Image quality measures based on intuitionistic fuzzy similarity and inclusion measures

Abstract

Objective image quality assessment plays an important role in various computer vision and image processing applications. The most widely used image quality measures are the classical mean squared error (MSE), computed by averaging the squared intensity differences of distorted and reference image pixels, and its related quantity of the peak signal-to-noise ratio (PSNR). Unfortunately, these measures are not very well matched to perceived visual quality. In this paper, we propose a measure based on the intuitionistic fuzzy set theory, whose performance would be more closely related to the human perception of visual quality. The proposed measure provides a flexible mathematical framework for modelling of imprecise or/and imperfect information often present in digital images. Furthermore, we show how the neighborhood-based intuitionistic fuzzy similarity measures can be combined with intuitionistic fuzzy inclusion measures for improving the perceptive behavior of intuitionistic fuzzy similarity measures. The performance of the proposed measure is evaluated on a set of real images with different distortion types and the obtained results demonstrate its advantages over the classical measures.

Keywords

Computer vision quality measure intuitionistic fuzzy set intuitionistic fuzzy similarity measure intuitionistic fuzzy inclusion measure

1 Introduction

Image quality measurement aims to assess the perceptual quality of image and plays a major role in numerous image processing and computer vision applications. It is usually adopted to measure the visual distortion of compressed images/video so as to evaluate the performance of lossy image compression; it can also be utilized to quantify the visual quality refinement after image enhancement [1, 2]. Thus, a central problem in these applications is determining the distance between images. It can be classified into objective and subjective methods [3]. Subjective methods relies on human diligence and work without reference to evident criteria [4] an the efficiency of these methods is measured by the extent of reconstruction errors visibility. It seems to be meaningful since there are no satisfactory mathematical models serving this task. As a result, a quality measure that predict the human perceptual sensitivity to distortion of images is beneficial to the design and evaluation of image processing algorithms. On the other hand, objective methods are often referred to as being based on comparisons using explicit numerical criteria. That is, the main goal of objective image quality assessment research is to design computational models that can predict perceived image quality accurately and automatically.

To this end, several objective quality measures are proposed in the literature [5, 6]. Among all of them, the mean squared error (MSE) and its related measures like peak signal-to-noise ratio (PSNR) are the most widely used measurements in image fidelity due to their analytic simplicity. But, these measures do not accurately measure the perceived distortion and their performances are not satisfactory in many scenarios [7]. Therefore, several image quality measures have been introduced in the literature to solve the problem of image comparison [8 –12].

Recently, there have been strenuous efforts to design an objective quality measures based on fuzzy similarity measures [2 , 14]. One significant problem associated with these quality measures is the selection of applicable fuzzy similarity measures to images comparison. In [15], Weken et al. showed that several similarity measures used to compare fuzzy sets can also be applied successfully in image processing. However, the pixel-based similarity measures still have some flaws in the sense of image quality estimation [16]. To cope with this drawback, fuzzy similarity based on homogeneity have been used in the image quality evaluation procedure for better observation of image’s relevant structures [17].

The intuitionistic fuzzy set (IFS) theory [18] is utilized in measuring the similarity between images. Essentially, the similarities/distances between IFSs are the extensions of conventional similarity/distance functions [19]. Several similarity measures based on the IFSs are surveyed in [20]. For instance, Farhadinia [21] introduced a similarity measure based on a distance defined on an interval using convex combination of endpoints that considers the property of min and max operators. While, a set of intuitionistic fuzzy similarity measures based on min max operators as a generalization of some known measures is proposed in [22]. Another similarity measure with high capacity for discriminating IFSs is introduced in [23] based on the direct operation on the membership, non-membership and hesitation functions as well as the upper bound of two IFSs membership functions. Nguyen [24] introduced a similarity measure based on the concept of a knowledge measure of information conveyed by the IFS. It is shown that this measure overcomes the drawbacks of existing measures on the problem of pattern recognition. Motivated by the success of IFS-based similarity measure in a variety of applications, this paper presents a set of image quality measures which are based on a twofold usage of IFS-based similarity measures introduced in [20] and inclusion measures which can be applied to image’s histograms introduced in [25]. The performance of the proposed measures is tested on a set of natural images with different distortion types and we conduct a comparison between the IFS-based measures and the classical MSE/PSNR metrics.

