A framework for objective image quality measures based on intuitionistic fuzzy sets

Accepted Manuscript Title: A Framework for Objective Image Quality Measures Based on Intuitionistic Fuzzy Sets Author: M. Hassaballah A. Ghareeb PII: DOI: Reference:

S1568-4946(17)30170-9 http://dx.doi.org/doi:10.1016/j.asoc.2017.03.046 ASOC 4131

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

19-7-2016 21-3-2017 30-3-2017

Please cite this article as: M. Hassaballah, A. Ghareeb, A Framework for Objective Image Quality Measures Based on Intuitionistic Fuzzy Sets, (2017), http://dx.doi.org/10.1016/j.asoc.2017.03.046 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.

Applied Soft Computing

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Applied Soft Computing 00 (2017) 1–24

M. Hassaballaha , A. Ghareebb,∗

Science Department, Faculty of Computers and Information, South Valley University, Luxor, Egypt. Department, Faculty of Science, South Valley University, Qena 83523, Egypt. The authors contributed equally to this work

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a Computer

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A Framework for Objective Image Quality Measures Based on Intuitionistic Fuzzy Sets

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b Mathematics

Abstract

c 2017 Published by Elsevier Ltd.

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Measuring the distance or similarity objectively between images is an essential and a challenging problem in various image processing and pattern recognition applications. As it is very difficult to find a certain measure that can be successfully applied to all kinds of images comparisons-related problems in the same time, it is appropriate to look for new approaches for measuring the similarity. Several similarity measures tested on numerical cases are developed in the literature based on Intuitionistic Fuzzy Sets (IFSs) without evaluation on real data. This paper introduces a framework for using the similarity measures on IFSs in image processing field, specifically for image comparison. First, some existing similarity measures are discussed and highlighted their properties. Then, modelling digital images using IFSs is explained. Moreover, the paper introduces an intuitionistic fuzzy based image quality index measure. Second, for improving the perceived visual quality of these IFS-based similarity measures, construction of neighborhood-based similarity is proposed, which takes into consideration homogeneity of images. Finally, the proposed framework is verified on real world natural images under various types of image distortions. Experimental results confirm the effectiveness of the proposed framework in measuring the similarity between images.

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Keywords: Image processing, Image comparison, Image similarity, Similarity measures, Intuitionistic fuzzy sets.

1. Introduction

In computer vision and image processing applications, measuring the similarity between images plays an important role. A visual task (e.g., object tracking, image classification) relies frequently on evaluation of similarity between objects within images [1–3]. Content-based image retrieval systems often look up in a database of images for all images that are close to a given query image based on some similarity measures and retrieve the most relevant ones as the result of this query. Thus, the performance of the retrieval system may depend on defining a appropriate similarity measure [4–6]. On the other hand, it is well known that images may be subject to several types of distortions during its acquisition, storage, transmission or compression [7], which in turns deteriorate the visual quality of these images. In this case, there is a must for efficient similarity measures for image processing/analysis tasks. For instance, similarity measures are utilized for evaluating compression algorithms as well as comparing image restoration methods. Also, similarity measures are helpful for the comparison of algorithms dedicated to noise reduction [8–10]. ∗ Corresponding

author. Email address: [email protected] (A. Ghareeb)

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Currently, the peak signal to noise ratio (PSNR) and the mean squared error (MSE) are still applied excessively for image quality assessment regardless of their questionable performance. As these methods are usually have low computational complexity and are independent of viewing conditions or visual observations. But, the PSNR and MSE have the defect of not very well matching the perceptual similarity. Several researchers have investigated the ability of developing new quantitative similarity measures that better adapted to the human perception similarity. For direct comparison of non-binary images without calculating visible features (e.g., edges, shapes,...), Di Gesu and V. Starovoitov [11] proposed image distance functions based on local distances combining intensity and structural image features. With a view to improve the perceptive behaviour of pixel-based similarity measures, neighbourhoodbased similarity measures that incorporates homogeneity in evaluating the image quality procedure are proposed in [12]. Wang and Bovik [13] designed a universal objective image quality index by modelling image distortion as a combination of three factors: loss of correlation, luminance distortion, and contrast distortion. Taking into account that in the case of using this measure in comparing color images, the correlation among the color image channels shall be considered and this measure cannot provide optimal performance [14]. Alternative methods based on the fuzzy sets theory and its counterparts are also investigated in the literature [15–17]. For example, following the procedure inspired in [13], Greˇcova and Morillas [14] proposed a method for perceptual color image similarity using fuzzy metrics. In [18], a Dengfeng-Chuntian’s operator-based similarity measure is introduced. While, in [19], a fuzzy image metric utilizing the Sugeno’s fuzzy integral is defined, where the fuzzy integral is utilized to bridge the gap between this metric model and pure subjective evaluation. Van der Weken et al. [20] investigated the applicability of several fuzzy similarity measures to images comparisons. Additionally, a large number of similarity measures owing different properties are defined on the intuitionistic fuzzy sets, introduced by Atanassov [21] as a generalization of the traditional fuzzy sets [22], to compare between two intuitionistic fuzzy sets [23, 24]. The present paper focuses exactly on this type of similarity measures defined on the intuitionistic fuzzy sets (IFSs). Broadly, the IFS may characterize the real status of the information more accurate than the traditional fuzzy set because it incorporates a second function, i.e., non-membership function along with the membership function of the conventional FS. That is, it allocates to each element in the set a membership degree, a non-membership degree and a hesitation degree. The IFS is considered a more effective way to deal with vagueness than fuzzy set. For these attributes, many researchers use the IFS instead of the original FS in various applications including: decision making [25], segmentation [26, 27], and noise reduction [28] as well as pattern recognition [29–31]. In the context of the paper subject, similarity measures on the IFSs, a comparative analysis from the pattern recognition point of view can be found in [32, 33]. While, a recent survey for these similarity measures between the IFSs is presented in [34]. Iancu [35] extended crisp cardinality measures to IFSs-based measures depending on the Frank t-norm family. In [36], a similarity measure is introduced by the direct operation on the membership, nonmembership and hesitation functions as well as the upper bound of two IFSs’ membership functions. In the same direction, a cosine similarity measure and its weighted version are proposed in [37] based on the concept of the cosine measure for fuzzy sets, considering also the information carried by the membership function and the non-membership function in the IFSs as a vector representation with the two elements. Farhadinia [38] developed another similarity measure based on a distance defined on an interval using convex combination of endpoints that considers the property of min and max operators. In spite of many similarity measures for the IFSs have been introduced in the previous studies, the definition of a similarity measure is still an open problem achieving more interest in several applications. Liang and Shi [39] reported with some examples that ” several of the existing similarity measures are not always effective in some cases”. In certain cases, these measures are inefficient and need more modifications for better results [35, 40]. More importantly, some of these similarity measures may not satisfy the well-known axioms of similarity and give counter-intuitive cases. Moreover, these measures were tested using only numerical examples and artificial cases from pattern recognition and did not test in real data. These numerical examples or artificial cases may be incorrect or inefficient to evaluate the performance of these similarity measures [41]. Motivated by these observation, we investigate applying the IFS-based similarity measures in image processing filed, specifically images comparison using real data rather than numerical examples. To this end, we conduct a comparison between these measures and the traditional PSNR/MSE measures using natural images, which illustrates the superiority of the IFS-based similarity measures. Modelling digital images using IFS theory (i.e., constructing IFS from the image pixels) is explained. Furthermore, a modification for the quality index measure [13] in the context of IFS is introduced as well as using the homogeneity of images in neighbourhood2

