In current study, image preprocessing takes place via three stages such as (i) color space conversion, (ii) filtering, and (iii) contrast enhancement. Initially, the colored DR image is converted into grayscale form. Then, the median filter is employed to reduce the speckle noise and salt and pepper noise. Finally, Contrast Limited Adaptive Histogram Equalization (CLAHE) technique is utilized to improve the local contrast of the image.

Once the DR images are preprocessed, INS approach is utilized to detect the infected regions in input fundus images. Neutrosophy is an original division of philosophy with new philosophical argument. In literature [14], Smarandache initially established the non-standard analyses, and Fuzzy Logic (FL) which are comprehensive to neutrosophic logic. Then, the fuzzy set is generalized for Neutrosophic Set (NS).

Definition 1. (NS). Assume X as a universe of discourse and NS A is determined on universe, X. The component x in set A is distinguished by x(t, i, f) , A and is written as follows A={[x, (TA(x), IA(x), FA(x))]|x∈X}

where, T,  I, and F denote real/non-standard sets of ]⁻0,  1⁺[ with sup T=t−sup, infT=t−infsupI=i−sup , infI=i−inf , supF=f_sup , infF=f_inf and n−sup=t−sup+i−sup+f_sup , n−inf=t−inf+i−inf+f_inf , so 0⁻ ≤ T_A(x) + I_A(x) + F_A(x) ≤ 3⁺. T,  I, and F are known as neutrosophic modules. In neutrosophic domain, the image is defined as Neutrosophic Image (NI).

Definition 2. (Neutrosophic image). Assume U as a universe of discourse. W = w * w is the group of image pixels, where W ⊆ U and w define the arguments and is computed as concrete state and personality [15]. Next, NI is categorized by two membership sets, I, and F.

In the image provided for analysis, all the pixels P(i, j) are converted from image domain to neutrosophic domain. Afterwards, all the pixels P(i, j) from NI are termed as PNS(i, j) , and PNS(i, j)={T(i, j), I(i, j), F(i, j) . Thus, the membership degree T(i, j), I(i, j), and F(i, j) are determined as follows

(1) T(i, j)=g¯(i, j)−g¯ming¯max−g¯min

(2) g¯(i, j)=1w×w∑m=i−w/2i+w/2⁡∑n=j−w/2j+w/2⁡g(m, n)

(3) I(i, j)=δ(i, j)−δminiδmaxi−δmini    

where,

(4) δ(i, j)=abs(g(i, j)−g¯(i, j))

(5) F(i, j)=1−T(i, j)

where, g(i, j) implies the grayscale value of pixel P(i, j) whereas g¯(i, j) denotes the regional mean value.

If NS theory is used for image processing, I, and F are going to to be ]⁻0,  1⁺[. Since the details regarding membership and non-membership degree are signified as individual values from NS, data ambiguity cannot be well-stated. To define the NI in the most exact form, INS is presented.

Definition 3. (INSt). Assume X as a universe of discourse, and A as an INS that is determined based on universe X. An element x from the set A is noticed as x(t, i, f) , A is written as follows A={[x, (TA(x), IA(x), FA(x))]|x∈X}

For all points x from the set A, the T_A(x), I_A(x) and FA(x)⊂[0, 1], TA(x) indicates the membership degree of point x from the set A.

If X is continuous, INS A is written as A=∫X<T(x) , I(x), F(x) > |x,  x ∈ X. If X is separate, then INS A is determined as =∑i=1n<T(xi) , I(x_i), F(x_i) > |x_i,  x_i ∈ X.

An element x is named after interval wise number and is written as x=([T1, T2], [I1, I2], [F1, F2]) from the INS.

Based on the benefits of INS, it is determined as the Interval Neutrosophic Image (INI).

Definition 4. (INI). For a provided image, all the pixels P(i, j) can be changed from image domain to INS domain. The pixel P(i, j) from an image is called as P_INS, and PINS(i, j)=T(i, j), I(i, j), F(i, j) . Now PINS(i, j) is an interval number set which are is written as follows.(6) PINS(i, j)={[T1(i, j), T2(i, j)], [I1(i, j), I2(i, j)], [F1(i, j), F2(i, j)]}

Here, T,  I,  F represent the hesitancy degree, non-membership degree, and membership degree correspondingly.

Intensification Operation. The intensification operation for fuzzy set is provided and the membership values are implemented to modify the intensifier operator. This operator uses the square root of maximal/minimal to stretch out the contrast value with the help of the equation. Later, image conversion occurs during when a few areas are blurred. So, intensification operation is used for image model since it makes the images clear. The outcomes of image segmentation is optimal.

Definition 5. (Intensification operation). In fuzzy set, the intensification function is determined as given below. μ(i,j)={M.μ2(i,j),0≤μ(i,j)≤t1−2⋅(1−μ(i,j))2,t≤μ(i,j)≤1

Score Function (SF). SF is the main portion of probability concepts. For optimal image segmentation, a novel SF is established to express the pixel level characteristics based on the ambiguity interval number from INI.

