Uncertainty representation using fuzzy-entropy approach: Special application in remotely sensed high-resolution satellite images (RSHRSIs)

Accepted Manuscript Title: Uncertainty Representation using Fuzzy-Entropy Approach: Special Application in Remotely Sensed High-Resolution Satellite Images (RSHRSIs) Author: PII: DOI: Reference:

S1568-4946(18)30426-5 https://doi.org/doi:10.1016/j.asoc.2018.07.038 ASOC 5006

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

24-8-2017 16-7-2018 19-7-2018

Please cite this article as: Pritpal SinghGaurav Dhiman, Uncertainty Representation using Fuzzy-Entropy Approach: Special Application in Remotely Sensed High-Resolution Satellite Images (RSHRSIs), (2018), https://doi.org/10.1016/j.asoc.2018.07.038 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Uncertainty Representation using Fuzzy-Entropy Approach: Special Application in Remotely Sensed High-Resolution Satellite Images (RSHRSIs)

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Pritpal Singh & Gaurav Dhiman

Abstract

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Remotely sensed high-resolution satellite images contain various information in context of

changes. By analyzing this information very minutely, changes occurred in various atmospheric

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phenomena can be identified. Therefore, in this study, a novel change detection method is proposed using the fuzzy set theory. The proposed method represents the uncertain changes in the form of a fuzzy set using the corresponding degree of membership values. By using the fuzzy set operators,

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such as max and min functions, this study derives very useful information from the images. This study also proposes a new function to identify the boundary of uncertain changes. Further, this study is propagated to identify the similarity or dissimilarity between different images of the same

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event that contain various uncertain changes. To recognize the changes in a fine-grained level, this study introduces a way to represent the fuzzy information in a granular way. The utilization of the proposed method is shown by recognizing changes and retrieving information from the remotely

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study.

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sensed high-resolution satellite images. Various experimental results exhibit the robustness of the

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Index Terms

Fuzzy sets; Change Recognition; Entropy; Probability; Granularization; Convex Set; Remotely

Sensed High-Resolution Satellite Images (RSHRSIs).

I. I NTRODUCTION

In the universe, changes always take place due to the occurrence of various uncertain events in the atmosphere. The intensity of these events may be moderate or severe. For example, earthquake, cyclone, flood, etc., always make the changes in the earth’s surface [1]. By simply monitoring, these changes can’t be observed. Therefore, researchers use the spatio-temporal images to detect temporal effects in these uncertain events. Hence, change detection is a technique to find out any changes in the surface of the universe by analyzing remotely-sensed digital images, which are captured at different time stamps. Identifying the changes are one of the most challenging tasks in the domain of pattern recognition and

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machine learning [2]. Roy et al. [3] stated that during change detection analysis, only two group of pixels are formed, which can be classified, as changed class and unchange class. Changes in the images can be identified by analyzing the corresponding pixels in a very granular level. Therefore,

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during analysis, changes can be classified into various groups of pixels, such as a low changed class, moderate changed class, high changed class, and unchanged class. To study change detection, three different kinds of techniques are used by the researchers, as: (a) supervised

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learning [4]–[6], (b) unsupervised learning [7]–[14], and (c) semi-supervised learning [3], [15].

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Recently, researchers have focussed on fuzzy based approaches in change detection. For example, Ghosh et al. [13] have used fuzzy C-means (FCM) and Gustafson–Kessel clustering

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(GKC) algorithms in change detection in multi-temporal remote sensing images. Gong et al. [16] deals with the identification of changes in synthetic aperture radar (SAR) images by introducing a novel image fusion approach and modified FCM clustering algorithm. Gong

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et al. [17] proposed a new approach for change detection in SAR images by classifying changed and unchanged regions using the FCM algorithm along with a new markov random

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field (MRF) energy function. Li et al. [18] proposed a multi-objective fuzzy clustering method for change detection in SAR images. Shi et al. [19] introduced a novel framework to resolve

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change detection problems in difference image (DI) based on fuzzy topology approach.

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A. Problem Statements and Research Contributions The study shows that the change detection domain is mainly constrained in recognizing land cover changes by utilizing multi-temporal remote sensing images. Researchers in this domain have not yet focused on identifying the changes in atmospheric phenomena using remotely sensed high-resolution satellite images (RSHRSIs), which are occurring due to dynamic events. The changes in these atmospheric phenomena (e.g., cloud density, snow cover, precipitation, etc.) can be characterized in terms of “low change”, “moderate change”, “severe change”, and so on. If such changes can be detected w.r.t. their shifting from one direction to another, then it will be easy to take a major initiative to stop the havoc caused by them. It will also be helpful to preserve the flora and fauna of a particular region, where destruction can be caused by such changes and their shifting. Therefore, in this study, emphasize has been put on detecting changes and their shifting, which may occur due to various kinds of uncertain atmospheric phenomena. July 21, 2018

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The application of conventional methods (i.e., supervised, unsupervised and semisupervised methods) in detecting changes in the atmospheric phenomena is very restricted. This may be due to the fact that these methods have not been found to be suitable for representing the uncertainty associated with them. Furthermore, from the review of literature,

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it is also obvious that there is no any uncertainty based method is proposed so far to quantify those changes.

A large number of studies demonstrate that the fuzzy set theory is an effective mathematical

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tool to deal with various kinds of uncertainties [20]. This theory is based on the assumption that each element that belongs to the fuzzy set has a degree of membership value having the

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range [0, 1]. But, in case of change detection, when this theory is applied to represent the uncertainty associated with the changes, a separate set of degree of membership is always

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required to maintain for each of the changes. This representation also suffers from the problem that it doesn’t have any information about the basis of uncertainty, which is essential to locate changes in the RSHRSIs. If anyone wants to locate the changes in a granular way, there is

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also no way in the fuzzy set to represent those changes in a fine-grained level. Recent applications of the fuzzy set theory in change detection in land cover area motivate us to

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propagate our study further in the case of identifying changes in atmospheric phenomena are presented, as follows:

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through the RSHRSIs. Hence, the main objectives along the contributions made in this study

1) To identify a way to represent uncertain changes using the fuzzy set theory: For this

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purpose, this study is persuaded to present a new framework for the representation of uncertain changes inherited in the RSHRSIs in terms of their corresponding degree of memberships using the fuzzy set theory. In this representation, uncertain changes and their corresponding degree of memberships are incorporated together, which is called as a fuzzy information (FI). In the FI, the basis of uncertainty for the uncertain changes is included, so that it can be easy to identify which basis causes which kind of changes.

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

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derived, which is termed as an average fuzzy information-gain (AFIG). This function is used to compute the average amount of uncertain changes associated with the FI. This AFIG function further helps us to determine an interesting region or space in the RSHRIs, which is referred, as a fuzzy information convex region (FICR). This FICR is

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useful to detect the shifting of changes in the RSHRIs. Various properties of this region are studied and presented in this article. Using the FI representation of uncertain changes, this study also provides a similarity parameter to examine distinguishes between two

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atmospheric phenomena.

3) To identify a way to represent the uncertain changes in a granular way: This study

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further propagates to detect the changes in a granular way. For this purpose, this study suggests the use of granular computing (GrC) [22]. This GrC approach is applied in characterized in a fine-grained level.

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the representation of uncertain changes in a more granular way so that changes can be

Hence, in this study, a novel approach for change detection in the RSHRSIs is introduced,

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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

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performance of the proposed algorithm, initial experiments are conducted on two different land cover area images of the Bangong Lake and the greater Washington, D.C.. Then,

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the experiments are further carried out on three different satellite images of atmospheric phenomena, which include weather satellite images of India, supper Typhoon-Megi in the

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Philippines, and the Mars planet. Various empirical analyses conclude that the proposed algorithm is more robust in change detection and shifting in the atmospheric phenomena in comparison to the existing fuzzy set based approaches. B. Organization of the Article

The rest of this article is organized as follows. Section II establishes the background required for the study. The proposed method of information retrieval and change detection is presented in Section III. Descriptions of data sets along with experimental set up are discussed in Section IV. Various empirical results are discussed in Section V. Section VI is dedicated for the discussion and conclusions.

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II. F ORMALIZATION FOR THE P ROPOSED S TUDY In this section, we introduce the formal definitions of the fuzzy information (FI), measurement of FI, various definitions, theorems and corollaries. Various examples and

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theorems/corollaries associated with the study are presented in Appendix A. A. Fuzzy Information and Its Representation

In case of RSHRSI, if every change is considered as an individual uncertain information,

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then it can be represented by its corresponding degree of membership using the fuzzy set.

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For example, if U = {m1 , m2 , m3 , . . . , mn } be a universe of discourse for n number of changes, then various significant changes can be detectd by categorizing them, as “low

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change”, “moderate change”, “severe change”, and so on. For the representation of such ˜ based on the events that belong to the universe of changes, we can define the fuzzy set M discourse U , as:

(1)

M

˜ = µ(m1 )/m1 + µ(m2 )/m2 + . . . + µ(mn )/mn M

Here, µ represents the degree of membership function used in the fuzzy set theory.

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Hence, µ(mk ) gives the degree of membership value between range [0, 1] for the uncertain ˜ . Note that we will use µ as the degree of information mk , associated with the fuzzy set M

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membership function throughout the discussion of our proposed theories and formulas. Assume that the universe of discourse F is composed of various uncertain events E˜1 , E˜2 , E˜3 , . . ., E˜n , whose respective degree of memberships µ(E˜1 ), µ(E˜2 ), µ(E˜3 ), . . ., µ(E˜n ) are assumed to be known. These two sets can be represented in matrix form, as: ˜ = [E˜1 , E˜2 , E˜3 , . . . , E˜n ], where [E]

n [

E˜k = F

(2)

k=1

˜ = [µ(E˜1 ), µ(E˜2 ), µ(E˜3 ), . . . , µ(E˜n )], [µ(E)] where µ(E˜k ) ∈ [0, 1]

(3)

Now, Eqs. (2) and (3) contain all the information about the fuzziness of F. These two equations can now be expressed in the form of fuzzy information (FI), as: Definition 1: (Fuzzy Information (FI)). A FI is a pair set of elements {E˜i , µ(E˜i )}(i = 1, 2, . . . , n), which can be denoted by ℵ. Mathematically, it can be expressed, as: ℵ = {E˜i , µ(E˜i )}/F, ∀ E˜i ∈ F July 21, 2018

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The FI {E˜i , µ(E˜i )}/F, as defined above, often doesn’t give complete information about how they are collected or evolved. In the FI, the existence of each event depends on the degree of membership, and subsequently on the source of information, i.e., the universe of discourse. Hence, to distinguish the FI in terms of its source of information, it can be

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categorized, as: (a) undistinguished FI (UFI), and (b) distinguished FI (DFI). Both these are defined and explained next.

Definition 2: (Undistinguished FI (UFI)). A collection of information from a similar

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source, is called an undistinguished FI (UFI). It is denoted by ℵu . Mathematically, it can be expressed, as:

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ℵu = [{E˜1 , µ(E˜1 )}/F, {E˜2 , µ(E˜2 )}/F, . . . , {E˜n , µ(E˜n )}/F],

∀E˜i ∈ F

(5)

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Here, each {E˜i , µ(E˜i )}/F represents the individual FI w.r.t. the universe of discourse F, where E˜i ∈ F and µ(E˜i ) ∈ [0, 1].

