A novel approach using the local energy function and its variations for medical image fusion

ABSTRACT

Medical image fusion plays a pivotal role in facilitating clinical diagnosis. However, the quality of input medical images may be marred by noise, low contrast, and lack of sharpness, presenting numerous challenges for medical image synthesis algorithms. Additionally, several fusion rules may degrade the brightness and contrast of the fused image. To this end, this paper presents a novel image synthesis approach to tackle the aforementioned issues. First, the input images undergo pre-processing to enhance their quality. Subsequently, we introduce the three-layer image decomposition (TLID) technique, which decomposes an image into three distinct layers: the base layer ($L_B$), the small-scale structure layer ($L_{SS}$), and the large-scale structure layer ($L_{LS}$). Next, we synthesize the base layers utilizing adaptive rules based on the Marine predators algorithm (MPA), ensuring that the output image is not degraded. Finally, we propose an efficient synthesis method for $L_{SS}$ and $L_{LS}$ layers, based on combining the local energy function with its variations. This fusion technique preserves the intricate details present in the original image. We evaluated our approach on 156 medical images using six evaluation metrics and compared it with seven state-of-the-art image synthesis techniques. Our results demonstrate that our method successfully generates high-quality output images and preserves detailed information throughout the image synthesis process.

KEYWORDS:

• Marine predators algorithm (MPA)
• three-layer image decomposition (TLID)
• rolling guidance filter (RGF)
• weighted mean curvature filter (WMCF)

1. Introduction

Medical imaging is an essential component of clinical applications today. With the advent of advanced image acquisition devices, a variety of multimodal medical images can now be obtained. Human organ structures are highly complex, and lesions cannot be adequately described using a single type of multimodal medical imaging. Computed Tomography (CT), magnetic resonance imaging (MRI), positron emission tomography (PET), and single photon emission computed tomography (SPECT) are among the commonly used types of multimodal medical imaging in image fusion. SPECT imaging is particularly useful for diagnosing vascular conditions and detecting tumors, as it offers insights into metabolic activity. In contrast, MRI is renowned for its ability to capture high-resolution images of soft tissues. To provide clinicians with comprehensive diagnostic information, it is crucial to synthesize the necessary information from multimodal medical images.

Traditional approaches to image fusion involve a sequence of three fundamental steps: image decomposition, synthesis of the decomposed components, and transforming the resulting fused image back to the original domain. Presently, various image decomposition methods exist. The first group of such methods comprises multi-scale decomposition (MSD)-based techniques, such as Discrete wavelet transform (DWT), Stationary wavelet transform (SWT), and Laplacian pyramid (LP). For example, Wang et al. [1] utilized the DWT method to decompose input images before fusing them, while Dinh [2] applied the SWT method to obtain high and low-frequency components of input images. Fu et al. [3] employed the LP method to decompose input images. However, MSD-based methods have limited capacity to capture directional information, thus resulting in suboptimal image representation. The second group of methods is multi-scale geometric analysis (MGA)-based approaches, which overcome the limitations of MSD-based methods. These methods include curvelet transform, contourlet transform (CT), non-subsampled contourlet transform (NSCT), shearlet transform (ST), and non-subsampled shearlet transform (NSST). They provide complete information about both phase and direction. For example, Wang et al. [4] employed NSCT to decompose input images, while Gao et al. [5] utilized NSST to transform input images. Several other image synthesis studies based on MGA methods exist, including [6], [7], and [8]. However, these methods have high computational complexity. The third group of image transformation methods comprises base-detail decomposition (BDD) based techniques, which exhibit benefits over MSD-based methods in terms of both computational efficiency and fusion performance. BDD-based methods enable the decomposition of an input image into base and detail layers by employing filters. For instance, Dinh [9] proposed a novel image fusion approach based on two-scale image decomposition. Li et al. [10] applied three-layer decomposition to fuse medical images. The fourth group of image transformation methods consists of Sparse representation (SR)-based methods, which have proven to be highly effective in various image synthesis studies. Wang et al. [11] presented a new multi-focus image fusion approach based on multi-scale SR. Jie et al. [12] integrated the SR with adaptive energy-choosing schemes to fuse Tri-modal medical images. Total-variational decomposition (TVD) has also been leveraged in several studies for image synthesis. Liu et al. [13] proposed a novel image synthesis method based on the Robust spiking cortical model and TVD. Additionally, Liu et al. [14] utilized spectral total variation and local structural patch measurement to construct an image composite model.

Traditional methods of image fusion have limitations that still persist. One limitation is that these methods are bound to use the same image decomposition method to acquire features and ensure their combinability in the subsequent phase. This results in the disregard of source image differences, which ultimately leads to poor expressiveness of the extracted features. Additionally, traditional methods acquire few and insufficiently diverse features from image decomposition, which still poses limitations on their synthesis performance. Deep learning has proven to be a powerful tool for addressing image processing problems, including image enhancement [15], image denoising [16], and image fusion. In particular, deep learning-based approaches have been instrumental in overcoming the limitations of traditional image fusion methods. Firstly, these approaches can conduct differentiated feature extraction by utilizing diverse network branches. Secondly, the feature fusion strategy can be effectively learned by designing appropriate loss functions. As a result, deep learning-based methods have significantly contributed to image fusion. For instance, Hou et al. [17] proposed a novel method for synthesizing CT and MRI images by combining convolutional neural networks with a dual-channel spiking cortical model. Ding et al. [18] presented an image synthesis method based on Dual-Branch CNNs in the NSST domain. Additionally, Ding et al. [19] employed Siamese networks in conjunction with the multi-scale local extrema scheme. Wang et al. [1] combined Convolutional neural networks (CNN) with Discrete wavelet transform (DWT) to fuse Multi-focus images. Kaur et al. [20] proposed a medical image fusion based on deep belief networks (DBN). Other studies have also employed deep learning techniques to synthesize images, including [21–25].

In recent times, meta-heuristic optimization-based image synthesis methods have been proven effective. Optimization algorithms offer adaptive rules for the synthesis process, leading to an enhanced quality of the output composited image. For example, Dinh [26] proposed a new method that uses the Equilibrium optimizer algorithm (EOA) to synthesize medical images. Shilpa et al. [27] improved the JAYA optimization algorithm and applied it to fuse medical images in the NSST domain. Particle swarm optimization (PSO) was applied by Shehanaz et al. [28] to fuse high-frequency components in the DWT domain. Xu et al. [29] modified the shark smell optimization (SSO) algorithm and combined it with the World Cup Optimization (WCO) algorithm to synthesize low-frequency components in the DWT domain. Dinh [30] utilized the Chameleon swarm algorithm to synthesize the base components, ensuring the preservation of composite image quality. Further optimization techniques for producing medical images are discussed in the research papers [9,31–34].

