An Extended Membrane System Based on Cell-like P Systems and Improved Particle Swarm Optimization for Image Segmentation

Abstract

An extended membrane system with a dynamic nested membrane structure, which is integrated with the evolution-communication mechanism of a cell-like P system with evolutional symport/antiport rules and active membranes (ECP), and the evolutionary mechanisms of particle swarm optimization (PSO) and improved PSO inspired by starling flock behavior (SPSO), named DSPSO-ECP, is designed and developed to try to break application restrictions of P systems in this paper. The purpose of DSPSO-ECP is to enhance the performance of extended membrane system in solving optimization problems. In the proposed DSPSO-ECP, the updated model of velocity and position of standard PSO, as basic evolution rules, are adopted to evolve objects in elementary membranes. The modified updated model of the velocity of improved SPSO is used as local evolution rules to evolve objects in sub-membranes. A group of sub-membranes for elementary membranes are specially designed to avoid prematurity through membrane creation and dissolution rules with promoter/inhibitor. The exchange and sharing of information between different membranes are achieved by communication rules for objects based on evolutional symport rules of ECP. At last, computational results, which are made on numerical benchmark functions and classic test images, are discussed and analyzed to validate the efficiency of the proposed DSPSO-ECP.

1. Introduction

Membrane computing (MC), as an important branch of bio-inspired computing, initiated by Păun [1], is inspired by the structure, functioning, interaction, and cooperation of living cells in tissues, organs, and organisms, and the computing model of membrane computing is also called membrane systems or P systems. P systems, as a class of distributed parallel computing models, which consist of membrane structure, objects (or data), and rules. Three classic computing models of P systems based on different membrane structures or cell arrangements are generally reported in previous studies and research, including cell-like P systems, tissue-like P systems, and neural-like P systems [2]. Research shows that three classic P systems and their variants have the same computing power as Turing machines or are more efficient [3].

Cell-like P systems, as a class computing model of MC, are inspired by the hierarchical structure and communication mechanism of living cells. The underlying membrane structure of cell-like P systems can be abstracted to arbitrary graphs in mathematics [4]. Cell-like P systems have highly effective efficiency in solving optimization problems with linear or polynomial complexity due to the distributed parallel computing working manner. It has been applied to many fields due to the simplicity and parallelism of computing models of P systems [5]. However, the application of cell-like P systems has been limited by the incompleteness of fundamental computation and the complicacy of implementation. Due to the limitation we have mentioned above, evolutionary membrane computing (EMC) [6] become important research in the application of cell-like P systems, which are integrated with evolutionary computation (EC), including evolutionary algorithm (EAs) and swarm intelligence (SI). Particle swarm optimization (PSO), as a classic optimization algorithm of SI algorithms, is simple and easy to implement, and the evolutionary mechanism of PSO, also provides a new way for EAs based on cell-like P systems to solve complex problems.

This work focuses on the development of an extended membrane system in dynamic nested membrane structure (NMS)-based MIEAs using cell-like P systems and improved PSO for solving optimization problems, and simply named DSPSO-ECP, which is based on the evolution-communication mechanism of cell-like P system with evolutional symport/antiport rules and active membranes (ECP), and evolutionary mechanism of PSO and improved PSO with starling flock behavior (SPSO), Fitness-Euclidean distance ratio (FER) and global information. Two kinds of evolution rules for objects based on updated models of PSO and improved SPSO are introduced to evolve objects in different membranes, including basic evolution rules in elementary membranes and local evolution rules in sub-membranes. A group of sub-membranes for elementary membranes are designed to change the membrane structure dynamically through membrane creation and dissolution rules with promoter/inhibitor. The exchange and sharing of information between different membranes are achieved by communication rules of ECP for objects. At last, the proposed DSPSO-ECP is evaluated on eight classic numerical benchmark functions and compared with four PSO-based optimization approaches to evaluate the effectiveness. Furthermore, computational experiments compared with other segmentation approaches, which are made on eight classic test images, are conducted to verify the efficiency of the proposed DSPSO-ECP. The proposed DSPSO-ECP based on cell-like P systems and improved PSO is to try to break these restraints of cell-like P systems and expand to widen the application of cell-like P systems, and the modified update model of improved PSO gives a new evolution mechanism for P systems to evolve objects and membranes. In addition, the PSO algorithm still has some limitations, such as the easily trapped into local optima and high complexity for solving big datasets [7], the evolution-communication mechanism and distributed parallel computing framework of cell-like P systems obviously need to balance the exploration and exploitation and improve the performance of PSO algorithm in solving complex problems including image segmentation problems.

The rest of this paper is organized as follows. The related works of cell-like P systems and PSO algorithms are summarized and discussed in Section 2. The basic framework of ECP, and improved SPSO based on FER and global information, are described in Section 3. The extended membrane system based on the evolution-communication mechanism of ECP and the evolutionary mechanism of PSO and improved SPSO is proposed in Section 4. Some rules, including evolution rules for objects, evolution rules with promoter/inhibitor for membranes, and communication rules for objects are described in more detail in this section. Experimental results and analysis, which are made on eight classic numerical benchmark functions compared with four PSO-based optimization approaches are reported in Section 5. Section 6 gives experimental results and a discussion of the image segmentation problem on eight classic test images to evaluate the efficiency of the proposed DSPSO-ECP. Section 7 provides some conclusions and outlines future research directions.

2. Related Works

The studies of cell-like P systems and their variants mainly focused on the theoretical analysis and application of computing models [8]. A variety of cell-like P systems based on biological facts [9], the biochemical reaction of cells [10], mathematical biological cells [11], and theoretical computer science have been directly designed and developed for solving problems in the studies of theoretical analysis of cell-like P systems, which is recruited various ingredients, including energy, catalysts, and mitosis [12], such as cell-like P systems with active membranes inspired in the mitosis process [13]; polarizationless P systems with active membranes [14], to avoid polarizations; cell-like P systems with evolutional symport/antiport rules inspired by conservation law [15], and its variants [16]. The analysis of computing power and computational efficiency of cell-like P systems and their variants are also an important part of theoretical studies and works [17].

Furthermore, EMC [6], as an important research application for cell-like P systems, which is integrated with EC, including EAs and SI as we mentioned above, is designed and developed to break the limitation of the original computing model of P systems and to solve more complicated problems [18]. Membrane-inspired evolutionary algorithms (MIEAs) of EMC based on extended cell-like P systems [19], which are utilized various SI algorithms, such as genetic algorithm (GA) [20], differential evolution (DE) [21], particle swarm optimization (PSO) [22,23,24] and its variants [25], ant colony optimization (ACO) [26], artificial bee colony algorithm (ABC) [27] and biogeography-based optimization (BBO) [28], and clustering algorithms, such as hierarchical clustering algorithm [29], consensus clustering algorithm [30] and feature selection algorithm [31], have been widely used for solving more complex problems in real-world.

PSO, which was initiated by Kennedy and Eberhart in 1995 [32], is a classic optimization algorithm of SI algorithms. The basic principle of PSO is based on the group behavior of birds flocking and individual trajectory analysis. The individual experiences and the global experience of the population are adopted in the updated model of PSO to balance exploration and exploitation [33]. Therefore, it has been successfully applied to many difficult problems [34]. However, like most SI algorithms, PSO is easily trapped in local optima and appeared prematurity under the guidance of local and global experiences [7]. A lot of research and work has been done to improve the performance of standard PSO, which can be divided into mechanism analysis [35], improved PSO [33], and applications of PSO [36].

The studies of improved PSO are mainly related to modified PSO algorithms and hybrid PSO algorithms based on meta-heuristic approaches [37]. Modified PSO algorithms are based on the updated model of PSO and adopted some strategies and methods in the search process of particles, such as flight mechanisms of particles including levy flight [38,39,40], learning strategies for particles including cirssoss learning [41], cognitive learning [42] and comprehensive learning [40]; population topology including stochastic topology[7] and dynamic topology [43]; and optimization strategies including random walk strategy [44], chaos strategy [45] and synergistic strategy [46]; search strategies including local search [47,48] and charged system search [49,50]. Hybrid PSO algorithms are combined with some traditional and evolutionary optimization methods in order to utilized the advantages of both methods and improve the global search ability of PSO, such as simulated annealing (SA) [51], tabu search (TS) [52], BBO [53], artificial bee colony (ABC) [54], genetic algorithm (GA) [55] and differential evolution (DE) [56]. Speciality, multi-swarm PSO is an important field of improved PSO in recent years [57,58,59,60]. PSO has been widely used to solve practical engineering problems due to easy implementation and robust performance [61], including clustering problems [62], signalized traffic problems [63], image segmentation [64], feature selection [65], antenna synthesis [66] and fuzzy controlled systems [67].

From the related works of cell-like P systems and PSO, EMC based on cell-like P systems and swarm intelligence, including PSO algorithm, mainly focuses on the traditional or classic evolutionary mechanism of evolutionary algorithms with static membrane structure, and improved optimization algorithms are required not only improved the performance of algorithms but also a more suitable way for combining the evolution-communication mechanism of cell-like P systems. The dynamic membrane structure needs to be specially designed to enhance the global search ability and avoid prematurity. Although improved PSO algorithms based on other traditional optimization strategies and approaches have great potential in solving respective problems, the balance between exploration and exploitation of PSO remains an important problem in improved PSO. The improved PSO integration with the framework and computing features, including dynamic, indeterminacy, and flat maximal parallelism of cell-like P systems also provided a new way to improve the global search ability of PSO and solve more complex problems.

3. Proposed ECP and Improved SPSO

3.1. Cell-like P System with Evolutional Symport/Antiport Rules and Active Membranes

As a classic computing model of cell-like P systems, cell-like P systems with evolutional symport/antiport rules are proposed to modify and change the information of objects during the process of communication, which is based on the real biochemical reaction of living cells [16]. But the membrane structure of the P system will not be changed during computation. Therefore, active membranes are introduced to dynamically change the membrane structure of a cell-like P system with evolutional symport/antiport rules, and this P system is simply named ECP. Especially, ECP is described as a tuple as the following:

$$\prod =\left(\mathsf{\Gamma },\epsilon ,\mu ,{\omega }_1,\cdots ,{\omega }_m,\mathrm{R},{\sigma }_{out}\right),$$

where

(1)

$\mathsf{\Gamma }$ is a nonempty finite alphabet of objects;

(2)

$\epsilon$ is a set of initial objects located in the environment;

(3)

$\mu$ is the initial membrane structure of ECP, which contains $m$ membranes;

(4)

${\omega }_1,\cdots ,{\omega }_m$ are finite multisets of initial objects over $\mathsf{\Gamma }$;

(5)

$\mathrm{R}$ is multiple but finite sets of evolution rules, which consist of two kinds of evolution rules, including evolutional symport/antiport rules for objects and evolution rules for membranes, which are defined in the following.

