show the significant changes in the images between1998 and 2013. Then, the FICR is identified and located in theseimages by using the dark circle (represented by cyan color). TheFICRs are shown in Fig. 9(a) and (b). By comparing these twoimages in terms of the FICR, the shifting of changes is obviouslyobserved in the Bangong Lake between 1998 and 2013. Fromthe change detection map, as shown in Fig. 9(c), it is evidentthat the lake area has been expanded, and these changes areclearly shifting along the southwest and northern shorelines.

Similarly, for the images of greater Washington, D.C., the MAX,MIN, and AFIG regions are shown in Fig. 10(a) and (b) withthe green, red and blue color spots, respectively. These spotsclearly reflect the significant changes in the images between23/12/1972 and 08/05/2012. The FICRs, as located in these

images (represented by dark cyan color in Fig. 10(a) and (b))clearly depict the shifting of changes in the greater Washington,D.C. between 23/12/1972 and 08/05/2012. The change detec-tion map of the greater Washington, D.C. (refer to Fig. 10(c))

sed FIRCDA. The change detection map showing the changes between 24/08/1998d) the FCM S1 approach, (e) the MRFFCM approach, (f) the FCM approach, and (g)

signifies the significant urban growth in the greater Washing-ton, D.C. area.

For the Bangong lake and greater Washington, D.C. images,change detection maps are derived by the existing compet-ing approaches [17,18,50,51], and shown in Fig. 9(d)–(g) andFig. 10(d)–(g), respectively. By comparing these change detec-tion maps with the change detection maps obtained by theproposed algorithm (refer to Fig. 9(c) and Fig. 10(c)), it hasbeen noticed that some of the changes are wrongly identifiedby the existing approaches [17,18,50,51] in comparison to theproposed algorithm.

(2) In terms of atmospheric phenomena data set: In this categoryof data set, we have analyzed the changes from the imagesof India, supper Typhoon-Megi, and the Mars planet. For allthese images, various information has been retrieved in termsof the FIG, MAX, MIN, and AFIG functions. In Fig. 11(a)–(d),Fig. 12(a)–(c), and Fig. 13(a) and (b), the identified regions ofthe MAX, MIN, and AFIG are shown, using the green, red andblue color spots, for the images of India, supper Typhoon-Megi,and the Mars planet, respectively. In case of India image (referto Fig. 11(a)–(d)), changes in cloud density are clearly observedbetween 05:30 am and 07:00 am on date 01/08/2016. Simi-larly, these spots reflect the changes in the images of supperTyphoon-Megi (refer to Fig. 12(a)–(c)) in terms of the air massbetween 20/10/2010 and 22/10/2010, which rotates aroundthe center of low atmospheric pressure. Finally, changes aremarked in the images of the Mars planet (Fig. 13(a) and (b)),which demonstrates the variations of dust storms on the Marsplanet between 09/07/1997 and 10/07/1997. In all these atmo-spheric phenomena images, the FICRs are located by usingthe dark circle, which is represented by cyan color (refer toFig. 11(a)–(d), Fig. 12(a)–(c), and Fig. 13(a) and (b)). By com-paring the FICRs of these images, shifting of changes w.r.t. dateand time have been clearly observed.

The change detection maps for all these images are obtainedby the proposed algorithm (refer to Fig. 11(e), Fig. 12(d),and Fig. 13(c)), which provides the information about thechanged and unchanged pixels w.r.t. date and time. These

change detection maps are compared with the change detec-tion maps acquired by the existing approaches [17,18,50,51],and it has been found that the proposed algorithm is an edge

P. Singh, G. Dhiman / Applied Soft Computing 72 (2018) 121–139 131

Fig. 10. Changes showing in the greater Washington, D.C. based on: (a) and (b) the proposed FIRCDA. The change detection map showing the changes between 23/12/1972and 08/05/2012 in the greater Washington, D.C. obtained by: (c) the proposed FIRCDA, (d) the FCM S1 approach, (e) the MRFFCM approach, (f) the FCM approach, and (g) theMOFCM approach.

