**2. Prerequisites**

*Math Operators.* At first, a short introduction into the mathematical part of the work is presented; this is, mainly based on Rus [19] who introduced the theory of admissible perturbations of an operator. The admissible perturbation operator was also studied in [20].

*Demicontractive operators.* As already stated in our previous work [18], a *demicontractive operator* (*T*) is defined by *C*, a subset of R (domains and co-domains). For an existing *contraction coefficient* (*k* < 1), each fixed point (*p*) of the demicontractive operator and all numbers (*x* ∈ *C*) the inequality (1) is true.

$$\|Tx - p\|^2 \le \|x - p\|^2 + k\|x - Tx\|^2. \tag{1}$$

For a nonempty set ( *X*) and an admissible mapping, ( *G* : *X* × *X* → *X*), the following statements are true: for all *x* ∈ *X* G(x, x) = x and *<sup>G</sup>*(*<sup>x</sup>*, *y*) = *x* implies *y* = *x* [19].

The admissible perturbation of the operator *f* (*f* : *X* → *X*) [19] is the admissible mapping *fG* : *X* → *X* (*fG*(*x*) := *<sup>G</sup>*(*<sup>x</sup>*, *f*(*x*))).

*Krasnoselskij operator.* The *Krasnoselskij algorithm* [19], corresponding to an admissible mapping ( *G* : *X* × *X* → *X*) of an nonlinear operator ( *f* : *X* → *X*), is defined as an iterative algorithm {*xn*}*n*∈<sup>N</sup> with *x*0 ∈ *X* and *xn*+1 = *<sup>G</sup>*(*xn*, *f*(*xn*)), where *n* ≥ 0.

For further details and examples, see [18,20,21]. *The χ operator.* The *χ* operator *χ* : R × R −→ [0, 1) is defined as:

$$\chi(\mathbf{x}, \mathbf{y}) = \frac{\mathbf{x}^2 \cdot \mathbf{y}^2}{(1 + \mathbf{x}^2) \cdot (1 + \mathbf{y}^2)},\tag{2}$$

where *y* : R → R, *y*(*x*) = 23 *x* sin 1*x* , if *x* = 0.

 *Particular Math Functions.* The operators further used as *f*(·) in the considered methods (see (8)) are as follows:

> *f*(*x*) =

Sin:

> KH:

$$f(\mathbf{x}) = (1 - \lambda) \cdot \mathbf{x} + \lambda \cdot \frac{2}{3} \mathbf{x} \sin \frac{1}{\mathbf{x}} \text{ if } \mathbf{x} \neq 0;\tag{4}$$

 (3)

sin*<sup>π</sup><sup>x</sup>* 2*λ*, if 0 ≤ *x* ≤ *λ*;

$$\text{Chi: } \qquad f(\mathbf{x}) = (1 - \chi(\mathbf{x}, y(\mathbf{x}))) \cdot \mathbf{x} + \chi(\mathbf{x}, y(\mathbf{x})) \cdot y(\mathbf{x}) \text{ if } \mathbf{x} \neq \mathbf{0};\tag{5}$$

The functions (3)–(5) are zero in all other possible cases. The *λ* parameter from Equations (3) and (4) adjusts the operators used as test functions in [22]. In [23], the authors used two admissible perturbation operators for computing the heuristic value required within the ACO algorithm.

In the present article, where admissible perturbations of demicontractive mappings are utilized as test functions, and the *PSL* vector is utilized for each ant, a sensitivity to the artificial pheromone is introduced using a specific coefficient influenced by the image's intensity values for the edge detection problem.

The ants have different roles in edge extraction: some agents are explorers and others are exploiters; these roles are exchanged as the *PSL* vector updates during processes. The obtained results for CT and X-ray medical images and the comparison among the results using the proposed operators are made in Section 4.

#### **3. Problem and Methods**

#### *3.1. Medical Image Edge Detection Problem*

The problem to solve is the edge detection problem. The current work improves solutions of particular medical images due to their complex edges based on X-rays and tomographic images.

Image edge detection involves the detection of discontinuities in brightness while processing the image in order to find the boundaries of objects.

#### *3.2. Sensitive Ant Colony Optimization Method*

The method used is an improved version of *Ant Colony Optimization (ACO)* [1] called the *Sensitive Ant Colony Optimization (SACO)* [24,25]. The ant colony optimization sensitive approach for medical image edge detection is further presented. There is considered a colony of *K* ants engaged in a search within the graph space X , with *M*1 × *M*2 nodes.

SACO as well as ACO use artificial ants to move in a 2D image in order to build the pheromone matrix; each matrix element represents the edge information for every pixel in the image.

