Enhanced Growth Optimizer and Its Application to Multispectral Image Fusion

1  Introduction

As a consequence of the recent advances in computational science, the real-world problems to be solved are becoming increasingly complex [1]. These problems are typically characterized by nonlinearity, large scale, multimodality, and constraints. In response to this, metaheuristic algorithms have emerged as a promising solution to complex optimization challenges.

The growth optimizer (GO) is a population-based meta-heuristic algorithm that models the learning and reflection mechanisms employed by individuals during their social growth. The original GO algorithm was proposed by Zhang et al. [2] in 2023 for solving continuous and discrete global optimization problems. The search process of the original GO algorithm is divided into two phases: the learning phase and the reflection phase. In the learning phase, which is mainly based on the cooperative search mechanism, each individual learns and acquires knowledge from the knowledge gaps that exist between different individuals. In the reflection phase, individuals use different strategies to recognize and overcome their weaknesses. Growth Optimizer (GO) has demonstrated its effectiveness as an optimization algorithm for the resolution of continuous and discrete global optimization problems [2]. Compared to other metaheuristic methods, GO yielded more encouraging outcomes, particularly concerning solution quality and avoiding local optima. Many studies have demonstrated that GO has the capacity for extensive exploration and the ability to accommodate a wide range of applications. For instance, GO has been effectively employed in the identification of parameters associated with solar photovoltaic cells [3], the segmentation of multilevel threshold images and the deployment of wireless sensor network nodes [4], and the enhancement of intrusion detection systems within the context of IoT and cloud environments [5]. However, very recently, Gao et al. [4] proposed an improved GO algorithm (QAGO) and demonstrated its superiority over the original GO algorithm and some other meta-heuristics in a number of numerical and real-world optimization problems. Gao et al. [4] reported that the original GO has challenges in parameter tuning and operator optimization. In particular, since all hyperparameters of the original GO are fixed values, its performance may be significantly degraded if the parameters are not set properly. Conversely, the search operator structure of the original GO constrains the diversity of the search process. In another study, Fatani et al. [5] proposed an enhanced version of GO, designated MGO, and applied it to an intrusion detection system (IDS). In MGO, the operators of the whale optimization algorithm (WOA) are employed to augment the search process of the original GO. Moreover, Kaveh et al. [6] put forth an enhanced hybrid growth optimizer (IHGO) to address discrete structure optimization challenges, where four enhancements are incorporated into the IHGO relative to the initial GO. Firstly, the learning phase of GO is enhanced through the integration of an improved meta-heuristic exploration phase, namely IAOA, which avoids useless search and enhances exploration. Secondly, the replacement strategy of GO is modified to prevent the loss of the current best solution. Thirdly, the hierarchical structure of GO is modified. Fourthly, the reflection phase of GO is adapted to promote the development of promising regions. GO, one of the traditional optimization algorithms, typically necessitates the storage of a considerable number of solutions and the computation of the corresponding values for each solution. This extensive number of computations presents a challenge to the performance of computing devices, as the large number of computations can lead to conflicts and hinder efficiency. In numerous real-world application scenarios, it is unlikely that high-performance computing devices will be utilized consistently for a multitude of reasons. However, a multitude of optimization problems still necessitate resolution.

Compact algorithms provide an efficient solution by retaining some of the advantages of population-based algorithms without the need to store the entire population in memory. In essence, compact evolutionary algorithms optimize the problem in question using only a solution using the population’s statistical features. Consequently, the number of solutions that need to be stored in memory is greatly reduced. As a result, the memory requirements to run these algorithms are greatly reduced compared to population-based algorithms. Compact algorithms fall within the category of Estimation of Distribution Algorithms (EDAs), where the explicit representation of the population is substituted with a probability distribution, as discussed in reference [7]. The initial instantiation of compact algorithms is exemplified by the compact Genetic Algorithm (cGA). The cGA emulates the behavior of a conventional binary-encoded Genetic Algorithm (GA). The findings in Reference [8] reveal that the performance of cGA is nearly comparable to that of GA while significantly demanding less memory space. The convergence analysis of the cGA is established in Reference [9], while an expanded version, known as the extended compact genetic algorithm (ecGA), is detailed in Reference [10]. Research on the scalability of the ecGA is explored in Reference [11], while Reference [12] documents the utilization of the cGA in training neural networks. Additionally, a number of related algorithmic enhancements utilizing the compact strategy have been put forth, such as compact Bat Algorithm(cBA), compact Particle Swarm Optimization (cPSO) [13], compact Sine Cosine Algorithm(cSCA), etc.