Definition 6. SF: Assume A={TA, IA, FA} as an interval NS. Then, SF S(A) is determined as follows S(A)=12∑tA∈TA,iA∈IA,fA∈FA ⁡(tA−iA−fA)

where, t_A ∈ T_A,  i_A ∈ I_A and f_A ∈ F_A.

Definition 7. (Clustering Algorithm). The clustering technique is used for the classification of same points under a similar group. Assume X = X_i, {i=1, 2, …, n} as a dataset while the clustering provides a reason for the division C = C₁, C₂, …, C_m, that fulfills the following equation. X=∪i=1m⁡Ci

where, Ci≠∅, for i=1, 2, …, m Ci∩⁡Cj=∅, for i, j=1, 2, …, m, i≠j

It considers the image as a group of features. Among the clustering techniques, k-means technique is utilized in most of the cases. It can act as a technique that can combine the objects as to k groups depend on its features. In a recent study, k-means technique was utilized to segment the images. All the clusters must be as compact as probable so that it can be defined as an objective function to cluster the analyses model. The objective function of k-means can be computed as follows. J=∑j=1x⁡∑i=1k⁡‖Xij−Cj‖ where the ‖Xij−Cj‖ defines the distance measures between data point and the cluster center. Every step in INI segmentation are listed herewith.

Input the target image and examine the pixels

Feature extraction calculates the dimensional decrease from image processing stage. Then, based on this outcome, the image is utilized for classification. In this stage, the input data is reduced to a diminished illustrative group of features. The features are employed as supporting entities so that the classes are allotted for classification [16]. The feature extraction process includes histogram features, texture features, and wavelet features, which derive a set of features for classification.

At the time of classification, ODBN model is executed during when the parameter tuning of DBN model takes place using SFO algorithm. Restricted Boltzmann Machine (RBM) is energy-oriented probabilistic method which is otherwise called as a restricted form of Boltzmann Machine (BM), a log linear Markov Random Field. It has an observable node x which is equivalent to input whereas hidden node h corresponds to the latent feature. The joint distribution of observable node x ∈ ℝ^J and hidden variables h ∈ ℝ^I are determined as follows(9) p(x, h)=1Ze−E(x,h), E(x, h)=−hWx−ch−bx where ∈ ℝ^I×J,  b ∈ ℝ^J, and c ∈ ℝ^I signify the model varaibles, and Z represents the separation function. As units from the layer are independent of RBM, the subsequent conditional distribution procedure is followed(10) p(h|x)=∏i=1I⁡p(hi|x), p(x|h)=∏j=1J⁡p(xj|h)

For every binary unit, x∈{0, 1}J and h∈{0, 1}I , it is composed p(h_i = 1|h) = σ(c_i + W_ix) and p(x_j = 1|h) = σ(b_j + W_jx) where σ() implies the sigmoid function. In this method, RBM with binary unit is otherwise called as unsupervised Neural Network (NN) through sigmoid activation function [17]. The method calibration of RBM is complete with minimized negative log-probability and Gradient Descent (GD). RBM has the benefits of taking over conditional probability that permits the attained model instances to be simple with Gibbs sampling model.

DBN is a generative graphical method that contains stacked RBM. According to deep structure, DBN performs the hierarchical illustration of input data. Hinton et al. [18] established DBN with trained technique which greedily trains one layer at once. Joint dispersal is determined as given herewith, provided the observable unit x and ℓ hidden layers are as follows.(11) p(x, h1, ⋯, hℓ)=p(hℓ−1, hℓ)(∏k=1ℓ−2⁡p(hk|hk+1))p(x|h1)

As all the layers of DBN are created as RBM, all the layers of DBN are trained similar to trained RBM. Fig. 2 illustrates the structure of DBN. The classifier is directed as an initialized network with DBN training. With training data set D={(x(1), y(1)), …, (x(|D|), y(|D|))} that has input x and label y, the pre-trained phase resolves the subsequent optimization problems at all the layers k.(12) minθk1|D|∑i=1|D|⁡[−logp(xk(i); θk)] where θk=(Wk, bk, ck) refers to the parameter of RBM method that refers to weight, visible biases, and hidden biases from energy function, and xk(i) implies the observable input to layer k equivalent to input x⁽ⁱ⁾. Layer-wise upgradation model is required to resolve ℓ from bottom to top hidden layers. In order to fine tune the phase, subsequent optimization problem is resolved.(13) minϕ1|D|∑i=1|D|⁡[ℒ(ϕ, y(i), h(x(i)))] where ℒ() refers the loss function, h represents the last hidden feature at layer ℓ, and ϕ signifies the classification parameter. When simplified, it is composed of h(x(i))=h(xℓ(i)) . If DBN and feed forward network are integrated with sigmoid activation, every weight and hidden bias parameters, between input are hidden layers, communicate to the trained stages.