The rest of this paper is organized as follows. Section 2 presents the necessary basic concepts and provides a brief introduction to intuitionistic fuzzy theory. While, the details of the proposed measures based on combining the IFS-based inclusion and IFS-based similarity measures are presented in Section 3. Section 4 presents the experiment results and the associated discussions. Finally, Section 5 concludes the paper.

2 Preliminaries

This section presents the necessary basic concepts that will be used in this paper. Also, it provides a brief introduction to intuitionistic fuzzy image processing.

2.1 Intuitionistic fuzzy sets

Definition 2.1. [18] An IFS A defined on a universe X is proposed by A = {〈x, μ_A (x) , ν_A (x) 〉|x ∈ X} , such that μ_A : X ⟶ [0, 1] and ν_A : X ⟶ [0, 1] and 0 ≤ μ_A (x) + ν_A (x) ≤1 for all x ∈ X.

The values of μ_A (x) and ν_A (x) denote the belongingness and non-belongingness degrees of x in A, respectively. For an IFS A in X, the intuitionistic fuzzy index of an element x ∈ X in A is given by the following equation: $π_{A} (x) = 1 - μ_{A} (x) - ν_{A} (x) .$ (1)

Clearly, a fuzzy set F can be represented by using the notation of IFS as follows $F = {〈 x, μ_{F} (x), 1 - μ_{F} (x) 〉 | x \in X} .$

In what follows, IFS (X) refers to the family of all intuitionistic fuzzy sets on X. The following basic operations on IFSs will be needed in our paper.

Definition 2.2. [26] For any A, B ∈ IFS (X), we define the following relations and operations:

A ⊆ B iff μ_A (x) ≤ μ_B (x) and ν_A (x) ≥ ν_B (x) for each x ∈ X.

A = B iff A ⊆ B and B ⊆ A.

thicksimA = {〈x, ν_A (x) , μ_A (x) 〉|x ∈ X}.

A ∪ B = {〈 x, max { μ_A (x), μ_B (x) },min { ν_A (x), ν_B (x) } 〉 |x ∈ X}.

A ∩ B = { 〈 x,min { μ_A (x),μ_B (x) },max { ν_A (x),ν_B (x)} 〉|x ∈ X}.

2.2 Intuitionistic fuzzy images and histograms

An image A of M × N size and L levels of grayness can be considered as M × N array of fuzzy singletons relating the intensity values of the image pixels. In order to represent the image A in intuitionistic fuzzy set (IFS) theory, many techniques are used. In 2007, Vlachos and Sergiadis [27] used gamma membership function to construct the membership values of the pixels of an image and used cross entropy to find the optimum threshold. Later on, a different approach used by Couto et al. [28] in the calculation of hesitation degree and used intuitionistic fuzzy entropy to find the optimum threshold. Several techniques introduced by T. Chaira can be found in [29 –32]. In this paper, we construct the membership of IFS-image by setting the exponential fuzzifier F_e = 2 and calculating the denominational fuzzifier F_d from the following equation: $F_{d} = \frac{g_{max} - g_{min}}{(\frac{1}{2})^{\frac{- 1}{F_{e}}} - 1};$ (2) where g_max and g_min are the maximum and minimum gray values of A, respectively. Based on the calculated parameter F_d, the membership function μ_A (g_ij) is given by: $μ_{A} (g_{ij}) = {(1 + \frac{g_{max} - g_{ij}}{F_{d}})}^{- F_{e}} .$ (3) where g_ij is the (i, j) ^th-gray level of the image. Now, the non-membership function ν_A (g_ij) can be calculated by using the membership values μ_A (g_ij) as follows:

$ν_{A} (g_{ij}) = {\begin{matrix} \frac{1}{2} max {| 1 - μ_{A} (g_{ij}) |, | 0 - μ_{A} (g_{ij}) |}, & if μ_{A} (g_{ij}) \in [0, \frac{1}{2}]; \\ \frac{1}{2} min {| 1 - μ_{A} (g_{ij}) |, | 0 - μ_{A} (g_{ij}) |}, & if μ_{A} (g_{ij}) \in [\frac{1}{2}, 1] . \end{matrix}$ (4)

An M × N pixels image A of L gray level ranging between 0 and L - 1 is represented by the IFS as follows $A = {〈 g_{ij}, μ_{A} (g_{ij}), ν_{A} (g_{ij}) 〉 | g_{ij} = 0, 1, \dots, L - 1} .$ (5)

An example for construction of IFS images from gray scale images using the above method is shown in Fig. 1.

