The outbreak of novel coronavirus pneumonia (COVID-19) in late 2019 and early 2020 is an emerging acute respiratory disease, and diagnosis based on X-ray images is one of the effective complementary diagnostic methods for COVID-19. In clinical practice, imaging devices, such as chest X-ray and chest CT, can significantly help to screen for COVID-19 (Bernheim et al., 2020; Kanne, 2020; Xie et al., 2020). COVID-19 causes severe respiratory symptoms, and most patients diagnosed with COVID-19 have been found to have abnormal chest X-ray images in clinical management, and the X-ray imaging presentation is varied. The diagnostic process is laborious and time-consuming if it is based on the experience of pathologists. As a result, computer-aided diagnostic techniques are a key pre-condition for further X-ray image analysis, which is vital for early illness diagnosis and functional analysis of small parts of the lung, such as lung density analysis, airway analysis, and pulmonary septum mechanics. A weakly supervised deep active learning system called COVID-AL was suggested by Wu et al. (2021a) to diagnose COVID-19 using CT images and patient-level labels. In their research, Shorfuzzaman and Hossain (2021) suggested an artificial intelligence (AI) method based on deep meta-learning to speed up the interpretation of chest X-ray images in the automated detection of COVID-19 patients. Hu et al. (2021) analytical model for COVID-19 diagnosis and therapy is built on complex networks and machine learning methods. For the automated diagnosis of COVID-19, Chen et al. (2021) introduced a new deep learning technique that only needs a small number of training data. Jin et al. (2021) presented a self-correction approach based on domain adaptation for COVID-19 infection segmentation on CT images. Abdel-Basset et al. (2020) suggested a new hybrid solution for COVID-19 chest X-ray pictures based on the thresholding technique by mixing a slime mold algorithm with the whale optimization algorithm. It is also important to be able to separate and identify lesions from COVID-19 pathology images regardless of the computer-aided diagnosis techniques. Therefore, the introduction of image segmentation in the pre-processing stage of COVID-19 pathology images would be more helpful for effective analysis of COVID-19 pathology images.

In recent years, multi-threshold image segmentation (MIS) method is playing an increasingly important role in medical image processing, which can achieve highly effective pre-processing of pathological images and help to promote the development of related medical aid diagnosis technologies (Manikandan et al., 2014; Li et al., 2017; Kotte et al., 2018; Tarkhaneh and Shen, 2019; Hilali-Jaghdam et al., 2020). Medical information systems are a key requirement in the current medical sciences (Ban et al., 2022; Qin et al., 2022). Therefore, in recent years, many scholars have carried out research on MIS techniques based on swarm intelligence optimization algorithms. Zhao et al. (2021a) developed an image segmentation model that was proved by conducting an experiment on a standard image set, in which a modified continuous version of the ant colony optimizer was its core. Verma et al. (2021) presented a hybrid algorithm by combining the excellent features of fuzzy c-means and particle swarm optimization (PSO) to achieve MIS, which was proved by an experiment on a triangular dataset and publicly available real brain datasets. Zhao et al. (2021b) proposed an MIS method based on an improved continuous ant colony optimization algorithm, which was verified by a test on a standard image set. Rakesh and Mahesh (2021) proposed a cuckoo search optimization technique to achieve the initial segmentation of the lung portion. Vaze et al. (2021) developed quantum entanglement-inspired PSO, which was applied to MIS and employed eight gray-scale standard test pictures. In order to resolve pre-treatment and post-treatment organ segmentation difficulties, Chakraborty et al. (2021) proposed a novel dynamically learned PSO-based neighborhood-influenced fuzzy c-means clustering technique. Aranguren et al. (2021) reported an MIS methodology for magnetic resonance brain imaging segmentation based on the LSHADE algorithm. Huang et al. (2021) used the fruit fly optimization algorithm (FOA) for OTSU segmentation, resulting in an FOA-OTSU segmentation method that employed a classical picture to validate the proposed segmentation technique.