Evolutional symport rules are of the form: ${\left[u\right]}_i{\left[\right]}_j\to {\left[\right]}_i{\left[u^{\prime }\right]}_j$, where $1\le i\ne j\le m$, $u\in {\mathsf{\Gamma }}^{+}$, $u^{\prime }\in {\mathsf{\Gamma }}^{\ast }$, which means objects $u$ in membrane $i$ evolve to new objects $u^{\prime }$ and send to membrane $j$. Evolutional antiport rules are of the form: ${\left[u\right]}_i{\left[v\right]}_j\to {\left[v^{\prime }\right]}_i{\left[u^{\prime }\right]}_j$, where $0<i\ne j\le m$, $u,v\in {\mathsf{\Gamma }}^{+}$, $u^{\prime },v^{\prime }\in {\mathsf{\Gamma }}^{\ast }$, which means objects $u$ in membrane $i$ evolve to new objects $u^{\prime }$ and send to membrane $j$, and objects $v$ in membrane $j$ evolve to new objects $v^{\prime }$ and send to membrane $i$ at the same time. Noted that the evolutional symport/antiport rules of ECP only executed on the membranes with set membership.

Evolution rules for membranes contain membrane creation rules and membrane dissolution rules. Membrane creation rules are of the form: ${\left[u\right]}_i\to {\left[u\right]}_{i_1}\cdots {\left[u\right]}_{i_n}$, for $0<i\le m$, $u\in \mathsf{\Gamma }$, which means a group of sub-membranes of membrane $i$ are created and the total number of sub-membranes is $n$. Membrane dissolution rules are of the form: ${\left[u\right]}_i\to \lambda$, for $0<i\le m$, $u\in \mathsf{\Gamma }$, which means membrane $i$ is dissolved in a configuration, and $\lambda$ is empty symbol [17].

(6)

${\sigma }_{out}$ is output membrane or region in ECP, where ${\sigma }_{out}\in \{{\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_m\}$.

3.2. Improved Particle Swarm Optimization Inspired by Starling Flock Behavior

A novel PSO approach, which is inspired by the collective responses of starling birds, is proposed to enhance the local search ability of particles, and is simply named SPSO [68]. In SPSO, the update of velocity and position of particles are based on the collective information of seven neighbors, and the selection of neighbors for each particle is based on Euclidean distance. Thus, the impact of quality and global information can easily be overlooked and underestimated. Therefore, an improved SPSO approach, which is based on the Fitness-Euclidean distance ratio (FER) [69] and global best information is introduced to avoid restrictions as we mentioned above. For particle $i$, the value of FER with particle $i^{\ast }$ ($i\ne i^{\ast }$) at iteration $t$ is determined by (1) in the following

$$FER\left(i,i^{\ast }\right)=\alpha \ast \frac{f\left(p_i\left(t\right)\right)-f\left(p_{i^{\ast }}\left(t\right)\right)}{‖p_i\left(t\right)-p_{i^{\ast }}\left(t\right)‖} , \mathrm{for} i,i^{\ast }=1,2,\cdots ,N,$$

(1)

where $FER\left(i,i^{\ast }\right)$ is value of FER between particle $i$ and particle $i^{\ast }$. $f\left(\right)$ is the fitness function. $p_i\left(t\right)$ is local best position of particle $i$ at iteration $t$. $‖p_i\left(t\right)-p_{i^{\ast }}\left(t\right)‖$ is the Euclidean distance between local best position $p_i\left(t\right)$ and $p_{i^{\ast }}\left(t\right)$ at iteration $t$. $\alpha$ is adjusting parameter, where $\alpha =‖S‖/\left(f\left(p_w(t)\right)-f\left(p_g(t)\right)\right)$, $p_w(t)$ and $p_g(t)$ are worst and best local position for particles at iteration $t$. $S$ is the size of search sparce, where $S=\sqrt{{\displaystyle {\sum }_j^D{\left(s_j^u-s_j^l\right)}^2}}$, $s_j^u$ and $s_j^l$ are the maximum and minimum of $j$-th dimension of search sparce, $D$ is the dimension of search space. All particles are listed in decreasing order of FER, and seven particles ranked on the top are taken as neighbors of particle $i$. The new velocity $v_i\left(t+1\right)$ of $i$-th particle at iteration $t+1$ is determined by (2) in the following

$$v_i\left(t+1\right)=\chi \ast \left(v_i\left(t\right)+r_3\ast \left(p_{n^{\#}}\left(t\right)-x_i\left(t\right)\right)+r_4\ast \left(p_g\left(t\right)-x_i\left(t\right)\right)\right), \mathrm{for} i=1,2,\cdots ,N,$$

(2)

where $\chi$ is constriction coefficient. $v_i\left(t\right)$ is the velocity of $i$-th particle at iteration $t$. $r_3$ and $r_4$ are two random numbers. $p_{n^{\#}}\left(t\right)$ is collective local best position of seven neighbors for particle $i$ at iteration $t$, where $p_{n^{\#}}\left(t\right)=\frac{1}{7}{\displaystyle \sum _{j=1}^7{p}_j\left(t\right)}$. $x_i\left(t\right)$ is the position of $i$-th particle at iteration $t$. $p_g\left(t\right)$ is global best position of particles at iteration $t$.

4. The Proposed DSPSO-ECP

In this section, an extended membrane system based on the dynamic membrane structure of ECP, and the evolutionary mechanism of PSO and improved SPSO, is designed and proposed, which is simply named DSPSO-ECP. The updated model in PSO and improved SPSO, as evolution rules for objects, are adopted to evolve objects in membranes. Evolutional symport rules of ECP, as communication rules for objects, are introduced to the exchanging and sharing of information in different membranes. Membrane creation/dissolution rules of ECP, as membrane evolution rules, are used to generate and dissolve sub-membranes in the P system. Therefore, DSPSO-ECP has two evolutionary mechanisms which are respectively applied to objects and membranes, and the communication mechanism is achieved by executing evolutional symport rules for objects in different membranes. More details about DSPSO-ECP are given in the following.

4.1. The General Framework of DSPSO-ECP

The general framework of DSPSO-ECP can be completely described as a tuple which is given in the following

$$\prod =\left(\mathsf{\Gamma },\epsilon ,\mu ,{\omega }_1,\cdots ,{\omega }_m,\mathrm{R},{\mathrm{R}}^{\prime },{\sigma }_{out}\right),$$

where

(1)

$\mathsf{\Gamma }$ is a non-empty finite alphabet of objects;

(2)

$\epsilon$ is a finite multiset of initial objects located in the environment;

(3)

$\mu$ is the membrane structure of DSPSO-ECP, which contains $m$ membranes;

(4)

${\omega }_1,\cdots ,{\omega }_m$ are finite multisets of initial objects in membranes, with ${\omega }_i\in \mathsf{\Gamma }$, for $1\le i\le m$;

(5)

$\mathrm{R}$ represents finite sets of evolution rules in DSPSO-ECP, which consist of two kinds of evolution rules, including evolution rules for objects and evolution rules for membranes, where $\mathrm{R}=\left\{{\mathrm{R}}^{\ast },{\mathrm{R}}^{\#}\right\}$. ${\mathrm{R}}^{\ast }$ represents finite sets of evolution rules for objects, ${\mathrm{R}}^{\ast }=\left\{R_2^{\ast },R_3^{\ast },\cdots ,R_m^{\ast }\right\}$, where $R_i^{\ast }$ ($2\le i\le m$) represents a finite set of evolution rules for objects associated with membrane $i$. Evolution rules $R_i^{\ast }$ of membrane $i$ are of the form: $R_i^{\ast }=\left\{u\to v\right\}$, for $u,v\in {\mathsf{\Gamma }}^{+}$. Especially, when an evolution rule of $R_i^{\ast }$ in membrane $i$ is applied, objects $u$ evolve to objects $v$ in same membrane. ${\mathrm{R}}^{\#}$ represents finite sets of evolution rules for membranes, ${\mathrm{R}}^{\#}=\left\{R_2^{\#},R_3^{\#},\cdots ,R_m^{\#}\right\}$, where $R_i^{\#}$ ($2\le i\le m$) represents a finite set of evolution rules for membranes associated with membrane $i$, including membrane creation rules and membrane dissolution rules.

(6)

${\mathrm{R}}^{\prime }$ represents finite sets of commutation rules based on evolutional symport rules of ECP in DSPSO-ECP, where ${\mathrm{R}}^{\prime }=\left\{R_1^{\prime },R_2^{\prime },\cdots ,R_m^{\prime }\right\}$. $R_i^{\prime }$ ($1\le i\le m$) represents a finite set of communication rules for objects associated with membrane $i$. Especially, $R_{ij}^{\prime }$ ($1\le j\ne i\le m$) represents communication rules for objects associated with membrane $i$ and membrane $j$, noted that membrane $j$ must be the child or parent membranes of membrane $i$.

(7)

${\sigma }_{out}$ is the output membrane in DSPSO-ECP, where ${\sigma }_{out}\in \{{\sigma }_1,\cdots ,{\sigma }_m\}$. When the computation of DSPSO-ECP has been completed, objects in the output membrane ${\sigma }_{out}$ are transferred to the environment, which can be viewed as the computational results of DSPSO-ECP. The initial membrane structure of the proposed DSPSO-ECP is graphically depicted in Figure 1.

In DSPSO-ECP, all membranes are labeled from 1 to $m$, and the environment is labeled as 0, which can be denoted by ${\sigma }_0,{\sigma }_1,\cdots ,{\sigma }_m$. Membrane 1 is a skin membrane which is not contained in any other membranes. Membrane $i$ ($2\le i\le m$) is an elementary membrane that does not contain any other membranes. The number of elementary membranes in DSPSO-ECP is $m-1$ as shown in Figure 1. Especially, membrane 1 is the parent membrane for elementary membrane $i$, similarly, elementary membrane $i$ are also child membranes for membrane 1. Noted that the communication relationship only exists in the parent membrane and its corresponding child membranes.