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based similarity is investigated for providing meaningful comparison under different types of image distortions. The rest of this paper is organized as follows. In Section 2, some mathematical preliminaries to the basic notions of IFSs as well as construction methods of IFSs for image processing applications are provided. The IFS-based similarity measures which are applicable in image processing are discussed in Section 3. The proposed IFS-based image quality index measure is introduced in Section 4. In Section 5, we describe the scheme of constructing neighborhood-based similarity and suitable properties of images that may consider for evaluating similarity measures. The experimental evaluation with different gray scale images is carried out in Section 6. Finally, conclusions and potential future work are given in Section 7. 2. Preliminaries

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This section recalls basic definitions that will be used in the rest of this paper. Furthermore, it presents the principles of modelling digital images using intuitionistic fuzzy stes.

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2.1. Intuitionistic fuzzy sets

Let X be a fixed universe, an intuitionistic fuzzy set A in X is defined by Atanassov [21] as (1)

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A = {(xi , µA (xi ), νA (xi ))|xi ∈ X}

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where µA : X −→ [0, 1] and νA : X −→ [0, 1] are the membership and non-membership degrees, respectively. Moreover, 0 ≤ µA (xi ) + νA (xi ) ≤ 1 for each xi ∈ X. Atanassov also defined another degree πA : X −→ [0, 1] through πA (xi ) = 1 − µA (xi ) − νA (xi ), called hesitancy degree of xi to A and corresponding to the degree of uncertainty. Obviously, 0 ≤ πA (xi ) ≤ 1, and for any traditional fuzzy set A, πA (xi ) = 0, ∀xi ∈ X. By L, we denote the set {(x, y)|(x, y) ∈ [0, 1] × [0, 1] and 0 ≤ x + y ≤ 1}. It is clear that 0L = (0, 1) and 1L = (1, 0) are the smallest and the greatest element in L, respectively.

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For an intuitionistic fuzzy set A, the operator Da is defined by

Da (A) = {(xi , µA (xi ) + aπA (xi ), νA (xi ) + (1 − a)πA (xi ))|xi ∈ X}

(2)

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where a ∈ [0, 1]. Obviously, Da (A) is fuzzy set. The operator Da has been extended to another operator, namely, Fa,b as follows: Fa,b (A) = {(xi , µA (xi ) + aπA (xi ), νA (xi ) + b.πA (xi ))|xi ∈ X} (3)

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where a, b ∈ [0, 1] and a + b ≤ 1. The operators Da and Fa,b are not only significant from the theoretical point of view but they are also significant from the applications point of view. The Da has been applied for classifier recognizing tasks imbalanced classes constructing and the Fa,b operator for image recognition. Several interesting properties of these operators Da and Fa,b can be found in [42]. 2.2. Definition of a similarity measure

Regarding measuring the similarity between objects that can be identified with IFSs, there is no unique definition and several definitions for similarity measures are proposed. The most appropriate one for this study is that introduced by Mitchell [18], where a function S : IFS (X) × IFS (X) → [0, 1] is called a similarity degree between two IFS sets A ∈ IFS (X) and B ∈ IFS (X), if S(A, B) satisfies the following four properties: (SP1) (SP2) (SP3) (SP4)

0 ≤ S(A, B) ≤ 1; S(A, B) = 1 if and only if A = B; S(A, B) = S(B, A); If A ⊆ B ⊆ C, then S(A, B) ≥ S(A, C), and S(B, C) ≥ S(A, C).

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2.3. Modelling digital images using IFS theory The IFS theory is an adequate tool for dealing with the imperfect information. Construction of both membership and non-membership is a common task and challenge in its applications. Two methods for constructing IFSs membership and non-membership functions from digital images are investigated in this work. The first one is by Jurio et al. [43], which is based on the idea of using fixed indeterminacy index for each element. The method also considers the uncertainty of experts in selecting the membership and non-membership degrees of the element beside the indeterminacy index. From a fuzzy set F, we have the following intuitionistic fuzzy set: A = {hxi , I(µF (xi ), π(xi ), α(xi ))i|xi ∈ X}

I(a, b, α) = (Iµ (a, b, α), Iν (a, b, α)), and

Iν (a, b, α) = 1 − a(1 − αb) − αb

If α(xi ) = 1 for each xi ∈ X, the intuitionistic fuzzy set A takes the form

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Iµ (a, b, α) = a(1 − αb),

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where, π, α : X −→ [0, 1] and the mapping I : [0, 1]2 × [0, 1] −→ L defined by

A = {(xi , µF (xi )(1 − π(xi )), 1 − µF (xi )(1 − π(xi )) − π(xi ))|xi ∈ X}

(4)

(5) (6)

(7)

G(µF (xi )) =

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While, the second method is proposed by Chaira [27] based on the Sugeno intuitionistic fuzzy generator 1 − µF (xi ) , 1 + λµF (xi )

