An improved arithmetic optimization algorithm with multi-strategy for adaptive multi-spectral image fusion

Abstract

Panchromatic and multi-spectral image fusion, called panchromatic sharpening, is the process of combining the spatial and spectral information of the source image into the fused image to give the image a higher spatial and spectral resolution. In order to improve the spatial resolution and spectral information quality of the image, an adaptive multi-spectral image fusion method based on an improved arithmetic optimization algorithm is proposed. This paper proposed improved arithmetic optimization algorithm, which uses dynamic stochastic search technique and oppositional learning operator, to perform local search and behavioral complementation of population individuals, and to improve the ability of population individuals to jump out of the local optimum. The method combines adaptive methods to calculate the weights of linear combinations of panchromatic and multi-spectral gradients to improve the quality of fused images. This study not only improves the quality and effect of image fusion, but also focuses on optimizing the operation efficiency of the algorithm to have real-time and high efficiency. Experimental results show that the proposed method exhibits strong performance on different datasets, improves the spatial resolution and spectral information quality of the fused images, and has good adaptability and robustness. The source code is available at: https://github.com/starboot/IAOA-For-Image-Fusion.

Keywords

Image fusion multi-spectral image panchromatic image oppositional learning operator arithmetic optimization algorithm meta-heuristic

1 Introduction

Generally, a single imaging sensor cannot obtain complete information about a target scene [1]. Therefore, to obtain richer information, researchers have frequently used multiple sensors to collect two types of images simultaneously: single-band panchromatic (PAN) and multi-spectral (MS). Under the current conditions, PAN image has both low spectral resolution and high spatial resolution. However, the opposite is the MS image. Combining these two types of images to produce high spectral and high spatial resolution images has been widely used in military, remote sensing [2], medical imaging [3], industrial inspection and other fields. Compared with images captured by a single imaging sensor, multi-modal images can provide richer information.

Image fusion can be divided into four categories of methods [4], including component replacement multi-resolution analysis, based on component substitution (CS), hybrid and model-based methods. Among these methods, CS-based methods are one of the most classical methods, such as Intensity Hue Saturation (IHS), principal component analysis (PCA) and Gram-Schmidt (GS), which have the advantages of fast speed, simple implementation and high spatial resolution, but may suffer from spectral distortion and local artifacts. Multi-resolution analysis (MRA) methods, on the other hand, can retain richer spectral information.

In order to reduce the spectral distortion in the IHS method, an adaptive IHS method has been proposed by Rahmani et al. [5], which can adjust the band coefficients according to the actual situation to obtain a more accurate detail map. In addition, to improve the spectral fidelity at the edges, researchers have used edge-preserving filters and injected detail maps into each MS band. In contrast, another research team Leung et al. [6] found that it was not enough to inject detail maps using PAN images only, so they also used a linear combination of MS images and PAN images in combination. However, there are still some drawbacks in these methods, such as the detailed maps with fixed weights may lead to distortion problems such as local artifacts and local pseudo details. Existing methods may perform poorly in terms of image sharpness, contrast and color accuracy. Meanwhile, image fusion techniques are not limited to single modality, but can also be used for multimodal data fusion. However, existing methods have limitations in preserving multimodal information and improving target detection performance. In practical applications, image fusion techniques need to have real-time and high efficiency. However, traditional mathematical methods are often difficult to meet the real-time processing requirements in large-scale scenarios because they are based on the computation of deterministic solutions and the processing speed may be low. To address these issues, this study uses a combination of adaptive weight tuning band coefficients and edge preserving filters to improve the spectral fidelity. In addition, this study enhances the performance of the fusion results in detail maps by linearly combining MS images with PAN images. Also, a meta-heuristic algorithm is used to balance the fusion results with the computation time. With these improvements, this study expects to better address the challenges of image fusion techniques in practical applications. Therefore, in this study, an adaptive computational method for incoming detail maps is proposed for computing lower resolution (LR) MS detail maps. In this process, how to calculate the weights of the detail maps in a reasonable adaptive manner is a key issue, which will directly affect the degree of resolution and reduction of the image distortionproblem.

Although traditional mathematical methods are able to obtain optimal solutions to problems through strict mathematical logic, they are more suitable for solving small-scale problems due to long computation time. With the increasing complexity of real-life problems, approximation algorithms began to emerge, including heuristics and meta-heuristics. Compared to traditional mathematical methods, they may not necessarily find the optimal fitness value, but can significantly reduction of running time. The exploration and mining capabilities of the algorithm are improved based on its characteristics, and have better practicality in various practical engineering problems.

Many researchers have chosen to use meta-heuristic algorithms to solve various problems in engineering, such as, IoT smart home energy management [7], vehicle routing problems (VRP) [8], and automatic guided vehicle (AGV) energy-efficient scheduling problems [9], because of the advantages of shorter computation time, the ability to handle more complex large-scale problems, and a wide range of applications compared to traditional mathematical methods. However, the free lunch theorem has been proven to have both advantages and disadvantages for all meta-heuristic algorithms in any field. For any problem, all algorithms need to be improved or new algorithms proposed to solve the problem [10]. Usually, based on the laws of special problems, academics improve existing algorithms or propose algorithms for special effects [11]. These improved and innovative algorithms can often be better adapted to the complexity and special needs of the problem at hand, and improve the optimization effect and searchspeed.

Table 1
Calculation formulas of image fusion quality

$CC (\begin{matrix} R, F \end{matrix}) = \frac{\sum_{a = 1}^{M} \sum_{b = 1}^{N} (R (\begin{matrix} a, b \end{matrix}) - \bar{R}) (F (\begin{matrix} a, b \end{matrix}) - \bar{F})}{\sqrt{\sum_{a = 1}^{M} \sum_{b = 1}^{N} {(R (\begin{matrix} a, b \end{matrix}) - \bar{R})}^{2} {(F (\begin{matrix} a, b \end{matrix}) - \bar{F})}^{2}}}$

R and F are fused images and original

$\bar{R}$ and $\bar{F}$ are the means of R and F.

$SAM = {cos}^{- 1} \frac{d^{T} x}{(d^{T} d)^{1 / 2} (x^{T} x)^{1 / 2}}$

d is the given target array and x is the array to be measured.

$RMSE = \sqrt{{\frac{\sum_{a = 1}^{m} \sum_{b = 1}^{n} (R (\begin{matrix} a, b \end{matrix}) - F (\begin{matrix} a, b \end{matrix}))}{m \times n}}^{2}}$

Pixels n×m Image dimension.

$RASE = \frac{100}{N} \sqrt{\frac{1}{M} \sum_{m = 1}^{M} {RMSE}^{2} (m)}$

Where N represents mean radiance of M bands.

$UIQI (R, F) = \frac{4 σ_{RF} \bar{R} \bar{F}}{({\bar{R}}^{2} + {\bar{F}}^{2}) (σ_{R}^{2} + σ_{F}^{2})}$

$σ_{R}^{2}$ and $σ_{F}^{2}$ are the variance of R and F;

σ_RF is the covariance between R and F.

In general, meta-heuristic algorithms fall into four categories: those based on the laws of genetic evolution in nature, those based on population intelligence, those based on physical and chemical phenomena, and those based on human social behavior. The first category is based on genetic evolution, which implements search by simulating biological evolutionary processes. It represents the problem as a chromosome and generates new solutions through selection, crossover and mutation operations. The classical evolutionary algorithm is the genetic algorithm (GA) [12], evolutionary strategy (ES) [13], differential evolutionary algorithm (DE) [14], biogeography based optimization algorithm (BBOA) [15], evolutionary programming (EP) [16] and immune algorithm (IA) [17]. The second type of meta-heuristic algorithm based on population intelligence, implements search by simulating the behavioral processes of animals during hunting or migration. It represents the problem as a population of individuals and computes it through the collective behavior and properties of animals in nature, and generates new solutions for solving engineering optimization problems. The classical population intelligence algorithms are particle swarm optimization (PSO) [18]. Ant colony optimization (ACO) [19] and artificial bee colony (ABC) [20] are also popular swarm intelligence algorithms today. Other common group intelligence algorithms include the ant lion optimization (ALO) [21], grey wolf optimization (GWO) [22], squirrel optimization algorithm (SSA) [23], whale optimization algorithm (WOA) [24], the orchard algorithm (OA) [25], artificial hummingbird algorithm (AHA) [26], chimpanzee optimization algorithm (ChOA) [27], firefly algorithm (FA) [28], aquila optimizer (AO) [29], african vultures optimization algorithm (AVOA) [30], snake optimizer (SO) [31] and dung beetle optimizer (DBO) [32]. The third category of meta-heuristic algorithms is based on physical and chemical methods, which achieve search by simulating physical phenomena or laws of nature. For example, simulated annealing algorithm (SAA) [33], gravitational search algorithm (GSA) [34], golden section optimization algorithm (GSOA) [35], black hole algorithm (BHA) [36], sine cosine algorithm (SCA) [37] and equilibrium optimizer (EO) [38] have been widely recognized in academia. The artificial electric field algorithm (AEFA) [39] and Henley gas solubility optimizer (HGSO) [40] are based on Henley’s law, among others, and propose new solutions for solving engineering optimization problems. The fourth category of meta-heuristic algorithms is based on the principles of human social behavior, by modeling the way human individuals or groups behave in order to solve complex social problems. In recent years, some of them are widely recognized by academics, such as social learning optimization algorithm (SLOA) [41], volleyball super league algorithm (VPLA) [42], teaching optimization algorithm (TLBO) [43] and election algorithm (EA) [44]. Among the more famous ones are Moghdani et al. [45]. An improved election optimization algorithm by Kanwal et al. [46] developed which has been applied to IoT cloud data processing. An improved teaching optimization algorithm by Xu et al. [47] proposed which has been applied to distance fitness learning strategies and has some potential for solving optimization decision making and collaborative cooperation.