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Definition 3: (Distinguished FI (DFI)). A collection of events from different sources, is called a distinguished FI (DFI). It is denoted by ℵd . Mathematically, it can be expressed, as: (6)

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˜i , µ(H ˜i )}/J], ∀E˜i ∈ F, ∀H ˜i ∈ J ℵd = [{E˜i , µ(E˜i )}/F, {H

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˜i ) ∈ [0, 1]. µ(H

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˜i , µ(H ˜i )}/J represent the individual FI w.r.t. the universe of In Eq. 6, {E˜i , µ(E˜i )}/F and {H ˜i ∈ J, where discourses F and J, respectively. Here, E˜i ∈ F, where µ(E˜i ) ∈ [0, 1]; and H

B. A Measure of Fuzzy Information The measure of a certain event is a set function M , which assign a number M (x) to each set x in a certain class [23]. Some parameters must be imposed on the class of sets on which M is defined. Hence, probability can be used as a type of parameter, which can be used to measure the occurrence of any certain event. Based on this probability parameter, Shannon [24]–[26] has suggested the following expression for the measurement of expected amount of information, as: H(P1 , P2 , . . . , Pn ) = −

n X

Pi log2 (Pi )

(7)

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Here, Pi (i = 1, 2, . . . , n) represents the probability of each event. The function H is also known as the Shannon’s entropy-function. July 21, 2018

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Suppose, ξ is a set of elements Ai (i = 1, 2, . . . , n), which are generated with certain random experiment. Now, subsets of ξ can be called “events”, and assigned a probability. Let Ak is an event, and if the question “Does Ak ∈ ξ ?” has a definite answer, as yes or no, then Ak is called “certain event”. For any certain event, information can be measured

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utilizing Eq. (7). In the majority of the cases, the occurrence of events is not hundred percent sure. Additionally, day by day atmospheric phenomena (e.g., cloud density, snow cover, precipitation, etc.) gradually changes, which can’t be defined precisely based on probability

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measure. Therefore, to measure the uncertainty involved in such kind of events, the fuzzy set is considered as the appropriate technique, where uncertainty associated with any kind of

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event can be measured based on the degree of membership value [27]. Hence, in this study, a new function is defined to measure the fuzziness involved in such uncertain events, which

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is called as a fuzzy information-gain (FIG). It can be defined, as:

Definition 4: (Fuzzy Information-Gain (FIG)). It is the measure of uncertainty regarding which event of the FI {E˜i , µ(E˜i )}/F, where i = 1, 2, . . . , n, has occurred or will occur in

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terms of degree of membership. It is denoted as Gℵ . Mathematically, it can be expressed, as: n X Gℵ = − µ(E˜i ) log2 µ(E˜i ) (8) i=1

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information-gain (FIG).

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Here, E˜i ∈ F, and µ(E˜i ) ∈ [0, 1]. In Eq. 8, the function Gℵ can be referred as a fuzzy The basic difference between Eq. 7 and Eq. 8 is that the former takes probability value as

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an input, while another takes a degree of membership value as an input. The average amount of uncertainty involved in the FI can also measured using the functions of the fuzzy set theory, as:

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

Iℵ =

M AX(µ(E˜i )) + M IN (µ(E˜i )) 2

(9)

Here, the function Iℵ returns the AFIG value associated with any FI. In Eq. (9), each event E˜i belongs to the universe of discourse F, i.e., E˜i ∈ F, and µ(E˜i ) ∈ [0, 1]. The values of M AX(µ(E˜i )) and M IN (µ(E˜i )), defined in Eq.9, can be obtained, as:

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M AX(µ(E˜i )) = − max(µ(E˜i )) log2 [max(µ(E˜i ))]

(10)

M IN (µ(E˜i )) = − min(µ(E˜i )) log2 [min(µ(E˜i ))]

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Outcomes of the MAX, MIN and AFIG functions.

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Fig. 1.

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In Eqs. (10) and (11), max and min represent the minimum and maximum operations

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of the fuzzy sets, respectively [28]. Here, M AX and M IN can be termed as a minimum information-gain function and maximum information-gain function, respectively. The exis-

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tence of both these functions in Eq. (9) help to capture the average amount of uncertainty available in any FI.

Based on Example (3) (refer to Appendix A), the AFIG value for the ℵd is 0.497. The

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M IN and M AX values for this FI can be shown with the help of blue-cut and green-cut

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function.

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lines in Fig.1, respectively. In this figure, red-cut line represents the outcome of the AFIG

C. Fuzzy Information Convexity

In this sub-section, we have discussed various definitions and properties based on the convexity property exhibited by the AFIG function. These definitions and properties of the AFIG function are vary from ones presented in [29]–[31]. These are discussed as follows: Definition 6: (Convex Fuzzy Set)[29]–[31]. Let U be the universe of discourse, 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 two uncertain points in R, then every uncertain point p = ax + (1 − a)y, 0 ≤ a ≤ 1, must also be in the subset R. In the following, the convexity is discussed based on outcome of the AFIG function, which can be applied on any FI. Hence, this definition is termed as a FI Convex (FIC), and can be defined, as: July 21, 2018

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Definition 7: (FI Convex (FIC)). Since, the universe of discourse F is composed of various uncertain events, E˜1 , E˜2 , E˜3 , . . ., E˜n , and the Iℵ is the representation of AFIG function, where each E˜i ∈ F. Then, the outcome of Iℵ is said to be FIC iff satisfying the condition M IN (µ(E˜i )) ≤ Iℵ ≤ M AX(µ(E˜i )), for all Iℵ ∈ [0, 1].

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Mathematically, this definition implies that outcome of Iℵ is an uncertain point in a degree of membership function, and it satisfies the convexity if its value lies between M IN (µ(E˜i )) and M AX(µ(E˜i )), i.e., M IN (µ(E˜i )) ≤ Iℵ ≤ M AX(µ(E˜i )). The convex property shown

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by the AFIG function, is illustrated in Fig. 1.

Based on above discussion, we can now define FI Convex Region (FICR), as:

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Definition 8: (FI Convex Region (FICR)). Since, the universe of discourse F is composed of various uncertain events E˜1 , E˜2 , E˜3 , . . ., E˜n . For each E˜i , we can define a FICR based

[

E˜i ∈F

{Cr (E˜i ) : {αmin , Iℵ , αmax } ⊆ F}

(12)

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Cr (E˜i ) =

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on three functions, viz., αmin , Iℵ , and αmax . The FICR of F is determined by each event E˜i ∈ F, hence it can be denoted by Cr (E˜i ). Mathematically, it can be expressed, as:

Mathematically, this definition implies that FICR is the combination of these three uncertain

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points, viz., αmin , Iℵ , and αmax , where αmin ≤ Iℵ ≤ αmax . In Fig. 2, a FICR is shown. Here, outcome of Iℵ is shown, which is bounded between

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αmin and αmax . Therefore, this outcome is also included in the representation of FICR.

Fig. 2.

FICR for the FI.

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

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Fig. 3.

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The FICR for Iℵ can be represented, as: {αmin , Iℵ , αmax } ∈ F.

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The unbounded property of Iℵ (i.e., AFIG) function leads to development of a new definition, which is called as a FI Concave Region (FICOR), which is presented, as follows:

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Definition 9: (FI Concave Region (FICOR)). Since, the universe of discourse F is composed of various uncertain events E˜1 , E˜2 , E˜3 , . . ., E˜n . For each E˜i , we can define a

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FICOR for outcome of the function Iℵ , which is not bounded by M IN (µ(E˜i )) (or, αmin ) and M AX(µ(E˜i )) (or, αmax ). The FICOR of F can be denoted by Vr (E˜i ). Mathematically, it can be expressed, as:

[

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Vr (E˜i ) =

{Vr (E˜i ) : {Iℵ } ⊆ F}

(13)

E˜i ∈F

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The necessary and sufficient conditions for outcome of the function Iℵ , to be concave, are: 1) For each E˜i ∈ F, iff µ(E˜1 ) = µ(E˜2 ) = . . . = µ(E˜n ), and 2) For each E˜i ∈ F, iff αmin = αmax . The FICOR is shown in Fig. 3, where outcome of the function Iℵ , which is θ1 (say), is completely unbounded, i.e., Iℵ : [0, 1] →)θ1 (. This concave region can be expressed as the combination of {Iℵ }, i.e., {Iℵ } ⊆ Vr (E˜i ). D. Measurable Fuzzy Information Convex Region In this sub-section, we discuss the measurable fuzzy information convex region (FICR) in terms of basic integral process. Let H is a real-valued convex function, which is able to measure the region bounded between interval [αmin , αmin ]. Here, we develop more general integration process by applying Riemann integral [32] on H, by considering that H is continuous. This integration process is July 21, 2018

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differ from one presented in [33]. This integration can be applied to the function, provided that certain “measurability” is possible. It is shown that collection of information can be measured in terms of two functions, viz., FIG and AFIG. Here, the AFIG function relies upon two principle measures, as αmin and

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αmax . Here, the AFIG function generates a region, which is bounded between αmin and αmax . This region is referred as the FICR (refer to Definition (8)), which is a measurable quantity. Hence, this integration process is applied here to measure this FICR, which can be defined,

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as:

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Definition 10: (Measurable function for FICR). Since, F is composed of various uncertain events E˜1 , E˜2 , E˜3 , . . ., E˜n . For each E˜i , Cr (E˜i ) is the convex region, if it satisfies the Definition (8). Therefore, Cr (E˜i ) is a fuzzy-space, and H is a function that can be measurable.

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If outcome of the experiment is η, then we may capture the value for H(η). Here, H(η) represents the function of measurable region or space covered by the Cr (E˜i ). Mathematically,

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we wish to compute µ{η : H(η) ∈ L}, where L = [αmin , αmax ]. For this to be possible, L must belongs to Cr (E˜i ). Since, αmin and αmax are both measurable function, then H is a “measurable function”, i.e., region involving H can be computed.

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If Iℵ (i.e., AFIG) be a measurable function for the F, and H(η) be a function that measures the space covered by each information E˜i , where E˜i ∈ F; then we can define this fuzzy-space

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in terms of integral of αmin and αmax functions w.r.t. µ, as:

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Definition 11: (Integral of FICR). Integral of FICR bounded between αmin and αmax can be defined from a ˜ to ˜b, as: Z ˜b Z a˜ H(η) = αmax dµ − αmin dµ (14) 0

0

Here, a ˜ to ˜b can be defined, as:

˜b = max[µ(E˜1 ), µ(E˜2 ), . . . , µ(E˜n )]

(15)

a ˜ = min[µ(E˜1 ), µ(E˜2 ), . . . , µ(E˜n )]

(16)

Here, max and min represent the minimum and maximum operations of the fuzzy sets [28], respectively; and a ˜, ˜b ∈ [0, 1]. Remark 1: The larger is the value of H(η), more is the uncertainty involved in the FICR. E. Fuzzy Information Similarity Matrix This similarity matrix is used to study the similarity or the dissimilarity between different RSHRSIs that contain various uncertain information. This uncertain information is represented July 21, 2018

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as a FI. Based on this FI contained in different RSHRSIs, the similarity or the dissimilarity between them can be studied. In this study, this parameter is called as a fuzzy information similarity parameter (FISP), whose representation is given below, as: Definition 12: (Fuzzy Information Similarity Parameter (FISP)). Let two universe of

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discourses M and N are defined for the set of events {m ˜ 1, m ˜ 2, . . . , m ˜ m } and {˜ n1 , n ˜2, . . . , n ˜ n }, ˜i respectively. The degree of memberships of occurrence of each of the events, say m ˜ i and n are µ(m ˜ i )(i = 1, 2, . . . , m) and µ(˜ ni )(i = 1, 2, . . . , n), respectively. The similarity between

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each of the events in terms of m ˜ i and n ˜ j associated with the M and N, respectively, can be given, as:

nj )] M IN [µ(m ˜ i ), µ(˜ (17) M AX[µ(m ˜ i ), µ(˜ nj )] Here, the function S is called as the FISP. The values of M IN [µ(m ˜ i ), µ(˜ nj )] and

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S(m ˜ i, n ˜j ) =

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M AX[µ(m ˜ i ), µ(˜ nj )], defined in Eq. (17), can be obtained, as:

= − max[µ(m ˜ i ), µ(˜ nj )] log2 [max(µ(m ˜ i ), µ(˜ nj ))]

(18)

M IN [µ(m ˜ i ), µ(˜ nj )]

= − min[µ(m ˜ i ), µ(˜ nj )] log2 [min(µ(m ˜ i ), µ(˜ nj ))]

(19)

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M AX[µ(m ˜ i ), µ(˜ nj )]

In Eqs. 18 and 19, max and min represent the minimum and maximum operations of the

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fuzzy sets [28], respectively.