λ>0

(8)

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where, G(1) = 0 and G(0) = 1. With the help of the Sugeno intuitionistic fuzzy generator, the IFS A is given by Aλ = {hxi , µF (xi ), and the hesitation degree is

1 − µF (xi ) i|xi ∈ X} 1 + λµF (xi )

(9)

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1 − µF (xi ) (10) 1 + λµF (xi ) Generally, an image in fuzzy sets theory can be regarded as a matrix of fuzzy singletons such that each element within this matrix indicates for each gray level g a membership value µA (gi j ), corresponding to the (i, j)-th pixel, considering a predefined image property such as brightness or homogeneity. An M × N pixels image A of L gray level ranging between 0 and L − 1 is represented by the intuitionistic fuzzy set as follows

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πAλ (xi ) = 1 − µF (xi ) −

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A = {hgi j , µA (gi j ), νA (gi j )i|gi j = 0, 1, . . . , L − 1}

(11)

Where, µA (gi j ) and νA (gi j ) denote the degree of membership and non-membership of (i, j)-th pixel of the set A associated with the predefined image property, respectively, with i = 1, 2, . . . , M , j = 1, 2, . . . , N,. This scheme for intuitionistic fuzzy image processing is illustrated in Fig. 1. An illustration for construction of IFS images from gray scale images using these two methods are shown in Fig. 2 and Fig. 3, respectively.

Figure 1: An illustration for Modelling digital images using IFS theory.

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Figure 2: IFS images construction using Jurio et al. method [43] with α(gi j ) = 1 and π(gi j ) = 1: (a) Gray scale image, (b) Membership, (c) Non-membership, (d) Hesitancy components.

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Figure 3: IFS images construction using Chaira’s method [27] with λ = 1: (a) Gray scale image, (b) Membership, (c) Non-membership, (d) Hesitancy components.

In our experiments, it was observed that increasing the value of λ decreases the fuzzy complement or the Sugeno generator thus the non-membership value decreases and the hesitation degree increases; while the suitable value of λ is 1. Also, we have defined the intuitionistic fuzzy sets based on the second method [27] with respect to α(gi j ) = 1 and π(gi j ) = 0.2. 3. IFS-based similarity measures

This section provides comprehensive analysis of existing similarity measures between IFSs. Let S (A, B) be a similarity measure between two IFS sets A ∈ IFS s(X) and B ∈ IFS s(X). Where, each set represents an IFS image of size M × N. One of the pioneer similarity measures for IFSs is proposed by Chen [44] in the following form P (i, j)∈X |(µA (i, j) − νA (i, j)) − (µ B (i, j) − ν B (i, j))| S C (A, B) = 1 − (12) 2MN 6

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S M (A, B) =

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Hong and Kim [45] proved by some numerical examples that this similarity measure does not fit well in some cases and thus they introduced a modified measure in the form P (i, j)∈X (|µA (i, j) − µ B (i, j)| + |νA (i, j) − ν B (i, j)|) (13) S HK (A, B) = 1 − 2MN In [46], Denfeng and Chuntain proposed the following similarity measure between the IFSs rP p p (i, j)∈X |ϕA (i, j) − ϕ B (i, j)| S DC (A, B) = 1 − (14) MN where µB (i, j) + 1 − νB (i, j) µA (i, j) + 1 − νA (i, j) , ϕB (i, j) = , and 1 ≤ p < ∞. ϕA (i, j) = 2 2 To correct the counter-intuitive problem in the Denfeng and Chuntain’s measure, Mitchell [18] introduced a more pragmatic similarity measure utilizing a statistical method by interpreting the IFSs as ensembles of ordered fuzzy sets 1 (σµ (A, B) + σν (A, B)), 2

rP σµ = 1 −

p

(i, j)∈X

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where,

(15)

|µA (i, j) − µB (i, j)| p , MN

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and

|νA (i, j) − νB (i, j)| p . MN Liang and Shi [39] presented three similarity measures and proved that their measures are more informative for the nature of the IFSs than the previous measures s  p P ϕ (i, j) + ϕ (i, j) p µ ν (i, j)∈X S ep (A, B) = 1 − , (16) MN where, ϕµ (i, j) = |µA (i, j) − µB (i, j)|/2 and ϕν (i, j) = |(1 − νA (i, j)) − (1 − νB (i, j))|/2. Clearly, this similarity measure is another modification of the Denfeng and Chuntain’s measure [46], where the difference of the IFSs is discerned. s P
p p (i, j)∈X ϕ s1 (i, j) + ϕ s2 (i, j) p S s (A, B) = 1 − , (17) MN rP p

(i, j)∈X

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σν = 1 −

where ϕ s1 (i, j) = |mA1 − mB1 |/2, ϕ s2 (i, j) = |mA2 − mB2 |/2, mA1 (i, j) = |µA (i, j) + mA (i, j)|/2, mB1 (i, j) = |µB (i, j) + mB (i, j)|/2, mA2 (i, j) = |1 − νA (i, j) + mA (i, j)|/2, mB1 (i, j) = |1 − νB (i, j) + mB (i, j)|/2, mA (i, j) = |1 − νA (i, j) + µA (i, j)|/2, mB (i, j) = |1 − νB (i, j) + µB (i, j)|/2. rP p p (i, j)∈X (η1 (i, j) + η2 (i, j) + η3 (i, j)) p , (18) S h (A, B) = 1 − 3MN where η1 (i, j) = φµ (i, j) + φν (i, j), η2 (i, j) = |φA (i, j) − φB (i, j)|, φA (i, j) = (µA (i, j) + 1 − νA (i, j))/2, φB (i, j) = (µB (i, j) + 1 − νB (i, j))/2, η3 (i, j) = max(lA (i, j), lB (i, j)) − min(lA (i, j), lB (i, j)), lA (i, j) = (1 − µA (i, j) − νA (i, j))/2, lB (i, j) = (1 − µB (i, j) − νB (i, j))/2. Using a different approach employing the Hausdorff distance dH (A, B) between IFSs, Hung and Yang [47] suggested the next three similarity measures:

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S 1HY (A, B) = 1 − dH (A, B), e−dH (A,B) − e−1 , 1 − e−1 1 − dH (A, B) S 2HY (A, B) = , 1 + dH (A, B)