4.2. Evolution Rules for Objects

Two types of evolution rules, which are based on the evolutionary mechanism of classic PSO and improved SPSO, are adopted to achieve the evolution of objects in membranes, including basic evolution rules in elementary membranes and local evolution rules in sub-membranes. In DSPSO-ECP, each object $u_i$ consisting of two attribute vector, i.e., velocity and position, $u_i=\left\{V_i,X_i\right\}$, where $V_i$ is the velocity of object $u_i$, and $X_i$ is the position of object $u_i$.

4.2.1. Basic Evolution Rules for Objects

As basic evolution rules for objects, the evolutionary model of classic PSO is introduced to update the velocity and position of objects, which are only executed on elementary membranes ${\sigma }_o$ ($2\le o\le m$). Specially, the velocity $V_i\left(t+1\right)$ of $i$-th object $u_i$ in elementary membrane ${\sigma }_o$ at iteration $t+1$ is determined by (3) in the following

$$V_i(t+1)=\omega \ast V_i(t)+c_1\ast r_1\ast \left(X_i^{lbest}(t)-X_i(t)\right)+c_2\ast r_2\ast \left(X_o^{gbest}(t)-X_i(t)\right), \mathrm{for} i=1,2,\cdots ,n_o,$$

(3)

where $\omega$ is inertia weight. $c_1$ and $c_2$ are the local and global learning factors, which control the influence of local best and global best information. $r_1$ and $r_2$ are two random numbers that uniformly distributed between 0 and 1. $X_i^{lbest}\left(t\right)$ is the local best position of object $u_i$ at iteration $t$, which is also called local best of object $u_i$ at iteration $t$ and is denoted by $u_i^{lbest}\left(t\right)$. $X_o^{gbest}\left(t\right)$ is the global best position for all objects in elementary membrane ${\sigma }_o$ at iteration $t$, which called global best of elementary membrane ${\sigma }_o$ at iteration $t$ and is denoted by $u_o^{gbest}\left(t\right)$. $n_o$ is the number of objects in elementary membrane ${\sigma }_o$. The position $X_i\left(t+1\right)$ of $i$-th object $u_i$ at iteration $t+1$ is determined by (4) in the following

$$X_i\left(t+1\right)=X_i\left(t\right)+V_i\left(t+1\right), \mathrm{for} i=1,2,\cdots ,n_o,$$

(4)

The local best position $X_i^{lbest}\left(t+1\right)$ of $i$-th object $u_i$ at iteration $t+1$ is updated according to (5) in the following

$$X_i^{lbest}\left(t+1\right)=\left\{\begin{array}{l}{X}_i\left(t+1\right),\hspace{0.17em}\hspace{0.17em}\mathrm{if}\hspace{0.17em}f\left(X_i\left(t+1\right)\right)<f\left(X_i^{lbest}\left(t\right)\right)\\ X_i^{lbest}\left(t\right),\hspace{0.33em} \hspace{0.17em}\mathrm{otherwise}\end{array}\right., \mathrm{for} i=1,2,\cdots ,n_o,$$

(5)

The global best position $X_o^{gbest}\left(t+1\right)$ in elementary membrane ${\sigma }_o$ at iteration $t+1$ is updated according to (6) in the following

$$X_o^{gbest}\left(t+1\right)=\left\{\begin{array}{l}{X}_i^{lbest}\left(t+1\right),\hspace{0.17em}\hspace{0.17em}\mathrm{if}\hspace{0.17em}f\left(X_i^{lbest}\left(t+1\right)\right)<f\left(X_o^{gbest}\left(t\right)\right)\\ X_o^{gbest}\left(t\right),\hspace{0.33em} \hspace{0.17em}\hspace{1em}\mathrm{otherwise}\end{array}\right., \mathrm{for} i=1,2,\cdots ,n_o,$$

(6)

A linear increasing strategy is adopted to update dynamically the values of inertia weight $\omega$, which is given by (7) in the following

$$\omega \left(t\right)={\omega }_{\min }+\left({\omega }_{\max }-{\omega }_{\min }\right)\ast \left(t/t_{\max }\right),$$

(7)

where ${\omega }_{\min }$ and ${\omega }_{\max }$ are the minimum and maximum of inertia weight, and $t_{\max }$ is the maximum of iterations.

4.2.2. Local Evolution Rules for Objects

As local evolution rules, the evolutionary model of improved SPSO is adopted to update the velocity and position of objects, which are only executed on sub-membranes ${\sigma }_{o_h}$ ($1\le h\le H$) of elementary membrane ${\sigma }_o$ ($2\le o\le m$), where $H$ is the number of sub-membranes of elementary membrane ${\sigma }_o$. The velocity $V_i\left(t+1\right)$ of $i$-th object $u_i$ in sub-membranes ${\sigma }_{o_h}$ at iteration $t+1$, is determined by (8) in the following

$$V_i\left(t+1\right)=\chi \ast \left(V_i\left(t\right)+r_3\ast \left(X_{n^{\#}}^{nbest}\left(t\right)-X_i\left(t\right)\right)+r_4\ast \left(X_{o_h}^{gbest}\left(t\right)-X_i\left(t\right)\right)\right), \mathrm{for} i=1,2,\cdots ,n_{o_h},$$

(8)

where $\chi$ is constriction coefficient which is given by $\chi =2/\left|2-{\phi }_{\max }-\sqrt{{\phi }_{\max }^2-4{\phi }_{\max }}\right|$. $r_3$ and $r_4$ are two random numbers that uniformly distributed between 0 and ${\phi }_{\max }/2$. ${\phi }_{\max }$ is positive constant and is usually set to 4.1, where ${\phi }_{\max }=4.1$ [69]. $X_{n^{\#}}^{nbest}\left(t\right)$ is the mean of local best position for seven neighbors of object $u_i$ at iteration $t$, for $X_{n^{\#}}^{nbest}(t)=1/7\ast {\displaystyle {\sum }_{n=1}^7{X}_n^{lbest}\left(t\right)}$, $X_n^{lbest}\left(t\right)$ is the local best position of neighbors $u_n$ for object $u_i$ at iteration $t$. The selection of neighbors $u_n$ for object $u_i$ are based on FER according to Equatiion (1). The position $X_i\left(t+1\right)$ of $i$-th object $u_i$ in sub-membrane ${\sigma }_{o_h}$ at iteration $t+1$ is determined by Equatiion (4). The local best position $X_i^{lbest}\left(t+1\right)$ and global best position $X_{o_h}^{gbest}\left(t+1\right)$ at iteration $t+1$ are also determined by Equations (5) and (6).

4.3. Evolution Rules for Membranes

There are two types of evolution rules with promoter/inhibitor $p$ ($p\in \mathsf{\Gamma }$) for membranes, including membrane creation rules with promoter $p^{\ast }$ ($p^{\ast }\in \mathsf{\Gamma }$) and membrane dissolution rules with inhibitor $\neg p^{\ast }$ ($\neg p^{\ast }\in \mathsf{\Gamma }$). The evolution rules for membranes with promoter/inhibitor are only achieved on elementary membranes ${\sigma }_o$ ($2\le o\le m$) and its sub-membranes ${\sigma }_{o_h}$ ($1\le h\le H$) in the proposed DSPSO-ECP. When an evolution rule for the membrane is applied, skin membrane ${\sigma }_1$, and other membranes not involved in membrane evolution and environment will keep the same configuration at the current iteration or time.

4.3.1. Membrane Creation Rules with Promoter

Membrane creation rules with the promoter in the proposed DSPSO-ECP are of the form, ${\left[u\left|{}_{p^{\ast }}\right.\right]}_o\to {\left[u\right]}_{o_1}{\left[u\right]}_{o_2}\cdots {\left[u\right]}_{o_H}$, for $2\le o\le m$. It only can be executed on a time or iteration if there is an elementary membrane ${\sigma }_o$ contains multiset of promoter objects $p^{\ast }$ and objects $u$ ($u\in \mathsf{\Gamma }$). When a membrane creation rule associated with elementary membrane ${\sigma }_o$ is applied, a group of sub-membranes ${\sigma }_{o_h}$ for elementary membrane ${\sigma }_o$ are created, which consist of $H$ number of sub-membranes. All objects in elementary membrane ${\sigma }_o$ are copied into sub-membrane ${\sigma }_{o_h}$.

Figure 2 gives an example of membrane creation. When elementary membrane 3 contains a multiset of promoter objects $p^{\ast }$ and objects, a group of sub-membranes for elementary membrane 3, which consist of $H$ number of membranes, will be created. All objects in sub-membrane ${\sigma }_{o_h}$ ($1\le h\le H$), as shown in the shade area of Figure 2, are copied from elementary membranes 3.

4.3.2. Membrane Dissolution Rules with Inhibitor

Membrane dissolution rules with an inhibitor in the proposed DSPSO-ECP are of the form, ${\left[u\left|{}_{\neg p^{\ast }}\right.\right]}_j\to \lambda$, where $2\le j\le m$ or $1\le j\le H$. It only can be executed on a time or iteration if there is an elementary membrane ${\sigma }_o$ or its corresponding sub-membranes ${\sigma }_{o_h}$ contain multiset of inhibitor objects $\neg p^{\ast }$. When a membrane dissolution rule associated with membrane ${\sigma }_j$ is applied, membrane ${\sigma }_j$ is dissolved, and all objects in membrane ${\sigma }_j$ are decomposed at the current time or iteration. Especially, $\lambda$ is an empty symbol, where $\lambda \in \mathsf{\Gamma }$.

Figure 3 gives an example of membrane dissolution. When elementary membrane 3 and its sub-membranes 1 to $H$ contain multiset of inhibitor objects $\neg p^{\ast }$ and objects, elementary membrane 3 and corresponding sub-membranes will be dissolved. Especially, if sub-membrane 2 of elementary membrane 3 not contains any multiset of inhibitor objects, as shown in blue area of Figure 3. Then sub-membrane 2 will turn into new elementary membrane 3 and continue to perform subsequent computation, elementary membrane 3 and other sub-membranes are dissolved through membrane dissolution rules.