Fig.1

Representation of the intuitionistic fuzzy images: (a) Original image, (b) Membership function image, and (c) Non-membership function image.

Histogram-based techniques are characterized by their simplicity and speed compared to their pixel-based counterparts. In [33, 34], Vlachos and Sergiadis proposed an automated approach for constructing the intuitionistic fuzzy histogram (IFS-based histogram) of a gray-scale and color images based on the notion of intuitionistic fuzzy numbers (i.e., IFS-based numbers). Where, the value of the histogram denoted by h_A (g) of an image A in the gray value g is equal to the total number of pixels with the gray value g.

Definition 2.3. [33, 34] Let A be a gray-scale image. The lower and upper IFS-based histogram are given by

$\begin{matrix} {\hat{h}}_{A} (g_{ij}) & = & | | {〈 g_{ij}, μ_{A} (g_{ij}) 〉} | g_{ij} \\ \in {0, 1, \dots, L - 1} | |, \end{matrix}$ (6) and

$\begin{matrix} {\overset{ˇ}{h}}_{A} (g_{ij}) & = & | | {〈 g_{ij}, ν_{A} (g_{ij}) 〉} | g_{ij} \\ \in {0, 1, \dots, L - 1} | |, \end{matrix}$ (7) respectively.

Figure 2 depicts fuzzy histogram and IFS-based histogram of “Cat” image shown in Fig. 1, where the lower and upper normalized IFS-based histogram are in red and blue lines in Fig. 2(b), respectively.

Fig.2

(a) Normalized crisp histogram for the original image of Fig. 1, (b) Corresponding lower (red line) and upper (blue line) normalized IFS-based histograms.

3 The proposed IFS-based image quality measures

IFS-based inclusion measure is an important topic in IFS theory. It is a pairwise relation between two IFSs, which express the degree of the inclusion between two IFS. Xie et al. [25] redefined the IFS-based inclusion measure of [35, 36] and five IFS-inclusion measures are constructed by means of different operators. Moreover, they showed that the IFS-based similarity measures obtained from IFS-based inclusion measures hold the normal similarity measures’s properties.

Definition 3.1. [25] The mapping $I : IFS (X) \times IFS (X) ⟶ [0, 1]$ is said to be an IFS-based inclusion measure if it satisfies the following statements, for all A, B, C ∈ IFS (X):

$I (A, B) = 1$ iff A ⊆ B;

if μ_A (x) ≥ ν_A (x) for all x ∈ X, then $I (A, A^{citsc}) = 0$ iff $A = \underline{1}$ ;

if A ⊆ B ⊆ C, then $I (C, A) \leq I (B, A)$ , and if A ⊆ B, then $I (C, A) \leq I (C, B)$ .

The following IFS-based inclusion measures were defined in [25]: $I_{1} (A, B) = {\begin{matrix} 1, & if A = B = \underline{0}; \\ \frac{| B |}{| A \cup B |}, & otherwise . \end{matrix}$ (8) $I_{2} (A, B) = {\begin{matrix} 1, & if A = B = \underline{0}; \\ \frac{| A \cap B |}{| A |}, & otherwise . \end{matrix}$ (9) $I_{3} (A, B) = {\begin{matrix} 1, & if A = B = \underline{1}; \\ \frac{| A |}{| A \cup B |}, & otherwise . \end{matrix}$ (10) $I_{4} (A, B) = {\begin{matrix} 1, & if A = B = \underline{1}; \\ \frac{| A \cap B |}{| A |}, & otherwise . \end{matrix}$ (11)