The arithmetic optimization algorithm (AOA) [48] is an optimization algorithm based on arithmetic operations designed to solve mathematical optimization problems. Its design background draws on basic arithmetic operations and optimization ideas in mathematics to search for optimal solutions by performing stepwise addition, subtraction, multiplication and division operations on the values in the solution space. AOA utilizes basic arithmetic operations and is very simple and intuitive in terms of numerical computation, easy to implement and understand [49, 50]. Due to these advantages, AOA has been successfully applied to problems in many engineering fields. Among them are Talpur et al. for optimizing deep neuro-fuzzy classifiers [51], Rajagopal et al. for improving deep convolution spiking neural network models [52], Kharrich et al. for solving grid energy storage system design problems [53], Montano et al. for solving optimal tide problems in DC networks [54], Bahmanyar et al. for solving multi-objective problems in IoT smart homes [7], Abualigah et al. for implementing efficient text document clustering methods [55], etc. However, Wolpert et al. points out, no single algorithm is applicable to all domains. Academic researchers need to improve algorithms to obtain better performance and results in specific domains, depending on the characteristics of the problem at hand [11].

This paper further demonstrates the effectiveness of meta-heuristic algorithms and image fusion in solving practical problems by investigating the problem of adaptive multi-spectral image fusion. Based on the AOA, improved arithmetic optimization algorithm (IAOA) is proposed to improve the ability of population individuals to jump out of the local optimum through dynamic stochastic search technique and adversarial learning, which is based on local search and behavioral complementarily of population individuals. The contributions and work of this paper are as follows:

IAOA using dynamic stochastic search tech-niques and dyadic learning operators is presented.

An adaptive multi-spectral image fusion method based on the IAOA is proposed.

Combining IAOA with multi-strategy for adaptive multi-spectral image fusion problems.

Experimental results show that the proposed method improved the spatial resolution and spectral information quality of the fused images, and has good adaptability and robustness.

The rest of this article is organized as follows: Section 2 introduction related work and basic CS-based model approach. Section 3, the AOA and its improved strategies are presented, and the fusion method using IAOA for multi-spectral images is introduced. The experimental results are discussed and analyzed in Section 4. Section 5, summary and future works.

2 Related work

This section presents four types of work related to image fusion: the current state of the art in image fusion development, CS-based methods, image evaluation metrics, and arithmetic optimization algorithms. In addition, multi-spectral image fusion is a classical problem in image processing and plays a crucial role in computer vision and medical image analysis applications. In the real world, due to the physical characteristics of imaging sensors, a single sensor can usually only acquire data with specific attributes. However, with multiple sensors, image datasets with many different features are obtained, i.e., multiple datasets with different attributes are generated in the same scene. However, any individual attribute dataset cannot satisfy the needs of downstream work processing; therefore, fusion utilization of datasets from multiple imaging sensors becomes particularly important. It can provide richer and more accurate image data for various downstream applications and further promote the application of image processing and analysis techniques in different fields. Because of this, it is of great practical significance to study image fusion. This paper further demonstrates the effectiveness of meta-heuristic algorithms and image fusion research in solving real-world problems by investigating improved arithmetic optimization algorithms for adaptive multi-spectral image fusion problems.

By combining the adaptive weights to adjust the band coefficients and the edge preserving filter, this paper in proposes a method based on the meta-heuristic algorithm to optimize the adaptive weights and then adjust the band coefficients, so that the fusion results can better maintain the spectral fidelity. At the same time, the introduction of an edge preserving filter helps to improve the clarity and fidelity of the edges. This combination of methods can effectively improve the quality of the fusion results, making the image details clearer and more realistic.

Different from the traditional methods, IAOA further enhances the performance of the fusion results in detail maps by optimizing the adaptive weights to linearly combine MS images and PAN images. This fusion strategy makes full use of multi-spectral and high-resolution information, resulting in more detailed and accurate final fusion results. IAOA introduces the dynamic random search technique and the opposition-based learning operator, which are used to enhance the optimization and generalization capabilities of the solution. The dynamic random search technique searches for optimal solutions on both global and local scales and further optimizes the solution. And the opposition-based learning operator optimizes its own decisions by competing and gaming with the adversaries in a multi-party game process.

These innovative techniques enable IAOA to better adapt to uncertain environments and provide more efficient and accurate fusion results. IAOA compares with existing methods in terms of fusion strategy, it has the features of heuristic adaptive weight adjustment, edge preservation, and linear combination, as well as the addition of dynamic random search technique and opposition-based learning operators make it perform well in image fusion problems. IAOA algorithm is designed to improve the performance of image fusion problems by combining multiple techniques and methods, aiming to improve the effect and performance of imagefusion.

2.1 Image fusion

Current remote sensors are unable to provide high spectral resolution and high spatial resolution imagery due to various limitations. Common remote sensors, such as IKONOS, QuickBird and World-View, are only able to acquire both PAN and low spatial resolution multi-spectral (LR MS) images. While multi-spectral images contain valuable spectral information for distinguishing between different materials, panchromatic images contain important spatial information such as the geometry of the object. By combining spectral and spatial information, classification accuracy can be improved and better results can be achieved in applications such as target detection.

Typically, researchers fuse a LR MS image with the original pixel information in a panchromatic image, aiming to produce a fused image with high spatial and spectral resolution, a process known as panchromatic sharpening. The panchromatic sharpening technique can effectively integrate the superior information from multiple images for downstream work in computer vision, industrial applications, remote sensing processing and more. The use of such fusion techniques can better meet demand and improve image quality and usability, thereby advancing related fields.

Early in the field of image fusion were Burt et al. [56] who proposed a pyramid fusion method to achieve image fusion through multi-scale decomposition and reconstruction of images. Mallat et al. [57] proposed a wavelet packet fusion method based on wavelet transform, extending the traditional wavelet fusion. The fusion of images is achieved by analyzing and fusing the wavelet coefficients. Li et al. [58] proposed a non-parametric image fusion method based on information theory. By maximizing the mutual information of the fused images, a better image fusion effect is achieved. Some authors have also done some work on image fusion through neural network models in recent years. Zhang et al. [59] used convolution neural networks for multi-focal image fusion task to improve the quality of multi-focal images through effective feature extraction and fusion. Singh et al. [60] proposed a feature-level image fusion and image segmentation method for night vision context enhancement based on arithmetic optimization to achieve low light image enhancement and fusion.

Finally, this study employs a combination of meta-heuristic algorithms and CS-based methods, while utilizing adaptive weights to adjust the band coefficients and edge keepers to improve the problems in image fusion. Specifically, the arithmetic optimization algorithm is improved by introducing dynamic random search technique operator and opposition-based learning operator to enhance the ability of the algorithm to jump out of the local optimal solution and the search capability. In solving the adaptive multi-spectral image fusion problem, several techniques are combined to simultaneously compute linear combinations of gradient weights to better fuse features from multiple images and reduce the effects of spectral distortion and local artifacts. These improved methods are expected to improve the performance and effectiveness of image fusion techniques and provide better solutions for practical applications.

2.2 CS-based method

For CS-based methods, the spatial and spectral information in the MS image can be considered as separable. Therefore, a spectral transformation of the MS image is generally performed to separate the component containing the most spatial information from the spectral component in order to replace the spatial component in the up-sampled LR MS image directly with the corresponding PAN image. Following this principle, the up-sampled LR MS image first needs to be projected into the new space by an appropriate transformation to produce the spatial components. These spatial components are then replaced, in whole or in part, with histogram-matched PAN images. Finally, the fused high-resolution (HR) MS image is generated by an inverse transformation. There are some classical and common transformations in this type of method, such as the IHS, PCA and GS transform [61].

CS methods have a number of advantages. They are simple and fast to run, and they retain the spatial information in the fused product [62]. Due to incomplete matching of spectral channels between MS and PAN images, local differences may lead to unwanted spectral distortion. Therefore, spectral distortion needs to be dealt with by histogram matching before replacement, ensuring that the histogram of the PAN image matches the histogram of the corresponding component MS image. The CS method is a premium solution when fusing highly correlated MS and PAN images.

Current research articles on image fusion show that CS-based methods excel in enhancing the spatial detail of fused images. These methods are simple to operate and easy to implement. Although in terms of accuracy, they may not be comparable to other methods. However, their appeal to researchers lies in their simplicity and effectiveness, and still makes them worthy of further research.

In mathematics, CS-based image sharpening methods have been summarized by several scholars and can usually be written as ${MS}_{k}^{H} = {MS}_{k}^{L} + g_{k} \times D$ (1) where ${MS}_{k}^{H}$ is represented the high-resolution multi-spectral band that completes the image fusion results, ${MS}_{k}^{L}$ is the data set of the original low-resolution multi-spectral image, and g_k is the injected gain value, usually a real number. The kth band in the MS image is usually represented by k. In the current experiment, we can assume that k belongs to the set 1, 2 and 3. And the spatial details of the detail map can be denoted by D. $D = P - I$ (2)

Where P refers to the PAN image and I is the intensity component of the IHS method and has spatial detail information. The formula is as follows. $I = \sum_{i = 1}^{3} α_{i} {MS}_{i}^{L}$ (3)

In this case, the variable α is used to define the band coefficients, which are non-negative real numbers less than 1. In order to evaluate the calculation of these coefficients, the following optimization problems need to be solved in the study to reduce the effect of the low frequency component: ${∥ D ∥}^{2} = {∥ P - \sum_{i = 1}^{3} α_{i} {MS}_{i}^{L} ∥}^{2}$ (4)

The optimal solution for the coefficient d, which can be calculated by using the least squares method (LSM), is expressed in the following form: $α_{i} = {({({MS}_{i}^{L})}^{T} {MS}_{i}^{L})}^{- 1} {({MS}_{i}^{L})}^{T} P$ (5)

At this point, Equation 2 and Equation 5 are used to calculate the primitive detail map.

2.3 Evaluation indexes and calculation formulas of image fusion quality

Several evaluation metrics can be used when objectively evaluating the experimentally obtained panchromatic sharpened images. In addition to visual inspection of the color sharpness and naturalness of the images, the following six commonly used metrics can be used for comparison to more fairly and objectively assess the fusion performance of different optimization algorithms.