4.4. Communication Rules for Objects

In DSPSO-ECP, communication rules based on evolutional symport rules of ECP are introduced to the exchanging and sharing of global information for different membranes. There are two kinds of communication rules for objects based on the set membership, including communication rules from elementary membrane ${\sigma }_o$ ($2\le o\le m$) to non-elementary membrane ${\sigma }_1$, and communication rules from non-elementary membrane ${\sigma }_1$ to elementary membrane ${\sigma }_o$. More details are given in the following.

4.4.1. Communication Rules from Elementary Membrane to Non-Elementary Membrane

Communication rules from elementary membrane ${\sigma }_o$ to non-elementary membrane ${\sigma }_1$ are of the form, $R_{o1}^{\prime }:{\left[u_o^{gbest}\left(t\right)\right]}_o{\left[\lambda \right]}_1\to {\left[\lambda \right]}_o{\left[u_o\left(t\right)\right]}_1$, for $2\le o\le m$. It only can be executed on a time or iteration if there is an elementary membrane ${\sigma }_o$ in a configuration contains multiset of global best $u_o^{gbest}$. When a communication rule at iteration $t$ is applied, global best $u_o^{gbest}\left(t\right)$ in elementary membrane ${\sigma }_o$ evolve to position of object $u_o\left(t\right)$ and send to non-elementary membrane ${\sigma }_1$. Thus, non-elementary membrane ${\sigma }_1$ only contains $m-1$ number of objects coming from elementary membranes ${\sigma }_2$ to ${\sigma }_m$. Objects with the best fitness values among these transferred objects will be selected in non-elementary membrane ${\sigma }_1$ as the global best at iteration $t$, which can be denoted by $u_1^{gbest}\left(t\right)$.

4.4.2. Communication Rules from Non-Elementary Membrane to Elementary Membrane

Communication rules from non-elementary membrane ${\sigma }_1$ to elementary membrane ${\sigma }_o$ are of the form, $R_{1o}^{\prime }:{\left[\lambda \right]}_o{\left[u_1^{gbest}\left(t\right)\right]}_1\to {\left[u_o^{gbest}\left(t\right)\right]}_o{\left[\lambda \right]}_1$, for $2\le o\le m$. It only can be executed on a time or iteration if there is a non-elementary membrane ${\sigma }_1$ in a configuration contains multiset of global best $u_1^{gbest}\left(t\right)$. When a communication rule at iteration $t$ is applied, global best $u_1^{gbest}\left(t\right)$ in non-elementary membrane ${\sigma }_1$ evolve to global best $u_o^{gbest}\left(t\right)$ and send to elementary membrane ${\sigma }_o$. At the same time, global best $u_1^{gbest}\left(t\right)$ in non-elementary membrane ${\sigma }_1$ is sent to environment ${\sigma }_0$, which is regarded as computational result of DSPSO-ECP at iteration $t$. The communication relationship in the proposed DSPSO-ECP is shown in Figure 4, and the direction of information transmission between different membranes is graphically depicted as the direction of black arrow.

4.5. Computation Procedure of DSPSO-ECP

(1)

Initialization mechanism

<1> Parameters initialization

In the proposed DSPSO-ECP, the total number of initial objects is denoted by $N$. Each elementary membrane contains the same number of initial objects as we mentioned above, where $n_o=n$. The number of elementary membranes is denoted by $m-1$, and the number of sub-membranes for the elementary membrane is denoted by $H$, noted that each elementary membrane has the same number of sub-membranes. The initial membrane structure of DSPSO-ECP is depicted in Figure 1;

<2> Velocity and position initialization

The velocity and position of initial objects in each elementary membrane are randomly generated from search space. The objective of the optimization problem is to minimize the fitness function;

<3> Update local best and global best

Update local best $u_i^{lbest}$ ($1\le i\le n_o$) and global best $u_o^{gbest}$ for all initial objects in each elementary membrane according to Equations (5) and (6). Noted that evolution rules with promoter/inhibitor for membranes only executed on a configuration at a moment when promoter and inhibitor conditions are satisfied;

(2)

Evolution mechanism

<1> Basic evolution mechanism for elementary membranes

Step 1: Update velocity and position

Basic evolution rules for objects are adopted to update the velocity and position of objects in each elementary membrane according to Equations (3), (4) and (7);

Step 2: Update local best and global best

Update local best $u_i^{lbest}$ and global best $u_o^{gbest}$ for all objects in each elementary membrane according to Equations (5) and (6);

<2> Local evolution mechanism for sub-membranes

Step 1: Create sub-membranes

In particular, promoter $p^{\ast }$ is regarded as judgment condition, which can be described as follows. When global best in the elementary membrane ${\sigma }_o$ cannot be further improved for $limit$ iterations, where $p^{\ast }=\left\{limit>2\right\}$, a group of sub-membranes ${\sigma }_{o_h}$ ($1\le h\le H$) for elementary membrane ${\sigma }_o$ are created through membrane creation rules;

Step 2: Update velocity and position

Local evolution rules for objects are adopted to update velocity and position of objects in each sub-membrane according to Equations (1), (4) and (8);

Step 3: Update global best

Update global best $u_{o_h}^{gbest}$ in each sub-membrane according to Equation (6);

Step 4: Dissolve membranes

Inhibitor $-p^{\ast }$ is regarded as a comparison condition, which is used to compare the fitness values of global best in elementary membrane ${\sigma }_o$ and its corresponding sub-membranes ${\sigma }_{o_h}$, where $-p^{\ast }=\left\{f\left(u_{o_h}^{gbest}\right)>f\left(u_o^{gbest}\right)\right\}$, the corresponding sub-membranes are dissolved through membrane dissolution rules. Otherwise, the sub-membrane ${\sigma }_{o_h}$ will replace its corresponding elementary membrane ${\sigma }_o$ and continue to perform the next evolution;

(3)

Communication mechanism

<1> Communication rules from elementary membranes to non-elementary membrane

For each elementary membrane, the first kind of communication rules based on evolutional symport rules of ECP is adopted to send global best $u_o^{gbest}$ of elementary membrane ${\sigma }_o$ to non-elementary membrane ${\sigma }_1$, and to evolve into the position of object $u_o$. Especially, if a group of sub-membranes are created, for each sub-membrane, the global best $u_{o_h}^{gbest}$ of sub-membrane ${\sigma }_{o_h}$ is also sent to non-elementary membrane ${\sigma }_1$ at the same time. The best object among these transferred global best is stored as global best $u_1^{gbest}$ in non-elementary membrane ${\sigma }_1$;

<2> Communication rules from non-elementary membrane to elementary membranes

For each elementary membrane, the second kind of communication rules based on evolutional symport rules of ECP is used to send global best $u_1^{gbest}$ of non-elementary membrane ${\sigma }_1$ to elementary membrane ${\sigma }_o$, and to evolve into global best $u_o^{gbest}$ of elementary membrane ${\sigma }_o$. At the same time, global best $u_1^{gbest}$ of non-elementary membrane ${\sigma }_1$ is also sent to environment ${\sigma }_0$ as the computational result of DSPSO-ECP for each iteration;

(4)

Halting and output

The evolution and communication mechanism of the proposed DSPSO-ECP are repeatedly implemented with an iteration form until the termination criterion is satisfied. The termination criterion of DSPSO-ECP is stopped to whether the maximum number of iterations is reached. When the system halts, the last global best, which is stored in the environment, is viewed as the final computational result of the proposed DSPSO-ECP. The pseudo-code of computation for the proposed DSPSO-ECP is given in the following.

Input: $N$, $m$, $t_{\max }$, $c_1$, $c_2$, $r_1$, $r_2$, ${\phi }_{\max }$, $Ne$, ${\omega }_{\min }$, ${\omega }_{\max }$, $H$, $limit$;

(1) Initialization mechanism  <1> Velocity and position initialization    for $i=1$ to $N$      Velocity of object $u_i$: $\hspace{0.33em}{V}_i=rand\left(s^l,s^u\right)$;      Position of object $u_i$: $X_i=rand\left(s^l,s^u\right)$;    end

<2> Update the local and global best;

$\mathrm{for} o=2$ to $m$

for $i=1$ to $n_o$ ($n_o=N/\left(m-1\right)$)

if $f\left(X_i\right)<f\left(X_i^{lbest}\right)$        Local best of $u_i^{lbest}$: $X_i^{lbest}=X_i$;

if $f\left(X_i^{lbest}\right)<f\left(X_i^{gbest}\right)$

Global best of $u_o^{gbest}$: $X_i^{gbest}=X_i^{lbest}$;

end if

end if

end    end

for $t=1$ to $t_{\max }$

(2) Evolution mechanism

<1> Basic evolution mechanism for elementary membranes   for $o=2$ to $m$

for $i=1$ to $n_o$

Step 1: Update velocity and position;    $\omega \left(t\right)={\omega }_{\min }+\left({\omega }_{\max }-{\omega }_{\min }\right)\ast \left(t/t_{\max }\right)$;        $V_i(t+1)=\omega \ast V_i(t)+c_1\ast r_1\ast \left(X_i^{lbest}(t)-X_i(t)\right)+c_2\ast r_2\ast \left(X_o^{gbest}(t)-X_i(t)\right)$;        $X_i\left(t+1\right)=X_i\left(t\right)+V_i\left(t+1\right)$;    Step 2: Update local best and global best;    <2> Local evolution mechanism for sub-membranes    if $p^{\ast }=\left\{limit>2\right\}$    Step 1: Membrane creation, ${\sigma }_{o_1}$ to ${\sigma }_{o_H}$ of ${\sigma }_o$ are created;    Step 2: Update velocity and position;    $V_i\left(t+1\right)=\chi \ast \left(V_i\left(t\right)+r_3\ast \left(X_{n^{\#}}^{nbest}\left(t\right)-X_i\left(t\right)\right)+r_4\ast \left(X_{o_h}^{gbest}\left(t\right)-X_i\left(t\right)\right)\right)$;    $X_i\left(t+1\right)=X_i\left(t\right)+V_i\left(t+1\right)$;    Step 3: Update global best;    Step 4: Membrane dissolution, ${\sigma }_{o_1}$ to ${\sigma }_{o_H}$ are dissolved;    for $j=1$ to $H$       if $-p^{\ast }=\left\{f\left(u_{o_j}^{gbest}\right)>f\left(u_o^{gbest}\right)\right\}$        sub-membrane ${\sigma }_{o_j}$ is dissolved;       else         Global best: $u_o^{gbest}=u_{o_j}^{gbest}$;         Sub-membrane ${\sigma }_{o_j}$ replace elementary membrane ${\sigma }_o$;        end       end      end (3) Communication mechanism <1> Communication rules from ${\sigma }_o$ to ${\sigma }_1$      for $o=2$ to $m$      Position of object $u_o$: $X_o=u_o^{gbest}$;      if $f\left(X_o\right)<f\left(u_1^{gbest}\right)$        Global best of $u_1^{gbest}$: $u_1^{gbest}=X_o$;        end if      end   <2> Communication rules from ${\sigma }_1$ to ${\sigma }_o$ and ${\sigma }_0$      for $o=2$ to $m$      Global best of $u_o^{gbest}$: $u_o^{gbest}=u_1^{gbest}$;      end      Global best of $u_0^{gbest}$: $u_0^{gbest}=u_1^{gbest}$; end

(4) Halting and output

if $t>t_{\max }$

Best Position of P system: $u_0^{gbest}$;

Best Fitness of P system: $f\left(u_0^{gbest}\right)$;

end

Output: Best Position of P system, Best Fitness of P system;

4.6. Complexity Analysis

In this subsection, the complexity of proposed the DSPSO-ECP is discussed and analyzed. $N$ is the total number of initial objects located in elementary membranes. $m-1$ is the number of elementary membranes. $n_o$ ($2\le o\le m$) is the number of initial objects in elementary membrane ${\sigma }_o$, where $n_o=n$, and $N={\displaystyle {\sum }_{o=2}^m{n}_o}=n\ast \left(m-1\right)$. $t_{\max }$ is the maximum number of iterations. $D$ is the dimension of search space.