IFS-based inclusion measures can be combined together by using a t-norm to construct a similarity measure. Suppose that $I_{iitsc}$ is IFS-based inclusion measure where i = 1, 2, 3, 4, then we obtain the following IFS-based similarity measure based on the product t-norm $T$ : $H_{i} (A, B) = T (I_{i} (A, B), I_{i} (B, A)) .$ (12)

This relation based on the crisp expression $A = B \Leftrightarrow A \subseteq B and B \subseteq A,$ and its fuzzification [37] $S (A, B) = T (I (A, B), I (B, A)),$

In our experiment, we will use the product t-norm $T (a, b) = a \cdot b$ as a special case.

IFS-based similarity measures is a relation which can be seen as an intuitionistic fuzzification of crisp equivalence relation, and it is an important tool for determining the degree of similarity between two objects. Many different similarities between IFSs have been proposed in the literatures. Let S (A, B) be a similarity measure between two IFSs A, B ∈ IFS (X). Where, each set represents an IFS image of size M × N. The following IFS-based similarity measures were defined in [14 , 38–42]: $S_{1} (A, B) = 1 - \frac{\sum_{(i, j) \in X} | (μ_{A} (i, j) - ν_{A} (i, j)) - (μ_{B} (i, j) - ν_{B} (i, j)) |}{2 MN}$ (13) $S_{2} (A, B) = 1 - \frac{\sum_{(i, j) \in X} (| μ_{A} (i, j) - μ_{B} (i, j) | + | ν_{A} (i, j) - ν_{B} (i, j) |)}{2 MN}$ (14) $S_{3} (A, B) = \frac{1}{2} (σ_{μ} (A, B) + σ_{ν} (A, B)),$ (15) where, $σ_{μ} = 1 - \sqrt[p]{\frac{\sum_{(i, j) \in X} | μ_{A} (i, j) - μ_{B} (i, j) |^{p}}{MN}},$ and

$\begin{matrix} σ_{ν} = 1 - \sqrt[p]{\frac{\sum_{(i, j) \in X} | ν_{A} (i, j) - ν_{B} (i, j) |^{p}}{MN}} . \\ S_{4} (A, B) = 1 - \frac{1}{2 MN} \sum_{(i, j) \in X} (| δ_{A} (i, j) - δ_{B} (i, j) | \\ + | α_{A} (i, j) - α_{B} (i, j) |), \end{matrix}$ (16) where, $\begin{matrix} δ_{A} (i, j) & = & μ_{A} (i, j) + π_{A} (i, j) μ_{A} (i, j), \\ δ_{B} (i, j) & = & μ_{B} (i, j) + π_{B} (i, j) μ_{B} (i, j), \\ α_{A} (i, j) & = & ν_{A} (i, j) + π_{A} (i, j) ν_{A} (i, j), and \\ α_{B} (i, j) & = & ν_{B} (i, j) + π_{B} (i, j) ν_{B} (i, j) . \end{matrix}$

$\begin{matrix} S_{5} (A, B) = \frac{1}{MN} \\ \sum_{(i, j) \in X} \frac{μ_{A} (i, j) μ_{B} (i, j) + ν_{A} (i, j) ν_{B} (i, j)}{\sqrt{μ_{A} (i, j)^{2} + ν_{A} (i, j)^{2}} \sqrt{μ_{B} (i, j)^{2} + ν_{B} (i, j)^{2}}} \end{matrix}$ (17)

$\begin{matrix} S_{6} (A, B) = \frac{1}{2 MN} \sum_{(i, j) \in X} (\sqrt{μ_{A} (i, j) μ_{B} (i, j)} \\ + 2 \sqrt{ν_{A} (i, j) ν_{B} (i, j)} + \sqrt{π_{A} (i, j) π_{B} (i, j)} \\ + \sqrt{(1 - ν_{A} (i, j)) (1 - ν_{B} (i, j))}) \end{matrix}$ (18)