The correlation coefficient (CC) is a measure of the correlation between the sharpening result and the reference, with a larger CC value indicating better panchromatic fusion and less distortion. The optimal value for CC is 1.

Root mean square error (RMSE) is a commonly used evaluation metric to measure the degree of difference between the reconstructed image and the original image, and thus to assess the quality of the reconstructed image.

Relative global dimensional synthesis error (ERGAS) is often used to measure the overall quality of the sharpening result, where an optimal value of 0 for ERGAS indicates optimal image fusion.

The universal image quality index (UIQI) is used to reflect the degree of correlation loss, the degree of luminance distortion and the degree of contrast distortion; A higher value of UIQI indicates better retention of spectral quality. When the value of UIQI reaches 1, the spectral quality is optimal.

The spectral angle mapper (SAM) is typically used to indicate the absolute value of the spectral angle between the fused result and the reference vector. The optimal value for SAM is 0.

The relative average spectral error (RASE) is typically used to represent the average performance within a spectral band. Where a lower value of RASE indicates a higher spectral quality. Here the optimum value ofRASE is 0.

2.4 The AOA Algorithm

Abualigah et al. were inspired by the principles of mathematical operators and proposed AOA using properties and laws of mathematical operators plus optimizers and other methods. The operational process of AOA includes parameter initialization, exploration and optimization in the development phase. In the design of the algorithm, the basic principles of meta-heuristic primitives are fully considered and inspired in this study and have been inspired by them. By modeling the behavior of arithmetic operators, AOA is able to select, update and move individuals in the population in order to find a more optimal solution. By continuously iterating and updating the population, AOA is able to progressively improve the optimization results and demonstrate good performance in solving various optimization problems.

A population needs to be randomly generated before the initiation of AOA, where each individual is a set of candidate solutions. Assume that the dimension space of the real problem is n and that there are N individuals within the population. The candidate solution matrix is shown below: $X = [\begin{matrix} x_{1, 1} & \dots & \dots & x_{1, j} & \dots & x_{1, n} \\ x_{2, 1} & \dots & \dots & x_{2, j} & \dots & x_{2, n} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ x_{N - 1, 1} & \dots & \dots & x_{N - 1, j} & \dots & x_{N - 1, n} \\ x_{N, 1} & \dots & \dots & x_{N, j} & \dots & x_{N, n} \end{matrix}]$ (6)

The AOA is developed mathematical optimizer acceleration (MOA) function for adaptive selection of exploration and exploitation behavior, the mathematical expression of which is:

$\begin{matrix} MOA (CurrentIter) = Min + CurrentIter \\ \times (\frac{Max - Min}{MaxIter}) \end{matrix}$ (7)

Where MaxIter is the pre-set number of population change updates and CurrentIter denotes the number of iterations currently in place with a value between 0 and Max. According to Abualigah et al. Min and Max are usually set to 1 and 0.2. Another important function as an AOA is the mathematical optimizer probability (MOP), where the mathematical expression is $MOP (CurrentIter) = 1 - \frac{{CurrentIter}^{\frac{1}{β}}}{{MaxIter}^{\frac{1}{β}}}$ (8)

where β is used is expressed as a sensitive parameter that usually indicates the development accuracy during the operation of the algorithm. Again this parameter was specified as 5.2021 in the experiments in the article by Abualigah et al.

In each iteration of AOA, three random numbers (r₁, r₂ and r₃) are generated. In this case, the size of the random number r₁ and MOA are compared to determine whether the algorithm enters the exploration or exploitation phase. When MOA is less than r₁, the AOA is selected to enter the exploration part D (exploration) or M (movement). In the exploration phase, the update scheme for the population individuals is shown below:

$\begin{matrix} x_{i, j} (CurrentIter + 1) = \\ {\begin{matrix} \begin{matrix} best X (x_{j}) \div (MOP + ℓ) \times \\ (({ub}_{j} - {lb}_{j}) \times μ + {lb}_{j}), r 2 < 0.5 \end{matrix} \\ \begin{matrix} bestX (x_{j}) \times MOP \times \\ (({ub}_{j} - {lb}_{j}) \times μ + {lb}_{j}), otherwise \end{matrix} \end{matrix} \end{matrix}$ (9)

Where x denotes the solution in the next iteration in dimension j for the ith individual in the population, and bestX is the best individual obtained in the previous iteration. To effectively prevent the denominator from being zero, Abualigah et al. avoid this by including a very small integer c. and by controlling the accuracy of AOA’s search for the best by μ. The formulae ub and lb denote the upper and lower bound values of the space to which the problem belongs, respectively.

The AOA decides whether to explore by comparing the random number r₁ with the MOA. If the conditions for entering exploration are not met, the AOA will automatically enter the exploitation phase. By performing S (exploration) or A (adaptation), AOA is able to effectively improve the accuracy of the search for excellence in complex problems. The following is a concrete implementation of the population location update formula for the exploitation phase:

$\begin{matrix} x_{i, j} (CurrentIter + 1) = \\ {\begin{matrix} \begin{matrix} bestX (x_{j}) - MOP \times \\ (({ub}_{j} - {lb}_{j}) \times μ + {lb}_{j}), r 3 < 0.5 \end{matrix} \\ \begin{matrix} bestX (x_{j}) + MOP \times \\ (({ub}_{j} - {lb}_{j}) \times μ + {lb}_{j}), otherwise \end{matrix} \end{matrix} \end{matrix}$ (10)

2.5 Pseudo-code for AOA

Algorithm 1 Pseudo-code for AOA
Initialize population matrix N_i (i = 1, 2 ... n) and parametersα, μ.
For (CurrentIter = 1: MaxIter)
Solve the overall fitness value of the current population.
The latest solution of MOA and MOP can be obtained by Eqs. 7 and 8.
While (ith < N)
While (jth < Dim)
Generate random numbers r₁, r₂ and r₃
between 0 and 1.
IfMOA < r₁
If0.5 < r₂
The “÷” operator is used for operation calculate ith position by Equation 9.
Else
The “×” operator is used for operation calculate ith position by Equation 9.
End If
Else
If0.5 < r₃
The “-” operator is used for operation calculate ith position by Equation 10.
Else
The “+” operator is used for operation calculate ith position by Equation 10.
End If
End If
End While
End While
End For
Return bestX.

3 Proposed AOA with OBL and dynamic random search for image fusion

3.1 Mathematical model

A schematic diagram for solving the adaptive multi-spectral image fusion problem is shown in Fig. 1. A simple modification of Equation (11) helps us to understand it better, and Equation (11) is modified as follows:

${MS}_{k}^{H} = {MS}_{k}^{L} + g_{k} \times Γ_{k} \otimes D$ (11)

Where Γ_k is a weighting matrix described as follows: $Γ_{k} = β (k) Γ_{p} + (1 - β (k)) Γ_{{MS}_{k}}$ (12)

$\begin{matrix} Γ_{p} = exp (- \frac{λ}{{| \nabla P |}^{4} + ɛ}), Γ_{{MS}_{k}} \\ = exp (- \frac{λ}{{| \nabla {MS}_{k} |}^{4} + ɛ}) \end{matrix}$ (13)

Fig. 1

Block diagram of the proposed method.

Fig. 2

Flowchart of the IAOA

Several variables and metrics can be defined here to illustrate the problem. P denotes the gradient of the PAN image and MS denotes the gradient of each multi-spectral band. A is a parameter used to control the smoothness of the image and e is a minimal value used to avoid the denominator being zero. As different multi-spectral bands have different spectral characteristics, this may lead to the appearance of local artifacts, which means that different multi-spectral bands do not have the same detail map. To solve this problem, we can calculate parameter B to adaptively refine the detail map by using the ERGAS index between the fused image and the low-resolution MS image.

The ERGAS index can be defined as follows:

$ERGAS (A, B) = 100 \frac{K}{L} \sqrt{\frac{1}{N} \sum_{i = 1}^{N} \frac{{RMSE}^{2} (A_{i} - B_{i})}{M_{i}^{2}}}$ (14)

In Equation 14 A and B represent the MS image and the fused image and are used to represent the relationship between them respectively. K represents the resolution of the high-resolution image and L represents the resolution of the low-resolution image. M is the average radiance of the only band involved in the fused image, while RMSE represents the root mean square error and is used to measure the degree of difference between the two images.

3.2 The dynamic random search technique (DRST)

The DRST is a novel random search method introduced by Hamzacebi et al. [55]. The DRST method consists of two phases: a regular search phase and a local search phase.

In the regular search phase, a global search is performed by randomly generating candidate solutions, similar to the traditional random search method. However, unlike the traditional method, DRST saves the information related to the location of the found fitness for use in the subsequent local search phase. In the local search phase, the saved aptitude location information is used as a base point by searching for a better solution within its surrounding area. In this way, DRST not only searches for the optimal solution at a global scale, but is also able to further optimize the solution at a local scale. By combining global and local searches, DRST is able to make full use of known information about the location of the fitter during the search process in order to approximate the optimal solution as efficiently as possible. This strategy can improve search efficiency and help overcome the limitations of traditional stochastic search methods.

In this study, the local search phase is the focus of the research. The mining capacity of AOA is enhanced by its search capability. Therefore, local search techniques are presented in this section. A random search is performed in the vicinity of the best solution of the mining phase of the algorithm, which in turn reveals whether a better solution exists in the vicinity of the historical optimal solution.

Algorithm 2 Dynamic random search technique
Initialize the number of iterations (MaxIter) and the fitness function (Fobj);
CurrIter = 0;
Construct a random vector dx range [lb, ub]
Select an x from the candidate solutions as the starting point and set it to the current solution.
While (CurrIter < MaxIter) do
Generate a random vector dx from the parameter space and add it to the current solution.
By computing the value of Fobj(x + dx), obtain the new solution (x + dx).
IfFobj(x + dx) < Fobj(x) do
The solution (x + dx) is retained as the current solution (x = x + dx).
End If
End While

3.3 The Opposition-based learning (OBL)

The OBL is a method based on adversarial. The basic principle of OBL is to introduce adversarial and to transform the learning problem into a multi-party game process [64]. In OBL, the model no longer optimizes alone, but competes and confronts one or more adversaries. By interacting with the adversaries, the model can gain feedback and inspiration from their behavior and further optimize its own decision-making strategy and performance.