S 2HY (A, B) =

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(19)

where,

1 X max(|µA (i, j) − µB (i, j)|, |νA (i, j) − νB (i, j)|) MN (i, j)∈X

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dH (A, B) =

P S HY (A, B) = 1 − L p (A, B)

where,

(20)

X (|µA (i, j) − µB (i, j)| p + |νA (i, j) − νB (i, j)| p ) 1p , p≥1 MN (i, j)∈X

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L p (A, B) =

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They proved that the similarity measures generated based on the Hausdorff distance are suited to be used in the linguistic variables. Also, Hung and Yang [48] proposed other similarity measures induced by L p metric rather than Hausdorff distance in the form

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Wang and Xin [49] introduced a similarity measure depending on the distance between the IFSs of the following form

(21)

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( 1 X |µA (i, j) − µB (i, j)| + |νA (i, j) − νB (i, j)| S WX (A, B) = 1 − MN (i, j)∈X 4 ) max {|µA (i, j) − µB (i, j)|, |νA (i, j) − νB (i, j)|} + . 2

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Similarity measures on three different kinds of fuzzy sets are proposed by Zhang and Fu in [15], which for the case of the IFSs takes the form S ZF (A, B) = 1 −


1 X |δA (i, j) − δB (i, j)| + |αA (i, j) − αB (i, j)| , 2MN (i, j)∈X

(22)

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where, δA (i, j) = µA (i, j) + πA (i, j)µA (i, j), δB (i, j) = µB (i, j) + πB (i, j)µB (i, j), αA (i, j) = νA (i, j) + πA (i, j)νA (i, j), and αB (i, j) = νB (i, j) + πB (i, j)νB (i, j). The cosine similarity measure between the IFSs proposed by Ye [37] takes the form S Y (A, B) =

1 X µA (i, j)µB (i, j) + νA (i, j)νB (i, j) p p MN (i, j)∈X µA (i, j)2 + νA (i, j)2 µB (i, j)2 + νB (i, j)2

(23)

Recently, Song et al. [50] proposed the following similarity measure between the IFSs p 1 X p µA (i, j)µB (i, j) + 2 νA (i, j)νB (i, j) 2MN (i, j)∈X  p p + πA (i, j)πB (i, j) + (1 − νA (i, j))(1 − νB (i, j))

S S (A, B) =

(24)

All these IFSs based similarity measures are investigated for images comparison, where the two IFSs A and B are the compared images. 8

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4. Intuitionistic fuzzy quality index Wang and Bovik [13] introduced a universal objective image quality index measure based on the idea of modelling any image distortion as a combination of loss of correlation (LC), luminance distortion (LD), and contrast distortion (CD). In this section, we extend this universal image quality index measure to the space of IFSs to increase its discriminative power via increasing its ability in measuring information loss (vague) occurred during the degradation processes of digital images.

1 [Ψ(µA , µB ) + Ψ(νA , νB )], 2

where, for any U and V,

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QIFS (A, B) =

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Let A = {hgi j , µA (gi j ), νA (gi j )i : gi j = 1, 2, . . . , p} be the original image signals and B = {hgi j , µB (gi j ), νB (gi j )i : gi j = 1, 2, . . . , p} be the test image signals (processed image) in intuitionistic fuzzy form. The intuitionistic fuzzy quality index between the images A and B can be defined as

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4CU,V U V  , 2 2 + S V2 ) U + V Pp Pp gi j =1 U(gi j ) gi j =1 V(gi j ) U= , V= , p p 2 2 Pp  Pp  U(g ) − V(g ) − U V i j i j g =1 g =1 ij ij S U2 = , S V2 = p−1 p−1 and the correlation in the IFS space can be computed based on [51] using   Pp  U(g ) − U V(g ) − V /(p − 1) i j i j gi j =1 CU,V = SU · SV (S U2

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Ψ(U, V) =

(25)

(26)

(27)

(28)

(29)

Equation (26) can be represented as the product of three terms as follows

with

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Ψ(U, V) = L(U, V).D(U, V).C(U, V),

L(U, V) = CU,V ,

2U V (U)2

+

(V)2

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and

C(U, V) =

2S U S V S U2 + S V2

(31)

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From equation (30), we have

D(U, V) =

(30)

LC = 12 (L(µA , µB ) + L(νA , νB )) is the correlation coefficient between the images A and B, LC tells us not only the strength of relationship between A and B but also the fact that the two images are positively or negatively related, and its dynamic range is always [−1, 1]. Moreover, if the image A and B are linearly related, then LC = 1. LD = 21 (D(µA , µB ) + D(νA , νB )) with a range of [0, 1] measures the luminance closeness degree between the two images. The superior value 1 is occurred when µA = µB and νA = νB . CD = 12 (C(µA , µB ) + C(νA , νB )) measures the contrast similarity between the images. The range of CD is also [0, 1] and its superior value occurs when S µA = S µB and S νA = S νB . It is obviously that the dynamic range of the QIFS (A, B) index is [−1, 1] and its best value QIFS (A, B) = 1 occurs if and only if µB (gi j ) = 2µA − µA (gi j ) and νB (gi j ) = 2νA − νA (gi j ) for each gi j = 1, 2, . . . , p. Additionally, the proposed intuitionistic fuzzy quality index Eq. (26) is still similar to the original index [13] regarding modelling any image distortion as a summation of the three factors: loss of correlation, luminance distortion and contrast distortion as illustrated in Eq. (30). Note that Eq. (26) represents simply the fuzzy version of the proposed IFS quality index. The advantages of the IFS quality index and its fuzzy version are discussed in the experimental results. 9

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5. Homogeneity neighborhood-based similarity construction

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Figure 4: Intuitionistic fuzzy model of an image: each pixel consists of membership and non-membership degree of intensity value.

