Glaucoma occurs due to damage to the nerve that connects the eye to the brain, known as the optic nerve. Glaucoma is one of the silent disorders that human beings can get without previous physical signs or symptoms. Tjandrasa et al. [1] assumed that individuals with glaucoma could lose their optic nerve tissues which could lead to losing their vision. Unfortunately, there is no cure for a glaucomatous eye. With that being said, early discovery of the glaucomatous eye can slow down the damage to the optic nerve which can fortunately save the affected individual’s vision. The Cup to Disk Ratio (CDR) is the primary tool used to check whether the individual is at hazard for glaucoma or not. The CDR and ISNT (inferior (I) > superior (S) > nasal (N) > temporal (T)) are the two key diagnostic tools used to evaluate the Optic Nerve Head (ONH). The CDR is a rule that was developed through manual estimation by an ophthalmologist. Despite its benefits, manually monitoring ONH adjustment is a time-consuming process. Since the early diagnosis of this disease is necessary to prevent vision loss, numerous efforts have been made to the programmed discovery of glaucoma at an early stage. Apart from this, optimized medical-aided diagnosis and treatment were analyzed by Zhang et al. [2].

This paper is structured as follows Section 2 provides a literature review of what has been previously done in this field of research. Section 3 discusses the concepts of intuitive fuzzy sets. Section 4 outlines the approach used in retinal image enhancement using fuzzy concepts and the implementation through CNN. This section also analyzes fuzzy manipulation and defuzzification techniques related to CDR. Section 5 deals with fuzzy image segmentation which is used to integrate all the detected regions and partition the image based on similar and dissimilar concentration levels. Spatial frequency fuzzy metric concepts are also discussed for generating a threshold image. In Section 6, numerical illustrations and the various results that were obtained are analyzed.

Haubecker et al. [3] proposed a neuro-fuzzy inference system to detect glaucoma using higher-order spectra (HOS) and discrete wavelet transform (DWT) which was based on the criterion of diffuse expulsion. The proposed system was tested on a series of actual ophthalmic images of normal and glaucomatous cases. The most important parameters were then extracted, after having recognized the elliptical form of both the optic disc and boundary. After extracting the highlights, a factual analysis was performed on the partitioned separation. Finally, Dehghani et al. [4] proposed an assortment of algorithmic rules based on fuzzy logic approaches to determine the condition of the patient’s eye. The condition of the eye was then distinguished into three categories including normal, mild glaucoma, and direct/severe glaucoma. Naik et al. suggested a fuzzy set theory-based image enhancement method on HSV (hue, saturation, value) images by applying Gaussian distribution on S and V parameters [5].

The optic disc cup and the parameters of the optic plate were used in the assessment of the glaucoma clutter. Fard et al. [6] utilized a Support Vector Machine (SVM) when calculating the F-cut done through the use of random mass transformation. Among the various detection techniques, background image-based detection of the retina plays an important role as it was subjected to non-invasive detection methods. Then, they made the categorization after selecting the shape. Detecting glaucoma disorder could be achieved by determining the shape of the retina and checking various medical characteristics of the eyes, such as measurements of the thickness of the retinal nerve fiber layer, the relationship between the edge and the disc, the optic disc and the head of the nerve. Rahman et al. [7] investigated the distinct techniques of glaucoma detection using retinal base images. Sharma et al. [8] studied retinal blood vessel segmentation like glaucomatous eye image segmentation.

Soltani et al. [9] predicted glaucoma in the early stage by identifying several characteristics of glaucoma which could be represented as (i) open-tangent glaucoma, (ii) closed-angle Glaucoma, (iii) normal-tension glaucoma, and (iv) connective glaucoma. Welfer et al. [10] tested tonometry (measures the pressure within the eye), ophthalmoscopy (inside the fundus of the eye and other structures), perimetry (systematic calculation of the visual field), gonioscopy (anatomical angle developed between the eye’s cornea and iris) and pachymetry (measures the thickness of the eye’s cornea) and noted that these tests need to be conducted to detect glaucoma. Recently, the detection of glaucoma has been achieved using Random-Forest classifiers with a 91% accuracy. Blurred convergence of inbreeding glaucoma was used to locate the optical resolution in the retinal frame. A test for glaucoma was performed using a digital eye camera to check for optical coherence tomography (OCT), measurement of intraocular pressure (IOP), and evaluations of optic nerve hypoplasia (ONH), retinal nerve fiber layer (RNFL), and visual field defects [11].

The SVM is a supervised machine learning algorithm that can be used for both classification and regression challenges which can be utilized in the process of glaucoma detection [12]. This method is based on the adaptive threshold and it improves the image quality by removing the noises in the image. On retinal images, Li et al. [13,14] and Vostatek et al. [15] proposed vessel segmentation and amplitude estimation utilizing large-scale production of matching filtered responses.