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ig. 11. Changes showing in the weather satellite images of India based on: (a)–(d)

nd 07:00 am in the weather satellite images of India obtained by: (e) the proposedi) the MOFCM approach.

over in distinguishing the changes in comparison to the existingapproaches [17,18,50,51].

.2. Information retrieval through the change detection maps

Various information retrieved from the experimental data setssing the proposed functions (i.e., FIG, MAX, MIN, AFIG, FICR, and(�)) are listed in Table 1. In this table, results are also compared

oposed FIRCDA. The change detection map showing the changes between 05:30 amA, (f) the FCM S1 approach, (g) the MRFFCM approach, (h) the FCM approach, and

with the existing approaches [17,18,50,51]. In Table 1, the functionH(�) represents the amount of space covered by the FICR region.From the comparison, it is obvious that the large value of H(�) forthe existing approaches [17,18,50,51] indicate that they consider a

large number of pixels for detecting the changes. However, the pro-posed FIRCDA can take granular level of decisions by consideringactual pixels that represent the changes.

132 P. Singh, G. Dhiman / Applied Soft Computing 72 (2018) 121–139

Fig. 12. Changes showing in the weather satellite images of supper Typhoon-Megi based on: (a)–(c) the proposed FIRCDA. The change detection map showing the changesbetween 20/10/2010 and 22/10/2010 in the weather satellite images of supper Typhoon-Megi obtained by: (d) the proposed FIRCDA, (e) the FCM S1 approach, (f) the MRFFCMapproach, (g) the FCM approach, and (h) the MOFCM approach.

F a) and0 by: (cF

tm

ig. 13. Changes showing in the weather satellite images of Mars planet based on: (9/07/1997 and 10/07/1997 in the weather satellite images of Mars planet obtainedCM approach, and (g) the MOFCM approach.

The similarity or dissimilarity in the experimental data sets inerms of changes are also studied, and presented in the form of

atrices, as shown in Eqs. (35)–(39), respectively.

(35)

(36)

(b) the proposed FIRCDA. The change detection map showing the changes between) the proposed FIRCDA, (d) the FCM S1 approach, (e) the MRFFCM approach, (f) the

(37)

(38)

P. Singh, G. Dhiman / Applied Soft Computing 72 (2018) 121–139 133

Table 1Information retrieved through the change detection maps for the experimental data sets.

Data set Information retrieved based on FIG MAX MIN AFIG FICR H(�)

Bangong Lake Image FIRCDA 63.5332 0.4663 0 0.2769 {0, 0.2769, 0.4663} 0.4633FCM S1 [50] 65.9987 0.4869 0 0.2819 {0, 0.2819, 0.4869} 0.4869

MRFFCM [17] 65.4312 0.4850 0 0.2803 {0, 0.2803, 0.4850} 0.4850FCM [51] 63.9671 0.4692 0 0.2786 {0, 0.2786, 0.4692} 0.4692

MOFCM [18] 70.0047 0.5128 0 0.3027 {0, 0.3027, 0.5128} 0.5128

Washington, D.C. Image FIRCDA 33.2684 0.2669 0 0.1527 {0, 0.1527, 0.2669} 0.2669FCM S1 [50] 36.8691 0.2768 0 0.1853 {0, 0.1853, 0.2768} 0.2768

MRFFCM [17] 34.5538 0.2684 0 0.1684 {0, 0.1684, 0.2684} 0.2684FCM [51] 38.9580 0.3028 0 0.2536 {0, 0.2536, 0.3028} 0.3028

MOFCM [18] 36.4427 0.2752 0 0.1847 {0, 0.1847, 0.2752} 0.2752

India Image FIRCDA 71.4981 0.5299 0 0.2649 {0, 0.2649, 0.5299} 0.5299FCM S1 [50] 72.5241 0.5365 0 0.2756 {0, 0.2756, 0.5365} 0.5365

MRFFCM [17] 71.9585 0.5311 0 0.2698 {0, 0.2698, 0.5311} 0.5311FCM [51] 73.4521 0.5323 0 0.2721 {0, 0.2721, 0.5323} 0.5323