The computation of the proposed DSPSO-ECP contains three steps. At the first step of initialization computing, the computation time for all objects in an elementary membrane is denoted by $nD$. The complexity of initialization mechanism in DSPSO-ECP is $O\left(nD\right)$ due to the distributed parallel computing model of P systems. At the second step of evolution computing, the evolution rules for objects in elementary membranes and its sub-membranes are executed in parallel. The computation time needed by executing a basic evolution rule for all objects in an elementary membrane is $nD$, and the computation time needed by executing a local evolution rule for all objects in a sub-membrane is $n\left(n+D\right)$. Thus, the time needed by one evolution for all objects in system is $n\left(n+D\right)$, and the complexity of evolution mechanism in DSPSO-ECP is $O\left(n\left(n+D\right)\right)$. At the third step of communication computing, the computation time needed by executing a communication rule between elementary membranes and non-elementary membrane is set to 2, and the time needed by one communication in system is $2\left(m-1\right)$. Thus, the complexity of the communication mechanism in DSPSO-ECP is $O\left(2\left(m-1\right)\right)$. Therefore, the computation time of the whole P system needed by once iteration is $n\left(n+D\right)+2\left(m-1\right)$, and the total computation time of DSPSO-ECP is $nD+t_{\max }\left(n\left(n+D\right)+2\left(m-1\right)\right)$. Due to $m\ll n$, the complexity of the proposed DSPSO-ECP can be simplified to $O\left(n{t}_{\max }\left(n+D\right)\right)$.

5. Experimental Results and Analysis

In this section, computational experiments which are made on eight classic numerical benchmark functions are conducted to evaluate the efficiency of the proposed DSPSO-ECP. More details about these classic benchmark functions are introduced first. The effectiveness of DSPSO-ECP is compared with four existing PSO approaches, including classic PSO, Fitness-Euclidean distance ratio (FER-PSO) [69], particle swarm optimization with multiple subpopulations (MPSO) [70], and SPSO [68]. Especially, all comparative approaches including DSPSO-ECP, are implemented on MATLAB (2016b) and all experiments are conducted on a DELL desktop computer, which made in Dell Technologies, Xiamen, China, with an Intel 8.00 GHZ i7-8550U processor and 16GB of RAM in a Window 10 Environment.

5.1. Benchmark Functions

In this subsection, eight classic numerical benchmark functions are adopted to evaluate the exploitation efficiency and exploration efficiency of competitive approaches, which are reported in previous studies and works [71], including unimodal and multimodal functions. Especially, the objective of these benchmark functions is minimized, and their domains are shown in

Table 1. Benchmark Functions.

Benchmark Functions	Function Expression	Domain	${\mathit{F}}_{\mathbf{min}}$
Griewank	$f_1\left(x\right)=\frac{1}{1000}{\displaystyle {\sum }_{i=1}^D{x}_i^2}-{\displaystyle {\prod }_{i=1}^D\cos \left(\frac{x_i}{\sqrt{i}}\right)+1}$	$x_i\in {\left[-600,600\right]}^D$	0
Ackley	$\begin{array}{l}{f}_2\left(x\right)=20+e-20\exp \left(-0.2\sqrt{\frac{1}{D}{\displaystyle {\sum }_{i=1}^D{x}_i^2}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}-\exp \left(\frac{1}{D}{\displaystyle {\sum }_{i=1}^D\cos 2\pi x_i}\right)\end{array}$	$x_i\in {\left[-32.768,32.768\right]}^D$	0
Tablet	$f_3\left(x\right)={10}^6{x}_1^2+{\displaystyle {\sum }_{i=2}^D{x}_i^2}$	$x_i\in {\left[-100,100\right]}^D$	0
Schwefel 2.22	$f_4\left(x\right)={\displaystyle {\sum }_{i=1}^D\left|x_i\right|}+{\displaystyle {\prod }_{i=1}^D\left|x_i\right|}$	$x_i\in {\left[-10,10\right]}^D$	0
Zakharov	$\begin{array}{l}{f}_5\left(x\right)={\displaystyle {\sum }_{i=1}^D{x}_i^2}+{\left({\displaystyle {\sum }_{i=1}^{D}0.5i{x}_i}\right)}^2\\ \hspace{0.33em}\hspace{0.33em}\hspace{1em}\hspace{0.33em}\hspace{1em}\hspace{0.33em}+{\left({\displaystyle {\sum }_{i=1}^{D}0.5i{x}_i}\right)}^4\end{array}$	$x_i\in {\left[-5,10\right]}^D$	0
Sphere	$f_6\left(x\right)={\displaystyle {\sum }_{i=1}^D{x}_i^2}$	$x_i\in {\left[-5.12,5.12\right]}^D$	0
Schwefel 2.21	$f_7\left(x\right)=\max \left|x_i\right|$	$x_i\in {\left[-100,100\right]}^D$	0
Dixon−Price	$f_8\left(x\right)={\left(x_1-1\right)}^2+{\displaystyle {\sum }_{i=2}^{D}i{\left(2x_i^2-x_{i-1}\right)}^2}$	$x_i\in {\left[-10,10\right]}^D$	0

.

The domains and minimum of eight classic numerical benchmark functions, including Griewank, Ackley, Tablet, Schwefel 2.22, Zakharov, Sphere, Schwefel 2.21, and Dixon−Price, are depicted in Table 1. The dimension is set to 2 and 10, where $D=2$ and $D=10$. The shape and ranges of these benchmark functions with $D=2$ are depicted in Figure 5.

5.2. Comparison with Other Exising Approaches

The classic PSO and three improved PSO approaches, such as FER-PSO, MPSO, and SPSO, are used as comparison approaches as we have mentioned above, which have been reported in previous literature and studies. The neighbor’s information based on FER is adopted to guide the trajectory of particles in FER-PSO. The population of particles is divided into multiple subpopulations in MPSO, and a migration strategy for subpopulation is adopted to enhance the global search ability. The values of adjusting parameters in these comparative approaches, including the proposed DSPSO-ECP, are the best ones which are listed in

Table 2. Parameter settings in the experiment.

Parameters	PSO	FER-PSO	MPSO	SPSO	DSPSO-ECP
$t_{\max }$	100	100	100	100	100
$N$	100	100	100	100	100
$c_1$$,c_2$	(2, 2)	(0.68, 0.68)	(0.5, 2.5)	(2, 2)	(2, 2)
$r_1$$,r_2$	(0, 1)	(0.2, 0.5)	(0.2, 0.5)	(0, 1)	(0, 1)
${\phi }_{\max }$	−	4.1	−	−	4.1
$\chi$	−	0.708	−	−	0.708
$r$	−	−	0.4	−	−
$mi$	−	−	5	−	1
${\omega }_{\min }$$,{\omega }_{\max }$	(0.4, 1.2)	−	(0.4, 0.9)	(0.2, 0.4)	(0.4, 1.2)
$Ne$	−	−	−	7	7
$m$	−	−	4	−	5 [72]
$H$				14	14
$limit$	−	−	−	2	2

.

DSPSO-ECP and other comparative approaches are run 50 independent times in order to eliminate the effects of random factors. Simple statistics, including worst values (Worst), best values (Best), mean values (Mean), and standard deviations (S.D.) of the fitness function, are introduced to measure the effectiveness of comparative approaches. Convergence results obtained by these approaches on eight benchmark functions with $D=2$ are depicted in Figure 6.

Compared with the classic PSO, FER-PSO, MPSO, and SPSO approach, the convergence curve of the proposed DSPSO-ECP declined quickly during the iterations as shown in Figure 6. The computational results obtained by these comparative approaches on eight benchmark functions are reported in

Table 3. Computational results of comparative approaches on eight benchmark functions ($D=2$).