Xing (2020) proposed and evaluated a MIS method based on improved Emperor Penguin Optimizer, utilizing Berkeley pictures, satellite images, and plant canopy pictures. Dinkar et al. (2021) suggested a modified equilibrium optimizer for segmenting gray-scale images with MIS and tested their proposed technique by utilizing some standard images. Wu B. et al. (2020) proposed a MIS approach using an improved teaching–learning-based optimization algorithm, which is successfully applied in casting X-ray image segmentation for MIS. The Harris Hawks optimization (HHO) technique was used by Rodríguez-Esparza et al. (2020) to suggest an effective solution for MIS and test it on specific digital mammography medical pictures. Abd Elaziz et al. (2021) presented a modified version of the manta ray foraging optimizer algorithm to deal with MIS problems, which is proved by a test on some standard images. Radha and Gopalakrishnan (2020) provided an intelligent fuzzy-level set approach for medical picture segmentation and an overall search proficiency of enhanced quantum PSO.

Kalyani et al. (2020) devised and evaluated an efficient exchange market algorithm for image segmentation utilizing the minimal cross-entropy thresholding approach on brain pictures with varying threshold values. Elaziz et al. (2020) introduced an improved Harris hawks optimizer for global optimization and determining the best threshold values for MIS situations. With the help of a greedy snake model and fuzzy C-means optimization, Sheela and Suganthi (2019) suggested an effective automated brain tumor segmentation. A brand-new multi-objective optimization strategy for segmenting magnetic resonance imaging of the human brain was introduced by Pham et al. (2019). Wang et al. (2022) developed a unique, consistent perception generative adversarial network for semi-supervised stroke lesion segmentation that may eliminate the need for wholly labeled data. Authors in You et al. (2022) proposed fine perceptive generative adversarial networks, which are built to deal with the low-frequency and high-frequency components of MR images separately and concurrently by using the divide-and-conquer strategy (Wang et al., 2022) presented a 3D end-to-end synthesis network dubbed Bidirectional Mapping Generative Adversarial Networks for brain magnetic resonance imaging and positron emission tomography synthesis and segmentation. Through the study of various MIS methods proposed in recent years, it is found that most MIS techniques are based on swarm intelligence optimization algorithms. In the MIS method based on swarm intelligence algorithm, the swarm intelligence optimization algorithm is the core of the segmentation, and the suitability of the optimization algorithm directly determines the good or bad segmentation effect. Due to the characteristics of the swarm intelligence algorithm itself, it is difficult for the algorithm to converge to the optimal solution of the problem in complex practical problems, so the segmentation method based on the optimization algorithm still has more room for improvement to a certain extent, and the image segmentation effect can be improved as much as possible by proposing a suitable optimization algorithm.

In terms of swarm intelligence optimization algorithms, not only a series of basic algorithms have been proposed, such as multi-verse optimizer (MVO) (Mirjalili et al., 2016), ant colony optimization for continuous domains (ACOR) (Socha and Dorigo, 2008), bat-inspired algorithm (BA) (Yang, 2010), different evolution (DE) (Storn and Price, 1997), firefly algorithm (FA) (Yang, 2009), gray wolf optimization (GWO) (Mirjalili et al., 2014), moth-flame optimization (MFO) (Mirjalili, 2015), PSO (Kennedy and Eberhart, 1995), sine cosine algorithm (SCA) (Mirjalili, 2016), slime mold algorithm (Chen H. et al., 2022), whale optimizer (WOA) (Mirjalili and Lewis, 2016), and HHO (Heidari et al., 2019c), but also many variant versions based on the basic algorithms have been proposed by many scholars, such as boosted GWO (OBLGWO) (Heidari et al., 2019a), opposition-based SCA (OBSCA) (Abd Elaziz et al., 2017), ant colony optimizer with random spare strategy and chaotic intensification strategy (RCACO) (Zhao et al., 2021a), hybrid bat algorithm (RCBA) (Liang et al., 2018), A-C parametric WOA (ACWOA) (Elhosseini et al., 2019), enhanced WOA with associative learning (BMWOA) (Heidari et al., 2019b), bat algorithm based on collaborative and dynamic learning of opposite population (CDLOBA) (Yong et al., 2018), enhanced whale optimizer with new communication mechanism and biogeography-based optimization (EWOA) (Tu et al., 2021), hybridized gray wolf optimization (HGWO) (Zhu et al., 2015), modified SCA (m_SCA) (Qu et al., 2018), biogeography-based learning PSO (BLPSO) (Chen X. et al., 2017), comprehensive learning PSO (CLPSO) (Liang et al., 2006), enhanced GWO with a new hierarchical structure (IGWO) (Cai et al., 2019), improved WOA (IWOA) (Tubishat et al., 2019) and so on. Swarm intelligence algorithms have been applied to solve many problems such as bankruptcy prediction (Zhang et al., 2021), feature selection (Xue et al., 2019, 2022a; Liu et al., 2022b), economic emission dispatch (Dong et al., 2021), dynamic multiobjective optimization (Yu et al., 2022), constrained multiobjective optimization (Liang et al., 2022), global optimization (Deng et al., 2022), large-scale complex optimization (Huang et al., 2023), and feed-forward neural networks (Xue et al., 2022b).