MOFCM [18] 73.5441 0.5310 0 0.2673 {0, 0.2673, 0.5310} 0.5310

Typhoon Image FIRCDA 31.0035 0.5293 0 0.2646 {0, 0.2646, 0.5293} 0.5293FCM S1 [50] 32.1125 0.5316 0 0.2679 {0, 0.2679, 0.5316} 0.5316

MRFFCM [17] 31.0190 0.5298 0 0.2683 {0, 0.2683, 0.5298} 0.5298FCM [51] 36.9965 0.5423 0 0.2967 {0, 0.2967, 0.5423} 0.5423

MOFCM [18] 34.0635 0.5386 0 0.2702 {0, 0.2702, 0.5386} 0.5386

Mars Image FIRCDA 70.5241 0.5340 0 0.2683 {0, 0.2683, 0.5340} 0.5340FCM S1 [50] 75.0063 0.5569 0 0.2756 {0, 0.2756, 0.5569} 0.5569

MRFFCM [17] 71.9967 0.5392 0 0.2697 {0, 0.2697, 0.5392} 0.5392FCM [51] 70.8693 0.5366 0 0.2703 {0, 0.2703, 0.5366} 0.5366

MOFCM [18] 73.0039 0.5409 0 0.2729 {0, 0.2729, 0.5409} 0.5409

Table 2The statistical results obtained by five approaches.

Data sets Indexes FIRCDA FCM S1 [50] MRFFCM [17] FCM [51] MOFCM [18]

Bangong Lake Image PCC 0.4253 0.5631 0.8562 0.6775 0.5967Kappa 0.5226 0.5777 0.6348 0.7564 0.6628

Washington, D.C. Image PCC 0.5069 0.8500 0.7569 0.6152 0.5576Kappa 0.6951 0.7002 0.7569 0.7013 0.8864

India Image PCC 0.3554 0.4527 0.9857 0.6671 0.3957Kappa 0.4397 0.5791 0.6841 0.6097 0.5980

Typhoon Image PCC 0.2239 0.3679 0.5473 0.6987 0.4328Kappa 0.3978 0.3986 0.5798 0.4570 0.5699

Mars Image PCC 0.5702 0.6397 0.7891 0.5990 0.9873Kappa 0.2869 0.3003 0.6798 0.4573 0.5090

0.5740.511

Iir

5

Pbewieof

Average PCC 0.4163

Kappa 0.4684

(39)

n Eqs. (35)–(39), Ii and Ij represent two different images, wherendices i and j represent their corresponding capturing dates,espectively.

.3. Statistical analysis and comparison study

For showing the efficiency of the proposed FIRCDA in terms ofCC and Kappa coefficient statistics, this study has adopted fourenchmark approaches [17,18,50,51], for the comparison study. Forach of the satellite images, the box plots are depicted in Fig. 14,hich show the statistical results of the PCC and Kappa over 30

ndependent runs. In each box plot, red line represents the median,dges are upper and lower quartiles, and + symbol denotes theutliners. The results reveal that the proposed approach outper-orms the competitive approaches [17,18,50,51]. Table 2 shows

7 0.7870 0.6515 0.59402 0.6671 0.5963 0.6452

the calculated average values of PCC and Kappa over 30 indepen-dent simulation runs. It has been concluded that the proposedalgorithm is statistically significant and perform better than com-petitive approaches in terms of PCC and Kappa.

The quantitative analysis of the proposed FIRCDA along with thecompeting approaches [17,18,50,51], is also carried out in termsof overall error (OE), the number of false alarms (FA) and missedalarms (MA). The false alarm occurs when the unchanged pixels areidentified as changed pixels, whereas missed alarm occurs whenthe changed pixels are identified as unchanged pixels. We havecompared the change detection maps obtained by the proposedFIRCDA with their corresponding reference maps. Table 3 showsthe change detection errors obtained by the proposed and compet-ing approaches [17,18,50,51] for the experimental data sets. FromTable 3, we can see that the MA, FA and OE are minimum for theFIRCDA in comparison to competing approaches. From this compar-ison, it is obvious that the wrongly detected unchanged pixels (i.e.,

false alarms) are minimum in the change detection maps obtainedusing the proposed FIRCDA. Hence, it can be concluded that thechange detection maps acquired by the proposed FIRCDA is veryclosely co-related with their corresponding reference maps.