Image fusion remains a cutting-edge topic in the fields of image processing and computer vision exploration. Image fusion refers to the fusion of information from multiple images together to generate a new image, thereby ensuring that the new image contains all the key information and features of the original image. Image fusion techniques can effectively combine image information from different sources to improve the quality and information content of the image. Rahmani et al. introduced an adaptive LHS approach to dynamically calculate band coefficients to achieve more precise detail maps. To improve the accuracy of the spectral representation in non-edge regions, they utilized an edge-holding filter on the Pan image to incorporate the detailed map into each MS band. To enhance the effectiveness of pansharpening, Leung et al. concluded that using Pan images to inject detail maps is not sufficient. To address this, they also employed a linear combination of MS images and Pan images with fixed weights. However, there is a limitation in this approach. The utilization of detail maps with fixed weights may also give rise to distortions, like localized artifacts. To mitigate these impacts, we propose an adaptive framework for the computation of the injected detail maps for each low-resolution (LR) MS band. Different algorithms are used to adaptively compute the weights of the detail maps, and the outcomes are examined and compared. The examination of the experimental findings indicates that the detail map weights obtained adaptively by the eGO algorithm are more appropriate for image fusion compared to other algorithms.

In this paper, we present a novel growth optimizer that employs a compact mechanism and Lévy flight [14] to enhance the optimization and convergence capabilities of the algorithm and reduce its memory requirements. Through the CEC2014 and CEC2017 [15] function sets test, the proposed eGO not only takes less memory but also has higher optimization and convergence capabilities. In comparison to other compact algorithms, the proposed eGO algorithm is also capable of demonstrating considerable competitiveness.

2  Growth Optimizer

The growth optimizer [2] is a robust metaheuristic algorithm that has been designed with inspiration from the learning and reflection mechanisms observed in individuals during their growth processes in society. The process of learning is defined as the acquisition of knowledge from external sources, which enables an individual to grow and develop. Reflection is the process of identifying and addressing the individual’s own shortcomings in order to adjust their learning strategies in a way that facilitates their growth. The individual in GO refers to a problem’s solution. It is employed to solve both continuous and discrete global optimization problems [16]. The GO algorithm comprises the following steps.

2.1 Initialization

Xi(0)|xi,j(0)∈[xi,jmin,xi,jmax],i∈[1,NP],j∈[1,D], where Xi(0) represents the ith individual of the 0th generation, and xi,jmin represents the lower bound of the population, while xi,jmax represents the upper bound of the population. The i denotes the ith individual, while the j denotes the jth dimension. Np represents the size of the population, while D represents the number of dimensions. The initialisation of populations is achieved through the application of the uniform distribution method Eq. (1).

xi,j(0)=unifrnd([xi,jmin,xi,jmax],Np,D)(1)

2.2 Learning Phase

The process of identifying and addressing gaps between individuals can significantly contribute to their personal growth. There are four gaps between individuals: The difference between the leader and the elite Ga→p1, the difference between the leader and the lowest-ranked individual Ga→p2, the difference between the elite and the lowest-ranked individual Ga→p3, and the disparity between the characteristics of two randomly selected individuals Ga→p4. The mathematical model is defined by Eq. (2).

Ga→p1=x→best−x→betterGa→p2=x→best−x→worseGa→p3=x→better−x→worseGa→p4=x→L1−x→L2(2)

where x→best signifies the society leader, while x→better signifies one of the next P1−1 best individuals, who are called an elite. In the context of GO, the leader and the elites collectively constitute the top tier of the society. x→worse denotes an individual from the population who is one of the P1 individuals and is positioned at the lowest level of the social class hierarchy. Both x→L1 and x→L2 are randomly chosen individuals distinct from ith individual. Ga→pk(k=1,2,3,4) represents the distinction between two individuals. Ga→pk enables learners to comprehensively grasp the dissimilarities between two individuals to their advantage.