To enhance the perceptive behavior of similarity measures, Hasaballah and Ghareeb [20] introduced a neighborhood-based similarity scheme by combining image’s homogeneity in the different neighborhoods. Firstly, the compared images A and B are divided into disjoint 4 × 4 parts and we measure the similarity between the asymmetric parts of these two images using pixel-based similarity measures. Suppose that each image is divided into P parts of size 4 × 4, and let the restricted similarity S (A_i, B_i) between two parts A_i and B_i of A and B, respectively; then the total similarity S^P between A and B is the weighted average of the similarities in asymmetric disjoint image parts. That is, $S^{P} (A, B) = \frac{1}{P} \sum_{p = 1}^{P} w_{p} . S (A_{i}, B_{i})$ (19) where, the weight w_p is the similarity between the homogeneity h_{A
_i} of part A_i of the image A and the homogeneity h_{B
_i} of part B_i of the image B. In order to compute the homogeneity of any part 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. 3).

Fig.3

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

Clearly, 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. Moreover, 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 $R (x, y) = {\begin{matrix} 1 - \frac{d (x, y)}{a}, & if d (x, y) < a; \\ 0, & otherwise . \end{matrix}$ (20) where d is the Hausdorff distance and $a = \frac{1}{2}$ . The function $R$ computes 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

$\begin{matrix} h_{A} & = & R ((max_{(i, j) \in A} μ_{A}, min_{(i, j) \in A} ν_{A}), \\ (min_{(i, j) \in A} μ_{A}, max_{(i, j) \in A} ν_{A})) . \end{matrix}$ (21)

The weight w_p is calculated using $w_{p} = R (h_{A}, h_{B}),$ (22)

It should be note that the weight w_p is a similarity measure between the homogeneity of corresponding image parts, where the distance d in Equation (20) is L₁-distance between the two values h_A and h_B.

Historically, image quality is known as the visibility of image’s distortion, like colour shifts, blurriness, blockiness and Gaussian noise. Therefore, the most used technique for modeling an image quality measure is by quantification of the visibility of these distortions. In this paper, we present a set of image quality measures which are rely on a twofold usage of ne-based similarity measures introduced in [20] and inclusion measures introduced in [25] which can be applied to image’s histograms. The proposed IFS-based image quality measures are simply given by combining the neighborhood-based similarity measures and the IFS-based similarity measures $H_{iitsc}$ as follows:

$\begin{matrix} Q_{i, j} (A, B) & = & S_{i}^{P} (A, B) \cdot H_{j} (A, B); \\ i = 1, \dots, 6, j = 1, \dots, 4 . \end{matrix}$ (23)

Equation 23 represents the proposed mathematical universal image quality measures. The term “universal” means that the measurement independent of tested images, viewing conditions, or individual observers.

4 Experimental results

To test the efficiency of the proposed IFS-based image quality measures, a set of experiments is carried on “Monkey” image shown in Fig. 4. Five kinds of distortions are applied on the image and we compare the results with pixel-based, neighborhood-based similarity measures, MSE, and PSNR. The image is affected by the following types of distortions such that the distorted images have the same MSE and PSNR relative to original image:

Implusive salt & pepper noise.

Multiplicative speckle noise.

Enlightening.

Blurring.

JPEG compression.

Fig.4

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

Figure 4 displays the original and distorted images. Table 1 shows the poor performance of pixel-based similarity, MSE and PSNR measures compared to neighborhood-based similarity measures. While, the results of the proposed IFS-based image quality measures Q_i,j where i = 1, …, 6 and j = 1, …, 4 compared to MSE and PSNR are displayed in Table 2.

Table 1

Performance of pixel-based and neighborhood-based IFS-similarity measures compared to MSE/PSNR for ‘Monkey’ image