Among them, ACOR is an algorithm proposed by Socha and Dorigo (2008) to apply it to solve problems in the continuous domain, which not only retains the original ACO characteristics but also overcomes the drawback that it can only be applied to discrete problems. Therefore, ACOR has a high research value and has been studied by many scholars. By combining the basic ant colony optimizer with chaotic intensification and random spare strategies, Zhao et al. (2021a) presented a unique variation of ACOR and put it to the test using 30 benchmark functions. Kumar et al. (2018) proposed a new ant colony optimization algorithm that incorporated Laplace distribution-based interaction scheme among the ants. A hybrid form of ant colony optimization for continuous domains was presented by Karakonstantis and Vlachos (2018), which can deal with continuous optimization issues with or without constraints. An innovative population-based elite-mixed continuous ant colony optimization with central initialization was developed by Chen and Shen (2018). In order to solve issues involving continuous variables, Wu et al. (2017) suggested a dynamic-edge ant colony systems technique that could produce edges between two nodes and provide pheromone measurements in a continuous solution space. A cooperative continuous ant colony optimization approach was put out by Juang et al. (2013) and used to solve accuracy-focused design issues for fuzzy systems. A novel classifier based on ant colony optimization in continuous domains was proposed by Shahraki and Zahiri (2017), and it was utilized to identify the decision hyperplanes between various classes. Falcon-Cardona and Coello (2017) developed an ant colony optimizer-based multi-objective ant colony optimizer for continuous search spaces. Chen Z. et al. (2017) published a robust ant colony optimization for continuous functions that employed self-adaptive approaches for domain modification, pheromone increment, domain division, and ant size. Zhang et al. (2016) introduced a novel hybrid ant colony optimization and PSO technique to solve the slow convergence of the ant colony optimization strategy for continuous domain difficulties. In order to solve a continuous space optimization problem, Huang (2016) suggested an ant colony algorithm that enhanced the fundamental algorithm in terms of ant colony initialization, information density function, distribution methods, the direction of ant colony motion, and other areas. Three crossover approaches are used in the enhanced continuous ant colony optimization with crossover operators developed by Chen and Wang (2017) to provide a new set of probability density functions. By enhancing the selection mechanism and incorporating horizontal and vertical crossover search into the ant colony optimization, Zhao et al. (2021b) proposed an improved ant colony optimization algorithm. This algorithm was tested in a series of comparative experiments using 30 benchmark functions from IEEE CEC 2014. Wu et al. (2019) developed an effective optimization technique for local search operations and established a multimodal continuous ant colony optimization algorithm. Gao (2015) proposed an unique immunological continuous ant colony approach to improve the efficacy of the standard evolutionary algorithm. Liu et al. (2014) provided an ant colony foraging distribution model for continuous domains based on their study of the interplay between position distribution and food supply in the process of ant colony foraging.