In the optimization problem, candidate solutions and their opposite solutions are played as adversaries in a multi-party game. This allows for faster convergence of the search algorithm and near-global optimal solutions. By selecting the better fitter as the initial solution through a process of competition and rivalry, it helps to bring the optimization algorithm closer to the globally optimal solution. By introducing adversarial competition and games, OBL can help algorithms learn to adapt to uncertain environments, optimize decision strategies, and have good generalization capabilities.

Assuming that $\bar{x}$ is the opposing solution to the original solution x, $\bar{x}$ is given: ${\bar{x}}_{(i, j)} = {UB}_{j} + {LB}_{j} - x_{(i, j)}, i = 1, 2, . . ., n .$ (15) where x_(i,j) denotes the jth dimension in the opposing solution of individual i, where UB and LB denote the upper and lower bounds of the solution space, and j denotes the jth dimension of the solution. The OBL method integrates the candidate and opposing solutions, and therefore performs the A or S operator of AOA and uses a greedy strategy to select the optimal value. By combining OBL with AOA, the search ability of the population in the solution space is significantly improved.

Algorithm 3 Pseudo-code of the IAOA
Initialize candidate x of IAOA
ForCurrIter = 1: MaxIterdo
Solve the overall fitness value of the current population.
Updating populations by the D, M, S and A operators.
Calculating opposing solutions using Eq. 15.
Comparing candidate and opposing solutions to obtain the optimal solution
Update to population individuals by greedy strategy
End For
Return the X_best.

3.4 The computational complexity of AOA

In this section, space complexity and time complexity will be used to illustrate the computational complexity of IAOA, which is discussed next by analyzing them separately. The complexity of AOA is mainly affected by the number of iterations of population updating (T), the number of individuals in the population (N), and the problem dimensions of multi-spectral adaptive fusion (D).

3.4.1 Time complexity

The IAOA requires these operations in each generation (iteration) when solving the multi-spectral image fusion problem. Firstly, update the position of the individual. Secondly, the fitness value of the individual is calculated (to assess the quality of the solution.). Finally update the individual optimal solution and the global optimal solution. The time complexity of updating the positions of the individuals is O (N×D). The time complexity of calculating the fitness value of an individual is O (N×D). The time complexity of updating the individual optimal solution and the global optimal solution is O (N). Therefore, the time complexity of each generation is O (N×D). The number of iterations of the OBL process is T and the number of iterations of the global and local search processes in the DRST operator is T. Thus the total time complexity is O (T×N×D).

3.4.2 Space complexity

IAOA needs to store the location information of individuals when solving the image fusion problem and requires a space of O (N×D). The DRST operator needs to store the individual optimal solution and the global optimal solution, which requires a space of O (D). Therefore, the total space complexity is O (N×D).

4 Experimental results and analysis

In this section, an evaluation of the effectiveness and feasibility of IAOA in solving adaptive multi-spectral image fusion models is carried out. In order to validate IAOA’s ability to solve the adaptive multi-spectral image fusion problem, experimental data from two satellites and with different themes are used as a benchmark. The dataset used for the experiments in this section is NBU_PansharpRSData, available at https://github.com/starboot/NBU_Panshar~pRSData. Where this dataset contains data from the QuickBird, IKONOS, GF and WorldView satellites, this experiment focuses on data from the commonly used QuickBird and GF satellites to obtain four classic scenes for experimentation. This section is presented through the following five aspects.

The dataset details and algorithm parameters are shown.

Experimental results of 10 well-known algorithms are presented and compared in the dataset with the theme Urban.

Experimental results of 10 well-known algorithms are presented and compared in the dataset with the theme Green Vegetation.

The experimental results of 10 well-known algorithms are presented and compared in the dataset on the theme of water scenes.

The experimental results of 10 well-known algorithms are presented and compared in the dataset on the theme of coast.

4.1 Experimental setup

The proposed algorithm is compared with 10 algorithms, such as PSO, ACO, DE, GA, GWO, WOA, EO, AHA, AO and AOA. The MATLAB source codes for PSO, DE, GA, GWO, WOA, AOA, EO, and AHA are available from https://seyedalimirjalili.com/, the ACO in MATLAB source code can be obtained at https://yarpiz.com/67/ypea104-acor [11] and the MATLAB source code for AO proposed by Abualigah L and Yousri D et al. can be downloaded from https://ww2.mathworks.cn/matlabcentral/fileexchange/89381-aquila-optimizer-a-meta-heuristic-optimization-algorithm. The parameter settings for each of these algorithms refer to the published articles, and the specific parameters and their values are shown in Table 3. The same data set was used for all experiments, and to eliminate the effect of randomness on this experiment, all algorithms were run 30 times independently, with 10 iterations as the dimensionality of the problem was not very large. The mean value obtained for each function in the experiment is recorded, where bold indicates superiority over the other algorithms. This paper effectively demonstrates the validity and feasibility of IAOA by comparing convergence curves, analysis of variance tests and radar plots. In addition, SAM is measured in degrees.

Considering the fairness and credibility of the experimental results, the system environment of all meta-heuristic algorithms in this experiment remains consistent. The experiments will be conducted in MATLAB 2018a with a device equipped with an Intel Core(TM) i7-9700 processor at 3.00 GHz, an operating system of Windows 10 and 16GB of RAM. This configuration will provide sufficient computing resources to run the experiments and ensure the accuracy and reproducibility of the results.

The open database NBU_PansharpRSData contains 2270 pairs of PAN/LR MS remote sensing images. In addition, the authors group the dataset according to different thematic surface features, including urban areas, green vegetation, and water scenes.

Table 2 clearly shows the details of the NBU_PansharpRSData database.

Table 2
Spatial and spectral resolutions of different satellites

Satellite Spectral Spatial (m) size Launch time

QuickBird Pan 1 0.61 1024×1024 2001

MS 4 2.44 256×256×4

IKONOS Pan 1 1 1024×1024 1999

MS 4 4 256×256×4

GF Pan 1 2 1024×1024 2013

MS 4 8 256×256×4

WorldView-2 Pan 1 0.5 1024×1024 2009

MS 8 2 256×256×8

WorldView-3 Pan 1 0.31 1024×1024 2014

MS 8 1.24 256×256×8

WorldView-4 Pan 1 0.31 1024×1024 2016

MS 4 1.24 256×256×4

Satellite	Spectral	Spatial (m)	size	Launch time
QuickBird	Pan	1	0.61	1024×1024	2001
	MS	4	2.44	256×256×4
IKONOS	Pan	1	1	1024×1024	1999
	MS	4	4	256×256×4
GF	Pan	1	2	1024×1024	2013
	MS	4	8	256×256×4
WorldView-2	Pan	1	0.5	1024×1024	2009
	MS	8	2	256×256×8
WorldView-3	Pan	1	0.31	1024×1024	2014
	MS	8	1.24	256×256×8
WorldView-4	Pan	1	0.31	1024×1024	2016
	MS	4	1.24	256×256×4

Table 3

Parameter variables involved in algorithms

Algorithms	Parameter	Value
PSO	Cognitive and social constant	C₁ and C₂ is 2
	Inertia weight	0.9 to 0.2
GA	Mu	0.05
	Crossover	0.8
	Elite fraction	0.05
DE	Scaling factor	0.5
	Crossover	0.5
ACO	heuristic factors	1
	pheromone	0.1
	pheromone evaporation	1
GWO	convergence	2 – 0
WOA	α	2 – 0
	β	1
EO	GP	0.5
AHA	/	/
AO	α	0.1
	δ	0.1
	μ	0.0265
IAOA	α	5
	μ	0.5

4.2 Experiments in urban scenarios

4.2.1 Experiments in urban scenarios

The urban scene is used as a test in this section. The urban scene includes imaging data from two detectors, GF and QuickBird, respectively. Comparison will be made with the improved AOA by 10 popular meta-heuristic algorithms such as PSO, ACO, GA, DE, GWO, WOA, EO, AHA and AO. And the performance metrics of different algorithms for different fusion results are demonstrated in Table 4, where black bold indicates the optimal results. The convergence curves of the different algorithms are displayed by Fig. 3 and Fig. 4 displays the variance of metrics of different algorithms for ERGAS in urban image fusion is used to reflect the stability of the algorithms. Figure 5 clearly shows the radar chart of different algorithms ranked in different metrics, and finally the fusion results of IAOA in urban scenarios are compared through Fig. 6.

Fig. 3

The Convergence curves of different algorithms for urban scenarios.

Fig. 4

The ANOVA test of different algorithms for urban scenarios.

Fig. 5

The radar map of different algorithms for urban scenarios.

Fig. 6

Visual assessment for marked region in Urban dataset.