Measures	(a) vs. (a)	(a) vs. (b)	(a) vs. (c)	(a) vs. (d)	(a) vs. (e)	(a) vs. (f)
S₁	1	0.98844	0.98020	0.99943	0.97777	0.98078
S₂	1	0.98844	0.98020	0.99943	0.97777	0.98078
S₃	1	0.89402	0.86131	0.97647	0.85306	0.86336
S₄	1	0.98907	0.98068	0.99944	0.97836	0.98101
S₅	1	0.99886	0.99932	1	0.99925	0.99945
S₆	1	0.99948	0.99936	1	0.99923	0.99941
$S_{1}^{P}$	1	0.97121	0.88455	0.99649	0.86949	0.22599
$S_{2}^{P}$	1	0.97121	0.88455	0.99649	0.86949	0.22599
$S_{3}^{P}$	1	0.93050	0.68218	0.95083	0.66096	0.17989
$S_{4}^{P}$	1	0.97283	0.89170	0.99671	0.87795	0.22768
$S_{5}^{P}$	1	0.98960	0.95545	0.99994	0.94759	0.24104
$S_{6}^{P}$	1	0.99241	0.97008	0.99997	0.96573	0.24460
MSE	0	242.29	242.39	241.89	242.74	242.32
PSNR	—	24.287	24.286	24.295	24.279	24.287

Table 2

Performance of the new IFS-based image quality measures compared to MSE/PSNR for ’Monkey’ image

Measures	(a) vs. (a)	(a) vs. (b)	(a) vs. (c)	(a) vs. (d)	(a) vs. (e)	(a) vs. (f)
Q _1,1	1	0.97280	0.90183	0.99649	0.88582	0.22988
Q _1,2	1	0.97348	0.89508	0.99649	0.83255	0.22274
Q _1,3	1	0.92946	0.82141	0.99430	0.79682	0.21020
Q _1,4	1	0.92872	0.81995	0.99430	0.79492	0.20967
Q _2,1	1	0.97280	0.90183	0.99649	0.88582	0.22988
Q _2,2	1	0.97348	0.89508	0.99649	0.83255	0.22274
Q _2,3	1	0.92946	0.82141	0.99430	0.79682	0.21020
Q _2,4	1	0.92872	0.81995	0.99430	0.79492	0.20967
Q _3,1	1	0.93202	0.69551	0.95083	0.67338	0.18299
Q _3,2	1	0.93268	0.69030	0.95083	0.63289	0.17731
Q _3,3	1	0.89050	0.63349	0.94873	0.60572	0.16732
Q _3,4	1	0.88979	0.63236	0.94873	0.60428	0.16690
Q _4,1	1	0.97442	0.90912	0.99671	0.89444	0.23160
Q _4,2	1	0.97511	0.90231	0.99671	0.84066	0.22440
Q _4,3	1	0.93102	0.82805	0.99451	0.80457	0.21177
Q _4,4	1	0.93028	0.82658	0.99451	0.80266	0.21123
Q _5,1	1	0.99122	0.97411	0.99994	0.96538	0.24519
Q _5,2	1	0.99191	0.96682	0.99994	0.90733	0.23758
Q _5,3	1	0.94706	0.88725	0.99774	0.86839	0.22420
Q _5,4	1	0.94631	0.88567	0.99774	0.86633	0.22363
Q _6,1	1	0.99403	0.98903	0.99997	0.98387	0.24881
Q _6,2	1	0.99473	0.98163	0.99997	0.92470	0.24108
Q _6,3	1	0.94975	0.90084	0.99776	0.88501	0.22750
Q _6,4	1	0.94899	0.89924	0.99776	0.88291	0.22693
MSE	0	242.29	242.39	241.89	242.74	242.32
PSNR	—	24.287	24.286	24.295	24.279	24.287

The results shows poorness of MSE and PSNR performance in the sense that images with nearly identical MSE and PSNR are drastically different in perceived quality. In contrast, the proposed IFS-based image quality measures yields significantly better results in comparison with MSE and PSNR. Furthermore, it can be note that incorporating the homogeneity make the similarity values are more reliable, where these values are decreased according to the noise type as reported in Table 2. Thus, the construction of neighborhood-based similarity scheme considering homogeneity of images seems to be very useful for computing reliably the similarity value, specifically for eliminating the effect of simple structured regions within compared images.