In Definition (12), the FISP is introduced for the recognition of overall similarity or

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dissimilarity between different RSHRSIs. In this computation, rather than only focusing on

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the identified changes (that are marked with the dark circle, for example as in the case of Figs. 9-13), overall changes in the RSHRSIs have been considered. In some cases, detection of minimum or insignificant changes is not possible. In such situation, it is not possible to provide a region for that changes. Therefore, this study provides a boundary for that changes using the M AX and M IN functions. Hence, in the computation of similarity or dissimilarity (refer to Eq. (17)), only the degrees of memberships returned by the M AX and M IN functions are utilized.

˜ j ) gives the strength of the relationship between two uncertain events In this matrix, S(m ˜ i, n ˜ j ) lies between in terms of corresponding degree of memberships. Hence, the value of S(m ˜ i, n interval [0, 1]. F. Granular Representation of Fuzzy Information The objective of this study is to represent the FI in a granular way. This representation is only possible if the degree of membership based on a secondary domain (granular FI) July 21, 2018

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13

coincides with the degree of membership based on the primary domain (original FI). Granular computing (GrC) is an emerging technique for information processing [22]. GrC is used to extract fine-grained information from the complex information or data. It is applied in various domains, such as data discretization [34], data classification [35], partition of universe

ip t

to solve complex problems [36], analysis of non-geometric patterns [37], and decision-making [38]. GrC techniques can be regarded as an indispensable part of FI granulation, rough sets and interval computations [39]. GrC is also used in data mining for rule representation, rule

cr

mining, and soft computing (especially in fuzzy and rough sets) [40].

In fuzzy sets, information granulation implies discretization the basis of information or

us

source of information into a fine-grained level. Such information can be ordered, as primary domain based information, secondary domain based information, and so on. Following

an

definitions illustrate the concept of primary and secondary domains based information. Definition 13: (Primary Domain Based FI (PDBFI)). Each information E˜i , which is initially depends on the universe of discourse F, is called as a primary domain based FI

M

(PDBFI). Here, the F is referred as the primary domain, and this representation is called as the PDBFI. This representation of FI can be denoted as Gp . Mathematically, it can be m [

d

expressed, as: Gp =

{E˜i , µ(E˜i )}/F |∀E˜i ∈ F

(20)

te

i=1

Here, each E˜i is satisfying Eqs. 2 and 3. This representation can also be considered as a

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zero-order information granularization (0-OIG). Definition 14: (Secondary Domain Based FI (SDBFI)). Let F = {F1 , F2 , . . . , Fn }, where S each Fi , i = 1, 2, . . . , m, is a subset of F, in such a way that m i=1 Fi = 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 that an information E˜k = {˜ e1 , e˜2 , . . . , e˜n }, where each e˜i , i = 1, 2, . . . , m, is a subset of E˜k . Then, the elements of E˜k , which are subsets of the E˜k , is called a granularization of E˜k , and information e˜1 , e˜2 , . . . , e˜n are called as a granule-information. Now, if any granule-information e˜i , can be able to defined on Fi , where Fi ∈ F, and e˜i ∈ E˜k , then each Fi is called a secondary-domain. This representation of information is called a secondary domain based FI (SDBFI), and can

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be denoted as Gs . Mathematically, it can be expressed, as:    m  [  {e˜i , µ(e˜i )}/Fi  |∀e˜i ∈ Fi , E˜k ∈ F Gs = ˜ ˜   { E , µ( E )}/F  k k i=1 

(21)

ip t

Here, each E˜k is satisfying Eqs. 2 and 3. This representation can also be considered as a first-

order information granularization (1-OIG). By applying discretization techniques [41]–[43], one can achieve various levels of granularization.

cr

Remark 2: (i) More granularization reduces the fuzziness of the information that belongs to the universe of discourse [44], thereby increasing the FIG and AFIG values of any uncertain

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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 limit thinner at a specific level; otherwise,

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it will convert the fuzzy set into the crisp set [20].

Following are the challenges and issues in FI granularization, as: 1) The properties of granulation provide a dynamic approach to process the FI into small

M

sub-modules to solve the particular problem. In this situation, finding the optimal number of user-defined granules are a noteworthy issue for the researchers to solve the particular

d

problem.

2) If there is a static event, the granulation level is already characterized. For instance,

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the University examination system includes the well-defined universe of marks for each subject to generate the particular grade of a student. However, if there is an occurrence

Ac ce p

of a dynamic event, the given universe differs in a given phase. For example, the universe for an every day temperature, universe for closing price of a stock index, and so on. In this situation, choosing a specific granulation is significantly a complex task.

3) Representation of the FI into various level of granulation is computationally very complex and expensive. Therefore, to get the final decision from such granular FI, various levels of defuzzification are required.

III. T HE P ROPOSED A LGORITHM FOR I NFORMATION R ETRIEVAL AND C HANGE D ETECTION

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

It is applicable in retrieving various information from the RSHRSIs by employing the functions, such as FIG, MAX, MIN, AFIG, and FISP.

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•

15

This algorithm is useful to locate the regions of uncertain changes in RSHRSIs, which is called as the FICR (as defined in Definition (8)). This algorithm is also useful to visualize the shifting of changes w.r.t. time and date.

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A. Fuzzy-Information Retrieval and Change Detection Algorithm (FIRCDA) Each phase of the FIRCDA is explained as follows. Step 1. Read the RSHRSI, and convert it into gray form.

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Explanation: Initially, RSHRSI is uploaded into the system. Then, it is converted into the gray form.

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Step 2. Find the color intensities in each pixel of RSHRSI.

can be represented, as: c˜i,j

c˜i+1,j+1 .. .

c˜i+m,j+1

d

   c˜i+1,j ~ = C  ..  .  c˜i+m,j

c˜i,j+1

c˜i,j+2

...

c˜i+1,j+2 .. .

... .. .

c˜i+m,j+2

...

M



an

Explanation: For RSHRSI, color intensity values corresponding to each pixel is ~ This vector C, ~ found. We consider this set of color intensity values, as a vector C.

c˜i,j+n



  c˜i+1,j+n   ..   .  c˜i+m,j+n

(22)

Here, m and n represent the maximum number of rows and columns in any RSHRSI.

te

Here i, j = 1, 2, 3, . . . , n.

Ac ce p

Step 3. Define the universe of discourse for the color intensity values. ~ U for the matrix, as shown in Eq. (22), Explanation: The universe of discourse C ~ U = [cmin , cmax ]. Here, cmin and cmax represent the minimum can be defined, as C and maximum color intensity values from the matrix, as shown in Eq. (22).

Step 4. Discretize the universe of discourse. ~ U = [cmin , cmax ], into several Explanation: We discretize the universe of discourse C

equal lengths of intervals, as a1 = [lb1 , ub1 ], a2 = [lb2 , ub2 ], a3 = [lb3 , ub3 ], . . .,

an = [lbn , ubn ]. Here, each lbi and ubi represents the lower and upper bounds of an ~U . interval ai , where i = 1, 2, . . . , n. Here, ubn ≤ cmax , and ai ∈ C

Step 5. Define fuzzy set for each of the intervals, and fuzzify each of the color intensity values. Explanation: For n number of intervals, we can define total n number of fuzzy ~ U . For all these fuzzy sets, sets, as A˜1 , A˜2 , . . ., A˜n , on the universe of discourse C as defined above, we adopt the linguistic values, as A˜1 = (very low) ∈ a1 , A˜2 =(not July 21, 2018

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very low) ∈ a2 , . . ., A˜n =(very very high) ∈ an . Now, to fuzzify the set of color intensity values, find the interval ai , where each color intensity value c˜h,k (as shown in Eq. (22)), belongs to. For example, a color

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intensity value c˜1,1 belongs to the interval a1 = [lb1 , ub1 ]. For this interval, we have defined the fuzzy set, as A˜1 . Hence, this color intensity value is fuzzified, as A˜1 . In this way, all the color intensity values are fuzzified. Step 6. Define the FI for the color intensity values.

cr

Explanation: For the color intensity values, as shown in Eq. (22), where each ~ U , can be c˜h,k (i = 1, 2, . . . , n) initially belongs to the universe of discourse C

us

represented based on the definition of FI (refer to Definition (1)), as: ~ U | c˜h,k ∈ C ~U ch,k , µ(˜ ch,k )}/C ℵu = {˜

(23)

an

~ U represents the individual FI w.r.t. C ~U . Here, each {˜ ch,k , µ(˜ ch,k )}/C Step 7. Calculate the degree of membership value for each of the color intensity values.

M

Explanation: For each color intensity value c˜h,k , its corresponding degree of ~ U = [cmin , cmax ], can be membership value w.r.t. the universe of discourse C

d

computed, as:

(24)

te

µ(˜ ch,k ) = (˜ ch,k − cmin )/(cmax − cmin )

Here, each µ(˜ ch,k ) represents the degree of membership value for the color intensity

Ac ce p

value c˜h,k .

The nature of membership function depends on the context of the application [45]. In the proposed method, degree of membership of color intensity value c˜h,k is

determined using Eq. (24), whose computation depends on the limit of the universe ~ U = [cmin , cmax ]. However, one can use the other fuzzy membership of discourse C functions, for example, triangular function, Gaussian function, and so on [46].

Step 8. Compute the F IG value for each of the color intensity values. Explanation: For each color intensity value c˜h,k , its corresponding FIG value can be computed based on Eq. (8), as: Gℵ = −µ(˜ ch,k ) log2 µ(˜ ch,k )

(25)

Here, the function Gℵ is called the “FIG”. Step 9. Calculate the M AX and M IN values from the set of color intensity values. Explanation: Based on Eqs. (10) and (11), we can compute the M AX and M IN July 21, 2018

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~ as: values from the set of color intensity values C, ~ M AX(µ(C))

~ log2 [max(µ(C))] ~ = − max(µ(C))

(26)

~ M IN (µ(C))

~ log2 [min(µ(C))] ~ = − min(µ(C))

(27)

operations of the fuzzy sets, respectively [28].

cr

Step 10. Obtain the AF IG value for the set of color intensity values.

ip t

In Eqs. (26) and (27), the max and min represent the minimum and maximum

Explanation: Based on Eqs. (26) and (27), the AF IG value can be obtained, as: ~ + M IN (µ(C)) ~ M AX(µ(C)) 2

us

Iℵ =

(28)

an

Here, the function Iℵ returns the AFIG value associated with the set of color intensity ~ values C. Step 11. Compute the similarity among RSHRSIs. RSHRSIs can be computed, as:

M

Explanation: Using F ISP (as defined in Definition (12)), similarity among

(29)

d

cj )] M IN [µ(˜ ci ), µ(˜ S(I˜i , I˜j ) = M AX[µ(˜ ci ), µ(˜ cj )]

te

Here, I˜i and I˜j represent two different RSHRSIs, where indices i and j represent their

Ac ce p

corresponding capturing information (such as time, date, year, etc.), respectively. In Eq. (29), M AX and M IN values for the I˜i and I˜j can be obtained from Eqs. (26) and (27).