In order to reflect the aforementioned variability, a learning factor LFk is introduced in each of the four disparity measures. For the ith individual, LFk affects its learning effect on the kth set of disparities. LFk is therefore modeled as follows Eq. (3).

LFk=||Ga→pk||Σk=14||Ga→pk||(3)

where LFk∈[0,1] represents the Euclidean distance’s normalized ratio for Ga→pk. As Ga→pk increases, LFk also increases, indicating that the ith individual will acquire more knowledge from the kth gap.

The ith individual employs SFi to gauge the extent of knowledge deemed acceptable for personal improvement, as illustrated in (Eq. (4)). A higher SFi indicates that individual SFi needs to acquire additional knowledge to enhance itself.

SFi=GRiGRmax(4)

Regarding GRi, individual i assimilates certain knowledge from them, forming the kth group of knowledge acquisition (KA→k). For individual i, KA→k results from the application of LFk and SFi on the kth group gap, and LFk is modeled as follows Eq. (5).

KA→k=SFi⋅LFk⋅Ga→pk(5)

where KA→k is the knowledge acquired by the ith individual from Ga→pk. SFi reflects an evaluation of its personal circumstances, while LFk signifies an appraisal of the external conditions.

Through assimilating knowledge disparities among various individuals, the ith individual undergoes a comprehensive process of accumulating substantial knowledge. The ith the individual’s specific learning process is shown in Eq. (6).

x→iIt+1=x→iIt+KA→1+KA→2+KA→3+KA→4(6)

where It represents the current iteration count, and x→i signifies the ith individual who incorporates the acquired knowledge from the learning phase to facilitate personal growth.

After the phase of learning adjustment, the quality of each individual may either improve or deteriorate. Therefore, it’s crucial to verify whether genuine progress has been achieved. If progress is observed, the growth resistance (GRi) of the individual decreases, and their rank increases. In the event that the ith individual degenerates, there is a high probability that the ith individual will relinquish part of the learned knowledge. Nevertheless, it is conceivable that the acquired knowledge may be retained with a minimal probability, given that the learning process necessitates a considerable investment of time and effort on the ith individual. In this context, P2 is accountable for regulating the probability of retention. The process is detailed by Eq. (7).

x→iIt+1={x→iIt+1iff(x→iIt+1)<f(x→iIt){x→iIt+1ifr1<P2x→iItelseelse(7)

where r1∈[0, 1] represents a uniformly distributed random number, ind(i) denotes the ith individual’s ranking is determined by the ascending order of GR, and P2 indicates whether, in the event that the ith individual neglects to update, the just learned knowledge is kept. In this case, P2 is equal to 0.001. The whole conditional judgment statement for maintaining newly acquired information (in case of an individual update failure) is included below, because to the limited space in Eq. (7): r1<P2 and ind(i)∼=ind(1). This indicates that in the event of a failed individual update, there is a 0.001 the probability that the individual will be included in the subsequent generation of the population. Furthermore, it makes sure that the present global optimal individual can not be changed, since doing so could cause the algorithm to converge incorrectly.

2.3 Reflection Phase

The processes of learning and reflection are mutually reinforcing. Individuals should develop both learning and reflection skills. Individuals should examine for and correct inadequacies in all areas, and retain information. To make up for deficiencies, they can learn from successful persons while retaining their strengths. If a lesson cannot be repaired, it is recommended to forgo past information and re-learn systematically. Three distinct processing methods exist. The initial approach is to maintain the original dimension; the subsequent way involves selecting an upper-class individual to direct the jth dimension of the ith individual, and the third method entails reconstructing the jth dimension of an individual with a low probability based on the second method. Mathematical modeling of GO’s reflection process through Eqs. (8) and (9).