Table 4

Experimental results of eleven algorithms for urban scenarios

Satellite	Algorithm	ERGAS	SAM	RASE	RMSE	UIQI	CC
QuickBird (Image1)	PSO	383.1194	5.1889	21.1412	13.0979	0.9749	0.9679
	ACO	382.6635	5.3614	20.6077	12.7674	0.9762	0.9700
	DE	382.6774	5.2620	20.4635	12.6781	0.9765	0.9706
	GA	382.3836	5.3452	20.6370	12.7856	0.9761	0.9699
	GWO	382.5007	5.3143	20.5632	12.7399	0.9763	0.9703
	WOA	382.5840	5.2788	20.9063	12.9524	0.9755	0.9688
	EO	382.8715	5.3009	20.5534	12.7338	0.9763	0.9703
	AHA	382.6284	5.3880	20.5523	12.7331	0.9763	0.9702
	AO	382.6153	5.2517	20.5539	12.7341	0.9763	0.9703
	AOA	382.6096	5.3204	20.4187	12.6503	0.9766	0.9708
	IAOA	382.4931	5.2470	20.3907	12.6330	0.9766	0.9709
QuickBird (Image2)	PSO	25.8286	19.0606	103.0102	65.6455	0.6514	0.8154
	ACO	26.1100	19.1761	104.0620	66.3158	0.6466	0.8074
	DE	26.9851	19.0910	106.9779	68.1740	0.6339	0.7779
	GA	28.0885	19.1833	112.7690	71.8645	0.6091	0.7428
	GWO	30.4363	19.0697	121.3869	77.3564	0.5743	0.8258
	WOA	26.6272	19.0918	106.3137	67.7507	0.6370	0.8044
	EO	29.2655	19.1161	116.8155	74.4432	0.5924	0.7320
	AHA	26.0127	19.1362	103.8628	66.1888	0.6476	0.8060
	AO	26.3025	19.1274	104.8452	66.8149	0.6434	0.8147
	AOA	25.8912	19.1962	103.3050	65.8333	0.6501	0.8149
	IAOA	25.5262	19.0596	101.7859	64.8653	0.6568	0.8269
GF (Image1)	PSO	13.5768	7.4837	28.1333	29.2788	0.9212	0.9849
	ACO	13.4574	7.4907	27.9312	29.0685	0.9223	0.9845
	DE	13.4613	7.4933	27.9433	29.0810	0.9223	0.9843
	GA	13.4629	7.5115	27.9480	29.0859	0.9222	0.9843
	GWO	13.4619	7.5013	27.9420	29.0797	0.9222	0.9844
	WOA	13.4611	7.5033	27.9712	29.1100	0.9221	0.9839
	EO	13.4733	7.5155	28.0064	29.1467	0.9219	0.9837
	AHA	13.4562	7.4837	27.9241	29.0610	0.9224	0.9846
	AO	13.4583	7.4823	27.9363	29.0737	0.9223	0.9844
	AOA	13.4605	7.4971	27.9476	29.0855	0.9222	0.9843
	IAOA	13.4561	7.4742	27.9164	29.0530	0.9224	0.9847
GF (Image2)	PSO	16.3615	8.9070	47.8333	43.5862	0.8159	0.8366
	ACO	14.1019	8.9416	34.4854	31.4235	0.8953	0.9657
	DE	14.5059	8.9594	36.0496	32.8488	0.8866	0.9504
	GA	14.6461	9.0408	37.0483	33.7588	0.8809	0.9358
	GWO	14.4707	8.9182	37.4201	34.0976	0.8789	0.9262
	WOA	14.1177	8.9523	34.5688	31.4995	0.8948	0.9646
	EO	16.0308	8.9022	48.2390	43.9559	0.8132	0.8291
	AHA	15.5713	9.0211	38.6154	35.1868	0.8720	0.9457
	AO	14.7062	9.0963	39.0163	35.5520	0.8695	0.9054
	AOA	15.4427	8.8891	40.5035	36.9073	0.8610	0.9112
	IAOA	14.0392	8.8332	33.9659	30.9501	0.8983	0.9744

4.2.2 Analyses of the experiment

The fusion results of 10 different algorithms are compared in urban scenes, the results can be clearly obtained. In the fusion results of image 1 of QuickBird it can be clearly seen that IAOA is not as good as GA in ERGAS metrics but all other metrics are better than other algorithms and only CC metrics are worse in GF-image 1.

The CC metric is usually used as a metric to measure the correlation between the sharpening results and the reference image. In urban scenes, there may be some detailed parts such as buildings and roads, which will have a great impact on the sharpening results. The GA algorithm’s unique genetic characteristics lead to the algorithm’s historical memory and genetic variation characteristics, which makes it per-form better in the face of complex building clusters and road details during operation. However, IAOA is already ahead of most algorithms.

In image 2 of the QuickBird and image 2 of the GF detector, all the performance indexes of the IAOA are better than the other algorithms. This proves the feasibility of the improved results. All the algorithms are run independently for 30 times, and it is displayed in Fig. 3 that the convergence speed of IAOA is better than the other algorithms. The convergence speed of IAOA is faster than other algorithms, which means it is easier to find the global optimal solution. The stability of IAOA is also relatively better as can be obtained in Fig. 4 [65], indicating that it has strong anti-interference ability against random noise or initial parameter changes.

4.3 Experiments in green vegetation scenarios

4.3.1 Results of the experiment

In this section, the green vegetation scene was used as a test. The green vegetation scene includes imaging data from two detectors, GF and QuickBird, respectively. The IAOA is compared with another algorithm. And the performance metrics of different algorithms for different fusion results are shown in Table 5, where black bold indicates the optimal results. Figure 7 shows the convergence curves of different algorithms, and Fig. 8 shows the variance of ERGAS metrics of different algorithms in green vegetation image fusion for reflecting the stability of the algorithms. Figure 9 clearly shows the radar chart of different algorithms ranked in different metrics, and finally the fusion results of IAOA in green vegetation scene are compared by showing in Fig. 10.

