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Lord Buddha Education Foundation, Kathmandu, Nepal (When the paper was submitted, Dr. Kautish was in LBEF Campus, Nepal. But currently he is in Chandigarh Group of Colleges, Jhanjeri, Mohali, Punjab, India.)

The intuitive fuzzy set has found important application in decision-making and machine learning. To enrich and utilize the intuitive fuzzy set, this study designed and developed a deep neural network-based glaucoma eye detection using fuzzy difference equations in the domain where the retinal images converge. Retinal image detections are categorized as normal eye recognition, suspected glaucomatous eye recognition, and glaucomatous eye recognition. Fuzzy degrees associated with weighted values are calculated to determine the level of concentration between the fuzzy partition and the retinal images. The proposed model was used to diagnose glaucoma using retinal images and involved utilizing the Convolutional Neural Network (CNN) and deep learning to identify the fuzzy weighted regularization between images. This methodology was used to clarify the input images and make them adequate for the process of glaucoma detection. The objective of this study was to propose a novel approach to the early diagnosis of glaucoma using the Fuzzy Expert System (FES) and Fuzzy differential equation (FDE). The intensities of the different regions in the images and their respective peak levels were determined. Once the peak regions were identified, the recurrence relationships among those peaks were then measured. Image partitioning was done due to varying degrees of similar and dissimilar concentrations in the image. Similar and dissimilar concentration levels and spatial frequency generated a threshold image from the combined fuzzy matrix and FDE. This distinguished between a normal and abnormal eye condition, thus detecting patients with glaucomatous eyes.

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. [

This paper is structured as follows

Haubecker et al. [

The optic disc cup and the parameters of the optic plate were used in the assessment of the glaucoma clutter. Fard et al. [

Soltani et al. [

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 [

Gour et al. [

For a fixed set

Here

Here,

where

where

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 [

From

It was also applied to the three aforementioned categories, including glaucomatous, suspected to be glaucomatous, and non-glaucomatous [

Some numerical specifications were required to enhance the image. Here, we considered the main CDR, optic nerve head, and area of optic nerve values. In general, CDR ranged from 0.26 to 0.5 mm. The average optic nerve head diameter varied from 1.2 to 2.5 mm. The area of the optic nerve head in

The predicted retinal image’s cup to disc values for average vertical CDR ranged from 0.541 (±0.226) to 0.593 (±0.187) for average horizontal CDR. The retinal image’s cup to disc values for suspected glaucoma ranged from 0.657 (±0.149) for vertical CDR to 0.681 (±0.0715) for horizontal CDR. For glaucomatous eyes, the vertical CDR range was 0.776 (±0.250), and the horizontal CDR was 0.797 (±0.143) as shown in

Eye condition | Cup to disc ratio (CDR) | |||
---|---|---|---|---|

Average vertical CDR | Tolerance | Average horizontal CDR | Tolerance | |

Normal eyes | 0.541 | ±0.226 | 0.593 | ±0.187 |

Mild/suspicious glaucomatous eyes | 0.657 | ±0.149 | 0.681 | ±0.0715 |

Severe glaucomatous eyes | 0.776 | ±0.250 | 0.797 | ±0.143 |

Let us consider a retinal image

where

where

The first step for fuzzy computation of the retinal image

The membership degrees of the three fuzzy sets (three images to be analyzed) described by using

Then, an instruction parameter

The fuzzy degrees of belonging, together with the weighted values

Now, the fuzzy weighted regularization could be calculated for the second image using the attenuation parameter

Similarly, the fuzzy degrees of belonging together, along with the weighted values

The three sets of calibrated weighted fuzzy partitions that use the attenuation parameter

After changing the membership degree, the membership gradation for the sets

A similar approach was taken to obtain

The value

A similar approach was followed for

The selection of the new and modified degree of membership for each fuzzy series of the three partitions based on

Fuzzy manipulation functions were obtained by using

The enhanced retinal images represented for each of the three images by

where

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 [

Let

The calculation of fuzzy integral for image segmentation in

The predictable three fuzzy partitions sets were defined as

The three fuzzy density values were estimated to be

Now, the characteristic of image δ could be obtained using the three characterized fuzzy densities

Therefore, any of the skills in the set of measures

After measuring the fuzzy density in the set of {_{0}, a_{1}, a_{2}

The above three equations must satisfy

where _{0}, I_{1}, I_{2}}.

The degree of belonging could be classified as an object in two classes

The degree of belonging could be classified as an object in two classes

Therefore, an object could belong to the class

where

The segmented retinal image

where

According to the proposed algorithm, which is illustrated in

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

S. no. | Specifications | Mean | SD | SE | Range |
---|---|---|---|---|---|

Optic disc | |||||

1 | Area of the disc (30–40 years of age group) | 2.139 | 0.5221 | 0.1167 | |

2 | Area of the disc (41–50 years of age group) | 2.6675 | 0.8941 | 0.1999 | |

3 | Area of optic nerve head ( |
2.485 | 0.6608 | 0.1478 | |

Disc diameter (mm) | |||||

4 | Horizontal disc diameter | 1.798 | 0.2421 | 0.0489 | |

5 | Vertical disc diameter | 1.881 | 0.2626 | 0.0412 | |

Neuroretinal rim | |||||

6 | Temporal inferior | 0.445 | 0.1418 | 0.0222 | |

7 | Temporal superior | 0.457 | 0.1238 | 0.0224 | |

8 | Nasal | 0.591 | 0.1446 | 0.02515 |

Note:

In general, according to the algorithm of retinal image processing which was shown in

where

Similarly,

Thus,

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.

The authors express their appreciation to King Saud University for funding the publication of this research through the Researchers Supporting Program (RSPD2023R809), King Saud University, Riyadh, Saudi Arabia.

This research was funded by the Researchers Supporting Program at King Saud University (RSPD2023R809), Riyadh, Saudi Arabia.

Dorathy Prema Kavitha, L. Francis Raj, Sandeep Kautish and Abdulaziz S. Almazyad conceived of the presented idea. D. Dorathy Prema Kavitha and L. Francis Raj developed the theory and performed the computations. Sandeep Kautish, Abdulaziz S. Almazyad, Karam M. Sallam, and Ali Wagdy Mohamed verified the analytical methods. Sandeep Kautish, Abdulaziz S. Almazyad encouraged D. Dorathy Prema Kavitha and L. Francis Raj to investigate the proposed method and supervised the findings of this work. All authors discussed the results, contributed to, and approved the final manuscript.

Not applicable.

The authors declare that they have no conflicts of interest to report regarding the present study.