x→i,jIt+1={{lb+r4×(ub−lb)ifr3<AFx→i,jIt+r5×(Rj−x→i,jIt)elseifr2<P3x→i,jItelse(8)

AF=0.01+0.99×(1−FEsMaxFEs)(9)

In Eq. (8), the upper and lower bounds of the search domain are represented by ub and lb, respectively. The values of r2, r3, r4, and r5 are uniformly distributed random numbers within the range of [0, 1]. The probability of reflections is controlled by P3, which is typically set to 0.3. In Eq. (9), the attenuation factor (AF) is calculated based on the current number of evaluations (FEs) and the maximum number of evaluations (MaxFEs). During the iteration of the procedure, the value of AF gradually approaches 0.01. This indicates that as individuals make progress, they are able to avoid spending excessive time on frequent initializations. The attenuation factor (AF) plays a crucial role in regulating the balance between exploration and exploitation during the optimization process. Initial phase (high value of AF): At the beginning of the algorithm, the attenuation factor AF is close to 1. This means that the positions of the individuals are more often randomly initialized. The purpose of this random initialization is to improve the exploration of the search space and help the algorithm avoid falling into local optimal solutions too early. Later stages (smaller AF values): As the algorithm runs, the current number of evaluations (FEs) gradually approaches the maximum number of evaluations (MaxFEs), and the attenuation factor (AF) gradually decreases to 0.01. At this stage, the algorithm is more biased towards exploitation, i.e., optimizing and improving existing solutions rather than exploring new regions. During the phase of reflection, the jth dimension of the ith individual is influenced by an upper-level individual (R→). R→ represents the higher-level individual, which is the current reflective learning guide for individual x→i. Rj presents the knowledge of the R→ dimension of the R→ vector. The range of vector R→ is determined by selecting the top P1+1 individuals, who are considered leaders or elites, from the population. This implies that when individual x→i needs to learn from others, the jth dimension of their learning will come from the other individuals, influenced by the upper-level individual R→.

3  Enhanced Growth Optimizer

It would appear that the extant literature does not endeavor to put forth a mechanism that enhances GO in compact strategy, thereby reducing the algorithm’s memory requirements to a considerable extent. GO may potentially become trapped in local optima and exhibit a slow convergence when attempting to solve high-dimensional, complex optimization problems. In response to the aforementioned questions, this section outlines the methodology for modifying the GO algorithm using the compact strategy, as well as subsequent enhancements to the compact strategy.

3.1 Compact Strategy

The core objective of implementing a compact strategy is to decrease memory consumption while maintaining, or potentially enhancing, the performance of the original algorithm. Enhanced growth optimizer (eGO) is a framework grounded in a growth optimizer which incorporates concepts and designs from GO algorithms. In this section, we will delve into further details regarding the eGO algorithm. The objective of eGO is to emulate the operational characteristics of the underlying GO algorithm while reducing the consumption of memory resources. A virtual population is used to convert the underlying GO individuals into a concise algorithm. The virtual population can be thought of as a probabilistic model of the solution population. It is stored in a data structure called the perturbation vector (PV). This approach permits the efficient representation and processing of potential solutions without the necessity of maintaining an actual population of individuals. In this framework, the perturbation vector captures the statistical properties of the population, defining the distribution of solutions by encoding the mean, variance, or other relevant parameters. We utilize perturbation vector (PV) to generate new candidate solutions through perturbation.

PVt=[μt,σt](10)

In the Eq. (10), μt and σt represent two parameters corresponding to the mean and standard deviation of the PV vector. The parameter t signifies the current iteration count. The values of μt and σt are limited to the range of probability density functions (PDF) [17], typically adjusted to fall within the interval of [−1, 1]. The PDF’s magnitude is normalized to maintain an area of 1, ensuring a uniform distribution with a complete shape, which aids in achieving adequate convergence.

The process of virtual population initialization is as follows: Each parameter is set such that μt = 0 and σt=λ, where λ represents a large constant (in this paper, we set λ =10). This initial setting ensures the establishment of a normal distribution characterized by truncated and broad shapes.