Functions	Statistics	Comparative Approaches
		PSO	FER-PSO	MPSO	SPSO	DSPSO-ECP
Griewank	Worst	4.70 × 10−2	1.33 × 10−2	1.02 × 10−2	7.40 × 10−3	8.88 × 10−16
	Best	3.18 × 10−5	5.49 × 10−7	8.30 × 10−9	3.80 × 10−9	0.00 × 10−0
	Mean	1.11 × 10−2	5.00 × 10−3	2.89 × 10−3	1.83 × 10−3	1.78 × 10−17
	S.D.	9.02 × 10−3	4.23 × 10−3	3.53 × 10−3	3.17 × 10−3	1.26 × 10−16
Ackley	Worst	5.09 × 10−3	5.30 × 10−4	1.63 × 10−4	2.89 × 10−5	0.00 × 10−0
	Best	1.53 × 10−4	3.55 × 10−5	1.64 × 10−6	5.56 × 10−7	0.00 × 10−0
	Mean	1.22 × 10−3	2.00 × 10−4	3.71 × 10−5	8.06 × 10−6	0.00 × 10−0
	S.D.	1.04 × 10−3	1.37 × 10−4	3.26 × 10−5	5.78 × 10−6	0.00 × 10−0
Tablet	Worst	3.05 × 10−1	1.13 × 10−2	1.36 × 10−3	4.73 × 10−5	4.22 × 10−25
	Best	1.72 × 10−4	3.40 × 10−6	5.94 × 10−7	8.07 × 10−10	5.41 × 10−30
	Mean	3.02 × 10−2	1.36 × 10−3	9.43 × 10−5	5.84 × 10−6	1.98 × 10−26
	S.D.	4.15 × 10−2	2.06 × 10−3	2.00 × 10−4	1.01 × 10−5	6.54 × 10−26
Schwefel 2.22	Worst	1.19 × 10−3	5.70 × 10−5	1.84 × 10−5	4.85 × 10−6	1.16 × 10−16
	Best	2.07 × 10−5	5.23 × 10−7	2.76 × 10−7	5.50 × 10−8	2.77 × 10−19
	Mean	1.73 × 10−4	2.53 × 10−5	3.85 × 10−6	1.04 × 10−6	1.74 × 10−17
	S.D.	1.79 × 10−4	1.65 × 10−5	3.57 × 10−6	1.05 × 10−6	2.55 × 10−17
Zakharov	Worst	8.95 × 10−8	2.95 × 10−9	1.04 × 10−10	1.18 × 10−11	1.78 × 10−32
	Best	6.64 × 10−11	4.95 × 10−12	5.76 × 10−14	3.87 × 10−16	2.65 × 10−37
	Mean	1.09 × 10−8	4.74 × 10−10	1.60 × 10−11	1.57 × 10−12	7.31 × 10−34
	S.D.	1.59 × 10−8	6.87 × 10−10	2.29 × 10−11	2.55 × 10−12	2.67 × 10−33
Sphere	Worst	6.18 × 10−8	1.06 × 10−9	7.60 × 10−11	4.56 × 10−12	2.04 × 10−32
	Best	1.02 × 10−11	1.48 × 10−12	5.30 × 10−15	8.80 × 10−16	1.00 × 10−38
	Mean	8.81 × 10−9	1.57 × 10−10	5.56 × 10−12	3.67 × 10−13	5.45 × 10−34
	S.D.	1.20 × 10−8	2.56 × 10−10	1.29 × 10−11	7.05 × 10−13	2.88 × 10−33
Schwefel 2.21	Worst	1.19 × 10−3	5.70 × 10−5	1.84 × 10−5	4.85 × 10−6	1.16 × 10−16
	Best	2.07 × 10−5	5.23 × 10−7	2.76 × 10−7	5.50 × 10−8	2.77 × 10−19
	Mean	1.73 × 10−4	2.53 × 10−5	3.85 × 10−6	1.04 × 10−6	1.74 × 10−17
	S.D.	1.79 × 10−4	1.65 × 10−5	3.57 × 10−6	1.05 × 10−6	2.55 × 10−17
Dixon−Price	Worst	1.66 × 10−4	3.00 × 10−7	3.62 × 10−8	1.98 × 10−9	1.97 × 10−31
	Best	1.50 × 10−9	5.60 × 10−11	1.19 × 10−12	1.88 × 10−14	3.70 × 10−32
	Mean	5.65 × 10−6	2.79 × 10−8	2.20 × 10−9	8.75 × 10−11	5.03 × 10−32
	S.D.	2.49 × 10−5	4.80 × 10−8	5.22 × 10−9	2.97 × 10−10	3.32 × 10−32

.

As indicated by Table 3, statistical results obtained by DSPSO-ECP, such as Worst, Best, Mean, and S.D., are the minimum compared with other approaches. The mean time of computation obtained by these comparative approaches on eight benchmark functions is listed in

Table 4. Mean time obtained by comparative approaches on eight benchmark functions. (Units: s).

Functions	Comparative Approaches
	PSO	FER-PSO	MPSO	SPSO	DSPSO-ECP
Griewank	0.1480	0.1422	0.4000	11.9162	9.1202
Ackley	0.1315	0.1450	0.3872	7.8330	7.7486
Tablet	0.1368	0.1445	0.3844	8.7875	6.8740
Schwefel 2.22	0.1400	0.1546	0.4113	8.8056	6.9248
Zakharov	0.1770	0.1462	0.3910	8.0578	5.6925
Sphere	0.1419	0.1372	0.3718	8.1060	5.0243
Schwefel 2.21	0.1238	0.1299	0.2182	7.9111	5.8182
Dixon−Price	0.1245	0.1271	0.2145	7.7758	6.2489

, and the computation time of the proposed DSPSO-ECP is acceptable.

Furthermore, the convergence results obtained by these comparative approaches on eight benchmark functions with $D=10$ are depicted in Figure 7. Statistical results of the fitness function are reported in

Table 5. Computational results of comparative approaches on eight benchmark functions ($D=10$).

Functions	Statistics	Comparative Approaches
		PSO	FER-PSO	MPSO	SPSO	DSPSO-ECP
Griewank	Worst	2.51 × 10−1	1.65 × 10−1	1.92 × 10−1	1.40 × 10−1	0.00 × 10−0
	Best	2.22 × 10−2	1.08 × 10−13	3.20 × 10−2	1.23 × 10−2	0.00 × 10−0
	Mean	9.55 × 10−2	6.82 × 10−2	9.12 × 10−2	6.20 × 10−2	0.00 × 10−0
	S.D.	5.37 × 10−2	3.47 × 10−2	3.91 × 10−2	2.90 × 10−2	0.00 × 10−0
Ackley	Worst	9.66 × 10−9	6.93 × 10−9	1.16 × 10−0	6.22 × 10−15	0.00 × 10−0
	Best	1.07 × 10−10	1.36 × 10−10	6.22 × 10−15	2.66 × 10−15	0.00 × 10−0
	Mean	1.80 × 10−9	9.55 × 10−10	3.24 × 10−11	2.74 × 10−15	0.00 × 10−0
	S.D.	1.89 × 10−9	1.12 × 10−9	1.70 × 10−10	5.02 × 10−16	0.00 × 10−0
Tablet	Worst	2.03 × 10−15	4.55 × 10−16	1.07 × 10−21	1.26 × 10−51	7.71 × 10−18
	Best	1.20 × 10−18	2.88 × 10−19	1.15 × 10−25	1.95 × 10−54	6.00 × 10−100
	Mean	1.81 × 10−17	5.97 × 10−17	5.16 × 10−23	1.64 × 10−52	1.54 × 10−39
	S.D.	1.16 × 10−17	9.85 × 10−17	1.74 × 10−22	4.28 × 10−53	1.09 × 10−23
Schwefel 2.22	Worst	3.04 × 10−10	4.48 × 10−10	1.84 × 10−10	1.35 × 10−29	7.62 × 10−69
	Best	5.19 × 10−12	6.81 × 10−12	3.06 × 10−14	4.68 × 10−31	1.72 × 10−87
	Mean	7.18 × 10−11	5.95 × 10−11	4.20 × 10−12	5.16 × 10−30	2.53 × 10−70
	S.D.	6.92 × 10−11	8.30 × 10−11	2.60 × 10−11	3.95 × 10−30	1.21 × 10−62
Zakharov	Worst	1.03 × 10−16	5.51 × 10−18	4.51 × 10−25	2.84 × 10−52	5.03 × 10−14
	Best	1.27 × 10−21	1.85 × 10−21	1.45 × 10−29	8.24 × 10−58	3.85 × 10−87
	Mean	5.79 × 10−19	4.36 × 10−19	4.26 × 10−26	2.49 × 10−54	1.01 × 10−57
	S.D.	8.14 × 10−19	8.63 × 10−19	8.76 × 10−26	5.57 × 10−55	7.12 × 10−23
Sphere	Worst	1.56 × 10−18	1.03 × 10−19	8.44 × 10−28	4.16 × 10−56	8.17 × 10−24
	Best	1.85 × 10−22	6.99 × 10−22	4.71 × 10−31	4.02 × 10−60	1.12 × 10−93
	Mean	5.92 × 10−20	1.82 × 10−20	1.32 × 10−28	4.72 × 10−57	1.63 × 10−62
	S.D.	2.31 × 10−19	2.28 × 10−20	2.18 × 10−28	8.95 × 10−57	1.16 × 10−46
Schwefel 2.21	Worst	5.75 × 10−05	4.94 × 10−5	6.01 × 10−7	4.03 × 10−20	8.11 × 10−67
	Best	7.83 × 10−7	7.39 × 10−7	1.23 × 10−9	8.61 × 10−23	1.79 × 10−94
	Mean	1.40 × 10−5	1.10 × 10−5	5.71 × 10−8	2.98 × 10−21	1.62 × 10−68
	S.D.	1.46 × 10−5	8.30 × 10−6	1.08 × 10−7	6.35 × 10−21	1.15 × 10−67
Dixon−Price	Worst	3.21 × 102	3.21 × 102	6.67 × 10−1	6.72 × 10−1	6.67 × 10−1
	Best	6.67 × 10−1	1.97 × 10−31	1.43 × 10−11	5.53 × 10−9	9.25 × 10−23
	Mean	4.74 × 101	7.03 × 10−0	6.13 × 10−1	6.40 × 10−1	5.75 × 10−1
	S.D.	8.89 × 101	4.53 × 101	1.83 × 10−1	1.32 × 10−1	2.19 × 10−1

. These computational results validate the efficiency of the proposed DSPSO-ECP in solving single-modal and multimodal function optimization problems with $D=2$ and $D=10$.

5.3. Friedman Test Statistics

In this subsection, the Friedman test is introduced to further investigate the performance of the proposed DSPSO-ECP. The Mean of fitness function obtained by these comparative approaches, as listed in Table 3 and Table 5, are used in the Friedman test. The null hypothesis is that all comparative approaches, including the proposed DSPSO-ECP, have the same performance for any benchmark function. Mathematically, the Friedman test works as follows [73].