The multiple study studies above discovered that when ACOR is used to tackle real issues, it is more prone to slip into local optimality, resulting in disappointing outcomes. Furthermore, when applying ACOR to MIS, we discovered a similar issue: ACOR is more prone to falling into local optima, resulting in pictures that are as prone to falling into local optima during the segmentation process, leading in worse segmentation outcomes. Therefore, we propose an enhanced version of ACOR (MGACO) to apply ACOR to MIS to avoid falling into local optima as much as possible and obtain high-quality segmentation results. In MGACO, not only a novel movement strategy is introduced, but also a Cauchy-Gaussian fusion strategy is incorporated after this movement strategy. The introduction of the novel movement strategy and the Cauchy-Gaussian fusion strategy has led to a speedup in the convergence rate of MGACO and has resulted in a significant enhancement in the ability of MGACO to jump out of the local optimum. To demonstrate these core advantages of MGACO, we have used the 30 benchmark functions of IEEE CEC2014 as a basis for qualitatively analyzing MGACO and ACOR in detail and comparing MGACO with ten traditional basic algorithms and ten variants of basic-based algorithms, respectively, in our experiments. We conducted a detailed statistical analysis of the obtained experimental results using mean, variance, Wilcoxon signed-rank test (García et al., 2010), and Friedman test (Derrac et al., 2011), all of which fully demonstrate that MGACO not only has a certain acceleration in convergence speed but also its ability to jump out of the local optimum is significantly enhanced. In addition, we developed an MIS method based on MGACO (MGACO-MIS) using the non-local means, 2D histogram as the basis, and 2D Kapur’s entropy as the fitness function. Intending to demonstrate that MGACO can obtain high-quality segmentation results during image segmentation, we conducted a comparison experiment between MGACO-MIS and eight other similar segmentation methods based on real pathology images of COVID-19 at threshold levels of 4, 5, 6, 15, 20, and 25, where 4, 5, and 6 represent low threshold levels and 15, 20, and 25 represent high threshold levels. First of all, three common and accepted evaluation methods were evaluated for obtaining segmentation results, including Peak Signal Noise Ratio (PSNR) (Huynh-Thu and Ghanbari, 2008), Structural Similarity Index (SSIM) (Zhou et al., 2004), and Feature Similarity Index (FSIM) (Zhang et al., 2011). Secondly, the evaluation results were further analyzed in detail using the mean, variance, Wilcoxon signed-rank test (Chen H. et al., 2022), and Friedman test (Mirjalili and Lewis, 2016). Finally, the evaluation and analysis result sufficiently demonstrate that the proposed MGACO-MIS can obtain high-quality segmentation results when performing image segmentation.

In summary, the main contributions of this paper are in the following aspects.

λ Taking ACOR as a basis, an enhanced version of ACOR combining a new mobility strategy and a Cauchy-Gaussian fusion strategy is proposed, called MGACO.

λ Based on IEEE CEC2014, the qualitative analysis of MGACO and ACOR is conducted, and MGACO is compared with many similar methods, which fully demonstrate the core advantages of MGACO.

λ An MIS based on MGACO, called MGACO-MIS, is developed using non-local means, 2D histogram, and 2D Kapur’s entropy as the fitness function.

λ Based on the real pathological images of COVID-19, the comparison experiments between MGACO-MIS and similar methods have been conducted at several threshold levels, and the segmentation effect of MGACO-MIS has been well demonstrated.

The other sections of this paper are organized as follows: Section “2. An overview of ACOR” provides a brief review of the ACOR basic theory. Section “3. Proposed MGACO” describes the proposed MGACO based on the new movement strategy and the Cauchy-Gaussian fusion strategy in detail. The experimental results and analysis are presented in Section “5. Experiments and results.” In Section “6. Discussion,” the main work of this paper is discussed, and Section “7. Conclusion and future works” gives a summary of the whole paper and future research directions.

In 2008, Socha and Dorigo (2008) proposed ACOR, which directly extends the ACO in the discrete domain to the continuous domain by treating the solutions in the solution archive as its pheromone, where the update of the pheromone is done by updating the solutions in the archive.