Based on our observations, the low efficiency of image fusion can be attributed to three main factors. Firstly, low-quality input images are a common issue, characterized by low brightness and contrast, noise, and lack of sharpness, which negatively impacts the quality of the composite image. Secondly, the average rule, a widely adopted method in several studies [35–37], has the advantage of simplicity and low computational complexity in synthesizing low-frequency components. However, its disadvantage is that it leads to a degradation in the brightness and contrast of the output composite image. Thirdly, the synthesis rules for high-frequency components have not been designed efficiently to capture full details from the input images, further impacting the quality of the composite image. In light of the aforementioned limitations, we propose the following approaches to overcome them. Firstly, we suggest enhancing the quality of input images using the Brighten low-light image (BLLI) method [38]. Secondly, we propose using the MPA optimization algorithm to generate adaptive rules for low-frequency components, thereby ensuring optimal brightness and contrast in the output image. Thirdly, we introduce a fusion rule based on the local energy function and its variations to generate an efficient fusion rule for detail layers. This is because the local energy function and its variations have proven effective in many image synthesis studies. For instance, Amini et al. [39] combined the local energy function with local variance fusion rules to synthesize MRI and PET images. Several variations of the local energy function have been proposed in recent years for medical image synthesis. Compass operators, such as Kirsch and Prewitt, have been combined with a local energy function to synthesize medical images [9, 26]. The structure tensor has been applied in several studies on image fusion [40–42]. Recently, Li et al. [43] combined the structure tensor salient detection operator with a local energy function to construct a synthesis rule for detail components.

In this work, we present a novel method to tackle the aforementioned limitations. Our principal contributions are summarized as follows:

• Firstly, The present study proposes a novel three-layer image composition (TLID) method for image decomposition into three layers, namely, the base layer ($L_B$), small-scale structure layer ($L_{SS}$), and large-scale structure layer ($L_{LS}$). This approach utilizes two filters, namely, the Rolling Guidance Filter (RGF) and the Weighted Median curvature filter (WMCF), to construct the TLID method.

• Secondly, we propose an efficient fusion method to merge the $L_{SS}$ and $L_{LS}$ layers, which involves the combination of the local energy function with its variations.

• Thirdly, in order to mitigate the loss of brightness and contrast during the image compositing process, we propose a novel method for fusing the base layers utilizing adaptive parameters.

The paper is structured as follows: Section 2 provides an overview of background methods including Rolling Guidance Filter (RGF), Weighted mean curvature filter (WMCF), Structure Tensor, Local Energy and its variations, and MPA algorithm. Section 3 introduces the TLID method, which combines variations of the local energy function (FR-CVLEF) to form a new fusion rule, as well as our proposed image fusion model. Section 4 contains experimental data and settings, results, and evaluations. Finally, Section 5 concludes the paper and presents future work.

2. Background

2.1. Rolling guidance filter (RGF)

The Rolling Guidance Filter (RGF) [44] is a digital image filtering technique that has gained popularity in recent years due to its ability to effectively denoise images while preserving their sharpness and details. RGF is a non-local means filter that is designed to exploit the global structures and textures of an image, which allows it to effectively remove noise without blurring the image or losing important information. The filter comprises of two primary stages: removing small structures and recovering edges.

Step 1: The small structure is removed by the GF.

The symbols I and $I_G$ refer to the input and output images obtained following filtration via the Gaussian filter, respectively. The computation of $I_G$ is performed in accordance with Equation (1(1) $I_G\left(u\right)=\frac{1}{H_u}\sum _{v\in N\left(u\right)}\exp (-\frac{{\Vert u-v\Vert }^2}{2{\sigma }_g^2}).I\left(v\right)$(1) ). (1) $I_G\left(u\right)=\frac{1}{H_u}\sum _{v\in N\left(u\right)}\exp (-\frac{{\Vert u-v\Vert }^2}{2{\sigma }_g^2}).I\left(v\right)$(1) where

• u and v are the position.

• ${\sigma }_g$ is the standard deviation of GF.

• $N\left(u\right)$ is the set of neighboring pixels whose center is u.

• $H_u$ is calculated according to Equation (2(2) $H_u=\sum _{v\in N\left(u\right)}\exp (-\frac{{\Vert u-v\Vert }^2}{2{\sigma }_g^2}).$(2) ).

(2) $H_u=\sum _{v\in N\left(u\right)}\exp (-\frac{{\Vert u-v\Vert }^2}{2{\sigma }_g^2}).$(2) Step 2: Edge is recovered by the guided filter.

The symbol $E^1$ denotes the GF output from the initial step. $K^{t+1}$ represents the outcome achieved at the t-th iteration. Computation of $K^{t+1}$ is executed in accordance with Equation (3(3) $K^{t+1}\left(u\right)=\frac{1}{H_u}\sum _{v\in N\left(u\right)}\exp (-\frac{{\Vert u-v\Vert }^2}{2{\sigma }_g^2}-\frac{{\Vert K(u{)}^t-K(v{)}^t\Vert }^2}{2{\sigma }_r^2}).I\left(v\right)$(3) ). (3) $K^{t+1}\left(u\right)=\frac{1}{H_u}\sum _{v\in N\left(u\right)}\exp (-\frac{{\Vert u-v\Vert }^2}{2{\sigma }_g^2}-\frac{{\Vert K(u{)}^t-K(v{)}^t\Vert }^2}{2{\sigma }_r^2}).I\left(v\right)$(3) where

• $H_u$ is calculated according to Equation (4(4) $H_u=\sum _{v\in N\left(u\right)}\exp (-\frac{{\Vert u-v\Vert }^2}{2{\sigma }_g^2}-\frac{{\Vert K(u{)}^t-K(v{)}^t\Vert }^2}{2{\sigma }_r^2}).$(4) ).

• ${\sigma }_r$ is the standard deviation of the Guided Filter.

(4) $H_u=\sum _{v\in N\left(u\right)}\exp (-\frac{{\Vert u-v\Vert }^2}{2{\sigma }_g^2}-\frac{{\Vert K(u{)}^t-K(v{)}^t\Vert }^2}{2{\sigma }_r^2}).$(4) The output images obtained from the RGF with several loops are illustrated in Figure 1.