It takes a thorough explanation to understand the sampling mechanism of a individual x→i associated with a generic candidate individual x from PV. A truncated Gaussian PDF with mean value μ and standard deviation σ is linked to each individual indexed by i. Eq. (11) describes the PDF.

PDF=e−(x−μ[i])22×σ[i]2×2πσ[i]×(erf(μ[i]+12×σ[i]−erf(μ[i]−12×σ[i])))(11)

The PDF represents the probability distribution function of PV, with parameters μ and σ being associated with a truncated Gaussian distribution. A novel candidate individual is generated through a process of iterative directing towards a favorable area of the most optimal individual. The PV components are updated by learning from previous generations. The error function erf, as defined in Reference [13], plays a crucial role. The PDF is related to the cumulative distribution function (CDF), constructed using a Chebyshev polynomial [18]. The CDF represents a random individual x that takes on real values and follows a probability distribution. In that case, the observed value is less than or equal to x→it. The connection between CDF and PDF is established as follows CDF=∫01PDF(x)dx. Operations on PV involve sampling the design variable by producing random values within the interval of (0, 1).

During the iterative process of the compact algorithm, this paper introduces a comparative function based on two parameters to facilitate the identification of superior individuals. The two modifiable parameters, pre-learning and post-learning, correspond to two sample individuals within the PV operation. The individual represented by the winner attains a higher fitness function value compared to another virtual members, while the individual represented by the loser falls short of meeting the fitness evaluation criterion. These two individuals, winner and loser, are derived from the objective function calculation and contribute to generating a new candidate individual. This new individual is then compared against the original global optimal individual, resulting in the identification of new winner and loser, When updating PV, adjustments to μ and σ can be made based on the following guidelines: If μ = 1, the update rule for every element within it, μt+1 and σt+1, is as follows (12) and (13):

μit+1=μit+winneri−loseriNp(12)

The virtual population size is denoted by NP, and the value of σ is specified accordingly. The Eq. (13) provides the updating rule for each element.

σit+1=(σit)2+(μit+1)2+winneri2−loseri2Np(13)

It should be remembered that Np is an empirical value that does not match the real population size, which is typically many times larger.

The proposed eGO utilizes a perturbation PVt=[μt,σt] with a structure identical to that presented in Eq. (10). At the commencement of the optimization process, the initialization of the perturbation vector (PV) is conducted according to the procedure outlined in reference [8]. Individuals need to learn from the disparities among leaders, elites, and inferior performers; therefore, it is necessary to generate three individuals using PV and identify leaders, elites, and inferior performers by comparing their fitness values.

The Eq. (14) for assimilating the knowledge gap among different individuals is modeled after the GO formula.

x→It+1=x→It+KA→1+KA→2+KA→3+KA→4(14)

What can be observed is that the Eq. (14) for assimilating the knowledge gap among different individuals shares the same structure as the Eq. (6) in GO. However, while index i seems to indicate the particle under investigation in the population-based GO, index i does not appear in its compact version.

Similarly, the other two Eqs. (15) and (16) used in the learning and reflection phases are replicated in the same manner.

x→It+1={x→It+1iff(x→It+1)<f(x→iIt){x→It+1ifr1<P2x→Itelseelse(15)

x→jIt+1={{lb+r4×(ub−lb)ifr3<AFx→jIt+r5×(Rj−x→jIt)elseifr2<P3x→jItelse(16)

In eGO, the compact strategy reduces memory usage by employing a probabilistic model representation, which is more memory efficient than the original GO. Traditional algorithms, such as GO, rely on maintaining a complete population, wherein each individual is required to store information such as their position, speed, fitness value, and so forth. For an optimization problem with problem dimension D, the size of the population is Np, and the memory footprint is O(Np×D). eGO, on the other hand, uses a probability distribution to show the information in the population. This means that only the probability values of three decision variables need to be stored, which cuts the memory overhead by three times D. This compact representation obviates the need to store specific individuals or populations, which markedly reduces memory usage and is particularly well-suited to memory-constrained scenarios.