In ACOR, the archive stores k solutions x_l(l=1,…,k), the fitness value of each solution, and the weight value corresponding to each solution, where each solution has n dimensions, which also represent the dimensions of the problem.xl=(xl1,xl2,…,xli,…,xln) denotes a solution of the problem, f(x_l) is its corresponding fitness value, and w_l is its corresponding weight value.

These k solutions are initially generated randomly, and then the solutions x_l are ordered according to the function value f(x_l) of the solutions.

Therefore, if the fitness value corresponding to each solution satisfies f(s₁)≤…f(s_l)≤…f(s_k), then the weight w_l corresponding to each solution satisfies w₁≥…w_l≥…w_k. Based on the above principle, the ACOR archive can be shown in Figure 1.

Therefore, the main mathematical model of the Gaussian kernel function corresponding to ACOR is shown in Eq. (1).

where w={w₁,…,w_l,…,w_k} is the weight vector corresponding to each solution, and w_l can be calculated by Eq. (2), σi={σ1i,…,σli,…,σki} is the vector of standard deviation corresponding to each solution, and σli can be calculated from Eq. (3), μi={μ1i,…,μli,…,μki} is the vector of means corresponding to each solution, and μli can be expressed as Eq. (4).

where q and ξ > 0 are the algorithmic parameters, and the weight w_l is essentially a Gaussian function with l as the mean and qk as the standard deviation. The sampling procedure is then carried out in practice so that the probability p_l is determined in the first stage using the weight size w_l and Eq. (5). In the second phase, a Gaussian function gli⁢(x) is picked using the probability p_l from the Gaussian kernel function Gⁱ(x), and the selected guiding solution x_l is calculated. The third phase uses the Gaussian function gli⁢(x) to sample each of the n dimensions of the guiding solution x_l.

After the completion of sampling, its pheromone updating process is represented by the updating process of the solutions in the archive. In each iteration, the m new solutions constructed by the ants are combined with the k old solutions in the archive, and the k better solutions from these m+k solutions are selected and sorted into the archive, while the remaining m worse solutions are discarded. Further, according to the algorithm principle of ACOR, Algorithm 1 gives its corresponding pseudo-code, and Figure 2 further gives its corresponding flowchart according to the pseudo-code.

In MGACO, not only a new movement strategy is introduced, but also the Cauchy-Gaussian fusion strategy is incorporated after the movement strategy. Due to the introduction of the new movement strategy and the Cauchy-Gaussian fusion strategy, MGACO has been accelerated in terms of convergence speed, and the ability of MGACO to jump out of local optimum has been significantly enhanced.

We employ 30 benchmark functions from IEEE CEC2014 as our starting point to qualitatively assess MGACO and ACOR in great depth as well as compare MGACO with 10 conventional basic algorithms and 10 variations based on basic algorithms in order to illustrate these essential benefits of MGACO firmly. The acquired experimental findings show conclusively that MGACO has improved leap out of local optimum capability in addition to a certain speedup in convergence. In image processing tasks, it is vital to utilize a valid dataset covering various features and properties to assess the method sincerely (Jin et al., 2022). We also performed comparison studies between MGACO-MIS and eight other comparable segmentation algorithms at various threshold levels using genuine pathological photos of COVID-19 in order to show that MGACO-MIS can provide high-quality segmentation results while doing image segmentation. The acquired experimental findings adequately show that the suggested MGACO-MIS may provide high quality segmentation results while conducting image segmentation.