Figure 1. RGF output images with some loops.

2.2. Weighted mean curvature filter (WMCF)

The WMCF is a technique used in image processing and computer vision. It was introduced by Gong et al. [45], and its advantages are scale invariance, sampling invariance, and contrast invariance. Several applications of this filter can be mentioned as multi-spectral and panchromatic image fusion [46], as well as medical image fusion [47]. We denote an input image as R, and define the symbol $m_i$ ($i=\overline{1,8}$) as the mask matrices of the WMFC filter, as illustrated in Figure 2.

Figure 2. Masks of the MWCF.

The WMCF can be calculated in two steps:

Step 1: Determine the distance $k_i$ as in Equation (5(5) $k_i=m_i\ast R,i=\overline{1,8}$(5) ). (5) $k_i=m_i\ast R,i=\overline{1,8}$(5)

Step 2: This filter is defined as Equation (6(6) $F\left(R\right)=k_m$(6) ). (6) $F\left(R\right)=k_m$(6) where $m=i\arg min\left(\right|k_i|i=\overline{1,8})$

Figure 3 illustrates the results of using the WMCF.

Figure 3. Some images obtained from the WMCF filter.

2.3. Structure tensor (ST)

The computation of the ST is based on the gradient of the gray-scale image, and finds wide-ranging applications such as hyperspectral and panchromatic image fusion [48], image denoising [49], and medical image fusion [43]. Let I be an input image. The ST can be calculated using Equation (7(7) $ST=\left[\begin{array}{cc}{\displaystyle \sum _w{K}_i^2}& {\displaystyle \sum _w{K}_i{K}_j}\\ {\displaystyle \sum _w{K}_i{K}_j}& {\displaystyle \sum _w{K}_j^2}\end{array}\right]$(7) ). (7) $ST=\left[\begin{array}{cc}{\displaystyle \sum _w{K}_i^2}& {\displaystyle \sum _w{K}_i{K}_j}\\ {\displaystyle \sum _w{K}_i{K}_j}& {\displaystyle \sum _w{K}_j^2}\end{array}\right]$(7) where,

• w is a local window.

• $K_i$ and $K_j$ are the gradients in the i-direction and j-direction, respectively.

Furthermore, the operator for detecting salient features using the structure tensor, referred to as the Structure Tensor Salient Detection Operator (STSDO) [50], is derived from the eigenvalues ($t_1$ and $t_2$) as per Equation (8(8) $S=\sqrt{{(t_1+t_2)}^2+0.5{(t_1-t_2)}^2}$(8) ). (8) $S=\sqrt{{(t_1+t_2)}^2+0.5{(t_1-t_2)}^2}$(8) Where $t_1$ and $t_2$ are determined according to Equations (9(9) $t_1=\frac{1}{2}\left(\begin{array}{rl}& \sum _w{K}_i^2+\sum _w{K}_j^2\\ & +\sqrt{{(\sum _w{K}_i^2-\sum _w{K}_j^2)}^2+4{\left(\sum _w{K}_i{K}_j\right)}^2}\end{array}\right)$(9) ) and (10(10) $t_2=\frac{1}{2}\left(\begin{array}{rl}& \sum _w{K}_i^2+\sum _w{K}_j^2\\ & -\sqrt{{(\sum _w{K}_i^2-\sum _w{K}_j^2)}^2+4{\left(\sum _w{K}_i{K}_j\right)}^2}\end{array}\right)$(10) ). (9) $t_1=\frac{1}{2}\left(\begin{array}{rl}& \sum _w{K}_i^2+\sum _w{K}_j^2\\ & +\sqrt{{(\sum _w{K}_i^2-\sum _w{K}_j^2)}^2+4{\left(\sum _w{K}_i{K}_j\right)}^2}\end{array}\right)$(9) (10) $t_2=\frac{1}{2}\left(\begin{array}{rl}& \sum _w{K}_i^2+\sum _w{K}_j^2\\ & -\sqrt{{(\sum _w{K}_i^2-\sum _w{K}_j^2)}^2+4{\left(\sum _w{K}_i{K}_j\right)}^2}\end{array}\right)$(10) Figure 4 illustrates the image obtained by applying the STSD operator.

Figure 4. The image obtained by the STSDO.

2.4. Local energy function and its variations

2.4.1. Local energy function

The Local Energy Function (LEF) has found numerous applications in studies related to image fusion [39, 51]. The computation of $LEF(i,j)$ is performed using Equation (11(11) $LEF(i,j)=\sum _{u=0}^{k-1}\sum _{v=0}^{k-1}{W}_{LE}(u,v)I^2(i+u,j+v)$(11) ). (11) $LEF(i,j)=\sum _{u=0}^{k-1}\sum _{v=0}^{k-1}{W}_{LE}(u,v)I^2(i+u,j+v)$(11)

• $W_{LE}$ is a unit window of size $k\times k$.

• I is the input image.

2.4.2. Local energy function using the Prewitt compass operator (LEF_PCO)

Dinh [26] proposed the LEF-PCO method for developing the synthesis rule for detail components. We denote an input image as I, and refer to the k-th mask of the Prewitt compass operator as $W_{PCO}^k$, with further clarification provided in Figure 5. The computation of LEF-PCO is carried out using Equation (12(12) $LEF\mathrm{\_}PC{O}^k(i,j)=\sum _{u=0}^{m-1}\sum _{v=0}^{m-1}{W}_{PCO}^k(u,v)I^2(i+u,j+v)$(12) ). (12) $LEF\mathrm{\_}PC{O}^k(i,j)=\sum _{u=0}^{m-1}\sum _{v=0}^{m-1}{W}_{PCO}^k(u,v)I^2(i+u,j+v)$(12)

Figure 5. An illustration of the masks employed by the Prewitt compass operator.

2.4.3. Local energy function combined with the structure tensor saliency

Dinh [52] introduced the Local Energy Function combined with the Structure Tensor Salient Detection Operator (LEF_STSDO) for generating fusion rules for detailed components.

The LEF_STSDO is defined as Equation (13(13) $LEF\mathrm{\_}STSDO=W_{STSDO}\odot LEF$(13) ). (13) $LEF\mathrm{\_}STSDO=W_{STSDO}\odot LEF$(13) where,

• $W_{STSDO}$ is defined as the STSDO in Equation (8(8) $S=\sqrt{{(t_1+t_2)}^2+0.5{(t_1-t_2)}^2}$(8) ).

• ⊙ represents entry-wise multiplication.