3.2 Lévy Flight

Lévy flight [14] represents a specific type of stochastic wandering with a probability distribution of step sizes proposed by the French mathematician Levy. The introduction of Lévy flight results in random movements with longer jumps and multiple abrupt changes in direction [19]. This implies that short local searches are interspersed with long jump global searches. In the algorithm space, this kind of flight finds farther-off solutions while improving the neighborhood search close to the ideal answer. This strategy can effectively increase the diversity of individuals and help the algorithm with escaping the local optimal solution [20]. Lévy flights introduce a probability distribution of step lengths, predominantly short steps but with occasional long jumps. This property is well suited for exploring unexplored regions of the search space. Long jumps allow the algorithm to jump out of the local optimal trap and explore more distant regions, increasing the chance of discovering a globally optimal solution. While long jumps make global exploration more efficient, short step sizes are used for local searches near the current optimal solution, which helps to refine the solution and thus speeds up convergence to the optimal solution. Many optimization algorithms are prone to falling into a local optimum, especially for complex multi-peaked functions. Long jumps at Lévy flight reduce this risk. These jumps break the algorithm’s stagnation in a locally optimal solution, allowing it to jump out of the local optimum and continue the global search. Such episodic long jumps disrupt the algorithm’s tendency to converge prematurely on a locally optimal solution, providing new opportunities for further search. In some cases, especially in complex search spaces with multiple local optima, random long jumps in Lévy’s flight can bring the algorithm directly close to the global optimum. This greatly speeds up the convergence process, as the algorithm is able to skip large regions of inefficiency. Once the algorithm has quickly reached the more optimal region, it can switch to fine-grained local search, further increasing the speed of convergence. Therefore, in this thesis, we consider adding Lévy flight to the end of the reflection phase of the algorithm and using a greedy strategy to select the new individuals obtained through Lévy flight and the individuals after the reflection phase.

The flowchart of the enhanced growth optimizer is displayed in Fig. 1.

Figure 1: Flowchart of enhanced growth optimizer

The pseudocode of enhanced growth optimizer is displayed in Algorithm 1.

4  Numerical Experimental Results and Analysis

To evaluate the performance of the proposed algorithm, we tested eGO using 30 test functions from CEC2014 and CEC2017. This paper compares eGO with eGO_l (the ehanced GO algorithm without Lévy flight), GO and other commonly used compact algorithms, including compact Cuckoo Search algorithm (cCS) [21], compact Pigeon-Inspired Optimization (cPIO) [22] and the above three compact algorithms cPSO, cBA and cSCA. CEC2014 and CEC2017 comprise various types of test functions, including single-objective optimization functions, multi-objective optimization functions, and constrained optimization functions, which can address a wide range of optimization problems. The included test functions have been meticulously designed and validated to ensure a fair evaluation of various optimization algorithms.

4.1 Experimental Parameter Settings

This paper uses the maximum number of evaluations (MaxFEs) as the termination criterion. The conditions for each algorithm to obtain relevant data include 60 benchmark problems, 30/50/100 dimensions, and each algorithm runs 20 times independently. The compact algorithms in the experiments were initialized using the same PV, while the original GO used random initialization in the search space. The size of the virtual population was analyzed in order to ascertain the sensitivity of the algorithm. The results of this analysis are presented in Table 1, which demonstrates that the algorithm performs similarly in terms of the quality of the final solution for virtual population sizes of 30, 60, 100, 150 and 200. This suggests that the virtual population size has a minimal impact on the algorithm’s performance within the specified range, indicating that the algorithm exhibits robust stability. Based on this outcome and in consideration of other compact literature, a smaller virtual population size (60) is recommended to effectively reduce the computational overhead while maintaining the algorithm’s performance. The original GO algorithm has a population size of 30, while all compact algorithms have a virtual population size of 60. The maximum number of evaluations is 5000.