Gour et al. [16] reported an approach to detect the order of glaucoma using eye-based images. This was done through a computerized diagnostic system that was used to find the malformation present in the fundus images. Fraz et al. [17] developed the method of perceiving glaucoma progression using Heidelberg Retinal Tomography (HRT) images. Fuzzy differential equations (FDE) along with intuitionistic fuzzy sets (IFS) are used to develop reliable traffic management systems and shift scheduling for supporting an automatic transmission along with artificial intelligence (AI). In robotics, FDE brings changes in the navigational directions to avoid the obstacle. Furthermore, it can be used in finding the difference between the crossover rate and mutation rate in the genetic algorithmic optimization process.

For a fixed set X1, the fuzzy set of A1 is defined as

A1={<x,μX1(x)>|x∈X1}

μA1 (x): X1→[0,1] defines the minimum and maximum degrees of the element x∈X1 to the set A1. For a fixed set X1, an IFS of A1 is an optimal identifier and it is classified as

A1={<x,μA1(x),vA1(x)>|x∈X1}

Here μA1 (x): X1→ [0, 1] and vA1 (x): X1→[0,1] defines the membership function degrees and the degrees of specific n-membership function whose element x∈X1 to the set A1. This IFS is used to identify the max-min-max of membership and non-membership functions and this leads to classifying the differences between the input values or functions. For every x∈X1, 0 ≤  μA1 (x) + vA1 (x) ≤ 1 and the quantity πA1(x) =  1−μA1 (x) − vA1 (x) is called the intuitive index or index of hesitation, which may require membership value, non-membership value, or both. Note that fuzzy sets are intuitive fuzzy sets (IFS), but the opposite is not necessarily true. Furthermore, A1 is an IFS of the set X1 and that R1 is a conditional relation of X1→Y1, then the composition B1 max-min-max of IFS X1 with the IF relation R1(X1→Y1) is defined as B1=R1,  A1 with membership and non-membership functions defined in Eqs. (1) and (2), respectively:

(1)μB1(y)=maxx∈X1⁡{min [ μA1(x) , μR1(x, y)]}

(2)nB1(y)=minx∈X1⁡{max [ vA1(x) , vR1(x, y)]}

Here, Eqs. (1) and (2) are used to identify the max-min (μB1(y)) and min-max (nB1(y)) values of membership and non-membership functions. The difference between max-min and min-max was measured through the fuzzy difference equation by Alamin et al. [18]. The relationship between symptoms and disease is classified in the following manner:

S1={s1,s2,…,sm}

D1={d1,d2,…,dn}

P1={p1,p2,…,pq}

where S, D, and P are the finite sets of symptoms, type of disease, and patients, respectively. Fuzzy Relations (FR) are defined as Q1 and R1 in Eqs. (3) and (4), respectively:

(3)Q1={<(p,s), μQ1(p,s),vQ1(p,s)>|(p,s)∈P1×S1}

(4)R1={<(s,d), μR1(s,d),vR1(s,d)>|(s,d)∈S1×D1}

where μQ1(p,s) indicates the degree to which symptoms appear inpatient p and vQ1(p,s) indicates the degree to which symptoms do not appear inpatient p. Similarly, μR1(s,d) indicates the degree to which symptoms confirm disease d and vR1(s,d) indicates the degree to which symptoms do not confirm disease d. Composition T1 of the IFR’s R1 and Q1(T1=R1Q1) describes the patient’s pi status in terms of P1 to D1. Glaucoma diagnosis was accomplished by utilizing a Convolutional Neural Network (CNN) which was used to classify glaucomatous and non-glaucomatous cases and this classification was made by the fuzzy difference Eq. (19). The proposed algorithm took an eye image as its input and assigned the values of membership function to relate its weights and biases using optimization function through deep learning [19]. Then, the CNN carefully separated and differentiated the eye images from one another [20]. This included for example differentiating an image with early stage of a glaucomatous eye from a glaucomatous eye where the case is more severe. This helped to enter deeply as

(5)μT1(pi, d)=maxs∈S1⁡{min [ μQ1(pi, d) , μR1(s, d)]}

(6)vT1(pi, d)=mins∈S1⁡{max [ vQ1(pi, d) , vR1(s, d)]}   ∀ pi∈P1and d∈D1

Based on the max–min consensus convergence algorithm, each stage was iteratively updated according to its weighted average together with its minimum and maximum stages from early stage to glaucomatous eyes [21].

From Q1 to R1, a new IFR T1 measurement could be calculated for which, in general, the patient’s diagnosis was labelled p for any type of disease d, to satisfy the following:

It was also applied to the three aforementioned categories, including glaucomatous, suspected to be glaucomatous, and non-glaucomatous [22]. This new measure T1 interprets the highest levels of association and the lowest degree of non-association of symptoms, as well as the lowest levels of the intuitionistic index in diagnoses. If almost equal values are obtained for a different diagnosis in T1, in this case, the minimal intuitionistic index is considered [23].