Table 5
Experimental results of eleven algorithms for green vegetation scenarios

Satellite Algorithm ERGAS SAM RASE RMSE UIQI CC

QuickBird (Image1) PSO 15.7384 18.2653 53.3681 36.6046 0.8480 0.9846

ACO 15.7256 18.3827 53.3278 36.5770 0.8484 0.9844

DE 15.7223 18.3560 53.3133 36.5671 0.8484 0.9845

GA 15.7163 18.2809 53.2839 36.5469 0.8485 0.9849

GWO 15.7243 18.3660 53.3199 36.5716 0.8483 0.9845

WOA 15.7204 18.3818 53.3106 36.5652 0.8485 0.9844

EO 15.7365 18.2627 53.3505 36.5926 0.8481 0.9849

AHA 15.7180 18.3637 53.2975 36.5562 0.8485 0.9846

AO 15.7232 18.3492 53.3120 36.5662 0.8484 0.9847

AOA 15.7186 18.2544 53.2942 36.5539 0.8484 0.9848

IAOA 15.7156 18.3476 53.2865 36.5486 0.8485 0.9847

QuickBird (Image2) PSO 16.5955 14.5448 57.7288 29.1575 0.8657 0.9531

ACO 16.6156 14.5612 57.7801 29.1834 0.8656 0.9521

DE 16.6285 14.6305 57.8774 29.2325 0.8652 0.9514

GA 16.6023 14.6041 57.7548 29.1706 0.8656 0.9530

GWO 16.5844 14.4642 57.6747 29.1301 0.8659 0.9535

WOA 16.6198 14.5630 57.8912 29.2395 0.8650 0.9514

EO 16.6290 14.5561 57.8321 29.2096 0.8654 0.9518

AHA 16.5895 14.4563 57.6856 29.1356 0.8659 0.9537

AO 16.5888 14.5592 57.7086 29.1473 0.8658 0.9531

AOA 16.5902 14.4562 57.6897 29.1377 0.8659 0.9537

IAOA 16.5758 14.4553 57.6228 29.1039 0.8662 0.9540

GF (Image1) PSO 11.4274 18.2263 69.8827 33.2268 0.8704 0.9690

ACO 11.1178 18.2113 68.7368 32.6819 0.8740 0.9795

DE 11.0551 18.0717 68.4231 32.5328 0.8748 0.9800

GA 11.0495 18.0596 68.4383 32.5400 0.8748 0.9797

GWO 11.0670 18.0990 68.4732 32.5566 0.8747 0.9794

WOA 11.0422 18.0640 68.4141 32.5285 0.8749 0.9802

EO 11.0772 18.0593 68.5106 32.5744 0.8746 0.9789

AHA 11.1527 18.1911 68.5913 32.6128 0.8742 0.9779

AO 11.1607 18.1683 68.6449 32.6383 0.8741 0.9776

AOA 11.0604 18.0647 68.4460 32.5437 0.8747 0.9798

IAOA 11.0382 18.0536 68.4078 32.5255 0.8749 0.9802

GF (Image2) PSO 4.9899 9.3890 30.8061 27.1067 0.9550 0.9841

ACO 4.9961 9.3692 30.7773 27.0814 0.9550 0.9839

DE 5.0176 9.3984 30.8062 27.1068 0.9550 0.9838

GA 4.9947 9.4014 30.8235 27.1220 0.9549 0.9840

GWO 5.0056 9.4401 30.9250 27.2113 0.9546 0.9840

WOA 5.0138 9.3933 30.8080 27.1084 0.9549 0.9837

EO 5.0141 9.3900 30.7934 27.0955 0.9550 0.9837

AHA 4.9928 9.3847 30.7996 27.1010 0.9550 0.9840

AO 4.9925 9.3621 30.7728 27.0774 0.9550 0.9839

AOA 5.0014 9.4084 30.8393 27.1359 0.9549 0.9839

IAOA 4.9800 9.3397 30.7594 27.0656 0.9551 0.9841

Satellite	Algorithm	ERGAS	SAM	RASE	RMSE	UIQI	CC
QuickBird (Image1)	PSO	15.7384	18.2653	53.3681	36.6046	0.8480	0.9846
	ACO	15.7256	18.3827	53.3278	36.5770	0.8484	0.9844
	DE	15.7223	18.3560	53.3133	36.5671	0.8484	0.9845
	GA	15.7163	18.2809	53.2839	36.5469	0.8485	0.9849
	GWO	15.7243	18.3660	53.3199	36.5716	0.8483	0.9845
	WOA	15.7204	18.3818	53.3106	36.5652	0.8485	0.9844
	EO	15.7365	18.2627	53.3505	36.5926	0.8481	0.9849
	AHA	15.7180	18.3637	53.2975	36.5562	0.8485	0.9846
	AO	15.7232	18.3492	53.3120	36.5662	0.8484	0.9847
	AOA	15.7186	18.2544	53.2942	36.5539	0.8484	0.9848
	IAOA	15.7156	18.3476	53.2865	36.5486	0.8485	0.9847
QuickBird (Image2)	PSO	16.5955	14.5448	57.7288	29.1575	0.8657	0.9531
	ACO	16.6156	14.5612	57.7801	29.1834	0.8656	0.9521
	DE	16.6285	14.6305	57.8774	29.2325	0.8652	0.9514
	GA	16.6023	14.6041	57.7548	29.1706	0.8656	0.9530
	GWO	16.5844	14.4642	57.6747	29.1301	0.8659	0.9535
	WOA	16.6198	14.5630	57.8912	29.2395	0.8650	0.9514
	EO	16.6290	14.5561	57.8321	29.2096	0.8654	0.9518
	AHA	16.5895	14.4563	57.6856	29.1356	0.8659	0.9537
	AO	16.5888	14.5592	57.7086	29.1473	0.8658	0.9531
	AOA	16.5902	14.4562	57.6897	29.1377	0.8659	0.9537
	IAOA	16.5758	14.4553	57.6228	29.1039	0.8662	0.9540
GF (Image1)	PSO	11.4274	18.2263	69.8827	33.2268	0.8704	0.9690
	ACO	11.1178	18.2113	68.7368	32.6819	0.8740	0.9795
	DE	11.0551	18.0717	68.4231	32.5328	0.8748	0.9800
	GA	11.0495	18.0596	68.4383	32.5400	0.8748	0.9797
	GWO	11.0670	18.0990	68.4732	32.5566	0.8747	0.9794
	WOA	11.0422	18.0640	68.4141	32.5285	0.8749	0.9802
	EO	11.0772	18.0593	68.5106	32.5744	0.8746	0.9789
	AHA	11.1527	18.1911	68.5913	32.6128	0.8742	0.9779
	AO	11.1607	18.1683	68.6449	32.6383	0.8741	0.9776
	AOA	11.0604	18.0647	68.4460	32.5437	0.8747	0.9798
	IAOA	11.0382	18.0536	68.4078	32.5255	0.8749	0.9802
GF (Image2)	PSO	4.9899	9.3890	30.8061	27.1067	0.9550	0.9841
	ACO	4.9961	9.3692	30.7773	27.0814	0.9550	0.9839
	DE	5.0176	9.3984	30.8062	27.1068	0.9550	0.9838
	GA	4.9947	9.4014	30.8235	27.1220	0.9549	0.9840
	GWO	5.0056	9.4401	30.9250	27.2113	0.9546	0.9840
	WOA	5.0138	9.3933	30.8080	27.1084	0.9549	0.9837
	EO	5.0141	9.3900	30.7934	27.0955	0.9550	0.9837
	AHA	4.9928	9.3847	30.7996	27.1010	0.9550	0.9840
	AO	4.9925	9.3621	30.7728	27.0774	0.9550	0.9839
	AOA	5.0014	9.4084	30.8393	27.1359	0.9549	0.9839
	IAOA	4.9800	9.3397	30.7594	27.0656	0.9551	0.9841

Fig. 7

The Convergence curves of different algorithms for green vegetation.

Fig. 8

The ANOVA test of different algorithms for green vegetation.

Fig. 9

The radar map of different algorithms for green vegetation.

Fig. 10

Visual assessment for marked region in green vegetation dataset

4.3.2 Analyses of the experiment

The fusion results of 10 different algorithms are compared in the green vegetation scene. The results can be clearly obtained, in the fusion results of the QuickBird image 1, it can be clearly seen that IAOA performs better in ERGAS and UIQI metrics but other metrics are inferior to other algorithms which is important because of the special background of the green vegetation, whereas all the metrics are better than the other algorithms in the other imaging image.

IAOA leads most algorithms in terms of the measure of correlation between the sharpening result and the reference image, the degree of difference between the reconstructed and original images, the absolute value of the spectral angle between the fusion result and the reference vector, and the average performance within the spectral bands, with only the first image data being less capable than GA in the test of the four green vegetation species. The reason for this has a lot to do with the dataset itself as well, but of course more to do with the fact that GA possesses greater reconstruction and distortion handling capabilities for this type of dataset. Because the meta-heuristic algorithm also suffers from a certain amount of randomness, the IAOA is considered to have a better performance capability all things considered.

In image 1 of the GF, only the UIQI and CC metrics are on par with the WOA, while the other metrics outperform all the algorithms. However, the fusion result of IAOA in the other image 2 outperforms the other algorithms. This proves the feasibility of the improved results. All the algorithms are run independently for 30 times, and the convergence speed of IAOA is shown to be better than other algorithms in Fig. 7. The stability of IAOA is also relatively better as can be obtained in Fig. 8.

4.4 Experiments in water scenarios

4.4.1 Results of the experiment

In this section, water scenes are used as tests. The water scene includes imaging data from two detectors, GF and QuickBird, respectively. The performance metrics of different algorithms for different fusion results are demonstrated by comparison in Table 6, where black bold indicates the optimal results.