134 P. Singh, G. Dhiman / Applied Soft Computing 72 (2018) 121–139

Fig. 14. Box plot analysis of the proposed and competing approaches.

P. Singh, G. Dhiman / Applied Soft Co

Table 3The obtained missed alarms, false alarms, and overall error for the experimentaldata sets.

Image Methods MA FA OE

FIRCDA 886 959 1779FCM S1 [50] 989 1607 2635

Bangong Lake Image MRFFCM [17] 1109 1513 2697FCM [51] 1002 1825 2852

MOFCM [18] 1236 1496 2676

FIRCDA 968 1063 1965FCM S1 [50] 1128 2041 2754

Washington, D.C. Image MRFFCM [17] 1090 1652 2885FCM [51] 1125 1967 2900

MOFCM [18] 1274 1528 2800

FIRCDA 836 928 1764FCM S1 [50] 985 1563 2548

India Image MRFFCM [17] 1072 1468 2540FCM [51] 961 1756 2717

MOFCM [18] 1168 1473 2641

FIRCDA 685 963 1648FCM S1 [50] 1258 1657 2915

Typhoon Image MRFFCM [17] 1008 1248 2256FCM [51] 981 1589 2570

MOFCM [18] 759 1176 1935

FIRCDA 985 1135 2120FCM S1 [50] 1374 1780 3154

Mars Image MRFFCM [17] 1058 1369 2427

6

rcmoufciimtoiiwuAcd

cnteFdss

liwea

FCM [51] 1483 1900 3383MOFCM [18] 1139 1878 3017

. Discussion and conclusions

This study shows that various uncertain changes can easily beepresented by the fuzzy set theory, and the fuzziness of thosehanges can be measured using their corresponding degree ofembership values. Further, this study demonstrates that based

n the degree of membership values, information inherited in anncertain event can be measured using the FIG function. Another

unction, i.e., AFIG, is able to quantify the average amount ofhanges inherited in RSHRSIs. Based on the FICR, this study hasdentified the shifting of changes w.r.t. time. In this study, a methods discussed to compute the area of FICR. This study shows that if

ore is the area of FICR, then more is the changes reflected byhe event. If uncertain changes are well defined, then it is easy tobtain information-gain value. Occasionally, uncertainties inherit

nside the uncertainties. To represent such kind of uncertaintiesn terms of changes, granular computing (GrC) approach is used,

here the basis of information is discretized to obtain the gran-lar level of information. Experimental results show that the FIG,FIG, FICR, area of FICR and granular representation of uncertainhanges have the abilities for the information retrieval and changeetection from the RSHRSIs.

The proposed algorithm can be applied to detect uncertainhanges and their shifting in an incremental learning way [52]. Ifew changes occur or they shift from one direction to another w.r.t.ime, then the proposed algorithm can represent such changes byxisting FI or by defining another FI. From such representation ofI, the proposed algorithm effectively has the capacity to retrieveiverse quantifiable information. Experimental results also demon-trate that the proposed algorithm is far superior to existing fuzzyet theory based approaches.

The proposed algorithm is applied on very high-resolution satel-ite images, where quality of images is very good in terms of colorntensity. The proposed algorithm is based on the fuzzy set theory,

hich can easily quantify the low quality pixels to high quality pix-ls by providing the class labels. Hence, it is easy for the proposedlgorithm to detect the changes and their shifting by representing

mputing 72 (2018) 121–139 135

those pixels in terms of fuzzy set. In spite of all these advantages, theproposed algorithm has certain shortcomings, which can be listed,as:

• In this study, computation of the degree of membership dependson the universe of discourse, which is defined by the users. Fur-ther, this universe is discretized into various user-defined values.

• For detecting changes in a more granular level, the basis of infor-mation (i.e., the universe of discourse) is required to discretizeinto a various number of intervals.

All these shortcomings can be rectified by integrating a robustmeta-heuristic optimization algorithm with the proposed algo-rithm. This meta-heuristic optimization algorithm can help toselect the optimal universe of discourse, as well as it can helpto select optimal interval lengths. However, these issues will beresolved in a future enhancement of this method. The efficiency ofthe proposed algorithm is also required to be explored, where it isdifficult to provide the class labels for the changes. Moreover, theproposed algorithm is also required to verify in images of differ-ent kinds of atmospheric phenomena to detect changes and theirshifting.