5 Conclusion

In this paper, we proposed new image quality measures based on a twofold usage of IFS-based similarity and IFS-based inclusion measures. We made use of the neighborhood-based similarity measures and employed the IFS-based inclusion measures, which can be applied to IFS-based histograms of images. From the obtained results, it can be conclude that the pixel-based similarity measures such as MSE and PSNR give wrong indication for measuring image’s quality. Compared to the existing relevant image quality measures, the proposed IFS-based metric has been shown to outperform the pixel-based similarity measures under different types of image distortions. The future work will be aimed at measuring the quality of color image and extending the proposed measures to assess other kinds of distortion as well as improving their performance to be able to produce quality scores that are more consistent with human perception.

Compliance with ethical standards

The authors declare that they haven’t received research grants from any institution and they have no conflict of interest. Moreover, this article does not contain any studies with human participants or animals performed by any of the authors.

Footnotes

Acknowledgments

The authors would like to express their sincere thanks to the associate editor Prof. Maria Guijarro. Also, the authors express their sincere thanks to the anonymous reviewers for their careful reading and constructive comments.

References

Wang

and Bovik

A.C.

, Modern image quality assessment, Synthesis Lectures on Image, Video, and Multimedia Processing 2 (2006), 1–156.

Grečova

and Morillas

, Perceptual similarity between color images using fuzzy metrics, Journal of Visual Communication and Image Representation 34 (2016), 230–235.

Guo

, Xiang

, and Li

, Image quality assessment based on multiscale fuzzy gradient similarity deviation, Soft Computing 21 (5) (2017), 1145–1155.

Farrell

J.E.

, Image quality evaluation, Colour Imaging: Vision and Technology (1999), 285–313.

Lin

and Kuo

C.-C.J.

, Perceptual visual quality metrics: A survey, Journal of Visual Communication andImage Representation 22 (4) (2011), 297–312.

Zhong

and Defée

, Performance of similarity measures based on histograms of local image feature vectors, Pattern Recognition Letters 28 (15) (2007), 2003–2010.

Wang

and Bovik

A.C.

, Mean squared error: Love it or leave it? A new look at signal fidelity measures, IEEE Signal Processing 26 (1) (2009), 98–117.

Wilson

D.L.

, Baddeley

A.J.

, and Owens

R.A.

, A new metric for grey-scale image comparison, International Journal of Computer Vision 24 (1) (1997), 5–17.

de Freitas Zampolo

and Seara

, A measure for perceptual image quality assessment, in: IEEE International Conference on Image Processing, 1, IEEE, 2003, pp. I–433.

10.

Hardeberg

J.Y.

, Bando

, and Pedersen

, Evaluating colour image difference metrics for gamut-mapped images, Coloration Technology 124 (4) (2008), 243–253.

11.

Lee

M.-S.

, Liu

L.-Y.

, and Lin

F.-S.

, Image similarity comparison using dual-tree wavelet transform, Advancesin Image and Video Technology (2006), 189–197.

12.

Allen

, Triantaphillidou

, and Jacobson

, Image quality comparison between jpeg and jpeg2000.i. psychophysical investigation, Journal of Imaging Science and Technology 51 (3) (2007), 248–258.

13.

Omran

, and Hassaballah

, A new class of similarity measures for fuzzy sets, International Journal of Fuzzy Logic and Intelligent Systems 6 (2) (2006), 100–104.

14.

Zhang

, and Fu

, Similarity measures on three kinds of fuzzy sets, Pattern Recognition Letters 27 (12) (2006), 1307–1317.

15.

der Weken

D.V.

, Nachtegael

and Kerre

E.E.

, An overview of similarity measures for images, in: IEEE International Conference on Acoustics, Speech, and Signal Processing, 4, 2002, pp. 3317–3320.

16.

der Weken

D.V.

, Nachtegael

and Kerre

, Using similarity measures and homogeneity for the comparison of images, Image and Vision Computing 22 (9) (2004), 695–702.

17.

Van der Weken

, Nachtegael

and Kerre

, Combining neighbourhood-based and histogram similarity measures forthe design of image quality measures, Image and Vision Computing 25(2) (2007), 184–195.

18.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1) (1986), 87–96.

19.

Milošević

, Petrović

and Jeremić

, Ifs-iba similarity measure in machine learning algorithms, Expert Systems with Applications 89 (9) (2017), 296–305.

20.