• LEF represents the local energy function and is defined by Equation (11(11) $LEF(i,j)=\sum _{u=0}^{k-1}\sum _{v=0}^{k-1}{W}_{LE}(u,v)I^2(i+u,j+v)$(11) ).

2.5. MPA algorithm

The MPA algorithm was initially proposed by Faramarzi et al. [53]. This algorithm has exhibited superior optimization performance compared to other algorithms such as Genetic algorithm (GA), Gravitational Search Algorithm (GSA) [54], and Salp Swarm Algorithm (SSA) [55]. The MPA algorithm has found numerous applications in various domains such as structural damage detection [56], image segmentation [57], and medical image fusion [52, 58]. The MPA algorithm can be outlined in three main steps as follows:

Step 1: During the first third of the loop, the prey exhibits faster movement than the predator. The calculation of stepsize ($\overrightarrow{X}i$) and prey ($\overrightarrow{Y}i$) is performed using Equations (14(14) ${\overrightarrow{X}}_i={\overrightarrow{R}}_B\otimes ({\overrightarrow{T}}_i-{\overrightarrow{R}}_B\otimes {\overrightarrow{Y}}_i)\mathrm{i}=\overline{1,\mathrm{n}}$(14) ) and (15(15) ${\overrightarrow{Y}}_i={\overrightarrow{Y}}_i+h.\overrightarrow{R}\otimes {\overrightarrow{X}}_i$(15) ). (14) ${\overrightarrow{X}}_i={\overrightarrow{R}}_B\otimes ({\overrightarrow{T}}_i-{\overrightarrow{R}}_B\otimes {\overrightarrow{Y}}_i)\mathrm{i}=\overline{1,\mathrm{n}}$(14) (15) ${\overrightarrow{Y}}_i={\overrightarrow{Y}}_i+h.\overrightarrow{R}\otimes {\overrightarrow{X}}_i$(15) Where

• $\overrightarrow{R}$ in [0,1].

• h = 0.5.

• ⊗ is entry-wise multiplication.

• ${\overrightarrow{R}}_B$ is selected randomly from a Brownian motion distribution.

• $\overrightarrow{T}$ holds the fitness solution.

Step 2: During the subsequent third of the loop, updates to $\overrightarrow{X}i$ and $\overrightarrow{Y}i$ are made using Equations (16(16) ${\overrightarrow{X}}_i={\overrightarrow{R}}_L\otimes ({\overrightarrow{T}}_i-{\overrightarrow{R}}_L\otimes {\overrightarrow{Y}}_i);i=\overline{1,n/2}$(16) ), (17(17) ${\overrightarrow{Y}}_i={\overrightarrow{Y}}_i+h.\overrightarrow{R}\otimes {\overrightarrow{X}}_i$(17) ), (18(18) ${\overrightarrow{X}}_i={\overrightarrow{R}}_B\otimes ({\overrightarrow{R}}_B\otimes {\overrightarrow{T}}_i-{\overrightarrow{Y}}_i);i=\overline{n/2,n}$(18) ), and (19(19) ${\overrightarrow{Y}}_i={\overrightarrow{T}}_i+h.a\otimes {\overrightarrow{X}}_i;$(19) ).

For the first half of the population: (16) ${\overrightarrow{X}}_i={\overrightarrow{R}}_L\otimes ({\overrightarrow{T}}_i-{\overrightarrow{R}}_L\otimes {\overrightarrow{Y}}_i);i=\overline{1,n/2}$(16) (17) ${\overrightarrow{Y}}_i={\overrightarrow{Y}}_i+h.\overrightarrow{R}\otimes {\overrightarrow{X}}_i$(17) For the second half of the population: (18) ${\overrightarrow{X}}_i={\overrightarrow{R}}_B\otimes ({\overrightarrow{R}}_B\otimes {\overrightarrow{T}}_i-{\overrightarrow{Y}}_i);i=\overline{n/2,n}$(18) (19) ${\overrightarrow{Y}}_i={\overrightarrow{T}}_i+h.a\otimes {\overrightarrow{X}}_i;$(19) where

• $a=(1-\frac{l}{l_{max}}{)}^{\frac{2\ast l}{l_{max}}}$.

• $\overrightarrow{R_L}$ is generated from the $L\acute{e}vy$ distribution.

Step 3: Updates to $\overrightarrow{X}i$ and $\overrightarrow{Y}i$ are made during the final third of the loop using Equations (20(20) ${\overrightarrow{X}}_i=R_L\otimes (R_L\otimes {\overrightarrow{T}}_i-{\overrightarrow{Y}}_i);i=\overline{1,n}$(20) ) and (21(21) ${\overrightarrow{Y}}_i={\overrightarrow{T}}_i+h.a\otimes {\overrightarrow{X}}_i$(21) ). (20) ${\overrightarrow{X}}_i=R_L\otimes (R_L\otimes {\overrightarrow{T}}_i-{\overrightarrow{Y}}_i);i=\overline{1,n}$(20) (21) ${\overrightarrow{Y}}_i={\overrightarrow{T}}_i+h.a\otimes {\overrightarrow{X}}_i$(21) Fish Aggregating Devices (FDAs) effect: ${\overrightarrow{Y}}_i$ is updated according to Equation (22(22) ${\overrightarrow{Y}}_i=\{\begin{array}{cc}& {\overrightarrow{Y}}_i+a\left(\right(X_{min}+\overrightarrow{R}\otimes (X_{max}-X_{min}))\otimes \overrightarrow{U})\\ & if(k\le FADs)\\ & {\overrightarrow{Y}}_i+(FADs\ast (1-k)+k)({\overrightarrow{Y}}_{k1}-{\overrightarrow{Y}}_{k2})\\ & if(k>FADs)\end{array}$(22) ). (22) ${\overrightarrow{Y}}_i=\{\begin{array}{cc}& {\overrightarrow{Y}}_i+a\left(\right(X_{min}+\overrightarrow{R}\otimes (X_{max}-X_{min}))\otimes \overrightarrow{U})\\ & if(k\le FADs)\\ & {\overrightarrow{Y}}_i+(FADs\ast (1-k)+k)({\overrightarrow{Y}}_{k1}-{\overrightarrow{Y}}_{k2})\\ & if(k>FADs)\end{array}$(22) where,

• $\overrightarrow{U}$ represents the binary vector array.

• k is a uniform random number in the range of [0,1].

• $k_1$ and $k_2$ represent random indexes of the prey matrix.