Acknowledgements

This research is supported by the Department of Scienceand Technology (DST)-SERB, Government of India, under GrantEEQ/2016/000021.

Appendix A. List of Examples and Theorems/Corollaries

Example 1. An example is illustrated here for the representa-tion of undistinguished FI (UFI). Let us consider that each of theevents E1, E2, . . ., En, are depended on a similar source, i.e., undis-tinguished universe of discourse F. Now, each of the events of F bethe members of FI, which can be represented, as ℵu = {Ei, �(Ei)}/F,where i = 1, 2, . . ., n. For example, disease “typhoid” can be regardedas the universe of discourse, whose different symptoms “fever”,“headaches”, and “diarrhea” can be characterized as a distinctinformation/event. Here, the source of information “typhoid”, isundistinguished. Hence, this information can be expressed in termsof UFI, as follows:

ℵu = [{fever, �(fever)}/typhoid,{headaches, �(headaches)}/typhoid,

{diarrhea, �(diarrhea)/typhoid}]

Example 2. An example is illustrated here for the representa-tion of distinguished FI (DFI). Let us consider that two differentevents Ei and Hi are gathered from two distinguished universe ofdiscourses, viz., F and J, respectively. These events be the mem-bers of FI, which can be represented in terms of F and J, asℵd = [{Ei, �(Ei)}/F, {Hi, �(Hi)}/J]. For example, obesity problem ofa patient might be because of imbalanced thyroid hormone, andfatigue problem for a similar patient might be because of diabetes.Here, “obesity” and “fatigue” can be characterized as a distinctinformation/event, and “thyroid” and “diabetes” can be called asthe distinguished universe of discourses. Hence, this informationcan be expressed in terms of DFI, as follows:

ℵd = [{obesity, �(obesity)}/thyroid, {fatigue, �(fatigue)}/diabetes]

Theorem 1. As the degree of membership of the correspondinginformation increases, the measure of information, i.e., FIG valuemonotonically increases, then monotonically decreases.

136 P. Singh, G. Dhiman / Applied Soft Computing 72 (2018) 121–139

d

ℵ 0

0)

Iooig(�

(

(

((

(

EX

ℵF

Nt

2

Fig. 15. A FIGC for the undistinguished fuzzy information (UFI).

Discussion. Assume an UFI (refer to Definition 2), which isefined on the universe of discourse U, as:

u = [{x1, 0.1}/U, {x2, 0.2}/U, . . ., {x10, 0}/U]∀xi ∈ U, i = 1, 2, . . ., 1

(4

n Eq. (40), different FIG values are obtained by arranging the degreef membership for each of the events x1, x2, . . ., x10 in ascendingrder. Now, a curve for the individual FIG values is fitted, which

s depicted in Fig. 15. This curve is entitled as a fuzzy information-ain curve (FIGC). This curve demonstrates how the FIG functioni.e., Gℵ) behaves when the input is a degree of membership, i.e.,(xi), i = 1, 2, . . ., 10. This curve indicates that:

1) If any uncertain information has the highest degree of mem-bership, i.e., �(x10) = 1, then Gℵ gives 0. It gives an idea thatcompletely uncertain event provides no any information.

2) When �(x4) = 0.4, then Gℵ reaches its maximum value, whichis equal to 0.53. In this case, the occurrence of uncertain eventprovides the maximum value.

3) From �(x4) = 0.4 onwards, the Gℵ value gradually decreases.4) Hence, this curve shows monotonic transformation property,

i.e., initially it increases up to �(x4) = 0.4, then monotonicallydecreases until �(x10) = 1.

5) This fuzzy information-gain curve (FIGC) (refer to Fig. 15) showsthat as the degree of membership associated with each of theevents increases, then corresponding FIG value initially mono-tonically increases then decreases.

xample 3. A DFI on three different universe of discourses, viz.,, Y and Z, is defined as follows:

d = [{x, 0.2}/X, {y, 0.3}/Y, {z, 0.4}/Z] (41)

ind the average uncertainty associated with this DFI.

