To aid with understanding of the notation, Table 2 contains a list of notation and the respective definitions.

Entropy is generally associated with the ratio of instability found within a system. According to Shannon Entropy (Lin et al., 2020), entropy is the ratio of chaos and relates to the information weight of the system. Shannon stated that when a physical system is taken apart as two free subsystems A and B, then the entropy value can be represented as

With consideration to Shannon's theory, a limited entropy concept was introduced by Vigelis et al. (2020) and expressed as

where T represents the system potentials and q represents the entropic index (Vigelis et al., 2020). Equation 2 will be equivalent to Shannon entropy when q → 1.

Tsallis cross-entropy (Vigelis et al., 2020), however, will be applied in this work. If the set of all probability that is on I_n = {1, …, n} by n={(p1,…,pn): ∑i=1npi=1 and pi≥0 for all i} for the two probability distributions p=(pi) and q= (qi) in Δn0. Also, if the subsystems A and B are two probability distributions, A = {a₁, a₂, …, a_N} and B = {b₁, b₂, …, b_N}, then Tsallis cross-entropy can be expressed as

Tsallis cross-entropy is selected as the objective function due to its non-extensive nature, which allows for a more flexible and sensitive measurement of information discrepancy than standard Shannon entropy. This sensitivity is crucial for capturing subtle diagnostic details and non-linear correlations in complex multimodal medical images.

Non-subsampled shearlet transform (NSST) is popular alongside some other tools used for multi-resolution processing, such as wavelet curvelet and non-subsampled contourlet transform, because of NSST's applications in the decomposition of signals or images into single low-frequency subbands and many high-frequency subbands (Mishra and Dhabal, 2020). However, NSST indicates some peculiarity over other analytical tools by involving a shear matrix which enables direction filtering.

Two source images A and B, as represented in this proposed approach, are fused to obtain Q, the final fused image. A specific integration of fusion is represented as Q = (A, B). The fusion process in low-frequency subbands can be described as follows.

The proposed model integrates the low-frequency sub-images using the averaging method (Hu et al., 2018) to reach the fused low-frequency coefficients. A fused low-frequency sub-image result is obtained as

where LFIGA and LFIGB are the two sub-images representing the two low-frequency sub-images from two modalities, and g = (1, …, G) denotes the scale of representation found in the transformed space at direction h(i, j) in alignment with a spatial location (i, j).

The multi-objective optimization minimization problem can be described mathematically as shown in the following equation (Zhu and Han, 2021):

where x = (x₁, x₂, …, x_D) is a D-dimensional decision variable, m is the number of objective functions in the multi-objective optimization problem, f_m(x) is the mth objective optimization problem, g_i(x) is the ith inequality constraint, p is the number of inequality constraints, h_j(x) is the jth equation constraint, and l is the number of equation constraints.

MMIF is treated as a complex multi-objective optimization problem (MOP) requiring simultaneous trade-off between conflicting goals. We will explicitly define the MOP objective function vector F(x) to maximize fusion quality while minimizing processing time: Optimize F(x) = [max[f₁(x)), min[f₂(x))]. Here, f₁(x) is the fusion quality maximized using Tsallis cross-entropy (as described in Section 4.1.1), and f₂(x) is the overall fusion processing time (T_run), the minimization of which validates the method's applicability for real-time surgical procedures.

Definitions 1–4 encompass the concept of multi-objective optimization (Han et al., 2021).

Definition 1 (Pareto dominance). The two solutions in the feasible region X_Ω are represented as x₁, x₂; x₁ can be seen as the solution that dominates x₂, and this relationship is mathematically denoted by x₁ > x₂ if and only if

In the context of multi-objective optimization, Pareto dominance is a rule used to compare two different solutions to determine which is superior when multiple goals are being balanced (such as image contrast vs. detail preservation).

A solution is said to dominate another if it meets two specific criteria:

Essentially, if a solution is “dominant,” it means you cannot improve it further in one area without making it worse in another. The goal of the FOMODPSO algorithm is to find a set of these “non-dominated” solutions, which together form what is known as the Pareto front.

Definition 2 (Pareto optimal). The solution x^* ∈ Ω is applied as the Pareto optimal of Equation 1 if and only if

Definition 3 (Pareto optimal set). The Pareto optimal set is put together by solutions that are finally obtained in the MOP, where each meets the conditions of Definition 2. It is defined as

Definition 4 (Pareto front). The region formed by the objective function values of all Pareto optimal solutions is called the Pareto front surface, which is defined as

The proposed FOMODPSO algorithm optimizes a multi-objective function vector, F(x), which is designed to balance image quality and efficiency. The two objectives are (1) maximize f₁: the Tsallis cross-entropy, which serves as the primary metric for fusion quality and information transfer, and (2) minimize f₂: the overall fusion processing time (T_run), ensuring the method's suitability for real-time clinical applications.