~ (as shown in Eq. (22)), whose Step 12. Find the columns’ index values (j th ) from the vector C c˜i,j value corresponds to M IN value. Locate all these M IN values in RSHRSIs, which occurs maximum number of times among all these values. Store all these ~ col , and can be defined, as: values in a vector, which is represented by a vector R ~ col = M AXoccur (j1 , j2 , j3 , . . . , jh ) R

(30)

Here, h defines the maximum number of index values, whose c˜i,j value corresponds to M IN value. ~ (as shown in Eq. (22)), whose Step 13. Find the rows’ index values (ith ) from the vector C c˜i,j value corresponds to M IN value, and its columns’ index values (j th ) are same as ~ col (refer to Step 1). Calculate previously obtained j th index value from the vector R

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the average absolute value of these rows’ index values (ith ), and store in a vector, ~ row . This vector can be defined, as: which is represented by R ~ row =| AV G(i1 , i2 , i3 , . . . , iq ) | R

(31)

ip t

Here, q defines the maximum number of index values, whose c˜i,j value corresponds

cr

to M IN value. ~ row (ith ) index and R ~ col (j th ) index of the vector C, ~ which are the co-ordinate Step 14. Locate R points of the M IN regions.

us

Step 15. Repeat Steps 12–14 for the co-ordinate points of the M AX and AF IG regions. Step 16. Make a region by connecting all these co-ordinate points of M IN , M AX and AF IG

an

regions. This region is called as the F ICR, as defined in Definition (8). B. Computational complexity

M

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

Time complexity for the proposed FIRCDA:

d

•

te

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.

Ac ce p

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 and N defines to discretize the universe of discourse.

4) The algorithm requires O(N ) time to compute the degree of membership and F IG value for each of the color intensity value.

5) The computational cost to calculate the M AX, M IN , AF IG values from the set of color intensity values, and similarity among RSHRSIs (using Eq. (29)), is O(N ) time.

6) It requires O(M × N ) time to compute the M IN , M AX, and AF IG regions, where M and N defines the row and column pixel values of a given RSHRSI, respectively. 7) The algorithm uses O(N ) time to make a region by connecting all the co-ordinates points. Therefore, the total computational complexity of all above steps for maximum number of iterations is O(M × N ).

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(a) Date: 24/08/1998

(b) Date: 02/09/2013

19

(c) the reference map

•

cr

and (c) the reference map for showing the changes in between 24/08/1998 and 02/09/2013.

ip t

Fig. 4. Images of the Bangong Lake, Himalayas: (a) Image is acquired on 24/08/1998, (b) Image is acquired on 02/09/2013,

Space complexity of the proposed FIRCDA:

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The space complexity of the proposed algorithm is the maximum amount of space, which is considered at any one time during its initialization process. Thus, the total

an

space complexity of the proposed FIRCDA is O(M × N ).

IV. D ESCRIPTIONS OF DATA S ETS

M

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

d

phenomena data set. A detailed description of data sets is provided next.

te

A. Data set corresponding to the land cover area In this category of the data set, two different categories of LANDSAT satellite images

Ac ce p

have been selected, as:

a) Images of Bangong Lake, Himalayas: Two LANDSAT satellite images of Bangong Lake in the Tibet Autonomous Region of China are acquired on 24/08/1998 and 02/09/2013. This data set is acquired from USGS (Source: https://remotesensing.usgs.gov/gallery/index. php). During this time interval, changes are observed in this lake, which have expanded the area of lake along the marshy southwestern and northern shorelines. These shoreline changes adversely affect the levels of local salinity, whose affect can be observed in the vegetation as well as in the living organisms. Fig. 4 (a) and (b) show the images of Bangong Lake for 24/08/1998 and 02/09/2013, respectively. The reference map is generated using these images, which is depicted in Fig. 4 (c). This reference map consists of 56,335 changed pixels and 4,62,065 unchanged pixels. This reference map is further used for detecting the changes in the Bangong lake.

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(a) Date: 23/12/1972

(b) Date: 08/05/2012

20

(c) the reference map

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and (c) the reference map for showing the changes in between 23/12/1972 and 08/05/2012.

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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,

b) Images of the greater Washington, D.C.: Two LANDSAT satellite images of the greater

us

Washington, D.C. area are acquired on 23/12/1972 and 08/05/2012. This data set is acquired from USGS (Source: https://remotesensing.usgs.gov/gallery/index.php). These two images are shown in Fig. 5 (a) and (b). In these images, highly red colors

an

indicate forests and large grassy region, while light colors indicate fields and the highly urban development areas. A comparison between these two images has been clearly

M

demonstrating the significant growth in the greater Washington, D.C. from the last 40 years. For further analysis, the reference map is generated using these images, which

te

3,92,777 unchanged pixels.

d

is depicted in Fig. 5 (c). This reference map consists of 1,25,623 changed pixels and

B. Data set corresponding to the atmospheric phenomena

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

a) Images of India: In this category of data set, four weather satellite images of India are included. These images are collected from the site of National Satellite Meteorological Centre, Ministry of Earth Sciences, Govt. of India [47], and depicted in Fig. 6 (a)-(d). The size of these four images are 720 × 720 pixels, which were captured on dated 01/08/2016 at four different time intervals, as 05:30 am, 06:00 am, 06:30 am, and 07:00 am. In these images, changes are observed continuously in terms of cloud density. In Fig. 6 (e), a reference map is given, which showing the changes between 05:30 am - 07:00 am in the atmospheric phenomena. This reference map is obtained from Fig. 6 (a)-(d). This reference map consists of 47,680 changed and 4,70,720 unchanged pixels. b) Images of supper Typhoon-Megi, Philippines: This data set consists of three images of typhoons with the size of 720 × 720 pixels, which were observed in the Philippines July 21, 2018

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on dates 20/10/2010, 21/10/2010 and 22/10/2010. These images are available on the website of the Earth Observatory Center, NASA [48]. These images are depicted in Fig. 7 (a)-(c). A typhoon is a mature tropical cyclone, which mostly occurs in the northwestern Pacific Ocean. In this part of the world, mostly Japan, the Philippines and Hong Kong

ip t

is drastically suffered by the typhoon. In Fig. 7 (d), a reference map is given, which showing the changes between 20/10/2010 and 22/10/2010 in the supper Typhoon-Megi. This reference map is obtained from Fig. 7 (a)-(c). This reference map consists of 38,966

cr

changed and 4,79,434 unchanged pixels.

c) Images of the Mars planet: Finally, two weather satellite images of the Mars planet

us

captured on dates 09/07/1997 and 10/07/1997 with the size of 720 × 720 pixels are used to analyze the behaviors and variations of dust on the Mars planet. These images are

an

available on the website of the Hubble European Space Agency, Germany [49]. These images are depicted in Fig. 8 (a)-(b). In Fig. 8 (c), a reference map is given, which showing the changes between 09/07/1997 and 10/07/1997 in the Mars planet. This reference

M

map is obtained from Fig. 8 (a)-(b). This reference map consists of 36,623 changed and

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te

d

4,81,777 unchanged pixels.

(a) Time: 05:30 am

(b) Time: 06:00 am

(c) Time: 06:30 am

(d) 07:00 am

(e) the reference map

Fig. 6. Images of India: (a)-(d) captured on dated 01/08/2016 at four different time stamps, and (e) the reference map for showing the changes in between 05:30 am - 07:00 am.

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(b) Date: 21/10/2010

(c) Date: 22/10/2010

(d) the reference map

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(a) Date: 20/10/2010

22

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Fig. 7. Images of supper Typhoon-Megi: (a)-(c) captured on dated 20/10/2010, 21/10/2010 and 22/10/2010, respectively,

d

M

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and (d) the reference map for showing the changes in between 20/10/2010 and 22/10/2010.

(b) Date: 10/07/1997

(c) the reference map

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(a) Date: 09/07/1997

Fig. 8. Images of the Mars planet: (a)-(b) captured on dates 09/07/1997 and 10/07/1997, respectively, and (c) the reference

Ac ce p

map for showing the changes in between 09/07/1997 and 10/07/1997.

C. Evaluation Indexes

The visualization approach is used to show the change detection in images. Apart from this, there are some other measurements to calculate the values of indexes for evaluation. In this work, percentage of correct classification (PCC) and Kappa coefficient are considered as evaluation indexes for experimentation [18]. The mathematical formulation of PCC and Kappa coefficients are described as follows: TP + TN N P CC − PV Kappa = 1 − PV P CC =

(32) (33)

where, PV = July 21, 2018

(TP + FP ) × RC + (TN + FN ) × RU N2

(34) DRAFT

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In Eq. (34), FP represents the number of pixels unchanged originally but falsely detected as changed, FN represents the number of pixels is changed but falsely detected as unchanged, TP represents the number of changed pixels, TN represents the number of unchanged pixels, RC represents the number of actual changed pixels, RU represents the number of actual

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unchanged pixels, and N is the total number of pixels. D. Experimental Setup

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The performance of the proposed algorithm is compared with various well-known approaches to demonstrate its applicability. The maximum number of iterations for the proposed

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algorithm is 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

an

the row-wise computation is 720, and for the column-wise computation is 720. The whole simulation process is carried out in Matlab R2017a environment operating on Microsoft Windows 8.1 with 64 bits on Core i-5 processor, and 2.40 GHz processor with 4 GB main

M

memory.

d

V. E MPIRICAL R ESULTS In this section, the effectiveness of the proposed algorithm (i.e., FIRCDA) is evaluated by

te

conducting experiments on two different kinds of data sets, as discussed in Section IV. As our proposed algorithm is based on the fuzzy set theory, therefore to assess the performance

Ac ce p

of the proposed algorithm, only this theory based methods have been adopted, which are FCM_S1 [50], MRFFCM [17], FCM [51], and MOFCM [18]. A. Change detection analysis

1) In terms of land cover area data set: In this category of data set, we have analyzed the changes from the images of the Bangong lake and greater Washington, D.C.. Initially, various information has been retrieved from these images using the proposed functions, viz., F IG, M AX, M IN , and AF IG. In Fig. 9 (a)-(b), the regions of M AX, M IN , and AF IG are identified in the images of Bangong Lake, using the green, red and blue color spots, respectively. These spots show the significant changes in the images between 1998 and 2013. Then, the FICR is identified and located in these images by using the dark circle (represented by cyan color). The FICRs are shown in Fig. 9 (a)-(b). By comparing these two images in terms of the FICR, the shifting of changes is obviously July 21, 2018

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(b) Date: 02/09/2013

(c)

(d)

(e)

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cr

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(a) Date: 24/08/1998

24

(f)

(g)

Fig. 9. Changes showing in the Bangong Lake, Himalayas based on: (a)-(b) the proposed FIRCDA. The change detection

an

map showing the changes between 24/08/1998 − 02/09/2013 in the Bangong Lake, Himalayas obtained by: (c) the proposed FIRCDA, (d) the FCM_S1 approach, (e) the MRFFCM approach, (f) the FCM approach, and (g) the MOFCM

M

approach.

observed in the Bangong Lake between 1998 and 2013. From the change detection map, as shown in Fig. 9 (c), it is evident that the lake area has been expanded, and these

d

changes are clearly shifting along the southwest and northern shorelines.

te

Similarly, for the images of greater Washington, D.C., the M AX, M IN , and AF IG regions are shown in Fig. 10 (a)-(b) with the green, red and blue color spots, respectively.

Ac ce p

These spots clearly reflect the significant changes in the images between 23/12/1972

and 08/05/2012. The FICRs, as located in these images (represented by dark cyan color in Fig. 10 (a)-(b)) clearly depict the shifting of changes in the greater Washington, D.C. between 23/12/1972 and 08/05/2012. The change detection map of the greater Washington, D.C. (refer to Fig. 10 (c)) signifies the significant urban growth in the greater Washington, D.C. area. For the Bangong lake and greater Washington, D.C. images, change detection maps are derived by the existing competing approaches [17], [18], [50], [51], and shown in Fig. 9 (d)-(g) and Fig. 10 (d)-(g), respectively. By comparing these change detection maps with the change detection maps obtained by the proposed algorithm (refer to Fig. 9 (c) and 10 (c)), it has been noticed that some of the changes are wrongly identified by the existing approaches [17], [18], [50], [51] in comparison to the proposed algorithm.

2) In terms of atmospheric phenomena data set: In this category of data set, we have July 21, 2018

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(b) Date: 08/05/2012

(c)

(d)

(e)

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cr

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(a) Date: 23/12/1972

25

(f)

(g)

Fig. 10. Changes showing in the greater Washington, D.C. based on: (a)-(b) the proposed FIRCDA. The change detection

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map showing the changes between 23/12/1972−08/05/2012 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) the MOFCM approach.