MAX(0.2, 0.3, 0.4) = − max(0.2, 0.3, 0.4) log2[max(0.2, 0.3, 0.4)]= −0.4 log2(0.4)= 0.529

MIN(0.2, 0.3, 0.4) = − min(0.2, 0.3, 0.4) log2[min(0.2, 0.3, 0.4)]= −0.2 log2(0.2)= 0.464

ow, using Eq. (9), the average uncertainty can be computed usinghe function AFIG, as:

Iℵ = 0.529 + 0.4642

= 0.497

Fig. 16. Convex property of the AFIG function.

Theorem 2. The outcome of the function AFIG, i.e., Iℵ, always liesbetween MIN(�(Ei)) and MAX(�(Ei)) functions.

Proof. Using Eqs. (10) and (11), we have:

MAX(�(Ei)) = − max(�(Ei)) log2[max(�(Ei))]

˛max(say)(42)

MIN(�(Ei)) = − min(�(Ei)) log2[min(�(Ei))]

˛min(say)(43)

Iℵ = ˛max + ˛min

2

⇒ 2[Iℵ] = &alphamax + ˛min

⇒ 2[Iℵ] − ˛min = ˛max

(44)

∵ 2[Iℵ] − ˛min ≤ ˛max (45)

Similarly, we can show that:

∵ 2[Iℵ] − ˛max ≥ ˛min (46)

From these two inequalities of Eqs. (45) and (46), we can say that:˛min ≤ Iℵ ≤ ˛max. This inequality relation is shown in Fig. 16, whichindicates that the outcome of the function Iℵ always lies between˛min and ˛max.

Corollary 1. The outcome of the function Iℵ has lost its convexity, ifeither of ˛min or ˛max is eliminated.

Proof. For the convexity property, outcome of the function Iℵalways exists between two functions, viz., ˛max and ˛min. If weeliminate either of ˛min or ˛max from Eq. (9), then outcome of thefunction Iℵ will never be bounded between ˛min and ˛max. Hence,it can be stated that the function Iℵ is lost its convexity due toelimination of either of the function, i.e., ˛min or ˛max.

Corollary 2. The outcome of the function Iℵ is no longer convexfor the information ℵu = {�(E1)/F, �(E2)/F, . . ., �(En)/F}, ∀Ei ∈ F,if �(E1) = �(E2) = . . . = �(En).

Proof. Using Eq. (9), we have:

Iℵ = MIN(�(E1), �(E2), . . ., �(En)) + MAX(�(E1), �(E2), . . ., �(En))2

(47)

From the problem definition, �(E1) = �(E2) = . . . = �(En). Hence,each degree of membership can be represented, as �(Ek). Now, Eq.(47) becomes:

Iℵ = MIN((�(Ek), �(Ek), . . ., �(Ek)) + MAX((�(Ek), �(Ek), . . ., �(Ek))2

⇒ Iℵ = [−�(Ek)log2(�(Ek))] + [−�(Ek)log2(�(Ek))]

⇒ Iℵ = �1 + �1

2, where �1 = −�(Ek)log2(�(Ek)) ∈ [0, 1]

⇒ Iℵ = 2�1

2⇒ Iℵ = �1

Soft Co

Ci

PI

d(I

Ci

P−bAlof

Ef

ℵ

D

�(x5

(x5)

Et

Ss�a

P. Singh, G. Dhiman / Applied

orollary 3. The outcome of the function Iℵ itself is always containedn FICR.

roof. By Theorem 2, it follows that the outcome of the functionℵ is convex, by assuming that it fulfill the necessary conditions asiscussed in Corollaries 1–2. Therefore, it can be said that Cr(Ei)refer to Definition 8) is a FICR for each information Ei ∈ F, whoseℵ always lies between ˛min and ˛max.

orollary 4. For the function Iℵ (i.e., AFIG), if ˛min = ˛max, then theres no region bounded by ˛min and ˛max.