We first conducted a comparison experiment based on the 30 test functions of IEEE CEC2014 between the new moving strategy and the variants created by the Cauchy-Gaussian fusion strategy, where MGACO obtained the minimum mean value the majority of the time on all function problems. MGACO also outperformed the other three variants of the algorithm in the analysis results. Additionally, the effect of the two new mechanisms on ACOR is thoroughly examined from a variety of angles, with an emphasis on diversity and balance. This analysis demonstrates that MGACO further optimizes the balance of exploration and development based on ACOR, enabling MGACO to handle more challenging issues. Then, to analyze the effectiveness of MGACO and ACOR more thoroughly, a comparison of the two programs is made across several aspects. The findings of this comparison conclusively show that MGACO can provide better outcomes and is more stable. MGACO has a superior capacity to resist entering a local optimum throughout the problem-solving process, which aids in developing a better solution. Finally, we evaluate MGACO against certain established, fundamental algorithms and some enhanced variations. The results of the Friedman test analysis, Wilcoxon signed-rank test analysis, and associated convergence curves clearly show that MGACO not only produces high-quality solutions but also has a little improvement in convergence speed and the capacity to depart from the local optimum. In conclusion, this paper’s proposal for the MGACO offers more advantages over the AOCR and a higher capacity for problem-solving. Additionally, we conducted MIS experiments based on actual COVID-19 pathology images, the evaluation results of PSNR, SSIM, and FSIM, and the additional Wilcoxon signed-rank test and Friedman test results can show that MGACO-MIS can achieve better segmentation results on the pathological images of COVID-19. The particular segmentation findings further demonstrate that MGACO-MIS outperforms its competitors in terms of detail preservation, local feature preservation, and overall segmentation outcomes.

As a result, it is undeniably established that MGACO is a very good swarm intelligence optimization algorithm and that MGACO-MIS is an even better segmentation technique when MGACO is used to segment problematic pictures from COVID-19. In future work, the proposed method can also be applied to more cases, such as the optimization of machine learning models iris segmentation and recognition (Chen et al., 2023), fine-grained alignment (Wang et al., 2023), remote pulse extraction (Zhao et al., 2022), Alzheimer’s disease identification (Yan et al., 2022), MRI reconstruction (Lv et al., 2021), renewable energy generation (Sun et al., 2022), power distribution network (Cao et al., 2022), retinal vessel segmentation (Li et al., 2022), privacy protection of personalized information retrieval (Wu Z. et al., 2020; Wu et al., 2021c,d, 2023), and privacy protection of location-based services (Wu et al., 2021b,2022).

An MIS approach (MGACO-MIS) based on an improved version of ACOR (MGACO) is developed in this study. Not only is a new move strategy included in MGACO, but also the Cauchy-Gaussian fusion approach. Due to the addition of the new movement strategy and the Cauchy-Gaussian fusion approach, MGACO’s convergence speed and capacity to jump out of the local optimum have been much improved. To highlight these fundamental benefits of MGACO, the 30 benchmark functions from IEEE CEC2014 are used to compare MGACO and ACOR with 10 conventional basic algorithms and 10 modifications. The outcomes show that MGACO increases convergence speed and considerably increases its capacity to exit the local optimum. In order to show that MGACO can produce high-quality segmentation results when performing image segmentation, a comparison experiment was carried out between MGACO-MIS and eight other comparable segmentation methods using real pathological images of COVID-19 at threshold levels 4, 5, 6, 15, and 25, where 4, 5, and 6 represent low threshold levels and 15, 20, and 25 represent high threshold levels. The final assessment and analysis findings conclusively show that the created MGACO-MIS can produce accurate image segmentation results. However, the proposed method achieves very good results, but the time consumption is still large. To better solve this problem, it will be solved by introducing parallel computing or high-performance computing in the future.

For future work, MGACO will be considered to be applied to more fields, such as bankruptcy prediction (Zhang et al., 2020), engineering design (Chen et al., 2019; Luo et al., 2019; Al-Betar et al., 2020; Tu et al., 2020; Wang et al., 2020), and financial stress prediction (Luo et al., 2018). Further, MGACO-MIS will be considered for the segmentation of more pathological images to achieve greater value and contribute to the advancement of medical diagnosis technology.

The original contributions presented in this study are included in the article/Supplementary material, further inquiries can be directed to the corresponding authors.

XZ, AH, YC, and BM: writing—original draft, writing—review and editing, software, visualization, and investigation. LL and HC: conceptualization, methodology, formal analysis, investigation, writing—review and editing, funding acquisition, and supervision. All authors contributed to the article and approved the submitted version.