M

analyzed the changes from the images of India, supper Typhoon-Megi, and the Mars planet. For all these images, various information has been retrieved in terms of the F IG,

d

M AX, M IN , and AF IG functions. In Fig. 11 (a)-(d), Fig. 12 (a)-(c), and Fig. 13 (a)(b), the identified regions of the M AX, M IN , and AF IG are shown, using the green,

te

red and blue color spots, for the images of India, supper Typhoon-Megi, and the Mars planet, respectively. In case of India image (refer to Fig. 11 (a)-(d)), changes in cloud

Ac ce p

density are clearly observed between 05:30 am and 07:00 am on dated 01/08/2016. Similarly, these spots reflect the changes in the images of supper Typhoon-Megi (refer to Fig. 12 (a)-(c)) in terms of the air mass between 20/10/2010 and 22/10/2010, which

rotates around the center of low atmospheric pressure. Finally, changes are marked in the images of the Mars planet (Fig. 13 (a)-(b)), which demonstrates the variations of dust storms on the Mars planet between 09/07/1997 and 10/07/1997. In all these

atmospheric phenomena images, the FICRs are located by using the dark circle, which is represented by cyan color (refer to Fig. 11 (a)-(d), Fig. 12 (a)-(c), and Fig. 13 (a)-(b)). By comparing the FICRs of these images, shifting of changes w.r.t. date and time have been clearly observed. The change detection maps for all these images are obtained by the proposed algorithm (refer to Fig. 11 (e), Fig. 12 (d), and Fig. 13 (c)), which provides the information about

July 21, 2018

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Page 25 of 46

FOURTH REVISION SUBMITTED TO: APPLIED SOFT COMPUTING

26

(b) Time: 06:00 am

(c) Time: 06:30 am

(e)

(f)

(g)

(d) Time: 07:00 am

Fig. 11.

an

(h)

M

(i)

us

cr

ip t

(a) Time: 05:30 am

Changes showing in the weather satellite images of India based on: (a)-(d) the proposed FIRCDA. The change

detection map showing the changes between 05:30 am - 07:00 am in the weather satellite images of India obtained by: (e)

te

approach.

d

the proposed FIRCDA, (f) the FCM_S1 approach, (g) the MRFFCM approach, (h) the FCM approach, and (i) the MOFCM

Ac ce p

the changed and unchanged pixels w.r.t. date and time. These change detection maps are compared with the change detection maps acquired by the existing approaches [17], [18], [50], [51], and it has been found that the proposed algorithm is an edge over in distinguishing the changes in comparison to the existing approaches [17], [18], [50], [51].

B. Information retrieval through the change detection maps Various information retrieved from the experimental data sets using the proposed functions (i.e., F IG, M AX, M IN , AF IG, F ICR, and H(η)) are listed in Table I. In this table, results are also compared with the existing approaches [17], [18], [50], [51]. In Table I, the function H(η) represents the amount of space covered by the FICR region. From the comparison, it is obvious that the large value of H(η) for the existing approaches [17], [18], [50], [51] indicate that they consider a large number of pixels for detecting the changes. However, the proposed July 21, 2018

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Page 26 of 46

27

(b) Date: 21/10/2010

(c) Date: 22/10/2010

(e)

(f)

(g)

(d)

us

cr

(a) Date: 20/10/2010

ip t

FOURTH REVISION SUBMITTED TO: APPLIED SOFT COMPUTING

(h)

an

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 changes between 20/10/2010 − 22/10/2010 in the weather satellite images of

Ac ce p

te

d

the FCM approach, and (h) the MOFCM approach.

M

supper Typhoon-Megi obtained by: (d) the proposed FIRCDA, (e) the FCM_S1 approach, (f) the MRFFCM approach, (g)

(a) Date: 09/07/1997

(b) Date: 10/07/1997

(e) Fig. 13.

(c)

(f)

(d)

(g)

Changes showing in the weather satellite images of Mars planet based on: (a)-(b) the proposed FIRCDA. The

change detection map showing the changes between 09/07/1997 − 10/07/1997 in the weather satellite images of Mars planet obtained by: (c) the proposed FIRCDA, (d) the FCM_S1 approach, (e) the MRFFCM approach, (f) the FCM approach, and (g) the MOFCM approach.

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28

TABLE I I NFORMATION RETRIEVED THROUGH THE CHANGE DETECTION MAPS FOR THE EXPERIMENTAL DATA SETS .

Typhoon Image

Mars Image

M IN

AF IG

F ICR

H(η)

0.4663

0

0.2769

{0, 0.2769, 0.4663}

0.4633 0.4869

65.9987

0.4869

0

0.2819

{0, 0.2819, 0.4869}

65.4312

0.4850

0

0.2803

{0, 0.2803, 0.4850}

0.4850

FCM [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

FIRCDA

33.2684

0.2669

0

0.1527

{0, 0.1527, 0.2669}

0.2669

FCM_S1 [50]

36.8691

0.2768

0

0.1853

{0, 0.1853, 0.2768}

0.2768

ip t

FCM_S1 [50] MRFFCM [17]

MRFFCM [17]

34.5538

0.2684

0

0.1684

{0, 0.1684, 0.2684}

0.2684

FCM [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

FIRCDA

71.4981

0.5299

0

0.2649

FCM_S1 [50]

72.5241

0.5365

0

0.2756

MRFFCM [17]

71.9585

0.5311

0

FCM [51]

73.4521

0.5323

0

MOFCM [18]

73.5441

0.5310

0

FIRCDA

31.0035

0.5293

0

FCM_S1 [50]

32.1125

0.5316

0

MRFFCM [17]

31.0190

FCM [51]

36.9965

MOFCM [18]

34.0635

FIRCDA

70.5241

FCM_S1 [50]

75.0063

MRFFCM [17]

71.9967

FCM [51]

0

0

{0, 0.2649, 0.5299}

0.5299

{0, 0.2756, 0.5365}

0.5365

{0, 0.2698, 0.5311}

0.5311

0.2721

{0, 0.2721, 0.5323}

0.5323 0.5310

0.2673

{0, 0.2673, 0.5310}

0.2646

{0, 0.2646, 0.5293}

0.5293

0.2679

{0, 0.2679, 0.5316}

0.5316

0.2683

{0, 0.2683, 0.5298}

0.5298

0.2967

{0, 0.2967, 0.5423}

0.5423

0.5386

0

0.2702

{0, 0.2702, 0.5386}

0.5386

0.5340

0

0.2683

{0, 0.2683, 0.5340}

0.5340

0.5569

0

0.2756

{0, 0.2756, 0.5569}

0.5569

0.5392

0

0.2697

{0, 0.2697, 0.5392}

0.5392

70.8693

0.5366

0

0.2703

{0, 0.2703, 0.5366}

0.5366

73.0039

0.5409

0

0.2729

{0, 0.2729, 0.5409}

0.5409

d

MOFCM [18]

0.5298

0.5423

0.2698

cr

India Image

M AX

63.5332

us

Washington, D.C. Image

F IG

FIRCDA

an

Bangong Lake Image

Information retrieved based on

M

Data set

the changes.

te

FIRCDA can take granular level of decisions by considering actual pixels that representing

Ac ce p

The similarity or dissimilarity in the experimental data sets in terms of changes are also studied, and presented in the form of matrices, as shown in Eqs. 35-39, respectively.

Bangong Lake Image, S(I˜i , I˜j ) =

"

Washington, D.C. Image, S(I˜i , I˜j ) =

July 21, 2018

"

24/08/1998

02/09/2013

1

0.2685

0.2692

1

#

23/12/1972

08/05/2012

1

0.4025

0.4103

1

24/08/1998

(35)

02/09/2013

#

23/12/1972

(36)

08/05/2012

DRAFT

Page 28 of 46

FOURTH REVISION SUBMITTED TO: APPLIED SOFT COMPUTING

06:00 am

06:30 am

07:00 am

1

0.0044

0.0044

0.0043

 0.0086    0.0114  

1

0.0083

0.0116

0.0116

1

0.0003

0.0003

0.0003

22/10/2010

1

0.9979

0.9952

1

1

1



 0.9973   1

09/07/1997

10/07/1997

1

0.1125

0.1133

1

M

"

1

06:00 am

(37)

06:30 am 07:00 am

cr

21/10/2010

Typhoon Image, S(I˜i , I˜j ) =   

Mars Image, S(I˜i , I˜j ) =

1

20/10/2010



05:30 am

ip t

0.0086



20/10/2010

21/10/2010

(38)

us

  ˜ ˜ India Image, S(Ii , Ij ) =    

05:30 am

an



29

#

22/10/2010

09/07/1997

(39)

10/07/1997

In Eqs. 35-39, I˜i and I˜j represent two different images, where indices i and j represent

te

d

their corresponding capturing dates, respectively. C. Statistical Analysis and Comparison Study

Ac ce p

For showing the efficiency of the proposed FIRCDA in terms PCC and Kappa coefficient statistics, this study has adopted four benchmark approaches [17], [18], [50], [51], for the comparison study. For each of the satellite images, the box plots are depicted in Fig. 14, which show the statistical results of the PCC and Kappa over 30 independent runs. In each box plot, red line represents the median, edges are upper and lower quartiles, and + symbol denotes the outliners. The results reveal that the proposed approach outperforms the competitive approaches [17], [18], [50], [51]. Table II shows the calculated average values of PCC and Kappa over 30 independent simulation runs. It has been concluded that the proposed algorithm is statistically significant and perform better than competitive approaches in terms of PCC and Kappa. The quantitative analysis of the proposed FIRCDA along with the competing approaches [17], [18], [50], [51], is also carried out in terms of overall error (OE), the number of false alarms (FA) and missed alarms (MA). The false alarm occurs when the unchanged pixels July 21, 2018

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Page 29 of 46

Fig. 14.

DA

0.9

0.8

0.7

0.6

July 21, 2018

Box plot analysis of the proposed and competing approaches.

M

M O FC M

M

M

M

M

M

C

O FC

FC

RF F

M

M

M

M

M

O FC

FC

C

M _S 1

RF F

M

M

FC

DA

us

cr

0.7

O FC

FC

C

M _S 1

C

0.8

M

RF F

FC

FI R

M

M

O FC

M

ip t M

FC

C

M _S 1

DA

RF F

FC

C

Statistical values of Kappa on Bangk. Lake dataset

FI R

M

M

M

M

C

O FC

FC

RF F

M

M

0.55

M

1 0.9

FC

0.4

M

0.5

M

0.6

C

0.7

RF F

0.8

M

0.9 0.4

M _S 1

0.6

FC

0.7 0.5

M _S 1

0.8 0.6

FC

1

an

0.4

DA

0.5

C

0.6

FI R

0.7

Statistical values of Kappa on Was., D.C. dataset

0.8

DA

0.9

M

0.9

C

d

M

M

M

C

O FC

FC

RF F

M

M

DA

M _S 1

C

0.4

FI R

M

M

FC

FI R

0.5

Statistical values of Kappa on India dataset

O FC

M

C

DA

M _S 1

C

0.6

Statistical values of Kappa on Typhoon dataset

M

FC

RF F

te M

FC

FI R

0.7

DA

DA

M _S 1

C

0.8

C

M

M

M

C

O FC

FC

RF F

M

M

FC

FI R

Statistical values of PCC on Bangk. Lake dataset 0.9

FI R

M

M

DA

M _S 1

C

Statistical values of PCC on Was., D.C. dataset

1

Statistical values of Kappa on Mars dataset

O FC

M

C

FC

FI R

0.4

FC

RF F

M _S 1

C

Statistical values of PCC on India dataset 0.5

M

M

FC

FI R

Statistical values of PCC on Typhoon dataset

Ac ce p

Statistical values of PCC on Mars dataset

FOURTH REVISION SUBMITTED TO: APPLIED SOFT COMPUTING 30

0.9

0.85

0.75

0.8

0.65

0.7

0.6

0.85 0.9

0.75

0.8

0.7

0.65

0.6

0.9

0.8

0.7

0.6

0.5

0.4

0.9

0.8

0.7

0.6

0.5

0.4

0.3

DRAFT

Page 30 of 46

FOURTH REVISION SUBMITTED TO: APPLIED SOFT COMPUTING

31

TABLE II T HE STATISTICAL RESULTS OBTAINED BY FIVE APPROACHES .