roof. We have ˛min = − min(�(Ei))log2[min(�(Ei))] and ˛max = max(�(Ei))log2[max(�(Ei))]. It is obvious that ˛min = ˛max wille possible, if �(E1) = �(E2) = . . . = �(En) for the function Iℵ (i.e.,FIG). By Corollary 2, it is proved that value of Iℵ (i.e., AFIG) is no

onger bounded between ˛min and ˛max, if each Ei have equal degreef membership. This special unbounded property of Iℵ (i.e., AFIG)unction is shown in Fig. 3.

xample 4. A DFI on the universe of discourse X, is defined asollows:

d = [{x1, 0.2}/X, {x2, 0.6}/X, {x3, 0.4}/X, {x4, 0.1}/X, {x5, 0.3}/X]

etermine the FICR for the ℵd.

˛max = − max[�(x1), �(x2), . . ., �(x5)]log2[max(�(x1), �(x2), . . .,= − max[�(0.2, 0.6, . . ., 0.3)]log2[max(�(0.2, 0.6, . . ., 0.3))]= −0.6 log2(0.6)= 0.44

˛min = − min[�(x1), �(x2), . . ., �(x5)]log2[min(�(x1), �(x2), . . ., �= − min[�(0.2, 0.6, . . ., 0.3)]log2[min(�(0.2, 0.6, . . ., 0.3))]= −0.1log2(0.1)= 0.33

Now, based on Eq. (9), the AFIG value is:

Iℵ = ˛max + ˛min

2= 0.44 + 0.33

2= 0.39

xample 5. Refer to Example (4), determine the space covered byhe information, which is defined on the universe of discourse X.

olution: From the data of the problem, the degree of member-hips are �(x1) = 0.2, �(x2) = 0.6, �(x3) = 0.4, �(x4) = 0.1, and(x5) = 0.3. From the solution of Example (4), we have ˛max = 0.44

nd ˛min = 0.33. Now, limits a and b can be obtained, as:

b = max[�(x1), �(x2), �(x3), �(x4), �(x5)]= max[0.2, 0.6, 0.4, 0.1, 0.3]= 0.6

a = min[�(x1), �(x2), �(x3), �(x4), �(x5)]

= min[0.2, 0.6, 0.4, 0.1, 0.3]= 0.1

mputing 72 (2018) 121–139 137

))]

)]

Using Eq. (14), the space covered by the information on X canbe measured, as:

H(�) =∫ b

0

˛max d� −∫ a

0

˛min d�

=∫ 0.6

0

0.44 d� −∫ 0.1

0

0.33 d�

= 0.44

∫ 0.6

0

d� − 0.33

∫ 0.1

0

d�

= 0.44 × [0.6 − 0] − 0.33 × [0.1 − 0]= 0.264 − 0.033= 0.231

Hence, the space covered by each information on X is: 0.231.In the following, we discuss various outcomes of the integration

process of FICR, as:

Theorem 3. If Iℵ (i.e., AFIG) be a measurable function for the universeof discourse F, H(�) be a function that measures the space covered byeach information Ei, where Ei ⊆ F on [a, b], and ˛min = ˛max for Iℵ,then measure of the FICR will be either H(�) = ˛max(b − a), or H(�) =˛min(b − a).

Proof. Using Eq. (14), we have:

H(�) =∫ b

0

˛max d� −∫ a

0

˛min d�

Given that ˛min = ˛max. Then, by above relation, we have:

H(�) =∫ b

0

˛max d� −∫ a

0

˛max d�

= ˛max

∫ b

0

d� − ˛max

∫ a

0

d�

= ˛max[b − 0] − ˛max[a− 0]= ˛max(b − a)

Similarly, we can show that H(�) = ˛min(b− a).

Theorem 4. If Iℵ (i.e., AFIG) be a measurable function for the universeof discourse F, H(�) be a function that measures the space coveredby each information Ei, where Ei ⊆ F on [a, b], and a = b for Iℵ, thenmeasure of the FICR will be either H(�) = b(˛max − ˛min) or H(�) =a(˛max − ˛min).

Proof. Using Eq. (14), we have:

H(�) =∫ b

0

˛max d� −∫ a

0

˛min d�

Given that a = b. Then, by above relation, we have:

H(�) =∫ b

0

˛max d� −∫ a

0

˛min d�

= ˛max

∫ b

0

d� − ˛min

∫ b

0

d�

= ˛max[b− 0] − ˛min[b− 0]= b(˛max − ˛min)

Similarly, we can show that H(�) = a(˛max − ˛min).