In the Friedman test, the benchmark function is treated as a random sample, which is denoted by $i$, for $1\le i\le 8$, and the comparative approach is viewed as treatment, which is denoted by $j$, for $1\le j\le 5$. The mean of the fitness function obtained by comparative approaches for each benchmark function are ranked from largest to smallest [74]. The rank of the comparative approach $j$ on the benchmark function $i$ is denoted by $r_{ij}$, the mean of rank is $\frac{1}{2}\left(p+1\right)$, in this case, three, where $p$ is the number of comparative approaches, in this case, five. The Friedman test statistic ${\chi }_r^2$ is given by (9) in the following

$${\chi }_r^2=\frac{12}{np\left(p+1\right)}{\displaystyle \sum _{j=1}^p{\left({\displaystyle \sum _{i=1}^n{r}_{ij}}\right)}^2}-3n\left(p+1\right)$$

(9)

where $n$ is the number of benchmark functions, in this case, eight. The Friedman test statistic follows a Chi-squared distribution with $p-1$ degrees of freedom. Ranks of comparative approaches on benchmark function with $D=2$ and $D=10$ are presented in

Table 6. Ranks of averages values for five comparative approaches (Mean) and computation of the Friedman test statistic ($D=2$).

Functions	PSO	FER-PSO	MPSO	SPSO	DSPSO-ECP
Griewank	5	4	3	2	1
Ackley	5	4	3	2	1
Tablet	5	4	3	2	1
Schwefel 2.22	5	4	3	2	1
Zakharov	5	4	3	2	1
Sphere	5	4	3	2	1
Schwefel 2.21	5	4	3	2	1
Dixon−Price	5	4	3	2	1
Total Rank	40	32	24	16	8
Average Rank	5	4	3	2	1
Deviation	2	1	0	–1	–2

and

Table 7. Ranks of averages values for five comparative approaches (Mean) and computation of the Friedman test statistic ($D=10$).

Functions	PSO	FER-PSO	MPSO	SPSO	DSPSO-ECP
Griewank	5	3	4	2	1
Ackley	5	4	3	2	1
Tablet	5	4	3	1	2
Schwefel 2.22	5	4	3	2	1
Zakharov	5	4	3	2	1
Sphere	5	4	3	2	1
Schwefel 2.21	5	4	3	2	1
Dixon−Price	5	4	3	2	1
Total Rank	40	31	25	15	9
Average Rank	5	3.875	3.125	1.875	1.125
Deviation	2	0.875	0.125	–1.125	–1.875

. With $p-1=4$ degrees of freedom, the critical value of statistic which is denoted by ${\chi }^2$ at the significance level $\alpha =5\%$, is set to 9.488, where ${\chi }^2=9.488$. The Friedman test statistics with different search dimensions are computed using the ranks of Table 6 and Table 7, and the results are, ${\chi }_r^2=32>9.488$ with $D=2$, and ${\chi }_r^2=30.6>9.488$ with $D=10$. Thus, the conclusion of the Friedman test is to reject the null hypothesis, i.e., the comparative approaches are significantly different. It was, in this comparison experiment, different optimization approaches have significantly different performances on eight benchmark functions with $D=2$ and $D=10$.

6. Image Segmentation

In this section, the experimental results obtained by existing segmentation approaches, including DSPSO-ECP, are reported and discussed in order to further evaluate the efficiency of the proposed DSPSO-ECP. All segmentation approaches are implemented on MATLAB (2016b) and all experiments are conducted on a DELL desktop computer with an Intel 8.00 GHZ i7-8550U processor and 16GB of RAM in a Windows 11 Environment.

6.1. Benchmark Images

Eight benchmark test images are adopted in the comparison experiment, including Stones, Island, Tiger, Worker, Surfer I, Teste4, Church, and Cow, which are provided by Berkeley Segmentation Dataset and Benchmark (BSDS300) [75], and the size of test images is set to 481 × 321. The original images are depicted in Figure 8.

6.2. Objective Function

Ostu’s threshold segmentation method is a classical approach for solving grey image segmentation problems, which is initiated by Ostu [76]. Discriminant criterion of Ostu’s method is introduced to evaluate the performance of the experimental approach according to maximize between-class variance. The discriminant criterion of Ostu’s method is described as follows. A grayscale image $I$ contains $N$ pixels is divided into $K+1$ classes, the values of the minimum and maximum of grey levels for image $I$ are set to 0 and $L$, usually, $L=255$. Then a group of thresholds $\left\{t_1,t_2,\cdots ,t_K\right\}$ contain $K$ thresholds are adopted to segment the original image $I$ into $K+1$ distinct classes $\left\{C_0,C_1,\cdots ,C_K\right\}$, which can be described by (10) in the following

$$\begin{gathered} C_0=\left\{p\left(x,y\right)\in I\left|0\le p\left(x,y\right)\le t_1\right.\right\}, \\ {C}_1=\left\{p\left(x,y\right)\in I\left|t_1+1\le p\left(x,y\right)\le t_2\right.\right\}, \\ ,\dots \dots , \\ {C}_K=\left\{p\left(x,y\right)\in I\left|t_K+1\le p\left(x,y\right)\le L\right.\right\} \end{gathered}$$

(10)

where $C_K$ is $K$-th class in image $I$. $t_K$ is $K$-th threshold of dividing thresholds $\left\{t_1,t_2,\cdots ,t_K\right\}$. $p\left(x,y\right)$ is the grey level of pixel located at $\left(x,y\right)$ in image $I$. The number of pixels which is included in $i$-th ($0\le i\le L$) grey level or frequency is denoted by $n_i$, and the probability of $i$-th grey level in image $I$ is denoted by $P_i$, where $P_i=n_i/N$ and $N={\displaystyle {\sum }_{i=0}^L{n}_i}$. The optimal thresholds $\{t_1^{\ast },t_2^{\ast },\cdots ,t_K^{\ast }\}$ are determined by (11), (12), (13), (14) and (15) in the following

$$\{t_1^{\ast },t_2^{\ast },\cdots ,t_K^{\ast }\}=\arg \hspace{0.33em}\max \{{\sigma }_B^2(t_1,t_2,\cdots ,t_K)\}$$

(11)

$${\omega }_0={\displaystyle \sum _{i=0}^{t_1}{P}_i}, {\omega }_1={\displaystyle \sum _{i=t_1+1}^{t_2}{P}_i}, \dots \dots ,{\omega }_k={\displaystyle \sum _{i=t_k+1}^{t_{k+1}}{P}_i}, \dots \dots , {\omega }_K={\displaystyle \sum _{i=t_K+1}^L{P}_i},$$

(12)

$${\mu }_0={\displaystyle \sum _{i=0}^{t_1}\left(i\ast P_i\right)/{\omega }_0}, {\mu }_1={\displaystyle \sum _{i=t_1+1}^{t_2}\left(i\ast P_i\right)/{\omega }_1}, \dots \dots ,{\mu }_k={\displaystyle \sum _{i=t_k+1}^{t_{k+1}}\left(i\ast P_i\right)/{\omega }_k}, \dots \dots , {\mu }_K={\displaystyle \sum _{i=t_K+1}^L\left(i\ast P_i\right)/{\omega }_K},$$

(13)

$${\sigma }_k^2={\omega }_k\ast {\left({\mu }_k-{\mu }_I\right)}^2, {\mu }_I={\displaystyle \sum _{i=0}^{L}i\ast P_i}, \mathrm{for} k=0,1,\cdots ,K,$$

(14)

$${\sigma }_B^2={\displaystyle \sum _{k=0}^K{\sigma }_k^2}={\sigma }_0^2+\cdots +{\sigma }_k^2+\cdots +{\sigma }_K^2$$

(15)

where ${\sigma }_B^2$ is between-class variance, which is also used as objective function of experimental approaches to measure the performance of segmentation approaches.

6.3. Comparison with Other Segmentation Approaches

In order to verify the efficiency of the proposed DSPSO-ECP for solving high dimensional multilevel thresholding image segmentation problems, the experimental experiments are conducted on eight benchmark test images with a different number of threshold values, i.e., $K=5,8,10$ [77]. Some nature-inspired approaches, such as PSO [32], DE [78], adaptive particle swarm optimization (APSO) [79], and complex chained P system based on the evolutionary mechanism (CCP) [80], are used as comparison approaches to validate the effectiveness of DSPSO-ECP. The main objective of these comparative approaches, including DSPSO-ECP, is to find a group of thresholds $\{t_1^{\ast },t_2^{\ast },\cdots ,t_K^{\ast }\}$ to maximize the between-class variance ${\sigma }_B^2$ according to Equations (11)–(15). Figure 9, Figure 10, Figure 11 and Figure 12 depicted the segmentation images of Stones, Island, Tiger, and Worker obtained by these comparative approaches.

It can be easily observed that the segmentation quality of test images has been gradually improved with the increasing number of threshold values. The grey-level histogram is also introduced to exhibit the threshold values obtained by comparative approaches more clearly. The grey-level histograms of the Worker image at a 10-threshold level are depicted in Figure 13.

The grey-level histograms of the Worker image with optimal threshold values obtained by all comparative approaches are shown in Figure 13, and the values of optimal thresholds obtained by comparative approaches are different. Simple statistics, including the Mean and S.D. of between-class variance obtained by comparative approaches on the different number of threshold values for 50 independent times are presented in

Table 8. Computational results of comparative approaches on 5, 8, and 10 number of threshold values.