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For further enhancement of the perceptive behavior of similarity measures, a neighborhood-based similarity scheme is introduced following the procedure inspired in [20]. The main idea of the neighborhood-based similarity measures is based on combining the homogeneity of images in the different neighborhoods. At the beginning, the compared images A and B are divided into disjoint 4 × 4 regions and we measure the similarity between the asymmetric regions of these two images using pixel-based similarity measures discussed in Section 3. Suppose that each image is divided into K regions of size 4 × 4, and let the restricted similarity S (Ai , Bi ) between two regions Ai and Bi of A and B, respectively; then the total similarity S˜ between A and B is the weighted average of the similarities in asymmetric disjoint image regions. That is, (32)

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K 1 X ˜ S (A, B) = ωk .S (Ai , Bi ) K k=1

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where, the weight ωk is the similarity between the homogeneity hAi of region Ai of the image A and the homogeneity hBi of region Bi of the image B. In order to compute the homogeneity of any region within the image in IFSs space, we have to understand well the intuitionistic fuzzy model of a digital image; where, each image pixel consists of two values the first is the membership degree of intensity while the second represents the non-membership degree of intensity (see Fig. 4). It is clear that a pixel has a maximum intensity if it has a maximum degree of membership and minimum degree of non-membership, while it has a minimum intensity if it has a minimum degree of membership and maximum degree of non-membership. Further, the homogeneity of the image can be calculated as the similarity between the gray-values of the pixel with maximum intensity and the gray-values of the pixel with minimum intensity using the following resemblance relation ( 1 − δ(a,b) α , if δ(a, b) < α; R(a, b) = (33) 0, otherwise. where δ is the Hausdorff distance and α = 12 . The function R evaluates the similarity between two intuitionistic values, where the distance commensurate with the similarity degree inversely. In this context, the homogeneity of the image A is computed as ! hA = R ( max µA , min νA ), ( min µA , max νA ) . (34) (i, j)∈A

The weight ωk is calculated using

(i, j)∈A

(i, j)∈A

ωk = R(hA , hB ),

(i, j)∈A

(35)

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6. Experimental results In order to investigate implementation possibility of the IFS-based similarity measures for images comparisons, we conduct several experiments using real images with various types of distortions. The following factors are considered in the experiments:

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• Reflexivity of a similarity measure, that is, the similarity measure value between the image and itself is equal to one. • Reaction to images distortion, If one added some noise to an image such as salt & pepper, enlightening, Gaussian noise,...etc, the similarity measure between the original image and the noisy one should not be affected too much, while it should show some decreasing with increasing in noise percentage.

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• Symmetry of a similarity measure, which means that the value of the similarity measure between two images not rely on their order.

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We also compare the performance of the IFS-based measures with Structural Similarity Index (SSIM) [52] and with the classical pixel-based images comparison metrics, namely mean square error (MSE) and peak signal to noise ratio (PSNR), where 1 X |A(i, j) − B(i, j)|2 (36) MS E(A, B) = MN (i, j)∈X and

PS NR(A, B) = 20 log10 √

255 MS E(A, B)

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6.1. Comparison with the MSE/PSNR and SSIM metrics To make the comparison, we use standard images of size 512 × 512 pixels such as ’Barbara’ and ’Lena’ as well as highly structured images collected from Internet called ’Zebra’, and ’Hiroshima Building’ with different kinds of deformations such as impulsive salt & pepper noise, multiplicative speckle noise, blurring, and JPEG compression. The original image and its distorted versions are shown in Fig. 5, Fig. 6 and Fig. 7 for ’Barbara’, Zebra’ and ’Hiroshima Building’ images, respectively. The performance of the IFS-based similarity measures against the MSE/PSNR metrics and the SSIM measure for these tested images is reported in Table 1, Table 2 and Table 3, respectively. It should be noted that the larger the value of a similarity measure, the more the similarity between the two images. It is clear that the reflexivity of the IFS-based similarity measures is achieved where the similarity between the image and itself is equal to 1 for all these IFS-based measures; while it is zero for MSE. On the other hand, both MSE and PSNR achieve poor performance compared to the IFS-based measures as shown in the three tables. Where, the values of MSE and PSNR are almost constants and the change is not obvious, while these values for the IFS-based measures express the actual difference in visual quality between the image and its distorted version. Besides, the obtained results by the IFS-based measures are comparable with those of SSIM; and in some case, the IFS-based measures outperform the SSIM measure as shown in Table 3 for case (a) vs. (d) the measures S y and S s achieve better performance than the SSIM. Also, for the case (a) vs. (f), it is clear that the IFS S M measure achieves better performance than the SSIM measure. Also, we note that the S c , S HK and S DC measures achieve similar performance with p = 1. While, they showed different performance on artificial classification problems using numerical examples form pattern recognition [37], [38], and [53]. Indeed, this contradictory performance supports our claim that ”testing the performance of a similarity measure using only numerical examples does not give reliable evaluation”, because numerical examples do not have any practical meaning. Furthermore, we tested the performance of the IFS-based measures in the case of medical images. The medical images of size 512 × 512 pixels shown in Fig. 8 are used in this experiment, where the JPEG compression distortion is replaced by Additive Gaussian because medical images are compressed with lossless compression rather than lossy compression (e.g., JPEG). The performance of the IFS-based measures compared to MSE/PSNR and SSIM measures is given in Table 4. One can note that the IFS-based measures performs favorably compared to the MSE/PSNR and SSIM measures, in some cases, they show some advantage in the performance. For instance, the SSIM measure achieves poorly for enlightened conditions (i.e. (a) vs. (d)), while most IFS-based measures give acceptable results for the same case as shown in Table 4. 11

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Figure 5: (a) The original ’Barbara’ image, (b) Impulsive salt & pepper12 noise, (c) Multiplicative speckle noise, (d) Enlightened, (e) Blurring, (f) JPEG compression.

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Figure 6: (a) The original ’Zebra’ image, (b) Impulsive salt & pepper 13 noise. (c) Multiplicative speckle noise, (d) Enlightened, (e) Blurring, (f) JPEG compression.

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Figure 7: (a) The original ’Hiroshima Building’ image, (b) Impulsive 14 salt & pepper noise, (c) Multiplicative speckle noise, (d) Enlightened, (e) Blurring, (f) JPEG compression.

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Figure 8: (a) The original ’Medical’ image, (b) Impulsive salt & pepper15noise, (c) Multiplicative speckle noise, (d) Enlightened, (e) Blurring, (f) Additive Gaussian noise.