In order to adjust the parameters of DBN technique, SSO algorithm is used [19]. It imitates the nature of shepherds. Shepherd places the horse or other animals in a herd based on the instinct of animal to find the optimum method for pasturing. Thus, the shepherd leads the sheep in the direction of horse. This phenomenon is the fundamental stimulation of SSO. In SSO, every candidate solution is assumed to be sheep. SSO is initiated with an arbitrarily-created sheep. Afterward, the following m sheep is placed around other residual sheep and is arbitrarily placed in the succeeding herd based on the quality similar to earlier created herd. This procedure is continued till every herd is created. While the procedure is named as ‘shuffling’, this process enhances the chance of survival by allocating the data in search procedure among the herds. Especially, the herds are developed with great potentials based on the replacement of data with other herds. Therefore, every shepherd contains a sheep and a horse. Later, for every shepherd, a horse and a sheep are arbitrarily chosen. The shepherd attempts at guiding the sheep against the horse. Next, the step size is evaluated depending upon their motion (motion of shepherd to horse and sheep).

When the novel location of shepherd is not worse compared to the prior one, the location gets upgraded (i.e., replacing approach). This procedure is repetitive for all the sheep in every herd. At last, the herd is combined with one another. Repetitively, the sheep are separated into herds and the above-discussed procedure is repeated till it attains the ending criteria. Shuffled Complex Evolution (SCE) is followed to ease the process of selecting a sheep and a herd so that it could denote a community and a member for every community, correspondingly. Next, SSO steps are provided concisely herewith.

Step 1: Initiation

SSO is initiated with randomly-produced initial Member Of Community (MOC) in the searching area using the formula given herewith. MOCij0=MOC min +rand×(MOCmax−MOCmin);  (14) i=1, 2, …, m and j=1, 2, …, n

Here, rand represents the arbitrary vector with every component created between [0, 1]; MOC_min and MOC_max represent the lower and upper bounds of the designed parameter, correspondingly; m indicates the amount of communities, and n denotes the amount of members that belong to every community.

Regarding this, it is assumed that the overall amount of community members is attained by the formula given herewith.(15) nMC=m×n

Step 2: Shuffling procedure

At the time of shuffling procedure, the process given in Eq. (16) is executed n times autonomously, till the MC matrix is created by [20]:(16) MC=[MOC1,1MOC1,2⋯MOC1,j⋯MOC1,nMOC2,1MOC2,2⋯MOC2,j⋯MOC2,n⋮⋮⋮⋮⋮⋮MOCi,1MOCi,2⋯MOCi,j⋯MOCi,n⋮⋮⋮⋮⋮⋮MOCm,1MOCm,2⋯MOCm,j⋯MOCm,n]

Step 3: Motion of community member

A single step size of motion is evaluated for every community member according to two vectors. The initial vector ( stepsizei,jWorse ) demonstrates the capacity for visiting a new region of search space (diversifying approach). Unlike, the next vector ( stepsizei,jBetter ) denotes the capacity for exploring the adjacent of formerly-visited significant search space region (intensified approach). The scientific equation of the step size is given herewith. stepsizei,j=stepsizei,jWorse+stepsizei,jBetter (17) i=1, 2, …, m and j=1, 2, …, n where stepsizei,jWorse and stepsizei,jBetter are determined as follows.(18) stepsizei,jWorse=α×rand1×(MOCi,w−MOCi,j) (19) stepsizei,jBetter=β×rand2×(MOCi,b−MOCi,j)

where rand₁ and rand₂ represent the arbitrary vectors with every component created between [0,1]; MOC_i,b (chosen horse) and MOC_i,w (chosen sheep) denote the worse and optimum members based on objective function relevant to MOC_i,j. It is significant that the primary member of (MOC_i,1) does not have a member more than itself. Therefore, the stepsizei,1Better is equivalent to 0. Alternatively, MOC_i,n does not have a member inferior to itself, since it is last member of i th community. Thus, the stepsizei,1Worse is also equivalent to 0. Furthermore, α and β indicate the factors that manage exploitation and exploration phases, correspondingly. This factor is determined as given herewith.(20) α=α0−α0×t; t=iteration  Max  iteration (21) β=β0+(βmax −β0)×t

It is clear that when the iteration number t raises, β and α decrease and vice versa. Consequently, the exploration rate becomes a failure when exploitation rate gets increased.

Step 4: Upgrade the location of every community members

As per earlier step, the novel place of MOC_i,j is evaluated by Eq. (22). Afterward, the place of MOC_i,j is upgraded or otherwise the older objective function value is given as follows.(22) newMOCi,j=MOCi,j+stepsizei,j

Step 5: Check end criteria

The SSO algorithm is stopped once the maximum iteration count is reached. Else, it is repeated to step 2 for a novel set of iterations. The overall processes are shown in Fig. 3 [21].