Data sets

Indexes

FIRCDA

FCM_S1 [50]

MRFFCM [17]

FCM [51]

Bangong Lake Image

PCC

0.4253

0.5631

0.8562

0.6775

0.5967

Kappa

0.5226

0.5777

0.6348

0.7564

0.6628

Typhoon Image

Mars Image

0.7569

0.6152

0.5576

0.7569

0.7013

0.8864

ip t

0.8500 0.7002

PCC

0.3554

0.4527

0.9857

0.6671

0.3957

Kappa

0.4397

0.5791

0.6841

0.6097

0.5980

PCC

0.2239

0.3679

0.5473

0.6987

0.4328

Kappa

0.3978

0.3986

0.5798

0.4570

0.5699

PCC

0.5702

0.6397

0.7891

0.5990

0.9873

Kappa

0.2869

0.3003

0.6798

0.4573

0.5090

PCC

0.4163

0.5747

0.7870

0.6515

0.5940

Kappa

0.4684

0.5112

0.6671

0.5963

0.6452

an

Average

0.5069 0.6951

cr

India Image

PCC Kappa

us

Washington, D.C. Image

MOFCM [18]

M

are identified as changed pixels, whereas missed alarm occurs when the changed pixels are identified as unchanged pixels. We have compared the change detection maps obtained by

d

the proposed FIRCDA with their corresponding reference maps. Table III shows the change detection errors obtained by the proposed and competing approaches [17], [18], [50], [51]

te

for the experimental data sets. From Table III, we can see that the MA, FA and OE are minimum for the FIRCDA in comparison to competing approaches. From this comparison,

Ac ce p

it is obvious that the wrongly detected unchanged pixels (i.e., false alarms) are minimum in the change detection maps obtained using the proposed FIRCDA. Hence, it can be concluded that the change detection maps acquired by the proposed FIRCDA is very closely co-related with their corresponding reference maps. VI. D ISCUSSION AND C ONCLUSIONS

This study shows that various uncertain changes can easily be represented by the fuzzy set theory, and the fuzziness of those changes can be measured using their corresponding degree of membership values. Further, this study demonstrates that based on the degree of membership values, information inherited in an uncertain event can be measured using the FIG function. Another function, i.e., AFIG, is able to quantify the average amount of changes inherited in RSHRSIs. Based on the FICR, this study has identified the shifting of changes w.r.t. time. In this study, a method is discussed to compute the area of FICR. This study shows July 21, 2018

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32

TABLE III

Washington, D.C. Image

India Image

Typhoon Image

FA

OE

886

959

1779

FCM_S1 [50]

989

1607

2635

MRFFCM [17]

1109

1513

2697

FCM [51]

1002

1825

2852

MOFCM [18]

1236

1496

2676

FIRCDA

968

1063

1965

FCM_S1 [50]

1128

2041

2754

MRFFCM [17]

1090

1652

2885 2900

FCM [51]

1125

1967

MOFCM [18]

1274

1528

2800

FIRCDA

836

928

1764

FCM_S1 [50]

985

1563

2548

MRFFCM [17]

1072

1468

2540

1756

2717

FCM [51]

961

MOFCM [18]

1168

FIRCDA

685

FCM_S1 [50]

1258

MRFFCM [17]

1008

FCM [51]

MOFCM [18]

2641

963

1648

1657

2915

1248

2256

981

1589

2570

1176

1935

2120

985

1135

FCM_S1 [50]

1374

1780

3154

MRFFCM [17]

1058

1369

2427

FCM [51]

1483

1900

3383

MOFCM [18]

1139

1878

3017

d

Mars Image

1473

759

M

FIRCDA

us

Bangong Lake Image

MA

cr

Methods FIRCDA

an

Image

ip t

T HE OBTAINED MISSED ALARMS , FALSE ALARMS , AND OVERALL ERROR FOR THE EXPERIMENTAL DATA SETS .

te

that if more is the area of FICR, then more is the changes reflected by the event. If uncertain changes are well defined, then it is easy to obtain information-gain value. Occasionally,

Ac ce p

uncertainties inherit inside the uncertainties. To represent such kind of uncertainties in terms of changes, granular computing (GrC) approach is used, where the basis of information is discretized to obtain the granular level of information. Experimental results show that the FIG, AFIG, FICR, area of FICR and granular representation of uncertain changes have the abilities for the information retrieval and change detection from the RSHRSIs. The proposed algorithm can be applied to detect uncertain changes and their shifting in an incremental learning way [52]. If new changes occur or they shift from one direction to another w.r.t. time, then the proposed algorithm can represent such changes by existing FI or by defining another FI. From such representation of FI, the proposed algorithm effectively has the capacity to retrieve diverse quantifiable information. Experimental results also demonstrate that the proposed algorithm is far superior to existing fuzzy set theory based approaches. The proposed algorithm is applied on very high-resolution satellite images, where quality of images is very good in terms of color intensity. The proposed algorithm is based on the July 21, 2018

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Page 32 of 46

FOURTH REVISION SUBMITTED TO: APPLIED SOFT COMPUTING

33

fuzzy set theory, which can easily quantify the low quality pixels to high quality pixels by providing the class labels. Hence, it is easy for the proposed algorithm to detect the changes and their shifting by representing those pixels in terms of fuzzy set. In spite of all these advantages, the proposed algorithm has certain shortcomings, which can be listed, as: In this study, computation of the degree of membership depends on the universe of

ip t

•

discourse, which is defined by the users. Further, this universe is discretized into various user-defined values.

For detecting changes in a more granular level, the basis of information (i.e., the universe

cr

•

of discourse) is required to discretize into a various number of intervals.

us

All these shortcomings can be rectified by integrating a robust meta-heuristic optimization algorithm with the proposed algorithm. This meta-heuristic optimization algorithm can help

an

to select the optimal universe of discourse, as well as it can help to select optimal interval lengths. However, these issues will be resolved in a future enhancement of this method. The efficiency of the proposed algorithm is also required to be explored, where it is difficult to

M

provide the class labels for the changes. Moreover, the proposed algorithm is also required to verify in images of different kinds of atmospheric phenomena to detect changes and their

d

shifting.

te

A PPENDIX A

Ac ce p

List of Examples and Theorems/Corollaries Example 1: An example is illustrated here for the representation of undistinguished FI (UFI). Let us consider that each of the events E˜1 , E˜2 , . . . , E˜n , are depended on a similar source, i.e., undistinguished universe of discourse F. Now, each of the events of F be the members of FI, which can be represented, as ℵu = {E˜i , µ(E˜i )}/F, where i = 1, 2, . . . , n. For example, disease “typhoid” can be regarded as the universe of discourse, whose different symptoms “fever”, “headaches”, and “diarrhea” can be characterized as a distinct information/event. Here, the source of information “typhoid”, is undistinguished. Hence, this information can be expressed in terms of UFI, as follows: ℵu = [{f ever, µ(f ever)}/typhoid, {headaches, µ(headaches)}/typhoid, {diarrhea, µ(diarrhea)/typhoid}]

July 21, 2018

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FOURTH REVISION SUBMITTED TO: APPLIED SOFT COMPUTING

34

Example 2: An example is illustrated here for the representation of distinguished FI (DFI). ˜i are gathered from two distinguished Let us consider that two different events E˜i and H universe of discourses, viz., F and J, respectively. These events be the members of FI, which ˜i , µ(H ˜i )}/J]. For example, can be represented in terms of F and J, as ℵd = [{E˜i , µ(E˜i )}/F, {H

ip t

obesity problem of a patient might be because of imbalanced thyroid hormone, and fatigue problem for a similar patient might be because of diabetes. Here, “obesity” and “fatigue” can be characterized as a distinct information/event, and “thyroid” and “diabetes” can be

cr

called as the distinguished universe of discourses. Hence, this information can be expressed

us

in terms of DFI, as follows:

ℵd = [{obesity, µ(obesity)}/thyroid, {f atigue, µ(f atigue)}/diabetes]

an

Theorem 1: As the degree of membership of the corresponding information increases, the measure of information, i.e., FIG value monotonically increases, then monotonically decreases.

M

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

d

ℵu = [{˜ x1 , 0.1}/U, {˜ x2 , 0.2}/U, . . . , {˜ x10 , 0}/U]∀˜ xi ∈ U, i = 1, 2, . . . , 10

(40)

te

In Eq. 40, different FIG values are obtained by arranging the degree of membership for each of the events x1 , x2 , . . . , x10 in ascending order. Now, a curve for the individual FIG

Ac ce p

values is fitted, which is depicterd in Fig. 15. This curve is entitled as a fuzzy informationgain curve (FIGC). This curve demonstrates how the FIG function (i.e., Gℵ ) behaves when the input is a degree of membership, i.e., µ(˜ xi ), i = 1, 2, . . . , 10. This curve indicates that: 1) If any uncertain information has the highest degree of membership, i.e., µ(˜ x10 ) = 1, then Gℵ gives 0. It gives an idea that completely uncertain event provides no any information.

2) When µ(˜ x4 ) = 0.4, then Gℵ reaches its maximum value, which is equal to 0.53. In this case, the occurrence of uncertain event provides the maximum value.

3) From µ(˜ x4 ) = 0.4 onwards, the Gℵ value gradually decreases. 4) Hence, this curve shows monotonic transformation property, i.e., initially it increases x10 ) = 1. upto µ(˜ x4 ) = 0.4, then monotonically decreases until µ(˜ 5) This fuzzy information-gain curve (FIGC) (refer to Fig. 15) shows that as the degree of membership associated with each of the events increases, then corresponding FIG value initially monotonically increases then decreases. July 21, 2018

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Page 34 of 46

35

A FIGC for the undistinguished fuzzy information (UFI).

an

Fig. 15.

us

cr

ip t

FOURTH REVISION SUBMITTED TO: APPLIED SOFT COMPUTING

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

M

ℵd = [{˜ x, 0.2}/X, {˜ y , 0.3}/Y, {˜ z , 0.4}/Z]

(41)

Find the average uncertainty associated with this DFI.

d

Solution: From the data of the problem, we have µ(˜ x) = 0.2, µ(˜ y ) = 0.3, and µ(˜ z ) = 0.4.

te

Now, using Eqs. (10) and (11), we have:

M AX(0.2, 0.3, 0.4) = − max(0.2, 0.3, 0.4) log2 [max(0.2, 0.3, 0.4)]

Ac ce p

= −0.4 log2 (0.4)

= 0.529

M IN (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

Now, using Eq. (9), the average uncertainty can be computed using the function AFIG, as: 0.529 + 0.464 2 = 0.497

Iℵ =

Theorem 2: The outcome of the function AFIG, i.e., Iℵ , always lies between M IN (µ(E˜i )) and M AX(µ(E˜i )) functions. July 21, 2018

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36

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

M AX(µ(E˜i )) = − max(µ(E˜i )) log2 [max(µ(E˜i ))] (42)

cr

M IN (µ(E˜i )) = − min(µ(E˜i )) log2 [min(µ(E˜i ))] ≃ αmin (say)

αmax + αmin 2

(43)

(44)

an

Iℵ =

us

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

ip t

≃ αmax (say)

⇒ 2[Iℵ ] = αmax + αmin

M

⇒ 2[Iℵ ] − αmin = αmax ∵ 2[Iℵ ] − αmin ≤ αmax

d

Similarly, we can show that:

(46)

te

∵ 2[Iℵ ] − αmax ≥ αmin

(45)

From these two inequalities of Eqs. (45) and (46), we can say that: αmin ≤ Iℵ ≤ αmax . This

Ac ce p

inequality relation is shown in Fig. 16, which indicates that the outcome of the function Iℵ always lies between αmin and αmax .