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Table 1: Performance of the IFS-based measures compared to MSE/PSNR and SSIM for ’Barbara’ image.

(a) vs. (e) 250.41 24.144 0.94648

(a) vs. (f) 252.67 24.105 0.89337

0.99412 0.99412 0.99412 0.92334 0.92334 0.93357 0.93561 0.99344 0.98966 0.98696 0.99086 0.99378 0.99412 0.99545 0.99582

0.95777 0.95777 0.95777 0.79486 0.79450 0.82555 0.82527 0.95064 0.92380 0.90592 0.99991 0.95420 0.95749 0.99628 0.99633

0.94199 0.94199 0.94199 0.75927 0.75914 0.78909 0.79626 0.93348 0.89819 0.87526 0.99975 0.93773 0.93913 0.99288 0.99558

0.96905 0.96905 0.96905 0.82411 0.82408 0.84867 0.85162 0.96490 0.94544 0.93218 0.99942 0.96698 0.96715 0.99387 0.99746

0.96011 0.96011 0.96011 0.80030 0.80028 0.82766 0.83099 0.95419 0.92917 0.91240 0.99953 0.95715 0.95859 0.99381 0.99667

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(a) vs. (d) 250.55 24.142 0.96349

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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(a) vs. (c) 250.37 24.145 0.87198

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S C (A, B) S HK (A, B) S DC (A, B) S M (A, B) S ep (A, B) S sp (A, B) S hp (A, B) S 1HY (A, B) S 2HY (A, B) S 3HY (A, B) p S HY (A, B) S WX (A, B) S ZF (A, B) S Y (A, B) S S (A, B)

(a) vs. (b) 250.38 24.145 0.86708

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Measures MSE PSNR SSIM

6.2. Reaction to images distortion

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In this experiment, we investigate the effect of adding different percentages of salt & pepper noise to images on the performance of the IFS-based similarity measures. The ’Cameraman’ image of size 256×256 pixels distorted by three different percentages of salt & pepper noise between, namely 3%, 6% and 30% is shown in Fig 9. One can perceive from the obtained results reported in Table 5 that the similarity values are relatively high with small noise percentage (i.e., 3%) where the image and its distorted version are visually similar. While they decrease slightly with respect to noise level increasing. In the same time, values of MSE and PSNR change significantly with change in noise level. For example, the value of MSE is changed from 599.18 to 6042.1 with change in noise from 3% to 30% as illustrated in Table 5. The S c , S HK and S DC measures achieve similar performance as well as the S M achieves performance similar to the S ep measure in this experiment. Which means that the discrimination capability of the these similarity measure are almost identical. Meanwhile, the performance of the SSIM measure in this case is questionable. Where the obtained similarity values are very small that means that there is no similarity between the image and its distorted versions, which is unreasonable. Reaction of enlightening noise on the performance of the similarity measures is also investigated. Where, the tested image ’Man’ of size 512 × 512 pixels is enlightened two times as shown in Fig 10. From Table 6, it is observed that the similarity values decrease with the increasing of the enlightening amount. The performance of the IFSbased similarity measures is competitive with that of the SSIM measure. The S c and S HK measures are still have similar performance. Meanwhile, the S M and S ep measures achieve a slight change in the performance. Moreover, the performance of the S M (A, B), S ep (A, B), S sp (A, B), and S WX (A, B) as well as SSIM is affect significantly with enlightening noise; while the other similarity measures give reasonable discrimination capability between the image and its enlightened versions. 6.3. Performance of homogeneity with neighborhood-based similarity The previous experiment of section 6.1 is repeated, where the similarity measures are computed considering the homogeneity of images explained in section 5. The obtained results for this experiment are presented in Table 7 and Table 8 for two images only; namely ’Barbara’ and ’Medical’ respectively. The performance of the SSIM and MSE/PSNR metrics are as in Table 1 and 4, thus it is omitted from Table 7 and 8. It can be note that incorporating the homogeneity make the similarity values are more reliable, where these values are decreased according to the noise 16

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Table 2: Performance of the IFS-based measures compared to MSE/PSNR and SSIM for ’Zebra’ image.

0.99507 0.99507 0.94397 0.92977 0.92977 0.93919 0.94090 0.99446 0.99125 0.98897 0.99305 0.99476 0.99507 0.99630 0.99652

0.96071 0.96071 0.95233 0.80222 0.80179 0.83213 0.83176 0.95438 0.92945 0.91274 0.99995 0.95755 0.96004 0.99693 0.99700

0.95127 0.95127 0.94096 0.77925 0.77925 0.80880 0.81453 0.94553 0.91614 0.89669 0.99987 0.94840 0.94720 0.99360 0.99722

0.96358 0.96358 0.94558 0.80919 0.80916 0.83579 0.83931 0.95896 0.93639 0.92115 0.99973 0.96127 0.96103 0.99473 0.99758

(a) vs. (f) 203.02 25.055 0.96879

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(a) vs. (e) 200.07 25.119 0.96042

0.96120 0.96120 0.94693 0.80309 0.80302 0.83097 0.83387 0.95601 0.93191 0.91572 0.99985 0.95860 0.95887 0.99501 0.99762

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(a) vs. (d) 200.38 25.112 0.98363

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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(a) vs. (c) 200.68 25.106 0.90877

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S C (A, B) S HK (A, B) S DC (A, B) S M (A, B) S ep (A, B) S sp (A, B) S hp (A, B) S 1HY (A, B) S 2HY (A, B) S 3HY (A, B) p S HY (A, B) S WX (A, B) S ZF (A, B) S Y (A, B) S S (A, B)

(a) vs. (b) 200.58 25.108 0.92573

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Table 3: Performance of the IFS-based measures compared to MSE/PSNR and SSIM for ’Hiroshima Building’ image.