4.2 eGO Performance

Tables 2–4 show the means of the eight algorithms on 30, 50, and 100 dimensions of CEC2014, Tables 5–7 show the means of the eight algorithms on 30, 50, and 100 dimensions of CEC2017, respectively, and the metrics of each algorithm at a significance level of α of 0.05 and Wilcoxon signed rank test. Tables A1–A3 show the standard deviations and minimum values of the eight algorithms on 30, 50, and 100 dimensions of CEC2014. Tables A4–A6 show the standard deviations and minimum values of the eight algorithms on 30, 50, and 100 dimensions of CEC2017. All algorithms are presented in a single table, thus facilitating a clear comparison between them. The symbol ‘+’ indicates that the eGO algorithm outperforms the other heuristics, while ‘=’ indicates that the eGO algorithm performs as well as the other heuristics. The symbol ‘−’ indicates that the eGO algorithm performs worse than the other heuristics. The summary row in each table shows that eGO outperforms the other algorithms. From the comparative results, it is evident that eGO has a significant advantage over the other compact algorithms and is highly competitive. It outperforms the other algorithms in most of the tested functions. In the comparison between eGO and GO, eGO obtain more ‘winning values’ in different dimensions. However, it is less effective than GO in optimizing certain functions.

To determine the effectiveness of the algorithm, it is essential to analyze both the optimal value achieved by the algorithm and the convergence of the algorithm. To better illustrate the data changes, we calculated the difference between the average value of the algorithm and the optimal value given by CEC for log. In this paper, a CEC2017-30-dimensional convergence graph is chosen as a representative for convergence analysis. The performance of the eGO algorithm proposed in this paper is compared with that of other algorithms in terms of convergence in Fig. 2. The horizontal axis represents the number of evaluations, while the vertical axis represents the logarithm of the difference between the function value f and the optimal value f∗(log(f−f∗)). A lower value on the vertical axis indicates better convergence, as it signifies a smaller difference between f and f∗. The convergence curves indicate that the eGO algorithm has a stronger convergence ability than the other algorithms in general.

Figure 2: F1–F30 convergence graph of CEC2017 benchmark function on 30-D

5  Case of Study: Enhanced Growth Optimizer for Image Fusion

This section implements image fusion using the eGO algorithm and compares it to traditional methods. Image fusion uses single-band panchromatic (PAN) images and multi-band multispectral (MS) images. The process of merging these two types of images to produce images with enhanced spectral and spatial resolution is commonly known as pan-sharpening. We use the Eq. (17).

MSkH=MSkL+gk×D(17)

where MSH represents the high-resolution multispectral band, MSL represents the low-resolution multispectral image, gk is a real number representing the mixing parameter, k represents the kth wave spectrum of the multispectral image, and D represents the spatial detail map in Eq. (18).

D=P−I(18)

where P is the panchromatic image and I is the intensity component in the IHS, calculated as follows Eq. (19).

I=∑i=1nαiMSkL(19)

where αi represents a non-negative coefficient less than 1, and it has an average value of 1/3 in RGB images; however, multispectral images have four bands, with RGB and an infrared band, so αi is often defined as 1/n for multispectral images, where n represents the number of channels. To minimize spectral distortion in IHS panchromatic sharpened images, the calculation of intensity bands is modified based on the original multispectral and panchromatic images. To minimize spectral distortion, the intensity bands should be as close as possible to the panchromatic image. Therefore, in this adaptive IHS method, we want to find the closest coefficient αi that satisfies the P≈∑i=1nαiMi [23,24]. Hence, it is necessary to solve the optimization problem stated in Eq. (20) in order to reduce the impact of low-frequency components.

‖D‖2=‖P−∑i=13αiMSiL‖2(20)

The optimal value of αi can be calculated using the Least Squares Method (LSM) as follows Eq. (21).

αi=((MSiL)TMSiL)−1(MSiL)TP.(21)

The seven algorithms described above are used to adaptively tune the αi. For ease of understanding, Eq. (17) is modified as follows Eq. (22).

MSkH=MSkL+gk×Γk⊙D(22)

where Γk is a weighting matrix defined as follows Eqs. (23) and (24).