In general, ACO and its versions builds a solution with the use of artificial ants; these agents search for the best path in a given space by depositing artificial pheromones [1,23]. These pheromone trails are updated during the search process.

The ACO and SACO general scheme includes an initialization process followed by *N* construction steps while creating and updating pheromone matrix, and, finally, performing the decision process to determine a beneficial solution.


*Initialization process:* In particular, for image edge detection with ACO and SACO during the initialization process, the entire ant colony (*K* ants) places ants randomly on the image matrix. Each image pixel is considered to be a node in a graph. Each initial pheromone matrix *<sup>τ</sup>*(0) has a constant *τinit* value. A constant value *L* defines the number of moves during the construction process.

For SACO in particular, each *PSL* vector component is initialized with 1, starting as the original ACO (see Figure 1).

**Figure 1.** Symbolic illustration of the sensitive ant model showing the ants' probability variation within the unit interval from the ACO probability when *PSL* = 1 to a random walk probability when *PSL* = 0.

*Construction phase:* at the *n*-th construction step, a randomly chosen ant will move from node *i* to *j* according to the transition probability in (6) for *L* steps.

$$p\_{ij}^n = \frac{\left(\pi\_{ij}^{(n-1)}\right)^\kappa \left(\eta\_{ij}\right)^\beta}{\sum\_{j \in \Omega\_i} \left(\pi\_{ij}^{(n-1)}\right)^\kappa \left(\eta\_{ij}\right)^{\beta'}} \qquad \text{if } j \in \Omega\_{i\prime} \tag{6}$$

where *τI J* is the pheromone value on (*i*, *j*); *ηI J* the heuristic value connecting nodes *i* and *j* the same for all *n* construction steps; *α* and *β* are the weighting factors for the pheromone and the heuristic; and Ω*i* includes the neighborhood nodes of node *i*.

The overall eight-connectivity neighborhood for each pixel *Ii*,*j* within the local configuration at the *Ii*,*j* pixel, *cIi*,*j*, for computing the variation value *Vc*(*Ii*,*j*) defined by (8) is illustrated in [18].

Here, we propose the computation of *ηi*,*j* according to the local statistic of the pixel (*i*, *j*) (Equation (7)).

$$\eta\_{i,j} = \frac{1}{Z} \cdot V\_{\mathfrak{c}}(I\_{i,j}) ,\tag{7}$$

where *Z* = *M*1 ∑ *i*=1 *M*2 ∑ *j*=1 *Vc*(*Ii*,*j*) is a normalization factor, *Ii*,*j* is the intensity value of the image

pixel (*i*, *j*); and the function *Vc*(*Ii*,*j*) processes the "clique" *cIi*,*<sup>j</sup>* [22].

The *Vc*(*Ii*,*j*) value at pixel *Ii*,*j* is influenced by the image's intensity values for *cIi*,*j*, and its value is [22]:

$$V\_{\mathbf{c}}(I\_{i,j}) = \begin{array}{c|c} \left| I\_{i-2,j-1} - I\_{i+2,j+1} \right| + \left| I\_{i-2,j+1} - I\_{i+2,j-1} \right| \\ + \left| I\_{i-1,j-2} - I\_{i+1,j+2} \right| + \left| I\_{i-1,j-1} - I\_{i+1,j+1} \right| \\ + \left| I\_{i-1,j} - I\_{i+1,j} \right| + \left| I\_{i-1,j+1} - I\_{i+1,j-1} \right| \\ + \left| I\_{i-1,j+2} - I\_{i+1,j-2} \right| + \left| I\_{i,j-1} - I\_{i,j+1} \right| \end{array} \tag{8}$$

In order to validate an edge within the solution, a decision is made for each image pixel by applying a threshold *T* (see [26]) to the final pheromone matrix *<sup>τ</sup>*(*N*).

The artificial pheromone matrix values are updated both locally and globally.

*Locally update the pheromone matrix τ.* The local pheromone matrix update is made after each ant moves within each construction step [22].

$$\text{Local update:} \quad \tau\_{ij}^{(n)} = \tau\_{ij}^{n-1} \cdot (1 - \rho) + \rho \cdot \Delta\_{ij}. \tag{9}$$

Notations: *ρ* is the pheromone evaporation rate, and <sup>Δ</sup>*ij* is the artificial pheromone laid on edge (*ij*).