3. Our approach

This section presents three algorithms. The first algorithm is the TLID method. The second algorithm introduces a new fusion rule that combines variations of the local energy function. The third algorithm proposes our image fusion method.

3.1. Three-layer image decomposition method

Image synthesis begins with image decomposition, which involves separating the source image into layers with complementary information. Typically, image decomposition algorithms create a base layer and one or more detail layers. In previous studies, a two-layer image decomposition method was commonly used, with the base layer obtained using the average [59] or low-pass filters [35]. However, these filters can cause loss of detailed information in the image, resulting in incomplete detail layers. To address these limitations, we propose a three-layer image decomposition method based on RGF and WMC filters. The algorithmic steps of this method are presented in Algorithm 1, and the process is illustrated in Figure 6. An illustrative example of a three-layer image decomposition is illustrated in Figure 7.

Figure 6. Diagram illustrating in detail how to decompose an input image into three layers.

Algorithm 1. The three-layer image decomposition

Input: I

Output: Three layers $L_B$, $L_{SS}$, and $L_{LS}$

Step 1: The generation of the base layer ($L_B$) involves applying the RGF filter to the image I.

Step 2: The WMC filter (WMCF) is employed to filter the image I, resulting in an image denoted as $I_W$.

Step 3: The layer containing small-scale structures ($L_{SS}$) is obtained by applying Equation (23(23) $L_{SS}=I-I_W$(23) ). (23) $L_{SS}=I-I_W$(23)

Step 4: According to Equation (24(24) $L_{LS}=I_W-L_B$(24) ), the large-scale structure layer ($L_{LS}$) can be obtained. (24) $L_{LS}=I_W-L_B$(24)

3.2. Fusion rules based on combining variations of the local energy function

In this subsection, we introduce a novel fusion rule, named FR-CVLEF, which combines variations of the local energy function to fuse the small-scale and large-scale structure layers.

Figure 7. Three resulting images obtained after decomposing an input image I.

Algorithm 2. FR-CVLEF

Input: Two layers ($L_1$, $L_2$)

Output: A fused layer $L_3$

Step 1: Calculate the local energy function for the $L_1$ and $L_2$ layers according to Equations (25(25) $LE{F}_1(i,j)=\sum _{u=0}^{k-1}\sum _{v=0}^{k-1}{W}_{LE}(u,v)L_1^2(i+u,j+v)$(25) ) and (26(26) $LE{F}_2(i,j)=\sum _{u=0}^{k-1}\sum _{v=0}^{k-1}{W}_{LE}(u,v)L_2^2(i+u,j+v)$(26) ). (25) $LE{F}_1(i,j)=\sum _{u=0}^{k-1}\sum _{v=0}^{k-1}{W}_{LE}(u,v)L_1^2(i+u,j+v)$(25) (26) $LE{F}_2(i,j)=\sum _{u=0}^{k-1}\sum _{v=0}^{k-1}{W}_{LE}(u,v)L_2^2(i+u,j+v)$(26)

Step 2: Calculate the LEF-PCO for the $L_1$ and $L_2$ layers according to Equations (27(27) $LEF\mathrm{\_}PC{O}_1^k(i,j)=\sum _{u=0}^{m-1}\sum _{v=0}^{m-1}{W}_{PCO}^k(u,v)L_1^2(i+u,j+v)$(27) ) and (28(28) $LEF\mathrm{\_}PC{O}_2^k(i,j)=\sum _{u=0}^{m-1}\sum _{v=0}^{m-1}{W}_{PCO}^k(u,v)L_2^2(i+u,j+v)$(28) ). (27) $LEF\mathrm{\_}PC{O}_1^k(i,j)=\sum _{u=0}^{m-1}\sum _{v=0}^{m-1}{W}_{PCO}^k(u,v)L_1^2(i+u,j+v)$(27) (28) $LEF\mathrm{\_}PC{O}_2^k(i,j)=\sum _{u=0}^{m-1}\sum _{v=0}^{m-1}{W}_{PCO}^k(u,v)L_2^2(i+u,j+v)$(28)

Step 3: Calculate the local energy function combined with the structure tensor salient detection operator for the $L_1$ and $L_2$ layers according to Equations (29(29) $LEF\mathrm{\_}STSD{O}_1=W_{STSDO}\left(L_1\right)\odot LEF\left(L_1\right)$(29) ) and (30(30) $LEF\mathrm{\_}STSD{O}_2=W_{STSDO}\left(L_2\right)\odot LEF\left(L_2\right)$(30) ). (29) $LEF\mathrm{\_}STSD{O}_1=W_{STSDO}\left(L_1\right)\odot LEF\left(L_1\right)$(29) (30) $LEF\mathrm{\_}STSD{O}_2=W_{STSDO}\left(L_2\right)\odot LEF\left(L_2\right)$(30)

Step 4: Calculate the maximum value of local energy functions and its variations for $L_1$ and $L_2$ layers corresponding to Equations (31(31) $M_1=Max(LE{F}_1,LEF\mathrm{\_}PC{O}_1^k,LEF\mathrm{\_}STSD{O}_1)$(31) ) and (32(32) $M_2=Max(LE{F}_2,LEF\mathrm{\_}PC{O}_2^k,LEF\mathrm{\_}STSD{O}_2)$(32) ). (31) $M_1=Max(LE{F}_1,LEF\mathrm{\_}PC{O}_1^k,LEF\mathrm{\_}STSD{O}_1)$(31) (32) $M_2=Max(LE{F}_2,LEF\mathrm{\_}PC{O}_2^k,LEF\mathrm{\_}STSD{O}_2)$(32)

Step 5: The FR-CVLEF is defined as Equation (33(33) $L_3(i,j)=\{\begin{array}{l}{L}_1(i,j)\mathrm{i}\mathrm{f}\left|M_1\right(i,j\left)\right|\ge \left|M_2\right(i,j\left)\right|\\ L_2(i,j)\mathrm{i}\mathrm{f}\left|M_1\right(i,j\left)\right|<\left|M_2\right(i,j\left)\right|\end{array}$(33) ). (33) $L_3(i,j)=\{\begin{array}{l}{L}_1(i,j)\mathrm{i}\mathrm{f}\left|M_1\right(i,j\left)\right|\ge \left|M_2\right(i,j\left)\right|\\ L_2(i,j)\mathrm{i}\mathrm{f}\left|M_1\right(i,j\left)\right|<\left|M_2\right(i,j\left)\right|\end{array}$(33)

Figure 8 illustrates the use of the FR-CVLEF.