Γk=B (k)Γp+(1−B (k))ΓMSk(23)

Γp=exp⁡(−λ|∇P|4+ϵ),ΓMSk=exp⁡(−λ|∇MSk|4+ϵ)(24)

The gradient of the panchromatic map (∇P) and the gradient of each spectral channel of the multispectral map (∇MS) are used in this equation, along with a parameter (λ) to control image smoothness. Additionally, a very small value (ϵ) is included to prevent division by zero in the denominator. This paper uses the relative dimensionless global error in synthesis (ERGAS) between hybrid and low-resolution multispectral images to compute the parameter β for adaptive Pan-Sharpening. The formula for ERGAS is Eq. (25).

ERGAS=100hl1N∑i=1n(RMSE(n)μ(n))2(25)

where h represents the resolution of the high resolution image, l represents the resolution of the low resolution image, μ is the mean value of the nth band, and N is the number of bands. RMSE is the root mean square error, the formula for calculating RMSE is as follows Eq. (26).

RMSE=1L×K∑i=1L∑i=1K(F(i,j)−MS(i,j))2(26)

The high-resolution hybrid image is represented by F, while the original low-resolution multispectral image is represented by MS. The size of the image is denoted by L×K. Experiments were conducted on a dataset collected by QuickBird, which includes a high-resolution Pan image and a low-resolution MS image composed of three different bands (red, green, and blue). The proposed algorithm’s performance is compared to six other algorithms mentioned above, as well as pso, using the ERGAS index as an evaluation function and calculating the coefficients in Eq. (21). In order to do a quantitative assessment of various picture fusion techniques, we employ five overarching metrics. These include the relative average spectral error (RASE) [25], correlation coefficient (CC) [26], universal image quality index (UIQI) [27], and spectral angle mapper (SAM). RASE represents the average performance of the spectral band, with lower values indicating higher spectral quality. Its calculation formula is as follows Eq. (27).

RASE=100M1N∑i=1NRASE2(n)(27)

CC is utilized to compare the degree of spectral distortion between the original multispectral bands and the final fused image bands. A higher CC value indicates better pan-sharpening and less distortion, with a maximum value of 1. UIQI quantifies the extent to which there is a lack of correlation, distortion in brightness, and distortion in contrast. A higher UIQI value indicates better maintained spectral quality, with a maximum value of 1. The SAM metric quantifies the magnitude of the spectral angle between the sharpened result and the reference vector. A smaller SAM indicates a better sharpened result, with a minimum value of zero.

The fused image was created by applying Eq. (22) to the refined detail maps. The evaluation results of the image fusion are presented in Table 8. Fig. 3a,b shows the original MS and PAN maps, and the other images in Fig. 3 show the images after fusion by each method. It is evident from the graph that the images fused by the eGO algorithm produce better results than those produced by other algorithms.

Figure 3: The MS and Pan image of QuickBird and the results of image fusion using various methods

6  Conclusions

This paper employs the compact strategy to improve the original GO algorithm and introduces Lévy flight to improve the eGO algorithm. The individuals generated in enhanced GO are based on a probabilistic model called perturbation vector (PV), which occupies less memory than an actual population. Meanwhile, the enhanced GO algorithm retains the search mechanism of the original GO algorithm, making it suitable for devices with limited memory. Experimental results on the CEC2017 test function demonstrate that eGO is efficient and effective compared to GO and other compact algorithms. When applied to image fusion, the αi found by eGO produces better fusion results compared to the previous PSO approach and other algorithms mentioned above. This has a beneficial effect on following image processing procedures.

Acknowledgement: The authors would like to express their gratitude to all the anonymous reviewers and the editorial team for their valuable feedback and suggestions.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: Study conception and design: Jeng-Shyang Pan, Shu-Chuan Chu, Junzo Watada; data collection: Wenda Li, Xiao Sui, Jeng-Shyang Pan; analysis and interpretation of results: Wenda Li, Xiao Sui, Shu-Chuan Chu; draft manuscript preparation: Wenda Li, Xiao Sui, Junzo Watada. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The source code of the GO can be found at https://github.com/tsingke/Growth-Optimizer (accessed on 03 August 2022). The source code for the eGO can be obtained from the corresponding author upon request.

Ethics Approval: Not applicable.

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

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Appendix A. Comparison of Standard Deviation and Minimum Value of eGO, eGO_l, GO, and Commonly Used Compact Algorithms