S C (A, B) S HK (A, B) S DC (A, B) S M (A, B) S ep (A, B) S sp (A, B) S hp (A, B) S 1HY (A, B) S 2HY (A, B) S 3HY (A, B) p S HY (A, B) S WX (A, B) S ZF (A, B) S Y (A, B) S S (A, B)

(a) vs. (b) 250.03 24.151 0.80627

(a) vs. (c) 250.02 24.151 0.82192

(a) vs. (d) 250.25 24.147 0.95575

(a) vs. (e) 250.04 24.151 0.92274

(a) vs. (f) 255.09 24.064 0.78133

0.99442 0.99442 0.93868 0.92532 0.92532 0.93529 0.93736 0.99380 0.99023 0.98768 0.99068 0.99411 0.99443 0.99560 0.99602

0.95956 0.95956 0.94695 0.79912 0.79889 0.82863 0.82823 0.95193 0.92575 0.90826 0.99988 0.95574 0.96055 0.99687 0.99562

0.93875 0.93875 0.92712 0.75290 0.75251 0.78170 0.79025 0.92927 0.89197 0.86788 0.99978 0.93401 0.93606 0.99275 0.99533

0.96961 0.96961 0.93733 0.82569 0.82567 0.84831 0.85298 0.96554 0.94642 0.93338 0.99901 0.96758 0.96770 0.99335 0.99692

0.95522 0.95522 0.93498 0.77843 0.78839 0.81551 0.82035 0.94795 0.91977 0.90105 0.99941 0.95159 0.95433 0.99332 0.99598

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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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Table 4: Performance of the IFS-based measures compared to MSE/PSNR and SSIM for ’Medical’ image.

(a) vs. (d) 250.31 24.146 0.78892

(a) vs. (e) 250.23 24.147 0.83443

(a) vs. (f) 250.02 24.151 0.76672

0.99456 0.99456 0.93827 0.92627 0.92627 0.93614 0.93828 0.99401 0.99055 0.98809 0.98971 0.99429 0.99457 0.99574 0.99581

0.96062 0.96062 0.94191 0.80158 0.80156 0.82909 0.83374 0.95645 0.93259 0.91654 0.99983 0.95854 0.95656 0.99328 0.99778

0.92888 0.92888 0.92034 0.73434 0.73331 0.76239 0.77131 0.91422 0.86996 0.84200 0.99968 0.92155 0.93370 0.99474 0.97500

0.95958 0.95958 0.93146 0.79911 0.79895 0.82355 0.83038 0.95411 0.92905 0.91225 0.99928 0.95685 0.95749 0.99251 0.99355

0.94971 0.94971 0.93082 0.77599 0.77575 0.80278 0.80967 0.94161 0.91027 0.88966 0.99961 0.94566 0.94922 0.99330 0.98928

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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(a) vs. (c) 250.16 24.149 0.92070

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S C (A, B) S HK (A, B) S DC (A, B) S M (A, B) S ep (A, B) S sp (A, B) S hp (A, B) S 1HY (A, B) S 2HY (A, B) S 3HY (A, B) p S HY (A, B) S WX (A, B) S ZF (A, B) S Y (A, B) S S (A, B)

(a) vs. (b) 250.44 24.144 0.83580

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Table 5: Performance of the IFS-based similarity measures with respect to salt & pepper noise of 3%, 6%, and 30%.

3% 0.98523 0.98523 0.98523 0.87847 0.87847 0.89478 0.89774 0.98340 0.97395 0.96734 0.97911 0.98431 0.98524 0.98891 0.98960

6% 0.97017 0.97017 0.97017 0.82728 0.82728 0.85043 0.85467 0.96647 0.94783 0.93511 0.95805 0.96832 0.97016 0.97750 0.97900

30% 0.85020 0.85020 0.85020 0.61295 0.61295 0.66480 0.67427 0.83151 0.75470 0.71162 0.79072 0.84085 0.85019 0.88735 0.89458

SSIM MSE PSNR

0.51114 599.18 20.355

0.40822 1207.8 17.311

0.19386 6042.1 10.319

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type. For example, in case of impulsive salt & pepper noise (i.e., (a) vs. (c) in the tables) the similarity values are small for ’Barbara’ image; while they are decreased significantly for the ’Medical’ images in the case (a) vs.(f) compared with the those shown in Table 4. Where, the effect of black regions within the ’Medical’ image (see Fig.8) is weighted more carefully in computing the total similarity value. Thus, combining homogeneity has a significant impact on computing the similarity value by these measures, especially for simple structured regions in images. 18

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Figure 9: The ’Cameraman’ image distorted by different percentages of salt & pepper noise.

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Figure 10: (a) The original ’Man’ image, (b) ’Man’ image enlightened one time, (c) ’Man’ image enlightened two times Table 7: Performance of the neighborhood-based similarity considering homogeneity of images for ’Barbara’ image.

Measures S˜ C (A, B) S˜ HK (A, B) S˜ DC (A, B) S˜ M (A, B) S˜ ep (A, B) S˜ sp (A, B) S˜ hp (A, B) S˜ 1HY (A, B) S˜ 2HY (A, B) S˜ 3HY (A, B) p S˜ HY (A, B) ˜

(a) vs. (a) 1 1 1 1 1 1 1 1 1 1 1

(a) vs. (b) 0.88798 0.88798 0.88798 0.88026 0.88022 0.88150 0.88163 0.88774 0.88670 0.88600 0.88771

(a) vs. (c) 0.61955 0.61955 0.61955 19 0.52518 0.52480 0.54122 0.54293 0.61611 0.60164 0.59201 0.64280

(a) vs. (d) 0.85366 0.85366 0.85366 0.70389 0.70323 0.72987 0.73433 0.84604 0.81607 0.79734 0.90489

(a) vs. (e) 0.72399 0.72399 0.72399 0.65612 0.65583 0.66663 0.66788 0.72242 0.71556 0.71091 0.73485

(a) vs. (f) 0.74667 0.74667 0.74667 0.63595 0.63546 0.65502 0.65694 0.74226 0.72423 0.71252 0.77609

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+0.2 0.88512 0.88512 0.86786 0.66112 0.66105 0.70457 0.71548 0.87203 0.80998 0.77310 0.99770 0.87857 0.87399 0.96760 0.98770

SSIM MSE PSNR

0.97397 242.95 24.28

0.91968 965.9 18.28

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Measures S C (A, B) S HK (A, B) S DC (A, B) S M (A, B) S ep (A, B) S sp (A, B) S hp (A, B) S 1HY (A, B) S 2HY (A, B) S 3HY (A, B) p S HY (A, B) S WX (A, B) S ZF (A, B) S Y (A, B) S S (A, B)

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Table 6: Performance of the IFS-based similarity measures with respect to image enlightening.