*Globally update the best* tour's *PSL* vector and pheromone matrix *τ*. The global update occurs after all ants finish all construction steps. Now, the *PSL* vector recording the pheromone sensitivity level for each ant is also updated according to a specified linear formula based on [24,25]; for this particular problem, the Equation (10) PSL update influenced by the image's intensity values is used.

$$PSL = ((1 - \rho) \ast PSL + \rho \ast \Delta\_{\bar{i}\bar{j}} \ast \upsilon(I\_{\bar{i}\bar{j}})) \ast \Delta\_{\bar{i}\bar{j}} + PSL \ast |1 - \Delta\_{\bar{i}\bar{j}}|.\tag{10}$$

Furthermore, the best tour is a user defined criterion; it can be the best tour found in the current construction step, or the best tour from the start of the ACO algorithm, or a combination of these two.

For ACO, global update of the pheromone matrix [1] is performed as in Equation (11).

$$\text{Global update ACO:} \quad \tau^{(n)} = (1-\psi) \cdot \tau^{(n-1)} + \psi \cdot \tau^{(0)},\tag{11}$$

where *ψ* is the pheromone decay rate.

For *SACO*, the global update is based on its sensitivity feature (Equation (12)).

$$\text{Global updateSACO:} \qquad \pi^{(n)} = \max\_{k=1:K} PSL(k) \cdot \pi^{(n-1)}.\tag{12}$$

The problem solution is obtained after reaching a stopping criteria, such as, for example, a maximal number of iterations.

#### **4. Experiments and Discussions**

The numerical experiments were carried out using Matlab on an AMD Rysen 5 2500U, 2GHz. The software is a version of the image edge detection using Ant Colony Optimization version 1.2.0.0. from MATLAB Central File Exchange [27]. The MATLAB implementation [28] of the Canny edge detection algorithm is based on [15].The software makes use of two thresholds in order to detect strong and weak edges; the weak edges are provided in the solution only if they are connected to the strongest ones: "a high threshold for low edge sensitivity and a low threshold for high edge sensitivity", as is specified in the software documentation [28]. In order to convert a gray-scale input image to a binary image, thresholding is used.

*Data set.* A dataset of medical images, free of copyright, was used for these experiments: *Brain CT* (could be provided by request from the authors), *Hand X-ray* [29], (reduced resolution from 225 × 225 to 128 × 128) and *Head CT* [30]. Several details are included in the Github page (Representation of results available at https://github.com/cristina-ticala/ Sensitive\_ACO; accessed on October 2021).

*Filtering.* In order to filter the medical images, the De-Noise convolutional neural network (DnCNN) was used in the present study, as well as in our previous related work [18]. The Image Processing Toolbox and Deep Learning Toolbox from Matlab [31] were used.

*Parameters.* Most of the parametric numbers are from [22]. In our previous work [18], we tested several parameters; in the present study, we used the best of them.

*Image-related parameters:* The image dimension influences and gives ACO and SACO a number of artificial ants *K* = √*M*<sup>1</sup> × *<sup>M</sup>*2!, where and ! are the left and right rounded values to the nearest integers less than or equal to *x*; e.g., for a 128 × 128 image resolution, the number of ants is considered 128.

*Iterations related parameters:* In [18] just 30,000 iterations for *L* = 100 steps were considered; here, we tested a smaller (1200 iterations for *L* = 4 steps) and a higher number of steps *L* = 1000 (300,000 iterations). An ant makes 300 moves at each step; e.g., for 128 ants (e.g. image resolution: 128 × 128), 38,400 moves are made during each step. Therefore, for *L* = 4, it is a total of 153,600 moves, 3,840,000 moves for *L* = 100 and a total of 38,400,000 ants' moves for *L* = 1000 steps.

*Connectivity-related parameters:* The connectivity neighborhood parameter Σ = 8 is based on the ants' movement range (Equation (6)).

*Pheromone trail parameters:* the value of each matrix component *τinit* = 0.0001; the weighting factors of pheromone information *α* = 1 and of heuristic information *β* = 0.1 (Equation (6)); the evaporation rate, *ρ*, is 0.1 (Equation (9)) and the value of the pheromone decay coefficient *ψ* is 0.001 (Equation (12)).

*Other parameters:* The adjusting factor *λ* of the functions (Equations (3)–(5)) is 10. The tolerance parameter (*ε* = 0.1) is used in the decision process. The stopping criterion is given by the maximal number of steps (*L*) set by the user.

*Comparison:* Beside the Canny algorithm, the Roberts, the Sobel and Prewitt edge algorithms were also used for comparison; the last two methods compute the horizontal and the vertical gradient of an image by using two orthogonal filter kernels, and after filtering, they compute the gradient magnitude and apply a threshold in order to find the regions of the image corresponding to the edges. Furthermore, the Roberts algorithm detects image edges at angles of 45 degrees and/ or 135 degrees from horizontal [32].