Fig. 16.

Convex property of the AFIG function.

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37

Corollary 1: The outcome of the function Iℵ has lost its convexity, if either of αmin or αmax is eliminated. Proof. For the convexity property, outcome of the function Iℵ always exists between two functions, viz., αmax and αmin . If we eliminate either of αmin or αmax from Eq. (9), then

ip t

outcome of the function Iℵ will never be bounded between αmin and αmax . Hence, it can be stated that the function Iℵ is lost its convexity due to elimination of either of the function, i.e., αmin or αmax .

cr

Corollary 2: The outcome of the function Iℵ is no longer convex for the information ℵu = {µ(E˜1 )/F, µ(E˜2 )/F, . . . , µ(E˜n )/F}, ∀E˜i ∈ F, if µ(E˜1 ) = µ(E˜2 ) = . . . = µ(E˜n ).

Iℵ =

us

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

M IN (µ(E˜1 ),µ(E˜2 ),...,µ(E˜n ))+M AX(µ(E˜1 ),µ(E˜2 ),...,µ(E˜n )) 2

(47)

M

M IN ((µ(E˜k ),µ(E˜k ),...,µ(E˜k ))+M AX((µ(E˜k ),µ(E˜k ),...,µ(E˜k )) 2

θ1 +θ1 , where 2

Ac ce p

⇒ Iℵ =

[−µ(E˜k ) log2 (µ(E˜k ))]+[−µ(E˜k ) log2 (µ(E˜k ))] 2

te

⇒ Iℵ =

d

Iℵ =

an

From the problem definition, µ(E˜1 ) = µ(E˜2 ) = . . . = µ(E˜n ). Hence, each degree of membership can be represented, as µ(E˜k ). Now, Eq. (47) becomes:

θ1 = −µ(E˜k ) log2 (µ(E˜k )) ∈ [0, 1]

⇒ Iℵ =

2θ1 2

⇒ I ℵ = θ1

Here, θ1 ∈ [0, 1], and outcome of the function Iℵ can lie on either side of θ1 , not between M IN (µ(E˜i )) and M AX(µ(E˜i )). Therefore, we can say that its convexity is lost. Corollary 3: The outcome of the function Iℵ itself is always contained in FICR. Proof. By Theorem 2, it follows that the outcome of the function Iℵ is convex, by assuming that it fulfill the necessary conditions as discussed in Corollaries (1)-(2). Therefore, it can be said that Cr (E˜i ) (refer to Definition 8) is a FICR for each information E˜i ∈ F, whose Iℵ always lies between αmin and αmax .

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38

Corollary 4: For the function Iℵ (i.e., AFIG), if αmin = αmax , then there is no region bounded by αmin and αmax . = − min(µ(E˜i )) log2 [min(µ(E˜i ))] and αmax = We have αmin − max(µ(E˜i )) log2 [max(µ(E˜i ))]. It is obvious that αmin = αmax will be possible, if µ(E˜1 ) = µ(E˜2 ) = . . . = µ(E˜n ) for the function Iℵ (i.e., AFIG). By Corollary 2, it is proved that value of Iℵ (i.e., AFIG) is no longer bounded between αmin and αmax , if each E˜i have

ip t

Proof.

equal degree of membership. This special unbounded property of Iℵ (i.e., AFIG) function is

cr

shown in Fig. 3.

us

Example 4: A DFI on the universe of discourse X, is defined as follows: ℵd = [{˜ x1 , 0.2}/X, {˜ x2 , 0.6}/X, {˜ x3 , 0.4}/X, {˜ x4 , 0.1}/X, {˜ x5 , 0.3}/X]

an

Determine the FICR for the ℵd .

Solution: From the data of the problem, the degree of memberships are µ(˜ x1 ) = 0.2, µ(˜ x2 ) = as defined in (42) and (43), we have:

M

x4 ) = 0.1, and µ(˜ x5 ) = 0.3. Now, from the definitions of αmax and αmax , 0.6, µ(˜ x3 ) = 0.4, µ(˜

αmax = − max[µ(˜x1 ), µ(˜x2 ), . . . , µ(˜x5 )] log2 [max(µ(˜x1 ), µ(˜x2 ), . . . , µ(˜x5 ))]

d

= − max[µ(0.2, 0.6, . . . , 0.3)] log2 [max(µ(0.2, 0.6, . . . , 0.3))]

te

= −0.6 log2 (0.6)

Ac ce p

= 0.44

αmin = − min[µ(˜x1 ), µ(˜x2 ), . . . , µ(˜x5 )] log2 [min(µ(˜x1 ), µ(˜x2 ), . . . , µ(˜x5 ))] = − min[µ(0.2, 0.6, . . . , 0.3)] log2 [min(µ(0.2, 0.6, . . . , 0.3))] = −0.1 log2 (0.1)

= 0.33

Now, based on Eq. (9), the AFIG value is: αmax + αmin 2 0.44 + 0.33 = 2

Iℵ =

= 0.39 Example 5: Refer to Example (4), determine the space covered by the information, which is defined on the universe of discourse X. July 21, 2018

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39

Solution: From the data of the problem, the degree of memberships are µ(˜ x1 ) = 0.2, µ(˜ x2 ) = 0.6, µ(˜ x3 ) = 0.4, µ(˜ x4 ) = 0.1, and µ(˜ x5 ) = 0.3. From the solution of Example (4), we have ˜ and ˜b can be obtained, as: αmax = 0.44 and αmin = 0.33. Now, limits a

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

cr

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

ip t

˜b = max[µ(˜ x2 ), µ(˜ x3 ), µ(˜ x4 ), µ(˜ x5 )] x1 ), µ(˜

us

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

an

Using Eq. (14), the space covered by the information on X can be measured, as: Z ˜b Z a˜ H(η) = αmax dµ − αmin dµ 0

=

0

0.6

0.44 dµ − 0

0.1

0.33 dµ

0.6

0

dµ − 0.33

0

d

= 0.44

Z

Z

M

Z

Z

0.1

dµ 0

te

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

Ac ce p

= 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 universe of discourse F, H(η) be a function that measures the space covered by each information E˜i , where E˜i ⊆ F on [˜ a, ˜b], and αmin = αmax for Iℵ , then measure of the FICR will be either H(η) = αmax (˜b − a ˜), or H(η) = αmin (˜b − a ˜). Proof. Using Eq. (14), we have: H(η) =

July 21, 2018

Z

˜b

αmax dµ − 0

Z

a ˜

αmin dµ 0

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40

Given that αmin = αmax . Then, by above relation, we have: Z a˜ Z ˜b αmax dµ − αmax dµ H(η) = 0

0

= αmax

Z

˜b

dµ − αmax 0

Z

a ˜

dµ 0

ip t

= αmax [˜b − 0] − αmax [˜ a − 0] ˜) = αmax (˜b − a

cr

Similarly, we can show that H(η) = αmin (˜b − a ˜).

us

Theorem 4: If Iℵ (i.e., AFIG) be a measurable function for the universe of discourse F, H(η) be a function that measures the space covered by each information E˜i , where E˜i ⊆ F on [˜ a, ˜b], and a ˜ = ˜b for Iℵ , then measure of the FICR will be either H(η) = ˜b(αmax − αmin )

an

or H(η) = a ˜(αmax − αmin ). Proof. Using Eq. (14), we have: ˜b

αmax dµ −

Z

M

H(η) =

Z

0

a ˜

αmin dµ

0

d

Given that a ˜ = ˜b. Then, by above relation, we have: Z ˜b Z H(η) = αmax dµ −

te

0

= αmax

Z

a ˜

αmin dµ 0

˜b

dµ − αmin 0

Z

˜b

dµ 0

Ac ce p

= αmax [˜b − 0] − αmin [˜b − 0]

= ˜b(αmax − αmin )

Similarly, we can show that H(η) = a ˜(αmax − αmin ). Theorem 5: Since, Iℵ (i.e., AFIG) be a measurable function for the universe of discourse F, and H(η) be a function that measures the space covered by each information E˜i , where a, ˜b]. If αmin = αmax and a ˜ = ˜b, then measure of the FICR will be H(η) = 0. E˜i ⊆ F on [˜ Proof. Using Eq. (14), we have:

H(η) =

Z

˜b

αmax dµ − 0

Z

a ˜

αmin dµ 0

Given that αmin = αmax and a ˜ = ˜b. Then, by above relation, we have: Z ˜b Z ˜b H(η) = αmax dµ − αmax dµ 0

0

= 0

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41

Hence, convexity of the function Iℵ is lost, if both the conditions of Theorems (3) and (4) are hold together. Example 6: The following table summarizes the degree of membership of each element

m ˜1

m ˜2

m ˜3

Degree of Membership:

0.20

0.33

0.40

n ˜1

n ˜2

n ˜3

G(2):

cr

G(1):

ip t

of FI, viz., G(1) and G(2), as:

us

Degree of Membership: 0.22 0.31 0.43 Find the similarities between these two FI (refer to Definition 12).

Solution: From the data of the problem, the degree of memberships for the G(1): µ(m ˜ 1 ) = 0.20, µ(m ˜ 2 ) = 0.33, and µ(m ˜ 3 ) = 0.40. Similarly, the degree of memberships

an

n2 ) = 0.31, and µ(˜ n3 ) = 0.43. Now, from the definitions of for the G(2): µ(˜ n1 ) = 0.22, µ(˜ nj )] and M IN [µ(m ˜ i ), µ(˜ nj )], as defined in Eqs. (18) and (19), respectively, M AX[µ(m ˜ i ), µ(˜

M

we have:

= − max[µ(m ˜ 1, n ˜ 1 )] log2 [max(µ(m ˜ 1, n ˜ 1 ))]

M AX[µ(m ˜ i ), µ(˜ nj )]

d

= − max[µ(0.20, 0.22)] log2 [max(µ(0.20, 0.22))]

te

= −0.22 log2 (0.22) = 0.481

Ac ce p

M IN [µ(m ˜ i ), µ(˜ nj )]

= − min[µ(m ˜ 1, n ˜ 1 )] log2 [min(µ(m ˜ 1, n ˜ 1 ))]

= − min[µ(0.20, 0.22)] log2 [min(µ(0.20, 0.22))] = −0.20 log2 (0.20) = 0.464

˜ 1 can be computed as: Now, Eq. (17), the similarity between m ˜ 1 and n M IN [µ(m ˜ 1 ), µ(˜ n1 )] M AX[µ(m ˜ 1 ), µ(˜ n1 )] 0.464 = 0.481 = 0.96

S(m ˜ 1, n ˜1) =

Other remaining similarities between two events can be measured in a similar way. Results are presented in the following matrix form, as: July 21, 2018

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S(m ˜ i, n ˜j )



S(m ˜ 1, n ˜1)

  = S(m˜ 2, n˜ 1) 

S(m ˜ 3, n ˜1)

S(m ˜ 1, n ˜2) S(m ˜ 2, n ˜2) S(m ˜ 3, n ˜2)

42





 0.96 0.89 0.89       S(m ˜ 2, n ˜ 3 ) = 0.91 0.99 1     S(m ˜ 3, n ˜3) 0.91 0.99 1 S(m ˜ 1, n ˜3)

ip t

˜ 3 ) and (m ˜ 3, n ˜ 3 ) have the In the above FI similarity matrix, two set of events, viz., (m ˜ 2, n similarity parameter value 1. This indicates that these events m ˜ 2 and n ˜ 3 ; and m ˜ 3 and n ˜ 3 are similar.

us

Proof. Using Eq. (17), two events m ˜ i and n ˜ j are similar, iff:

cr

˜ j ) is equal to 1, two fuzzy events m ˜ i and n ˜ j are similar. Theorem 6: When S(m ˜ i, n

nj )] = M IN [µ(m ˜ i ), µ(˜ nj )] M AX[µ(m ˜ i ), µ(˜

(48)

Equality of this Eq. (48) only holds, iff µ(m ˜ i ) = µ(˜ nj ). In Eq. (48), if both µ(m ˜ i ) = µ(˜ nj )

an

are equal, then similarity parameter (as defined in Eq. (17)) gives value 1. Corollary 5: More granularization of information leads to increase the FIG value.