Figure 8. Illustration for the FR-CVLEF method.

3.3. Our image fusion method

In this subsection, we present our approach, which involves three main steps. Firstly, we enhance the input image ($I_{MRI}$) using two methods: unsharp masking and the Brighten Low-Light Image (BLLI) method [38]. Next, the image $I_{PET}$ is transformed into the channels Y, U, and V. We then decompose $I_{MRI}$ and Y into base layers, small-scale structure layers ($L_{SS}$), and large-scale structure layers ($L_{LS}$) using the TLID method. After that, we fuse the base layers using an adaptive rule and the $L_{SS}$ and $L_{LS}$ layers using the FR-CVLEF to obtain the synthesized layers $L_B^F$, $L_{SS}^F$ and $L_{LS}^F$, respectively. We then calculate the composite gray image ($I^{F}Gray$) by summing the three layers, $L_B^F$, $L_{SS}^F$ and $L_{LS}^F$. Finally, we convert $I_{Gray}^F$, U, and V into a color composite image ($I_{Color}^F$). Our approach is illustrated in Algorithm 3 and Figure 9.

Figure 9. The diagram of our approach.

Algorithm 3. The proposed approach

Input: $I_{MRI}$, $I_{PET}(S\times T)$.

Output: $I_{Fusion}$

Step 1: Enhancing the input image $I_{MRI}$ using the Unsharp masking and the BLLI [38] method, obtaining the $I_M$ image.

Step 2: Convert $I_{PET}$ from RGB to YUV color space, obtaining three components Y, U, and V.

Step 3: Decompose $I_M$ and Y by TLID method, obtained layers, ($L_B^{MRI}$, $L_B^Y$), ($L_{SS}^{MRI}$, $L_{SS}^Y$), and ($L_{LS}^{MRI}$, $L_{LS}^Y$), respectively.

Step 4: ($L_{SS}^{MRI}$, $L_{SS}^Y$), and ($L_{LS}^{MRI}$, $L_{LS}^Y$) layers are fused by the FR_CVLEF method according to Equations (34(34) $L_{SS}^F=FR\mathrm{\_}CVLEF(L_{SS}^{MRI},L_{SS}^Y)$(34) ) and (35(35) $L_{LS}^F=FR\mathrm{\_}CVLEF(L_{LS}^{MRI},L_{LS}^Y)$(35) ). (34) $L_{SS}^F=FR\mathrm{\_}CVLEF(L_{SS}^{MRI},L_{SS}^Y)$(34) (35) $L_{LS}^F=FR\mathrm{\_}CVLEF(L_{LS}^{MRI},L_{LS}^Y)$(35)

Step 5: $L_B^{MRI}$ and $L_B^Y$ are fused by by adaptive parameters $({\rho }_1\in [0.8,1]$, ${\rho }_2\in [0,0.2]$, and ${\rho }_3\in [0.9,1])$ according to Equation (36(36) $L_B^F={\rho }_1{L}_B^{MRI}+{\rho }_2{L}_B^Y$(36) ). (36) $L_B^F={\rho }_1{L}_B^{MRI}+{\rho }_2{L}_B^Y$(36) Where the MPA algorithm is used to find the extrema of the fitness function according to Equation (37(37) $F=\frac{{\mu }_T}{R\times {\sigma }_T^2}(H_T-H_M)(H_T-H_Y).$(37) ). (37) $F=\frac{{\mu }_T}{R\times {\sigma }_T^2}(H_T-H_M)(H_T-H_Y).$(37) Where $R(I_{MRI},L_T^F)$, $R(Y,L_T^F)$, and R are determined as Equations (38(38) $R(I_M,L_T^F)=\frac{1}{S\times T}\sum _{s=1}^S\sum _{t=1}^T\left(L_T^F\right(s,t)-I_M(s,t){)}^2$(38) ), (39(39) $R(Y,L_T^F)=\frac{1}{S\times T}\sum _{s=1}^S\sum _{t=1}^T\left(L_T^F\right(s,t)-Y(s,t){)}^2$(39) ), and (40(40) $R={\rho }_{3}R(I_M,L_T^F)+(1-{\rho }_3)R(Y,L_T^F)$(40) ), respectively. (38) $R(I_M,L_T^F)=\frac{1}{S\times T}\sum _{s=1}^S\sum _{t=1}^T\left(L_T^F\right(s,t)-I_M(s,t){)}^2$(38) (39) $R(Y,L_T^F)=\frac{1}{S\times T}\sum _{s=1}^S\sum _{t=1}^T\left(L_T^F\right(s,t)-Y(s,t){)}^2$(39) (40) $R={\rho }_{3}R(I_M,L_T^F)+(1-{\rho }_3)R(Y,L_T^F)$(40) ${\mu }_T$ and ${\sigma }_T^2$ are mean and variance of $L_T^F$ ($L_T^F=L_B^F+L_{SS}^F+L_{LS}^F$) in each iteration of the MPA algorithm. $H_M$, $H_Y$, and $H_T$ are the Entropy of $I_M$, Y, and $L_T^F$, respectively.

Step 6: The image $L_T^F$ obtained after loop termination is a fusion gray image ($I_{Gray}^F$).

Step 7: $I_{Gray}^F$, U, and V are converted into $I_{Color}^F$.

4. Experimental setup and evaluation

4.1. Experimental data

A total of 156 images, including 78 pairs of MRI and PET images, were utilized in this study. The images were sourced from ‘The Whole Brain Atlas’ (http://www.med.harvard.edu/AANLIB/) and are described in detail in Table 1. The three pairs of images in the K4 dataset are illustrated in Figure 10.

Figure 10. Illustration of several pairs of MRI and PET images.

Table 1. Dataset.

Groups	Number of images	Description
K1	26 pairs (MRI (T2) – PET)	Slices from 61 to 86 along the Transaxial (T) axis.
K2	26 pairs (MRI (T2) – PET)	Slices from 61 to 86 along the Sagittal (S) axis.
K3	26 pairs (MRI (T2) – PET)	Slices from 61 to 86 along the Coronal (C) axis.
K4	3 pairs (MRI (T2) – PET)	Slice #78 along the T, S, and C axis.

4.2. Experimental setup

We design some experiments as follows:

Experiment #1

We selected several optimization algorithms for comparison with the MPA algorithm. The algorithms considered are described in Table 2, and the experimental data used in this study is the K4 dataset.

Table 2. Five optimization algorithms.