Table 8: Performance of the neighborhood-based similarity considering homogeneity of images for ’Medical’ image.

(a) vs. (c) 0.63079 0.63079 0.62763 0.57193 0.57180 0.58130 0.58393 0.62922 0.61978 0.61354 0.64673 0.63001 0.62906 0.64427 0.64602

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(a) vs. (b) 0.88296 0.88296 0.88099 0.87998 0.87997 0.88045 0.88053 0.88287 0.88245 0.88217 0.88259 0.88291 0.88296 0.88315 0.88315

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Measures S˜ C (A, B) S˜ HK (A, B) S˜ DC (A, B) S˜ M (A, B) S˜ ep (A, B) S˜ sp (A, B) S˜ hp (A, B) S˜ 1HY (A, B) S˜ 2HY (A, B) S˜ 3HY (A, B) p S˜ HY (A, B) S˜ WX (A, B) S˜ ZF (A, B) S˜ Y (A, B) S˜ S (A, B)

(a) vs. (d) 0.85901 0.85901 0.85857 0.68780 0.68640 0.71370 0.72123 0.84519 0.80564 0.78150 0.92471 0.85210 0.86390 0.92025 0.90082

(a) vs. (e) 0.69284 0.69284 0.68980 0.62324 0.62297 0.63431 0.63694 0.69037 0.67935 0.67224 0.71135 0.69161 0.69178 0.70889 0.70910

(a) vs. (f) 0.37390 0.37390 0.36828 0.30728 0.30697 0.31762 0.32023 0.37107 0.35919 0.35139 0.39317 0.37249 0.37327 0.39065 0.39024

6.4. Performance of the IFS-based quality index We conduct a similar experiment to that was carried out in [13] using same images and same distortions. The ’Lena’ image and its distortions are shown in Fig. 11. Thanks to [13] for free test images and efficient MATLAB implementation. The results of the proposed IFS-based quality index (IFSQI), Fuzzy quality index (FSQI) and their counterpart universal quality index (UQI) are shown in Table 9. It is clear that the IFS-based quality index compares favorably with the quality index of [13] and in some cases gives better performance than it. For example, for the JPEG compression (Fig. 11 (h) ) the obtained rank is better than UQI and mean subjective rank. Moreover, it very sensitive to image changes as in the cases of mean shift and contrast stretching shown in Fig. 11 (e) and Fig. 11 (f), respectively. Also, our obtained results are in a good agreement with what is reported in [13] that ”the performance 20

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of the traditional MSE is extremely poor in the sense that images with nearly identical MSE are drastically different in perceived quality”. Note that the values of UQI are obtained by applying it on gray scale images, while the FSQI values are obtained after normalize the images by dividing each pixel by 255 which is the simple fuzzification process used through this work. Table 9: Performance comparison of the universal, FS, and IFS quality index on ’Lena’ image with different deformations.

7. Conclusion

UQI 0.97856 0.94755 0.99959 0.99979 0.99953 0.99458 0.99264

FSQI [13] 0.98942 0.93719 0.64938 0.44076 0.38911 0.34612 0.28755

IFSQI 0.98389 0.93369 0.64950 0.44208 0.38737 0.34512 0.28667

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Distortion type Mean shift Contrast stretching Impulsive Salt-Pepper noise Multiplicative speckle noise Additive Gaussian noise Blurring JPEG compression

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The main goal of this paper is to introduce the application of the intuitionistic fuzzy set into image processing field, especially images comparisons due to its efficiency in modeling the uncertainties associated with image pixels. To this end, the paper investigates the applicability of several intuitionistic fuzzy sets-based similarity measures in images comparisons under different image distortions. A throughout comparison between these measures and the traditional PSNR/MSE as well as the state-of-the art measure ’SSIM’ using natural images ranging from simple to high structure is conducted, which shows the superiority of the intuitionistic fuzzy sets-based similarity measures over the traditional PSNR/MSE. While, they are competitive with the SSIM measure and show some advantages in several cases. In order to increase consistently of the IFS-based similarity measures with respect to the perceived visual quality, construction of neighborhood-based similarity is proposed considering homogeneity of images. This scheme seems to be very useful for computing reliably the similarity value, specifically for eliminating the effect of simple structured regions within compared images. Furthermore, a modification for the statistical quality index measure [13] on the space of the intuitionistic fuzzy sets theory is presented in this paper. The conducted experiments show that the IFS-based quality index compares favorably with statistical quality index measure. This simple modification proves that considering the intuitionistic fuzzy set theory in designing solutions for some applications may impact the performance of algorithms. Thus, we believe that the research introduced in this paper is a promising starting point for the future development of more successful images comparisons measures using intuitionistic fuzzy sets- based approaches. Also, we expect that the proposed framework for using intuitionistic fuzzy sets in real applications, such as those currently handled by the fuzzy set theory, will attract the interest of theoreticians and practitioners within a short time. In future works, more extensive investigations on the effect of membership and non-membership functions construction should certainly be conducted from the application point of view. Also, distance-based similarity measures and similarity measures for color images will be the subject of investigations. Acknowledgments

The authors would like to thank the Associate Editor, Professor Witold Pedrycz and the anonymous reviewers for their helpful and constructive comments, which considerably improved the quality of the paper.

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(b) Salt & pepper noise

(c) Additive Gaussian noise

(e) Mean shift

(f) Contrast stretching

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(d) Multiplicative speckle noise

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(a) Original image

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(g) Blurring noise

(h) JPEG compression

Figure 11: ’Lena’ image distorted by different types of noise [13] used in evaluation of the IFS-based quality index.

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*Highlights (for review)

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This paper introduces a framework for using Intuitionistic Fuzzy Sets (IFSs) theory in image processing, specifically in image comparison. Existing similarity measures on the IFSs space are discussed and highlighted their properties. This paper also introduces an intuitionistic fuzzy based image quality index measure. Besides, construction of neighborhood-based similarity is also proposed for improving the perceived visual quality of these IFS-based similarity measures. The proposed framework is tested on real world images under various types of image distortions and the obtained results are encourages.

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*Graphical abstract (for review)

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