M

Proof. When the discretization process is initiated, it brings the information representation level from the zero-order information granularization (0-OIG) (refer to Definition 13) to the first-order information granularization (1-OIG) (refer to Definition 14) or more higher

d

order. In each level of granularization, it brings the upper and lower bound of each interval

te

(granule) more closer. This decreases the degree of membership of the element residing in

Ac ce p

the corresponding granule interval, thereby increasing the FIG value. ACKNOWLEDGEMENT

This research is supported by the Department of Science and Technology (DST)-SERB, Government of India, under Grant EEQ/2016/000021. R EFERENCES

[1] A. Singh, “Digital change detection techniques using remotely-sensed data,” International Journal of Remote Sensing, vol. 10, no. 6, pp. 989–1003, 1989.

[2] C. Bishop, Pattern Recognition and Machine Learning. New York, USA: Springer, 2006. [3] M. Roy, S. Ghosh, and A. Ghosh, “A novel approach for change detection of remotely sensed images using semisupervised multiple classifier system,” Information Sciences, vol. 269, pp. 35–47, 2014. [4] F. Yuan, K. E. Sawaya, B. C. Loeffelholz, and M. E. Bauer, “Land cover classification and change analysis of the twin cities (minnesota) metropolitan area by multitemporal landsat remote sensing,” Remote sensing of Environment, vol. 98, no. 2, pp. 317–328, 2005. [5] G. Foody, “Monitoring the magnitude of land-cover change around the southern limits of the sahara,” Photogrammetric Engineering and Remote Sensing, vol. 67, no. 7, pp. 841–847, 2001. July 21, 2018

DRAFT

Page 42 of 46

FOURTH REVISION SUBMITTED TO: APPLIED SOFT COMPUTING

43

[6] G. Camps-Valls, L. Gómez-Chova, J. Muñoz-Marí, J. L. Rojo-Álvarez, and M. Martínez-Ramón, “Kernel-based framework for multitemporal and multisource remote sensing data classification and change detection,” IEEE Transactions on Geoscience and Remote Sensing, vol. 46, no. 6, pp. 1822–1835, 2008. [7] S. Ghosh, L. Bruzzone, S. Patra, F. Bovolo, and A. Ghosh, “A context-sensitive technique for unsupervised change detection based on hopfield-type neural networks,” IEEE Transactions on Geoscience and Remote Sensing, vol. 45, no. 3, pp. 778–789, 2007.

ip t

[8] S. Ghosh, S. Patra, and A. Ghosh, “An unsupervised context-sensitive change detection technique based on modified self-organizing feature map neural network,” International Journal of Approximate Reasoning, vol. 50, no. 1, pp. 37–50, 2009.

cr

[9] F. Melgani, G. Moser, and S. B. Serpico, “Unsupervised change-detection methods for remote-sensing images,” Optical Engineering, vol. 41, no. 12, pp. 3288–3297, 2002.

[10] D. Liu, K. Song, J. R. Townshend, and P. Gong, “Using local transition probability models in Markov random fields

us

for forest change detection,” Remote Sensing of Environment, vol. 112, no. 5, pp. 2222–2231, 2008.

[11] T. Kasetkasem and P. K. Varshney, “An image change detection algorithm based on Markov random field models,” IEEE Transactions on Geoscience and Remote Sensing, vol. 40, no. 8, pp. 1815–1823, 2002. Remote Sensing, vol. 23, no. 12, pp. 2513–2518, 2002.

an

[12] X. Liu and R. G. L. Jr., “Urban change detection based on an artificial neural network,” International Journal of [13] A. Ghosh, N. S. Mishra, and S. Ghosh, “Fuzzy clustering algorithms for unsupervised change detection in remote

M

sensing images,” Information Sciences, vol. 181, no. 4, pp. 699–715, 2011.

[14] G. Pajares, “A hopfield neural network for image change detection,” IEEE Transactions on Neural Networks, vol. 17, no. 5, pp. 1250–1264, 2006.

[15] O. Chapelle, B. Schölkopf, and A. Zien, Semi-supervised Learning. Cambridge: MIT Press, 2006.

d

[16] M. Gong, Z. Zhou, and J. Ma, “Change detection in synthetic aperture radar images based on image fusion and fuzzy clustering,” IEEE Transactions on Image Processing, vol. 21, no. 4, pp. 2141–2151, 2012.

te

[17] M. Gong, L. Su, M. Jia, and W. Chen, “Fuzzy clustering with a modified MRF energy function for change detection in synthetic aperture radar images,” IEEE Transactions on Fuzzy Systems, vol. 22, no. 1, pp. 98–109, 2014.

Ac ce p

[18] H. Li, M. Gong, Q. Wang, J. Liu, and L. Su, “A multiobjective fuzzy clustering method for change detection in SAR images,” Applied Soft Computing, vol. 46, pp. 767–777, 2016.

[19] W. Shi, P. Shao, M. Hao, P. He, and J. Wang, “Fuzzy topology–based method for unsupervised change detection,” Remote sensing letters, vol. 7, no. 1, pp. 81–90, 2016.

[20] P. Singh, Applications of Soft Computing in Time Series Forecasting: Simulation and Modeling Techniques. Springer, 2015, vol. 330.

[21] Y. Min and Z. Sen, “Generalized fuzzy entropy and its applications,” in Fourth International Conference on Signal Processing Proceedings, vol. 2, 1998, pp. 1197–1200.

[22] G. Peters and R. Weber, “DCC: a framework for dynamic granular clustering,” Granular Computing, vol. 1, no. 1, pp. 1–11, 2016.

[23] R. B. Ash and C. A. Dolans-Dade, Probability and Measure Theory, 2nd ed. Elsevier, 2005. [24] A. Kolmogorov, “Logical basis for information theory and probability theory,” IEEE Transactions on Information Theory, vol. 14, no. 5, pp. 662–664, 1968. [25] C. E. Shannon, “A mathematical theory of communication,” SIGMOBILE Mob. Comput. Commun. Rev., vol. 5, no. 1, pp. 3–55, 2001. [26] R. A. H. Lorentz, “On the entropy of a function,” Journal of Approximation Theory, vol. 158, no. 2, pp. 145–150, 2009.

July 21, 2018

DRAFT

Page 43 of 46

FOURTH REVISION SUBMITTED TO: APPLIED SOFT COMPUTING

44

[27] P. Singh, “Indian summer monsoon rainfall (ISMR) forecasting using time series data: A fuzzy-entropy-neuro based expert system,” Geoscience Frontiers, 2017. [28] D. P. Filev and R. R. Yager, “Operations on fuzzy numbers via fuzzy reasoning,” Fuzzy Sets and Systems, vol. 91, no. 2, pp. 137–142, 1997. [29] Y. ming Liu, “Some properties of convex fuzzy sets,” Journal of Mathematical Analysis and Applications, vol. 111, no. 1, pp. 119–129, 1985.

ip t

[30] X. M. Yanga and F. M. Yangb, “A property on convex fuzzy sets,” Fuzzy Sets and Systems, vol. 126, pp. 269–271, 2002.

[31] X.-H. Yuan and E. Lee, “The definition of convex fuzzy subset,” Computers & Mathematics with Applications, vol. 47,

cr

no. 1, pp. 101–113, 2004.

[32] G. B. Thomas, M. D. Weir, and J. R. Hass, Thomas’ Calculus, 12nd ed. Pearson, 2015. 430–434, 1982.

us

[33] M. L. Puri and D. A. Ralescu, “Integration on fuzzy sets,” Advances in Applied Mathematics, vol. 3, no. 4, pp. [34] P. Lingras, F. Haider, , and M. Triff, “Granular meta-clustering based on hierarchical, network, and temporal connections,” Granular Computing, vol. 1, no. 1, pp. 71–92, 2016.

an

[35] M. Antonelli, P. Ducange, B. Lazzerini, , and F. Marcelloni, “Multi-objective evolutionary design of granular rule-based classifiers,” Granular Computing, vol. 1, no. 1, pp. 37–58, 2016.

[36] L. A. Zadeh, “Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy

M

logic,” Fuzzy Sets and Systems, vol. 90, no. 2, pp. 111–127, 1997.

[37] L. Livi and A. Sadeghian, “Granular computing, computational intelligence, and the analysis of non-geometric input spaces,” Granular Computing, vol. 1, no. 1, pp. 13–20, 2016.

[38] Z. Xu and H. Wang, “Managing multi-granularity linguistic information in qualitative group decision making: an

d

overview,” Granular Computing, vol. 1, no. 1, pp. 21–35, 2016. Raton, 2013.

CRC Press / Francis Taylor, Boca

te

[39] W. Pedrycz, Granular Computing: Analysis and Design of Intelligent Systems.

[40] Y. Yao, “Granular computing for data mining,” in Proceedings of SPIE Conference on Data Mining, Intrusion Detection,

Ac ce p

Information Assurance, and Data Networks Security, 2006.

[41] P. Singh and B. Borah, “High-order fuzzy-neuro expert system for time series forecasting,” Knowledge-Based Systems, vol. 46, pp. 12–21, 2013.

[42] ——, “An efficient time series forecasting model based on fuzzy time series,” Engineering Applications of Artificial Intelligence, vol. 26, no. 10, pp. 2443–2457, 2013.

[43] P. Singh, “High-order fuzzy-neuro-entropy integration-based expert system for time series forecasting,” Neural Computing and Applications, pp. 1–18, 2016.

[44] ——, “A brief review of modeling approaches based on fuzzy time series,” International Journal of Machine Learning and Cybernetics, vol. 8, no. 2, pp. 397–420, 2017.

[45] S. Roy and U. Chakraborty, Soft Computing, 1st ed. Pearson Education India, 2013. [46] T. J. Ross, Fuzzy logic with engineering applications, 3rd ed. Wiley India Pvt. Ltd., 2013. [47] W. satellite images of India, “National Satellite Meteorological Centre, Ministry of Earth Sciences, Govt. of India,” http://satellite.imd.gov.in/insat.htm, 2017. [48] W. satellite images of typhoon, “Earth Observatory Center, NASA,” http://earthobservatory.nasa.gov/, 2017. [49] H. S. Telescope, “European Southern Observatory, Germany,” https://www.spacetelescope.org/, 2017. [50] S. Chen and D. Zhang, “Robust image segmentation using FCM with spatial constraints based on new kernel-induced

July 21, 2018

DRAFT

Page 44 of 46

FOURTH REVISION SUBMITTED TO: APPLIED SOFT COMPUTING

45

distance measure,” IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 34, no. 4, pp. 1907–1916, 2004. [51] N. R. Pal and J. C. Bezdek, “On cluster validity for the fuzzy c-means model,” IEEE Transactions on Fuzzy Systems, vol. 3, no. 3, pp. 370–379, 1995. [52] Y. Wang and M. Weyrich, “An adaptive image processing system based on incremental learning for industrial applications,” in Proceedings of the 2014 IEEE Emerging Technology and Factory Automation (ETFA).

Ac ce p

te

d

M

an

us

cr

ip t

Spain: IEEE, 2014, pp. 1–4.

Barcelona,

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*Highlights (for review)

Highlights

Ac ce p

te

d

M

an

us

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· This study helps to define a collection of various uncertain events/information as a fuzzy-information (FI). · A new measure, which is called “fuzzy information-gain (FIG)”, is defined in this study. · Using max and min operators of fuzzy sets, a new measure, which is called “average fuzzy information-gain (AFIG)”, is defined in this study. · We also define a convex property for the FI, which is called “FI Convex (FIC)”. · This study shows how FI can be represented in a granular manner. · Two case studies are discussed using high resolution weather satellite images.

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