Table 6
Experimental results of eleven algorithms for water scenarios

Satellite Algorithm ERGAS SAM RASE RMSE UIQI CC

QuickBird (Image1) PSO 170.9970 9.0692 36.0895 17.9781 0.9278 0.8854

ACO 170.9545 9.5414 36.3996 18.1326 0.9260 0.8807

DE 170.9540 9.1973 35.9079 17.8876 0.9281 0.8861

GA 170.8471 9.4802 36.1965 18.0314 0.9267 0.8824

GWO 171.0259 9.0836 35.9255 17.8964 0.9279 0.8857

WOA 170.9739 9.4252 36.2181 18.0421 0.9269 0.8833

EO 171.0083 9.1202 36.1861 18.0262 0.9274 0.8851

AHA 170.8619 9.1395 35.9429 17.9051 0.9276 0.8854

AO 170.9094 9.4346 36.2531 18.0596 0.9263 0.8815

AOA 171.0349 9.4807 36.3650 18.1153 0.9258 0.8802

IAOA 170.9289 9.0521 35.7693 17.8186 0.9285 0.8870

QuickBird (Image2) PSO 33.4788 8.2553 35.8430 18.5230 0.9291 0.9557

ACO 33.0547 8.2355 35.4364 18.3128 0.9306 0.9571

DE 33.0567 8.1622 35.3031 18.2440 0.9312 0.9584

GA 33.0762 8.2339 35.4185 18.3036 0.9307 0.9571

GWO 33.1116 8.2882 35.4109 18.2996 0.9309 0.9576

WOA 33.0539 8.1726 35.3253 18.2554 0.9311 0.9582

EO 33.0663 8.1664 35.2809 18.2325 0.9314 0.9587

AHA 33.0525 8.1718 35.2912 18.2378 0.9312 0.9586

AO 33.0771 8.2381 35.5592 18.3763 0.9301 0.9553

AOA 33.0559 8.2221 35.4250 18.3069 0.9307 0.9572

IAOA 33.0317 8.1558 35.2476 18.2153 0.9314 0.9588

GF (Image1) PSO 72.4035 4.9066 34.8038 10.4810 0.9683 0.9889

ACO 72.3467 4.9666 34.7987 10.4795 0.9683 0.9888

DE 72.3467 4.8870 34.7991 10.4796 0.9683 0.9888

GA 72.3514 4.8598 34.7824 10.4746 0.9684 0.9888

GWO 72.3468 4.8877 34.7941 10.4781 0.9683 0.9888

WOA 72.3474 4.9732 34.8414 10.4923 0.9683 0.9887

EO 72.3460 4.8537 34.7649 10.4693 0.9684 0.9889

AHA 72.3463 4.8737 34.7710 10.4711 0.9684 0.9889

AO 72.3495 5.1294 34.9395 10.5219 0.9681 0.9884

AOA 72.3461 4.8319 34.7586 10.4674 0.9684 0.9889

IAOA 72.3459 4.7844 34.7500 10.4648 0.9684 0.9889

GF (Image2) PSO 24.1441 2.9521 25.6865 10.9514 0.9730 0.9868

ACO 24.1281 3.0320 25.7017 10.9579 0.9730 0.9866

DE 24.1285 2.9583 25.6740 10.9461 0.9731 0.9868

GA 24.1252 3.0252 25.7003 10.9573 0.9730 0.9866

GWO 24.1120 3.0149 25.6783 10.9479 0.9731 0.9866

WOA 24.1207 3.0056 25.7185 10.9651 0.9730 0.9864

EO 24.1102 3.0357 25.6986 10.9566 0.9730 0.9865

AHA 24.1141 2.9090 25.6548 10.9379 0.9731 0.9868

AO 24.1191 2.9833 25.6705 10.9446 0.9731 0.9867

AOA 24.1049 2.9953 25.6639 10.9418 0.9731 0.9867

IAOA 24.1036 2.9070 25.6418 10.9323 0.9731 0.9868

Satellite	Algorithm	ERGAS	SAM	RASE	RMSE	UIQI	CC
QuickBird (Image1)	PSO	170.9970	9.0692	36.0895	17.9781	0.9278	0.8854
	ACO	170.9545	9.5414	36.3996	18.1326	0.9260	0.8807
	DE	170.9540	9.1973	35.9079	17.8876	0.9281	0.8861
	GA	170.8471	9.4802	36.1965	18.0314	0.9267	0.8824
	GWO	171.0259	9.0836	35.9255	17.8964	0.9279	0.8857
	WOA	170.9739	9.4252	36.2181	18.0421	0.9269	0.8833
	EO	171.0083	9.1202	36.1861	18.0262	0.9274	0.8851
	AHA	170.8619	9.1395	35.9429	17.9051	0.9276	0.8854
	AO	170.9094	9.4346	36.2531	18.0596	0.9263	0.8815
	AOA	171.0349	9.4807	36.3650	18.1153	0.9258	0.8802
	IAOA	170.9289	9.0521	35.7693	17.8186	0.9285	0.8870
QuickBird (Image2)	PSO	33.4788	8.2553	35.8430	18.5230	0.9291	0.9557
	ACO	33.0547	8.2355	35.4364	18.3128	0.9306	0.9571
	DE	33.0567	8.1622	35.3031	18.2440	0.9312	0.9584
	GA	33.0762	8.2339	35.4185	18.3036	0.9307	0.9571
	GWO	33.1116	8.2882	35.4109	18.2996	0.9309	0.9576
	WOA	33.0539	8.1726	35.3253	18.2554	0.9311	0.9582
	EO	33.0663	8.1664	35.2809	18.2325	0.9314	0.9587
	AHA	33.0525	8.1718	35.2912	18.2378	0.9312	0.9586
	AO	33.0771	8.2381	35.5592	18.3763	0.9301	0.9553
	AOA	33.0559	8.2221	35.4250	18.3069	0.9307	0.9572
	IAOA	33.0317	8.1558	35.2476	18.2153	0.9314	0.9588
GF (Image1)	PSO	72.4035	4.9066	34.8038	10.4810	0.9683	0.9889
	ACO	72.3467	4.9666	34.7987	10.4795	0.9683	0.9888
	DE	72.3467	4.8870	34.7991	10.4796	0.9683	0.9888
	GA	72.3514	4.8598	34.7824	10.4746	0.9684	0.9888
	GWO	72.3468	4.8877	34.7941	10.4781	0.9683	0.9888
	WOA	72.3474	4.9732	34.8414	10.4923	0.9683	0.9887
	EO	72.3460	4.8537	34.7649	10.4693	0.9684	0.9889
	AHA	72.3463	4.8737	34.7710	10.4711	0.9684	0.9889
	AO	72.3495	5.1294	34.9395	10.5219	0.9681	0.9884
	AOA	72.3461	4.8319	34.7586	10.4674	0.9684	0.9889
	IAOA	72.3459	4.7844	34.7500	10.4648	0.9684	0.9889
GF (Image2)	PSO	24.1441	2.9521	25.6865	10.9514	0.9730	0.9868
	ACO	24.1281	3.0320	25.7017	10.9579	0.9730	0.9866
	DE	24.1285	2.9583	25.6740	10.9461	0.9731	0.9868
	GA	24.1252	3.0252	25.7003	10.9573	0.9730	0.9866
	GWO	24.1120	3.0149	25.6783	10.9479	0.9731	0.9866
	WOA	24.1207	3.0056	25.7185	10.9651	0.9730	0.9864
	EO	24.1102	3.0357	25.6986	10.9566	0.9730	0.9865
	AHA	24.1141	2.9090	25.6548	10.9379	0.9731	0.9868
	AO	24.1191	2.9833	25.6705	10.9446	0.9731	0.9867
	AOA	24.1049	2.9953	25.6639	10.9418	0.9731	0.9867
	IAOA	24.1036	2.9070	25.6418	10.9323	0.9731	0.9868

Figure 11 shows the convergence curves of different algorithms, and Fig. 12 shows the variance of ERGAS metrics of different algorithms in water image fusion for reflecting the stability of the algorithms. Figure 13 clearly shows the radar chart of different algorithms ranked in different metrics, and finally the fusion results of IAOA in water scenario are compared by showing in Fig. 14.

Fig. 11

The Convergence curves of different algorithms for water scenario.

Fig. 12

The ANOVA test of different algorithms for water scenario.

Fig. 13

The radar map of different algorithms for water scenario.

Fig. 14

Visual assessment for marked region in water scenario dataset.

4.4.2 Analyses of the experiment

The fusion results of 10 different algorithms are compared in the water scene. The results can be clearly obtained, in the fusion results of the QuickBird image 1 it can be clearly seen that IAOA does not perform as well as GA in ERGAS metrics but all other metrics are better than other algorithms, and in the image 2 of the QuickBird the obtained performance reaches the optimum.

After comparing the fusion results of four image datasets from two different satellites, it is clearly seen that the IAOA algorithm is leading in the water scene. Although the best results may not be achieved in a small number of detail metrics, this is related to the stochastic nature of the meta-heuristic algorithm. In water scenes, the comprehensive quality assessment metric ERGAS is usually used to measure the overall quality of the sharpening results. Since the detail parts such as water surface ripples and underwater transmissibility have important impacts on the sharpening results, they may affect the results more. The GA algorithm is able to cope better with the complex water surface ripples and underwater imaging details through its unique genetic characteristics. However, the IAOA algorithm has outperformed the other algorithms in most cases in the comprehensive comparison. In short, the IAOA algorithm is ahead of most other algorithms in terms of comprehensive performance through the fusion comparison of image data in water scenes, although there may be cases where it is not as good as other algorithms in a few detail metrics.

This proves the feasibility of the improved results. All the algorithms are run independently for 30 times, and the convergence speed of IAOA is shown in Fig. 11 to be better than the other algorithms. The OBL operator introduced in the IAOA algorithm improves the search capability through a competitive mechanism. The game between competitors enables the algorithm to better explore the solution space and find a better solution. Such a competitive mechanism effectively avoids the algorithm from falling into the dilemma of local optimal solutions. This is the reason why it can jump out of the local optimal ability. In addition, the introduction of DRST operator allows the algorithm to maintain a balance between local and global search. By introducing randomness, the algorithm is able to jump out of the local optimum and continue the global search, thus avoiding falling into the local optimum. Such a balanced search strategy significantly improves the convergence speed of the algorithm. As shown in Fig. 11, it is able to find the optimal solution faster in most cases compared to other algorithms. The stability of IAOA can be obtained in Fig. 12.

4.5 Experiments in coastal scenarios

4.5.1 Experiments in coastal scenarios

In this section, the coastal scene is used as a test. The coastal scene includes imaging data from two detectors, GF and QuickBird, respectively. Eleven different meta-inspired algorithms are put together for adaptive multi-spectral image fusion. And the performance metrics of different algorithms for different fusion results are shown in Table 7, where black bold indicates the optimal results. Figure 15 shows the convergence curves of different algorithms and Fig. 16 shows the metrics variance of different algorithms for ERGAS in coastal image fusion used to reflect the stability of the algorithms. Figure 17 clearly shows the radar chart of different algorithms ranked in different metrics, and finally the fusion results of IAOA in coastal scenes are compared through Fig. 18.