**Table 1.** The best number of correctly identified number of pixels, standardized using the overall average and standard deviation for all considered medical images, with every considered operator on all considered algorithms results for sensitive (SACO) and original ACO with DnCNN.

**Table 2.** The best number of the correctly identified number of pixels, standardized using the overall average and standard deviation for all considered medical images, with every considered operator on all considered algorithms results for Canny edge detection [15], as well as the Prewitt, Sobel, and Roberts methods [32].


*Running time.* The average running time was around 4500 seconds for both ACO and SACO with the presented parameters on the utilized computer.

Table 1 shows the best, maximal results of the number of correctly identified pixels standardized using the overall average, *Avg* = 2107.030303, and standard deviation, *StdDev* = 563.50, for all considered medical images, and operators on the sensitive ant colony method (SACO) and ACO denoised with DnCNN. Table 2 shows the Canny [15], Prewitt, Sobel and Roberts methods results and Table 3 illustrates the original ACO and SACO results before post-processing with DnCNN.

**Table 3.** The best number of correctly identified number of pixels, standardized using the overall average and standard deviation for all considered medical images, with every considered operator on all considered algorithms; results for sensitive (SACO) and original ACO methods.


The best solutions obtained for the considered medical images (Head CT, Brain CT and Hand X-ray) while comparing ACO and SACO for 300,000 iterations and the considered demicontractive operators are included in Figure 2; in the last image, the original medical images are overlapped with the best solutions.

*Analysis.* The values are already standardized based on the denoised ACO and SACO, Canny, Prewitt, Sobel, and Roberts results; therefore, the difference between SACO and ACO is significant from an analytic perspective.


**Figure 2.** Successive illustrations of the best solutions obtained after 300,000 iterations with *Ant colony optimization (ACO)* and *Sensitive ACO (SACO)* post-processed with the *Denoise Convolutional Neural Network (DnCNN)* and the overlapped best solutions' edges over the original medical images for (**a**) Brain CT; (**b**) Head CT and (**c**) Hand X-ray.

**Figure 3.** Head CT, Brain CT and Hand X-ray results based on the difference between SACO and ACO standardized values, before (up) and after post-processing with the DeNoise convolutional neural network (DnCNN) (down).

*Stability & quality of the solutions.* The quality of the partial solution is influenced by the amount of modified pheromone of the ants' trail.

The stability of the global solution is influenced by used parameters. The included PSL parameter hopefully influenced the global solution for the better.

The global solution is found after the entire ant colony, based on the existing pheromone information, is guided to more promising regions in the search space.

The pheromone sensitivity factor balances the exploring and exploiting activities; its value is a number from [0, 1]. An ant ignores information when *PSL* = 0 and has the maximum pheromone sensitivity when *PSL* = 1.


In time, the process modifies ants' pheromone sensitivity (*PSL*). In the current work, the *PSL* is globally modified (increased or decreased) by the search space topology [17].

*SACO advantages.* By adding the PSL vector, the present algorithm offers the stability of its solutions; for the considered examples, after around 300 iterations, SACO generated edges which almost overlapped over the original images. As a plus, the image edge results are much more compact and close to the original when compared with the ACO results.

*SACO disadvantages.* As the number of parameters increases, the user should properly configure their values. This could take more time and resources, but the improved results are worth the effort.

Figure 4 shows the improvements of SACO-DnCNN compared to the Canny, Prewitt, Sobel and Roberts edge detection techniques [32]. The best SACO-DnCNN results, using *χ*, outperform the Canny Edge detector results by 37.76%; the Prewitt, Sobel and Roberts [32] methods were significantly outperformed by over 159%, 157% and 224%.

Future work will use images with higher resolutions, and hopefully the impact of sensitivity will improve the problem solutions.

Future work will also include implementing specific ACO and SACO features to solve publicly available medical datasets, including COVID- and SARS-Cov-2-related data sets. Furthermore, sensitivity for different artificial intelligence models could be involved within different domains, e.g., data mining [33] and similar.

Other improvements could utilize fuzzy techniques and multiple ant colonies for ACO, as in [34], which could be used to enhance the solutions for image edge detection.

Further work could make use of image segmentation with edge detection in order to obtain a more thorough edge [35]. As prerequisites for the development of knowledgebased applications, ontologies for the segmentation of radiological images [36] were proposed by the authors.

Other metaheuristics, mostly bio-inspired ones [37], could be further enhanced with sensitivity features in order to improve the results of complex problems. Human-in-theloop [38] could also enhance the problem results.