The PSO algorithm is generally given as Wu et al. (2021):

where n is the moving particle, xtn is the position of the particle, vtn is the velocity of the particle, xtn is the local best. ntn is the neighborhood best, gtn is the global best, ρ1, ρ2, and ρ3 assign weights to the global best, local best, and neighborhood best, respectively, when determining the new velocity, and γ1, γ2, and γ3 are random vectors with each component generally a uniform random number between 0 and 1.

DPSO is different from PSO only in the fact that some simple rules are followed to delete a swarm, delete particles, and spawn a new swarm and a new particle.

After the deletion of the particle, instead of being zero, the counter is reset to a value approaching the threshold number, according to Ahilan et al. (2019):

where N_kill is the number of particles deleted from the swarm over a period in which there was no improvement in fitness.

Fractional calculus (FC) is an essential mathematical tool in the applied sciences (Sabatier et al., 2007). FC has been influential in increasing the performance of several algorithms applied in modeling, curve-fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability, and robustness (Couceiro et al., 2012). FC has been applied to increase the convergence rate of PSO in some works, such as Pires et al. (2010) and Couceiro et al. (2012), where the convergence rate was improved for the velocity of PSO and DPSO, respectively. Fractional-order Darwinian particle swarm optimization (FODPSO) was applied in Ahilan et al. (2019) for multilevel thresholding of medical images, and the results obtained showed that the fractional coefficient gives the advantage of a greater level of exploration, thereby giving rise to the global solution of the algorithm. FODPSO was also applied in Waseem et al. (2020), where FODPSO was hybridized with a neural network to obtain better solutions for ordinary differential equations, and high-performance results were obtained.

The FC derived from the Grünwald–Letnikof definition based on the concept of fractional definition with fractional coefficient α ε c of a general signal x(t) is given by:

where γ is the gamma function.

The discrete-time implementation of Equation 12 is

where T represents the sampling period and r represents the truncation order.

The equation of fractional-order particle swarm optimization (FOPSO) is expressed as

FODPSO offers an O(r) memory need. For r = 4 the differential derivative is given as

where α is the FC and DPSO is an exceptional case of FODPSO with α = 1.

The FC operator is introduced into the DPSO velocity update formulas (Equation 15) to provide the particle velocity with a crucial “memory effect,” which fundamentally alters the algorithm's search behavior. This enhanced capability serves two functions: it significantly increases exploration, allowing the algorithm to escape common local optima and avoid premature convergence prevalent in image fusion tasks, and mathematically, it is proven to increase the fundamental convergence rate of the optimization process. This increased speed is the direct mechanism that ensures the high computational efficiency required for real-time application.

To clarify the methodological novelty, the proposed FOMODPSO represents a substantive algorithmic evolution from standard PSO variants. While MOPSO handles multiple objectives, it lacks the multi-swarm diversity of Darwinian principles, often leading to premature convergence. Conversely, DPSO excels at escaping local optima through natural selection but is traditionally limited to single-objective tasks. MODPSO bridges these by applying Pareto dominance to the Darwinian swarm-deletion logic. The final enhancement, FOMODPSO, introduces fractional-order velocity updates to add non-local memory, significantly accelerating convergence speed beyond integer-order methods. Table 3 summarizes these distinctions.

The multichannel PCNN (m-PCNN) is well known to be able to effectively manage different types of medical images (Zhao et al., 2014). The simplified m-PCNN is usually referred to as a three-dimensional PCNN because it has only three parameters (Raha et al., 2020). In this paper, we actualize the efficient optimization of parameters in the m-PCNN model. This is because it is essential to adjust the parameters to align with the input images for more accurate image fusion results. The simplified m-PCNN model can be seen in Figure 1.

The system utilizes a simplified m-PCNN, which is a zero-learning architecture that eliminates the need for large training datasets or traditional loss functions. Instead of manual tuning or backpropagation, the network's performance depends on the selection of optimal parameters (β, Θ, α_θ, V_θ) through the proposed FOMODPSO algorithm. This optimization process treats the Tsallis cross-entropy between source and fused images as the fitness function, ensuring that the network's firing activity, which drives the fusion of high-frequency subbands is maximized for information content. The architecture is applied to 14 groups of CT/MRI images (256 × 256 pixels), which are co-registered and normalized as a preprocessing step to ensure diagnostic accuracy.

The integration of these specific techniques is motivated by their complementary strengths: FOMODPSO is utilized to provide global optimization with fast convergence via fractional-order updates; Tsallis cross-entropy is selected as a robust objective function to measure multisource information transfer more effectively than standard entropy; and the PCNN is employed as the fusion rule for its biological pulse-synchronization property, which is superior at preserving fine spatial details and edges.