Table 7
Experimental results of eleven algorithms for coast scenarios

Satellite Algorithm ERGAS SAM RASE RMSE UIQI CC

QuickBird (Image1) PSO 6.1020 8.7529 26.9606 25.1109 0.9543 0.9926

ACO 6.1008 8.7669 26.9618 25.1121 0.9543 0.9926

DE 6.1132 8.8052 26.9939 25.1420 0.9542 0.9923

GA 6.1183 8.7819 27.0331 25.1785 0.9540 0.9924

GWO 6.1089 8.7369 26.9871 25.1356 0.9542 0.9927

WOA 6.1148 8.7828 27.0025 25.1499 0.9542 0.9922

EO 6.1144 8.7559 27.0069 25.1540 0.9542 0.9922

AHA 6.1076 8.8069 26.9817 25.1306 0.9542 0.9925

AO 6.1172 8.7874 27.0127 25.1595 0.9541 0.9923

AOA 6.1104 8.7354 26.9978 25.1455 0.9542 0.9925

IAOA 6.0944 8.7322 26.9347 25.0868 0.9543 0.9928

QuickBird (Image2) PSO 97.1138 6.3107 38.3659 18.3615 0.9217 0.9008

ACO 97.1332 6.4400 37.1650 17.7868 0.9260 0.9080

DE 97.1310 6.3141 35.7726 17.1203 0.9311 0.9151

GA 97.1552 6.3659 37.8764 18.1272 0.9234 0.9040

GWO 98.5931 6.2628 37.8131 18.0969 0.9241 0.9047

WOA 97.0195 6.3062 35.9302 17.1958 0.9305 0.9146

EO 97.1762 6.4253 37.1236 17.7670 0.9261 0.9082

AHA 97.0469 6.2457 36.0554 17.2557 0.9302 0.9132

AO 97.0993 6.2991 33.7320 16.1438 0.9382 0.9268

AOA 97.2478 6.2794 34.4280 16.4769 0.9358 0.9239

IAOA 97.0016 6.7724 30.0854 14.3985 0.9497 0.9495

GF (Image1) PSO 46.5828 3.3283 20.6930 14.0151 0.9754 0.9902

ACO 46.5815 3.4044 20.7320 14.0415 0.9753 0.9901

DE 46.5801 3.4072 20.7327 14.0419 0.9753 0.9900

GA 46.5819 3.2946 20.7637 14.0630 0.9752 0.9899

GWO 46.5800 3.4144 20.7345 14.0432 0.9753 0.9900

WOA 46.5813 3.4255 20.7896 14.0805 0.9752 0.9898

EO 46.5807 3.3927 20.7382 14.0457 0.9753 0.9900

AHA 46.5810 3.3304 20.7052 14.0233 0.9754 0.9902

AO 46.5822 3.4351 20.7735 14.0696 0.9752 0.9899

AOA 46.5816 3.2625 20.6996 14.0196 0.9754 0.9902

IAOA 46.5799 3.3223 20.6962 14.0172 0.9754 0.9902

GF (Image2) PSO 51.6962 2.8638 33.1775 18.5635 0.9468 0.9597

ACO 50.6891 2.8616 26.8770 15.0382 0.9644 0.9797

DE 50.8264 2.8678 28.3232 15.8474 0.9607 0.9744

GA 50.8378 2.8774 27.5282 15.4026 0.9627 0.9762

GWO 50.8298 2.8615 29.3771 16.4371 0.9578 0.9724

WOA 50.6099 2.8681 26.1553 14.6344 0.9663 0.9815

EO 50.8209 2.8848 26.3923 14.7670 0.9657 0.9809

AHA 50.9210 2.9319 30.1940 16.8942 0.9555 0.9673

AO 50.7379 2.8645 27.9868 15.6592 0.9615 0.9753

AOA 50.7471 2.8610 27.0687 15.1455 0.9640 0.9797

IAOA 50.5770 2.8594 25.5686 14.3061 0.9677 0.9846

Satellite	Algorithm	ERGAS	SAM	RASE	RMSE	UIQI	CC
QuickBird (Image1)	PSO	6.1020	8.7529	26.9606	25.1109	0.9543	0.9926
	ACO	6.1008	8.7669	26.9618	25.1121	0.9543	0.9926
	DE	6.1132	8.8052	26.9939	25.1420	0.9542	0.9923
	GA	6.1183	8.7819	27.0331	25.1785	0.9540	0.9924
	GWO	6.1089	8.7369	26.9871	25.1356	0.9542	0.9927
	WOA	6.1148	8.7828	27.0025	25.1499	0.9542	0.9922
	EO	6.1144	8.7559	27.0069	25.1540	0.9542	0.9922
	AHA	6.1076	8.8069	26.9817	25.1306	0.9542	0.9925
	AO	6.1172	8.7874	27.0127	25.1595	0.9541	0.9923
	AOA	6.1104	8.7354	26.9978	25.1455	0.9542	0.9925
	IAOA	6.0944	8.7322	26.9347	25.0868	0.9543	0.9928
QuickBird (Image2)	PSO	97.1138	6.3107	38.3659	18.3615	0.9217	0.9008
	ACO	97.1332	6.4400	37.1650	17.7868	0.9260	0.9080
	DE	97.1310	6.3141	35.7726	17.1203	0.9311	0.9151
	GA	97.1552	6.3659	37.8764	18.1272	0.9234	0.9040
	GWO	98.5931	6.2628	37.8131	18.0969	0.9241	0.9047
	WOA	97.0195	6.3062	35.9302	17.1958	0.9305	0.9146
	EO	97.1762	6.4253	37.1236	17.7670	0.9261	0.9082
	AHA	97.0469	6.2457	36.0554	17.2557	0.9302	0.9132
	AO	97.0993	6.2991	33.7320	16.1438	0.9382	0.9268
	AOA	97.2478	6.2794	34.4280	16.4769	0.9358	0.9239
	IAOA	97.0016	6.7724	30.0854	14.3985	0.9497	0.9495
GF (Image1)	PSO	46.5828	3.3283	20.6930	14.0151	0.9754	0.9902
	ACO	46.5815	3.4044	20.7320	14.0415	0.9753	0.9901
	DE	46.5801	3.4072	20.7327	14.0419	0.9753	0.9900
	GA	46.5819	3.2946	20.7637	14.0630	0.9752	0.9899
	GWO	46.5800	3.4144	20.7345	14.0432	0.9753	0.9900
	WOA	46.5813	3.4255	20.7896	14.0805	0.9752	0.9898
	EO	46.5807	3.3927	20.7382	14.0457	0.9753	0.9900
	AHA	46.5810	3.3304	20.7052	14.0233	0.9754	0.9902
	AO	46.5822	3.4351	20.7735	14.0696	0.9752	0.9899
	AOA	46.5816	3.2625	20.6996	14.0196	0.9754	0.9902
	IAOA	46.5799	3.3223	20.6962	14.0172	0.9754	0.9902
GF (Image2)	PSO	51.6962	2.8638	33.1775	18.5635	0.9468	0.9597
	ACO	50.6891	2.8616	26.8770	15.0382	0.9644	0.9797
	DE	50.8264	2.8678	28.3232	15.8474	0.9607	0.9744
	GA	50.8378	2.8774	27.5282	15.4026	0.9627	0.9762
	GWO	50.8298	2.8615	29.3771	16.4371	0.9578	0.9724
	WOA	50.6099	2.8681	26.1553	14.6344	0.9663	0.9815
	EO	50.8209	2.8848	26.3923	14.7670	0.9657	0.9809
	AHA	50.9210	2.9319	30.1940	16.8942	0.9555	0.9673
	AO	50.7379	2.8645	27.9868	15.6592	0.9615	0.9753
	AOA	50.7471	2.8610	27.0687	15.1455	0.9640	0.9797
	IAOA	50.5770	2.8594	25.5686	14.3061	0.9677	0.9846

Fig. 15

The Convergence curves of different algorithms coast.

Fig. 16

The ANOVA test of different algorithms for coast.

Fig. 17

The radar map of different algorithms for coast.

Fig. 18

Visual assessment for marked region in coast dataset

4.5.2 Analyses of the experiment

The fusion results of 10 different algorithms are compared in the coastal scene. The results can be clearly obtained, in the fusion results of QuickBird image 1 it can be clearly seen that IAOA’s performance in UIQI metrics is on par with PSO and ACO, but all other metrics are better than the other algorithms, and the SAM metrics in the other imaging data map are not as good as the AHA but ahead of the other algorithms, and the other metrics are the optimal results. In image 1 of the GF detector, IAOA’s SAM, RASE, and RMSE performance metrics are not as good as other algorithms because of the complexity of the coastal scene. Among them, SAM is commonly used to measure the absolute difference in spectral angle between the fusion result and the reference vector. RASE is commonly used to indicate the average performance within a spectral band. RMSE is a commonly used metric to measure the degree of difference between the reconstructed image and the original image in order to assess the quality of the reconstructed image. All these metrics are effective in demonstrating the dissimilarity and completeness of image fusion. But in another imaging data map IAOA achieves the optimal results in all metrics.

In the coastal scenario, the complex details of the urban and water scenarios are integrated and belong to the integrated application scenario. A comprehensive evaluation of the algorithm will be more in-depth. Although some of the metrics in the GF Image1 dataset have not reached optimal values, this is somewhat related to the dataset itself and the size of the IAOA algorithm for the size of the building complexes in the neighboring sea cities. However, in most cases, the IAOA algorithm has outperformed most other algorithms. This proves the feasibility of the improved results.

All the algorithms are run independently for 30 times, and the convergence speed of IAOA is better than the other algorithms as shown in Fig. 15. It is thus known that the introduction of these improved operators enables the IAOA algorithm to better explore the solution space during the search process and avoid falling into local optima, which accelerates the convergence of the algorithm. The stability of IAOA can be obtained in Fig. 16. In short, the IAOA algorithm is ahead of most other algorithms in terms of overall performance by comparing the fusion of image data from coastal scenes, although it may be inferior to other algorithms in a few detailed metrics.

4.6 Running time of the experiment

This section quantitatively compares the runtimes of 11 different algorithms using four datasets with different topics and analyzes the balance between time and fusion effectiveness. Figure 19 illustrates the average running time of different algorithms. The results show that IAOA has a longer runtime than AOA, in all scenarios. However, IAOA outperforms AOA and leads most other algorithms in fusion performance metrics for all topics.

Fig. 19

Running time of different algorithms.

Analyzing the improved algorithms by computational complexity, it is found that the running time does not increase exponentially. It is observed that this runtime is acceptable in domains such as remote sensing image processing, and thus the improvement effect can be shown to be effective as long as the runtime permits.

5 Conclusion and future work

In image fusion problems, the commonly used fixed weights lead to local artifacts and local pseudo-details, while the existing methods perform poorly in terms of image sharpness, contrast and color accuracy. In addition, image fusion techniques are not limited to single modality but can also be used for multimodal data fusion. However, existing methods have limitations in preserving multimodal information and improving target detection performance. In practical applications, image fusion techniques need to be real-time and efficient. Traditional methods may have low processing speed due to the computation based on deterministic solutions. To address these issues, in this study, we use a combination of adaptive weight tuning band coefficients and edge preserving filters to improve spectral fidelity and improve the performance of the detail map fusion results by linearly combining MS images with PAN images. In addition, a meta-heuristic algorithm is used in this study to balance the fusion results with the computation time. With these improvements, image clarity, contrast, and color accuracy can be improved while preserving details and multimodal information, and a faster processing speed can be achieved.

Improvements are made to the original algorithm with respect to its convergence speed as well as search accuracy. By introducing the OBL and DRST operators, the IAOA algorithm does improve the search capability and avoids falling into local optimal solutions, thus speeding up the convergence speed. These improved techniques allow the algorithm to show better performance on several metrics. The OBL operator increases the diversity of the search by introducing competitors. These competitors play games with the current solution and improve the search capability by selecting a better solution as a new candidate solution, which effectively motivates the algorithm to better explore in the solution space. At the same time, we also introduce the DRST operator as a dynamic stochastic search operator, which introduces randomness in the search process and increases the diversity of the algorithm’s search. In this way, even if the algorithm falls into local optimums in some local regions, there is still a chance to jump out of these local optimums through random search and continue to explore better solutions, thus avoiding falling into local optimums and improving the global search capability of the algorithm. The introduction of these improved operators enables the IAOA algorithm to better explore the solution space during the search process and avoid falling into local optimal solutions, thus accelerating the convergence speed and improving the convergence accuracy of the algorithm.

After employing the least squares method to calculate the initial detail image, a refined image is obtained by utilizing a linear combination of gradients from the PAN and MS images. The weights of the gradients are compared using both the IAOA and classical algorithms. Experimental results are obtained by comparing the six metrics and convergence graphs are used to verify the effectiveness of the improvements made.

The IAOA algorithm is able to find the global optimal solution faster than other algorithms by improving the search characteristics and the ability to avoid falling into local optima, improving the performance of the algorithm on most metrics. However, this advantage also comes with additional runtime consumption. The next step of our work will aim to improve the AOA algorithm to achieve a balance between convergence ability and runtime. Meanwhile, we will apply the improved algorithm and image fusion method to real engineering problems to further verify its effectiveness.

The fused images are finally analyzed for comparison. The experimental results are discussed and their advantages and disadvantages are analyzed. The experimental results demonstrate the feasibility and effectiveness of the proposed improved algorithm. In the future, IAOA will be further investigated to overcome the shortcomings and applied to research problems in last mile distribution, thermoelectric scheduling and path planning problems.

Fundings

This work was supported by the National Natural Science Foundation of China under Grant Nos. U21A20464, 62066005.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Author contributions

XM: Investigation, experiment, writing-draft; QL: Writing—review and editing. YZ: Supervision, Algorithm design, writing-review.

Ethical and informed consent for data used

This article does not contain any studies with human participants or animals performed by any of the authors.

Data availability and access

The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding authors.

Footnotes

Appendix

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