Theorem 5. Since, Iℵ (i.e., AFIG) be a measurable function for theuniverse of discourse F, and H(�) be a function that measures the space

covered by each information Ei, where Ei ⊆ F on [a, b]. If ˛min = ˛max

and a = b, then measure of the FICR will be H(�) = 0.

Proof. Using Eq. (14), we have:

1 Soft Co

H

G

HT

Eb

Sb00Ma

))]

)]

Na

sf

S

Iat

Ta

P

M

Ebi

Ct

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

38 P. Singh, G. Dhiman / Applied

(�) =∫ b

0

˛max d� −∫ a

0

˛min d�

iven that ˛min = ˛max and a = b. Then, by above relation, we have:

H(�) =∫ b

0

˛max d� −∫ b

0

˛max d�

= 0

ence, convexity of the function Iℵ is lost, if both the conditions ofheorems 3 and 4 are hold together.

xample 6. The following table summarizes the degree of mem-ership of each element of FI, viz., G(1) and G(2), as:

G(1): m1 m2 m3

Degree of Membership: 0.20 0.33 0.40G(2): n1 n2 n3

Degree of Membership: 0.22 0.31 0.43

olution: From the data of the problem, the degree of mem-erships for the G(1): �(m1) = 0.20, �(m2) = 0.33, and �(m3) =.40. Similarly, the degree of memberships for the G(2): �(n1) =.22, �(n2) = 0.31, and �(n3) = 0.43. Now, from the definitions ofAX[�(mi), �(nj)] and MIN[�(mi), �(nj)], as defined in Eqs. (18)

nd (19), respectively, we have:

MAX[�(mi), �(nj)] = − max[�(m1, n1)] log2[max(�(m1, n1))]= − max[�(0.20, 0.22)]log2[max(�(0.20, 0.22= −0.22 log2(0.22)= 0.481

MIN[�(mi), �(nj)] = − min[�(m1, n1)] log2[min(�(m1, n1))]= − min[�(0.20, 0.22)] log2[min(�(0.20, 0.22)= −0.20 log2(0.20)= 0.464

ow, Eq. (17), the similarity between m1 and n1 can be computeds:

S(m1, n1) = MIN[�(m1), �(n1)]MAX[�(m1), �(n1)]

= 0.4640.481

= 0.96

Other remaining similarities between two events can be mea-ured in a similar way. Results are presented in the following matrixorm, as:

(mi, nj) =

⎡⎣S(m1, n1) S(m1, n2) S(m1, n3)S(m2, n1) S(m2, n2) S(m2, n3)S(m3, n1) S(m3, n2) S(m3, n3)

⎤⎦ =

⎡⎣

0.96 0.89 0.890.91 0.99 10.91 0.99 1

⎤⎦

n the above FI similarity matrix, two set of events, viz., (m2, n3)nd (m3, n3) have the similarity parameter value 1. This indicateshat these events m2 and n3; and m3 and n3 are similar.

heorem 6. When S(mi, nj) is equal to 1, two fuzzy events mi and njre similar.

roof. Using Eq. (17), two events mi and nj are similar, iff:

AX[�(mi), �(nj)] = MIN[�(mi), �(nj)] (48)

quality of this Eq. (48) only holds, iff �(mi) = �(nj). In Eq. (48), if

oth �(mi) = �(nj) are equal, then similarity parameter (as defined

n Eq. (17)) gives value 1.

orollary 5. More granularization of information leads to increasehe FIG value.

[

[

mputing 72 (2018) 121–139

Proof. When the discretization process is initiated, it brings theinformation representation level from the zero-order informationgranularization (0-OIG) (refer to Definition 13) to the first-orderinformation granularization (1-OIG) (refer to Definition 14) or morehigher order. In each level of granularization, it brings the upper andlower bound of each interval (granule) more closer. This decreasesthe degree of membership of the element residing in the corre-sponding granule interval, thereby increasing the FIG value.

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