To denote the point of the input images by similar corresponding pixels, the source images are required to be co-registered for the proposed method to give efficient results. The implementation of the multimodal image fusion scheme is described in the following steps and Algorithm 1 shows the framework of FOMODPSO.

Figure 2 shows the schematic diagram of the steps of the proposed image fusion framework.

To investigate the accuracy and robustness of the proposed FOMODPSO–PCNN method, the experimental setup utilized 14 groups of public benchmark medical image pairs (28 images total), consisting of CT and magnetic resonance imaging (MRI) modalities with a resolution of 256 × 256 pixels. These images (Johnson and Becker, 1999) were sourced from the Harvard Medical Image Database (https://dataverse.harvard.edu/). For statistical validity, the multi-objective search quality (inverted generational distance (IGD) and HV) was evaluated based on 31 independent experimental runs, reporting the mean and standard deviation for each test instance. Additionally, a three-fold cross-validation protocol was implemented for the diagnostic performance evaluation (Table 4), ensuring that the reported accuracy (90.7%) and sensitivity results are consistent across multiple subsets of the data.

As seen in Figure 3, images A are the source CT images and images B are the source MRI images, which are to be fused. In total, we used 28 source images to conduct our experiments; 14 from each modality. The images have 256 × 256 pixel resolutions. All the images in the dataset are correctly preregistered. The simulation software is adopted by Matlab R2018a version 9.4.0.813654 64-bit for the search of the optimal values of the test functions, and the hardware used is 3.20 GHz, 4 GB RAM, and Windows 7.

The common parameters of the FOMODPSO, the population size and maximum iteration, are kept randomly and conveniently at 20 and 200, respectively; the stagnancy threshold is set randomly and conveniently at 15; we set the α parameter to 0.6 as it has been proven to give better results when applied (Chakraborty and Verma, 2021).

To evaluate the performance of the proposed method denoted as the FOMODPSO + PCNN MMIF method, some comparative experiments are conducted with eight image fusion algorithms on five multi-objective problems (test instances) f₁ − f₅ (Gao et al., 2014; Couceiro et al., 2012). The eight image fusion algorithms are QPSO+PCNN (Tian et al., 2016), MOPSO+PCNN (Wang et al., 2016), PSO+PCNN (Kumar et al., 2021), MFOA+PCNN (Tang et al., 2019), WOA + PCNN (Raha et al., 2020), PSO-TV+PCNN (Nie et al., 2021), PSO-DE+PCNN (Liu R. et al., 2022), and DE+PCNN (Das et al., 2022). QPSO+PCNN is an MMIF method that uses QPSO for the optimization of the PCNN to obtain good-performance fused images. MOPSO+PCNN is an MMIF method that applies MOPSO in the optimization of the PCNN to obtain fused images of very good visual quality. PSO+PCNN is an MMIF method that applies hybrid PSO for the optimization of the PCNN to obtain high-performance fused image results. MFOA+PCNN is an MMIF method that applies an MFOA for the optimization of the PCNN to obtain fused image results of high quality. WOA+PCNN is an MMIF method that optimizes PCNN using the WOA for fused image results of well-preserved image details. PSO-TV+PCNN is an MMIF approach that applies PSO and total vibration (TV) for the optimization of the PCNN to obtain fused image results of high visual quality. PSO-DE+PCNN is an MMIF method that optimizes PCNN using PSO and standard DE algorithm to obtain fused images of very good fusion effect. DE+PCNN is an MMIF method that applies the DE for the optimization of the PCNN to obtain fused images of both good visual quality and superior quantitative evaluation results.

Multi-objective optimization vs. single-objective optimization indicates how much more the complexity of multi-objective optimization is. Another difference is in the non-uniqueness of the solution of multi-objective optimization, which is a solution set. Table 5 shows the optimal solution set. The most suitable visual effect corresponding to the image fusion results is selected. They are the final optimal PCNN parameters: the connection weight W = [0.5, 1.0, 0.5;1.0, 0, 1.0;0.5, 1.0, 0.5], N = 100; β = 0.9950; θ = 0.1233; V_θ = 1.8119; α_θ = 0.1482.

The optimization parameters can be summarized as

To evaluate the computational efficiency, we compute and list the average time consumption on all image groups for different methods, as shown in Table 1. Our method takes the shortest time, while MFOA + PCNN has a high time consumption. The reason our method has low time consumption is because of an increased rate of PCNN parameter optimization. Increasing the convergence rate of MODPSO using the FC operator, and applying the improved optimization algorithm to Tsallis cross-entropy maximization, and the optimized entropy is used as input to the PCNN network causing the reduction of the average time consumption of our method in comparison to other comparative methods.

This computational efficiency confirms that our method can be used to carry out MMIF within a very short time, and this is the main criterion for effective real-time image fusion applications (Fang et al., 2023).