Num	Algorithms	Description
1	MVO [60]	Multi-Verse Optimizer
2	WOA [61]	Whale Optimization Algorithm
3	SSA [55]	Salp swarm algorithm
4	SCA [62]	Sine cosine algorithm
5	GWO [61]	Grey wolf optimizer

Experiment #2

Several fusion rules were used for comparison with the proposed rule (FR_CVLEF), as outlined below:

• Max selection rule ($R_{Max}$).

• Maximum local energy ($R_{MLE}$) [63].

• PA-PCNN ($R_{PA-PCNN}$) [64].

• Sum-modified laplacian ($R_{SML}$) [65].

Experiment #3

Our study involves a comparative analysis between our algorithm and seven contemporary image fusion algorithms, which are comprehensively described in Table 3.

Table 3. Seven image fusion algorithms.

Num	Algorithms	Years
1	PC-LLE-NSCT ($G1$) [66]	2019
2	TLD-SR ($G2$) [10]	2021
3	JBF-LGE ($G3$) [43]	2021
4	CSE (Contrast and structure extraction) ($G4$) [67]	2021
5	CNPS-NSST ($G5$) [68]	2021
6	DTNP-NSCT ($G6$) [69]	2021
7	ACO ($G7$) [70]	2022

We compare our algorithm with seven contemporary image fusion algorithms (as detailed in Table 3).

For Experiments #2 and #3, we utilized data sets K1, K2, and K3. We evaluated the performance of the image fusion algorithms using the following five indicators:

• Average light intensity ($Q_{ALI}$).

• Contrast index ($Q_{CI}$).

• Sharpness ($Q_S$).

• Edge-based similarity measure ($Q^{AB/F}$) [71].

• Feature mutual information [72] ($Q_{FMI}$).

The necessary parameters used in our model are set as follows:

• n = 50, $l_{max}=50$.

• h = 0.5; $FADs=0.2;k\in [0,1]$.

4.3. Results and evaluation

The results of the three experiments described in Section 4.2 are presented here. In the first experiment, we ran each optimization algorithm 30 times independently and evaluated the mean ($Q_M$) and standard deviation ($Q_{SD}$) using two metrics. The results are presented in Table 4 and Figure 11. The MPA algorithm outperformed the others, producing the highest $Q_M$ value and the lowest $Q_{SD}$ value. Therefore, it is the preferred choice for the proposed model. We also conducted a Wilcoxon rank-sum test [73] to assess the significance of the results, and the P-values were found to be above 0.05, indicating statistical significance, as shown in Table 5.

Figure 11. The fitness function values obtained from various optimization algorithms in 30 independent on dataset K4 runs are visually presented in the form of a box plot.

Table 4. $Q_M$ and $Q_{SD}$ from 30 different runs.

Dataset	Algorithms	$Q_M$	$Q_{SD}$
K4(#78T)	MPA	0.017876671935888	0.000009033932259
	MVO	0.016518230357696	0.001090932265423
	WOA	0.017277469311324	0.000534054285376
	SSA	0.016899097232778	0.001118836345425
	SCA	0.017217245270695	0.000289792732718
	GWO	0.017669608631168	0.000189772085703
K4(#78S)	MPA	0.029647134938738	0.000026405604409
	MVO	0.028356602594419	0.001161220657539
	WOA	0.028698747803551	0.000690713025417
	SSA	0.027886822958534	0.002416877831579
	SCA	0.029093335806103	0.000290569097601
	GWO	0.029300807571747	0.000241217843724
K4(#78C)	MPA	0.005885862817517	0.000003159852779
	MVO	0.005790921812435	0.000067984350292
	WOA	0.005828160907535	0.000074232890256
	SSA	0.005816524467147	0.000069518552257
	SCA	0.005774386053179	0.000043737425495
	GWO	0.005836384455938	0.000020781565237

Table 5. P-values from Wilcoxon test.

Dataset	Algorithms	P-values
K4(#78T)	MPA vs MVO	3.019859359162151e−11
	MPA vs WOA	3.689725853981014e−11
	MPA vs SSA	6.065757009046759e−11
	MPA vs SCA	3.019859359162151e−11
	MPA vs GWO	3.689725853981014e−11
K4(#78S)	MPA vs MVO	3.019859359162151e−11
	MPA vs WOA	1.464306887715034e−10
	MPA vs SSA	6.695518965500180e−11
	MPA vs SCA	3.019859359162151e−11
	MPA vs GWO	4.077164846825348e−11
K4(#78C)	MPA vs MVO	3.019859359162151e−11
	MPA vs WOA	8.993406027014880e−11
	MPA vs SSA	3.019859359162151e−11
	MPA vs SCA	3.019859359162151e−11
	MPA vs GWO	4.504322112705322e−11

In the second experiment, we compared our fusion rules with four other fusion rules for SS and LS layers. We evaluated the five indicators, and the results are presented in Table 6. Our FR_CVLEF rule performed the best as it obtained the highest scores for two evaluation indicators, $Q^{AB/F}$ and $Q_{FMI}$, compared to the other fusion rules.

Table 6. Two evaluation metrics ($Q^{AB/F}$ and $Q_{FMI}$) obtained from different fusion rules.

Dataset	Fusion rules	$Q^{AB/F}$	$Q_{FMI}$
K1	$R_{\mathrm{M}\mathrm{a}\mathrm{x}}$	0.4829	0.8144
	$R_{MLE}$	0.7124	0.8720
	$R_{PA-PCNN}$	0.6669	0.8658
	$R_{SML}$	0.6984	0.8632
	FR-CVLEF	0.7440	0.8737
K2	$R_{\mathrm{M}\mathrm{a}\mathrm{x}}$	0.4868	0.8092
	$R_{MLE}$	0.7271	0.8749
	$R_{PA-PCNN}$	0.6893	0.8689
	$R_{SML}$	0.6802	0.8619
	FR-CVLEF	0.7539	0.8782
K3	$R_{\mathrm{M}\mathrm{a}\mathrm{x}}$	0.4957	0.8073
	$R_{MLE}$	0.7454	0.8728
	$R_{PA-PCNN}$	0.7152	0.8664
	$R_{SML}$	0.7146	0.8600
	FR-CVLEF	0.7716	0.8743

For the third experiment, adaptive parameters were obtained and are listed in Table 7. A value of approximately 1 for ${\rho }_3$ indicates that the fused image has the highest similarity to the MRI image. Moreover, the ${\rho }_1$ value is considerably larger than the ${\rho }_2$ value, which suggests that the MRI images provide substantial information to the output image. The output images were generated using seven image fusion algorithms, including our own algorithm, which is depicted in Figures 12, 15, and 18. Visual inspection indicates that the output image produced by our algorithm exhibits high quality in terms of light intensity, contrast, and sharpness. A small area was cropped from the image to observe its details, as shown in Figures 13, 16, and 19. The details in these small areas are well preserved. In terms of quantitative analysis, Table 8 presents the five evaluation indicators obtained from various image fusion algorithms. The proposed method's output image has significantly higher quality than the other image fusion methods due to the adaptive rule used for the base layers. The FR_CVLRF method guarantees the output images retain the input image details.