Hassaballah

and Ghareeb

, A framework for objective image quality measures based on intuitionistic fuzzysets, Applied Soft Computing 57 (2017), 48–59.

21.

Farhadinia

, An efficient similarity measure for intuitionistic fuzzy sets, Soft Computing 18 (1) (2014), 85–94.

22.

Iancu

, Intuitionistic fuzzy similarity measures based on min– max operators, Pattern Analysis and Applications (2017), 1–10.

23.

Song

, Wang

, Lei

, and Xue

, A novel similarity measure on intuitionistic fuzzy sets with its applications, Applied Intelligence 42 (2) (2015), 252–261.

24.

Ngan

S.-C.

, An activation detection based similarity measure for intuitionistic fuzzy sets, Expert Systems with Applications 60 (2016), 62–80.

25.

Xie

, Han

L.-W.

, and Mi

J.-S.

, Inclusion measure and similarity measure of intuitionistic fuzzy sets, in: IEEE International Conference on Machine Learning and Cybernetics, 2009, pp. 700–704.

26.

Atanassov

K.T.

, On Intuitionistic Fuzzy Sets Theory, Springer, 283, Springer, pp. 2012.

27.

Vlachos

I.K.

and Sergiadis

G.D.

, Intuitionistic fuzzy information applications to pattern recognition, Pattern Recognition Letters 28 (2) (2007), 197–206.

28.

Couto

, Pagola

, Bustince

, Barranechea

, and Melo-Pinto

, Image segmentation using a-ifs, in: Processing of IPMU, Malaga, 2008, pp. 1620–1627.

29.

Chaira

, Medical image thresholding scheme using a tanassov intuitionistic fuzzy set, in: 2009 Proceeding of International Conference on Methods and Models in Computer Science (ICM2CS), 2009, pp. 1–5.

30.

Chaira

, Medical image enhancement using intuitionistic fuzzy set, in: 2012 1st International Conference on Recent Advances in Information Technology (RAIT), 2012, 54–57.

31.

Chaira

, Intuitionistic fuzzy segmentation of medical images, IEEE Transactions on Biomedical Engineering 57 (6) (2010), 1430–1436.

32.

Chaira

, Construction of intuitionistic fuzzy contrast enhanced medical images, in: 2012 4th International Conference on Intelligent Human Comuter Interaction (IHCI), 2012, pp. 1–5.

33.

Vlachos

and Sergiadis

, Intuitionistic fuzzy histograms of an image, in: Foundations of Fuzzy Logic and Soft Computing, the 12th International Fuzzy Systems Association World Congress, Springer, 2007, pp. 86–95.

34.

Vlachos

I.K.

and Sergiadis

G.D.

, Hesitancy histogram equalization, in: IEEE International Conference on Fuzzy Systems, IEEE, 2007, pp. 1–6.

35.

Cornelis

and Kerre

, Inclusion measures in intuitionistic fuzzy set theory, in: European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty, Springer, 2003, 345–356.

36.

Grzgorzewski

and Mrowka

, Subsethood measure of intuitionistic fuzzy sets, in: IEEE Int Conf on FuzzySystems, 2004, pp. 139–142.

37.

derWeken

D.V.

, Nachtegael

, and Kerre

, Combining similarity measures and inclusion measures for the design of image quality measures, in: WILF 2003 Workshop, 2003, pp. 1–6.

38.

Chen

S.M.

, Measures of similarity between vague sets, Fuzzy Sets and Systems 74 (2) (1995), 217–223.

39.

Hong

D.H.

and Kim

, A note on similarity measures between vague sets and between elements, Information Sciences 115 (1-4) (1999), 83–96.

40.

Mitchell

H.B.

, On the Dengfeng-Chuntian similarity measure and its application to pattern recognition, Pattern Recognition Letters 24 (16) (2003), 3101–3104.

41.

, Cosine similarity measures for intuitionistic fuzzy sets and their applications, Mathematical andComputer Modelling 53 (1-2) (2011), 91–97.

42.

Song

, Wang

, Lei

and Xue

, A new similarity measure between intuitionistic fuzzy sets and its application to pattern recognition, Abstract and Applied Analysis (2014), 11. Article ID 384241.