Fuzzy integrals based on the method of combining a fuzzy image region with different fuzzy characteristics have been used in nonlinear image processing (region, edge detection, histogram, and analyzer) for segmentation. These are used to recursively combine the regions based on the maximum and minimum fuzzy integral [26]. The goal of integrating the regions is to combine the fuzzy targets, and evidence of the fuzzy measurement is reflected in the image regions. The proposed retinal image processing algorithm is shown in Fig. 2 and the implementation of the proposed algorithm was achieved as depicted in Figs. 3–5.

Let X1 = {x1,x2, . . . ,xn } be a compilation of data sources, a fuzzy measure is a function of the real value μ defined as μIi:2x→[0, 1] in the set of the family μIi, μI2, and μI3 which is used to segment the images. The fuzzy density measure is μδ and it can be measured as

(27)δ+1=∏i=1n(1+δμIi)

The calculation of fuzzy integral for image segmentation in Eq. (22), δ represents the characteristics of the image as there are different intensities for the various regions [27]. The proposed algorithm is used to measure the fuzziness and the fuzzy integral to connect different regions of partitioning with similar and dissimilar concentration levels.

According to the proposed algorithm, which is illustrated in Figs. 1 and 2, the following were applied to the sample images. Canny edge detection, histogram equalized images, histogram of original images, and histogram of histogram equalized images were classified for normal eyes, early-stage glaucomatous eyes, and glaucomatous eyes as shown in Figs. 3–5.

To assess objective improvement, we used the fuzzy spatial frequency metric along with a canny edge detector to analyze the execution of the proposed strategy towards the parameters of the area of the disc of the age group of 30–40-year-olds and 41–50-year-olds. The corresponding disc diameters (both horizontal and vertical discs), temporal inferior, temporal superior, and nasal were taken as parameters. Sample data for 20 individuals were analyzed and the results are presented in Table 2 and illustrated in Fig. 6.

In general, according to the algorithm of retinal image processing which was shown in Fig. 2, the spatial recurrence was assessed based on the standard level of visual clarity. When the spatial recurrence was determined to be a high value, which indicated better vision,

(36)F=(RF2+CF2)

where R represents the row frequency (a horizontal measurement of CDR) and C represents the frequency column (a vertical measurement of CDR), M×N is defined by the matrix IF1 which is used to calculate the CDR value and the corresponding standard error as follows:

(37) RF1=1M(N−1)∑i=0M−1∑i=0N−1(IF1(i,j+1)−IF1(i,j))2

CF1=1M(N−1)∑i=0M−1∑i=0N−1(IF1(i+1,j)−IF1(i,j))2 

Similarly, IF2 and IF3 could also be better estimated using the following equations:

RF2=1M(N−1)∑i=0M−1∑i=0N−1(IF2(i,j+1)−IF2(i,j))2

(38)CF2=1M(N−1)∑i=0M−1∑i=0N−1(IF2(i+1,j)−IF2(i,j))2

RF3=1M(N−1)∑i=0M−1∑i=0N−1(IF3(i,j+1)−IF3(i,j))2

(39) CF3=1M(N−1)∑i=0M−1∑i=0N−1(IF3(i+1,j)−IF3(i,j))2 

Thus, Eqs. (37)–(39) along with Eqs. (24)–(26) could be used to categorize the eye condition of the patient. The validation of this categorization was then validated through hypothetical medical information, such as optic disc area, disc diameter, and neuroretinal rim values. Row and column frequencies were also used to obtain and classify the range of the region of convergence of the retinal images.

A novel approach for the detection of a glaucomatous eye using intuitive fuzzy sets with membership and non-membership functions was discussed in this study. This was accomplished by utilizing intuitive fuzzy (optimal classifier) sets in the region of convergence of the partitioned retinal images. Fuzzy degrees and fuzzy weighted values were then used as the concentration level in the retinal images. Once the concentration levels of the retinal images were found, the image was then classified as normal eyes, suspected glaucomatous eyes, or glaucomatous eyes. The measurement of the concentration level and checking for similarities in different regions of the images generated a threshold image from the combined fuzzy matrix along with the spatial frequency fuzzy metric which led to the precise classification of the eye condition. Canny edge detection, a histogram equalized image, a histogram of original images and a histogram of histogram equalized images were classified for normal eyes, early-stage glaucomatous eyes, and glaucomatous eyes. These techniques were assorted through the proposed deep learning algorithm using the Convolutional Neural Network. Based on this analysis, the intuitionistic fuzzy sets were carried out on a similar pattern identifier envisaged by models of human perception.