Figure 12. Results obtained from eight image fusion algorithms for MRI-PET image pair #068T in K1.

Figure 13. Frames cropped from Figure 12.

Figure 14. Five evaluation indicators are obtained from eight image synthesis algorithms on K1.

Figure 15. Results obtained from eight image fusion algorithms for MRI-PET image pair #068S in K2.

Figure 16. Frames cropped from Figure 15.

Figure 17. Five evaluation indicators are obtained from eight image synthesis algorithms on K2.

Figure 18. Results obtained from eight image fusion algorithms for MRI-PET image pair #068C in K3.

Figure 19. Frames cropped from Figure 18.

Figure 20. Five evaluation indicators are obtained from eight image synthesis algorithms on K3.

Table 7. Adaptive parameters obtained in three data sets K1, K2, and K3.

Dataset	${\rho }_1$	${\rho }_2$	${\rho }_3$
K1	0.9927	0.0453	0.999
K2	0.9829	0.0524	0.999
K3	0.9942	0.0575	0.999

Table 8. The evaluation indicators obtained from image fusion algorithms.

Dataset	Algorithms	$Q_{ALI}$	$Q_{CI}$	$Q_S$	$Q^{AB/F}$	$Q_{FMI}$
K1	G1	0.2492	0.2910	0.0528	0.6248	0.8569
	G2	0.2634	0.2997	0.0499	0.6786	0.8646
	G3	0.2634	0.2992	0.0511	0.6771	0.8682
	G4	0.2233	0.2479	0.0461	0.6552	0.8681
	G5	0.2060	0.2382	0.0546	0.6192	0.8583
	G6	0.2558	0.2922	0.0522	0.6376	0.8619
	G7	0.2415	0.2596	0.0470	0.6587	0.8562
	Ours	0.3131	0.3356	0.0829	0.7440	0.8737
K2	G1	0.2555	0.2816	0.0506	0.6432	0.8675
	G2	0.2700	0.2905	0.0474	0.6818	0.8740
	G3	0.2713	0.2920	0.0487	0.7099	0.8765
	G4	0.2317	0.2410	0.0435	0.6703	0.8762
	G5	0.2162	0.2305	0.0517	0.6344	0.8694
	G6	0.2626	0.2835	0.0497	0.6537	0.8714
	G7	0.2754	0.2727	0.0463	0.6996	0.8695
	Ours	0.3313	0.3217	0.0740	0.7539	0.8782
K3	G1	0.2060	0.2648	0.0452	0.6628	0.8674
	G2	0.2175	0.2757	0.0428	0.6987	0.8726
	G3	0.2181	0.2761	0.0435	0.7133	0.8766
	G4	0.1925	0.2380	0.0398	0.6914	0.8716
	G5	0.1789	0.2251	0.0463	0.6524	0.8665
	G6	0.2120	0.2684	0.0446	0.6732	0.8716
	G7	0.2103	0.2421	0.0400	0.6980	0.8644
	Ours	0.2778	0.3230	0.0693	0.7716	0.8743

Table 9 displays the average running times of the image fusion algorithms. The algorithm developed by us takes an average of 4.58 seconds to complete the fusion process. However, this time increases as the number of iterations increases, making it a disadvantage of our algorithm.

Table 9. Average running time (ART) of image fusion algorithms on the data set (K1&K2&K3).

Approaches	ART (s)
G1	3.02972
G2	14.92673
G3	0.26316
G4	0.10035
G5	3.36244
G6	17.90511
G7	1.05729
Ours (50 loops)	4.57611

5. Conclusion

In this paper, a new approach for fusing medical images is introduced, which combines a local energy function and variations of it with the MPA algorithm. The proposed method starts by introducing a three-layer image decomposition technique, which separates the input images into three distinct layers: the base layer ($L_B$), the small-scale structure layer ($L_{SS}$), and the large-scale structure layer ($L_{LS}$). Next, a fusion rule built on combining the local energy function with its variations is introduced to fuse the $L_{SS}$ and $L_{LS}$ layers. Thirdly, the base layers are synthesized based on the MPA algorithm. The method was tested using 156 MRI-T2 and PET images, along with five evaluation metrics and seven state-of-the-art image fusion algorithms. The proposed adaptive rules for the base layers resulted in an output composite image that had a high level of quality in terms of brightness intensity and contrast. For example, from Table 8, the $Q_{ALI}$ and $Q_{CI}$ indexes obtained from the proposed model are the highest, at 0.3131 and 0.3356 on data set K1. Furthermore, applying the FR_CVLEF method to the small-scale and large-scale structure layers has also shown significant efficiency in the detailed information of the output image. For instance, from Table 8, the $Q_S$, $Q_{AB/F}$, and $Q_{FMI}$ indexes obtained from the proposed model are the highest, at 0.0829, 0.7440, and 0.8737 on data set K1, respectively.

In the future, our goal is to enhance the performance of the proposed model in several aspects. First, apply new image enhancement methods [74–76] to replace the BLLI method. This allows for significantly improved performance of image synthesis. Second, select some recent optimization algorithms, such as the Chameleon swarm algorithm (CSA) [77, 78] and White Shark Optimizer (WSO) [79], to replace the MPA algorithm. This can improve the runtime of the proposed model.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Phu-Hung Dinh

Phu-Hung Dinh was born in Vietnam. Currently, he is a lecturer at the Faculty of Computer Science and Engineering, Thuyloi University, Hanoi, Vietnam. His research interests include signal processing, image processing, meta-heuristic optimization, and computer vision. Furthermore, he is also a reviewer for many prestigious journals, such as Expert systems with applications, Artificial intelligence review, Computers in biology and medicine, Biomedical signal processing & control, ISA transactions, BMC Bioinformatics, Applied Intelligence, Soft Computing, Imaging science journal, and Journal of Supercomputing.