Images	$\mathit{K}$	Statistics	Comparative Approaches
			PSO	DE	APSO	CCP	DSPSO-ECP
Stones	5	Mean	5339.8525	5340.1752	5339.3330	5340.1752	5340.7360
		S.D.	3.96 × 10−0	0.42 × 10−0	1.07 × 10−0	0.42 × 10−0	5.95 × 10−3
	8	Mean	5376.0678	5378.7184	5380.4948	5380.5312	5380.5419
		S.D.	2.73 × 10−0	0.61 × 10−0	0.02 × 10−0	0.01 × 10−0	2.35 × 10−2
	10	Mean	5385.9706	5387.5031	5388.8301	5388.9719	5389.2911
		S.D.	1.59 × 10−0	0.61 × 10−0	0.81 × 10−0	0.20 × 10−0	1.11 × 10−1
Island	5	Mean	1664.0941	1664.1178	1664.5995	1664.6022	1664.6061
		S.D.	4.65 × 10−1	1.95 × 10−1	1.41 × 10−2	9.37 × 10−3	2.33 × 10−13
	8	Mean	1706.1217	1705.7711	1707.2824	1707.3147	1707.3275
		S.D.	7.86 × 10−1	4.82 × 10−1	6.13 × 10−2	5.65 × 10−2	2.07 × 10−2
	10	Mean	1716.0441	1717.0603	1719.0759	1719.2548	1719.3554
		S.D.	1.91 × 10−0	4.95 × 10−1	6.01 × 10−1	4.15 × 10−2	2.14 × 10−2
Tiger	5	Mean	1558.0329	1557.9318	1558.4506	1558.4501	1558.4540
		S.D.	5.33 × 10−1	2.22 × 10−1	9.97 × 10−3	7.16 × 10−3	2.33 × 10−13
	8	Mean	1599.8831	1600.9186	1602.4582	1602.5154	1602.5726
		S.D.	1.49 × 10−0	4.69 × 10−1	3.31 × 10−1	4.20 × 10−2	1.03 × 10−3
	10	Mean	1611.4752	1612.0636	1613.9468	1614.0530	1614.1907
		S.D.	1.48 × 10−0	5.17 × 10−1	3.63 × 10−1	6.94 × 10−2	1.22 × 10−2
Worker	5	Mean	3615.3859	3616.0241	3616.8727	3616.8783	3616.8847
		S.D.	1.36 × 10−0	4.07 × 10−1	1.35 × 10−2	1.06 × 10−2	4.67 × 10−13
	8	Mean	3673.9643	3674.9757	3677.2924	3677.3820	3677.4455
		S.D.	2.58 × 10−0	6.99 × 10−1	1.08 × 10−1	6.38 × 10−2	9.84 × 10−3
	10	Mean	3687.2989	3689.9856	3692.5210	3692.6557	3692.7237
		S.D.	2.14 × 10−0	7.65 × 10−1	1.16 × 10−1	6.34 × 10−2	6.67 × 10−3
Surfer I	5	Mean	1712.5020	1712.7946	1713.3050	1713.3086	1713.3169
		S.D.	8.28 × 10−1	3.18 × 10−1	2.72 × 10−2	8.85 × 10−3	2.33 × 10−13
	8	Mean	1760.4105	1762.0123	1764.0041	1764.1581	1764.2167
		S.D.	2.16 × 10−0	6.11 × 10−1	1.79 × 10−1	4.76 × 10−2	8.97 × 10−2
	10	Mean	1774.9283	1776.3085	1779.0553	1779.1577	1779.3295
		S.D.	1.93 × 10−0	8.59 × 10−1	8.96 × 10−1	7.31 × 10−2	1.40 × 10−2
Teste4	5	Mean	5427.4074	5427.8418	5428.3645	5428.3695	5428.3717
		S.D.	1.20 × 10−0	3.73 × 10−1	1.74 × 10−2	9.78 × 10−3	1.87 × 10−12
	8	Mean	5471.5151	5471.9936	5473.8474	5473.8861	5473.9076
		S.D.	2.45 × 10−0	5.14 × 10−1	2.85 × 10−2	5.32 × 10−2	1.06 × 10−3
	10	Mean	5483.5430	5483.6152	5485.6993	5485.7677	5485.8291
		S.D.	1.33 × 10−0	7.08 × 10−1	6.33 × 10−2	9.38 × 10−2	7.46 × 10−3
Church	5	Mean	3381.9107	3381.4923	3381.9236	3381.9246	3381.9246
		S.D.	1.71 × 10−2	2.25 × 10−1	4.62 × 10−3	9.25 × 10−13	9.33 × 10−13
	8	Mean	3422.5621	3420.9578	3420.7834	3423.2006	3423.4542
		S.D.	2.66 × 10−0	9.18 × 10−1	2.87 × 10−0	1.23 × 10−0	8.34 × 10−2
	10	Mean	3429.5570	3432.8333	3434.3211	3435.2509	3435.3928
		S.D.	2.40 × 10−0	5.23 × 10−1	1.80 × 10−0	1.36 × 10−0	4.55 × 10−1
Cow	5	Mean	3965.2806	3964.3722	3965.2925	3965.2932	3965.2932
		S.D.	1.96 × 10−2	4.53 × 10−1	3.22 × 10−3	4.67 × 10−13	9.33 × 10−13
	8	Mean	4028.8447	4026.0490	4028.9299	4028.9450	4028.9562
		S.D.	7.58 × 10−2	8.64 × 10−1	5.48 × 10−2	1.07 × 10−2	1.39 × 10−2
	10	Mean	4038.3761	4040.5136	4042.9971	4043.3044	4043.7911
		S.D.	2.42 × 10−0	6.22 × 10−1	9.34 × 10−1	8.28 × 10−1	3.77 × 10−1

.

Compared to other segmentation approaches, the computational results demonstrate the efficiency of the proposed DSPSO-ECP on eight benchmark test images with a different number of threshold values as shown in Table 8. Furthermore, some existing segmentation approaches, including the whale optimization algorithm (WOA) [81], gray wolf optimizer (GWO) [82], whale optimization algorithm based on thresholding heuristic (WOA-TH) [77], and wolf optimizer based on thresholding heuristic (GWO-TH) [77], are adopted as comparison approaches to further evaluate the segmentation effectiveness of the proposed DSPSO-ECP. The maximum number of iterations is set to 3000, 3600, and 4000. Simple statistics, including the Mean and S.D., of between-class variance obtained by these segmentation approaches on a different number of threshold values for 100 independent times are presented in

Table 9. Computational results of comparative approaches on 5, 8 and 10 number of threshold values.

Images	$\mathit{K}$	Statistics	Comparative Approaches
			WOA	GWO	WOA-TH	GWO-TH	DSPSO-ECP
Stones	5	Mean	5340.7208	5339.9497	5340.7385	5340.7385	5340.7385
		S.D.	2.45 × 10−2	4.96 × 10−0	1.01 × 10−11	1.01 × 10−11	1.87 × 10−12
	8	Mean	5379.5706	5378.4169	5380.5360	5379.7390	5380.5429
		S.D.	3.04 × 10−0	1.49 × 10−0	7.15 × 10−3	2.87 × 10−0	1.58 × 10−3
	10	Mean	5388.1229	5387.1123	5389.3097	5389.1551	5389.3174
		S.D.	2.08 × 10−0	2.19 × 10−0	7.43 × 10−3	3.04 × 10−1	4.52 × 10−3
Island	5	Mean	1664.5952	1664.4986	1664.6061	1664.6052	1664.6061
		S.D.	1.56 × 10−2	9.76 × 10−2	1.60 × 10−12	5.31 × 10−3	2.33 × 10−13
	8	Mean	1707.3306	1706.9431	1707.3653	1707.2232	1707.3565
		S.D.	3.79 × 10−2	6.86 × 10−1	4.81 × 10−2	2.79 × 10−1	4.56 × 10−2
	10	Mean	1719.3382	1718.4319	1719.3669	1719.1169	1719.3623
		S.D.	3.10 × 10−2	6.47 × 10−1	2.32 × 10−3	4.83 × 10−1	9.82 × 10−3
Tiger	5	Mean	1554.2734	1554.1612	1554.2736	1554.2410	1558.4539
		S.D.	2.49 × 10−3	8.37 × 10−2	2.97 × 10−12	3.45 × 10−2	2.33 × 10−13
	8	Mean	1597.6955	1597.2803	1598.2458	1598.1909	1602.5731
		S.D.	2.19 × 10−0	1.30 × 10−0	1.29 × 10−3	5.88 × 10−2	7.20 × 10−4
	10	Mean	1609.4682	1608.1028	1609.8290	1609.6996	1614.1956
		S.D.	1.31 × 10−0	2.20 × 10−0	9.56 × 10−3	1.50 × 10−1	7.00 × 10−13
Worker	5	Mean	3616.3946	3616.7348	3616.8785	3616.8641	3616.8847
		S.D.	4.81 × 10−0	1.27 × 10−1	9.73 × 10−3	3.41 × 10−2	4.67 × 10−13
	8	Mean	3676.3050	3673.8170	3677.4476	3677.3291	3677.4483
		S.D.	3.62 × 10−0	2.38 × 10−0	6.66 × 10−3	8.68 × 10−2	9.33 × 10−13
	10	Mean	3691.5718	3689.2568	3692.6588	3692.6093	3692.7201
		S.D.	2.45 × 10−0	4.09 × 10−0	6.34 × 10−1	6.48 × 10−2	2.75 × 10−2
Surfer I	5	Mean	1713.3089	1713.2091	1713.3169	1713.2978	1713.3169
		S.D.	2.08 × 10−2	7.68 × 10−2	4.57 × 10−13	3.39 × 10−2	2.33 × 10−13
	8	Mean	1764.0838	1763.5736	1764.1274	1763.9812	1764.1835
		S.D.	2.12 × 10−1	1.04 × 10−0	1.83 × 10−1	1.71 × 10−1	1.45 × 10−1
	10	Mean	1779.1829	1778.1728	1779.3312	1778.7397	1779.3396
		S.D.	9.28 × 10−1	1.36 × 10−0	7.61 × 10−3	1.18 × 10−0	4.62 × 10−4
Teste4	5	Mean	5355.8520	5355.7733	5355.8555	5355.8555	5428.3717
		S.D.	1.21 × 10−2	8.47 × 10−2	5.48 × 10−2	5.48 × 10−12	1.87 × 10−12
	8	Mean	5399.6564	5399.0698	5399.6565	5399.5638	5473.9078
		S.D.	2.91 × 10−3	9.46 × 10−1	4.65 × 10−3	1.27 × 10−1	1.81 × 10−12
	10	Mean	5411.0984	5410.3541	5411.1016	5410.8512	5485.8309
		S.D.	7.79 × 10−3	7.74 × 10−1	5.41 × 10−3	1.86 × 10−1	5.03 × 10−3
Church	5	Mean	3375.8721	3374.2197	3375.9118	3371.9813	3381.9246
		S.D.	2.65 × 10−0	6.74 × 10−0	2.62 × 10−0	9.65 × 10−0	9.33 × 10−13
	8	Mean	3416.6815	3415.4415	3417.0876	3415.6558	3421.3577
		S.D.	2.40 × 10−0	1.74 × 10−0	1.78 × 10−0	3.51 × 10−0	2.83 × 10−0
	10	Mean	3429.4901	3426.6018	3429.7239	3428.1937	3434.8720
		S.D.	1.42 × 10−0	2.53 × 10−0	1.11 × 10−0	2.06 × 10−0	1.80 × 10−0
Cow	5	Mean	3965.2795	3965.1739	3965.2932	3964.6096	3965.2932
		S.D.	2.01 × 10−2	9.72 × 10−2	6.40 × 10−12	2.93 × 10−0	4.67 × 10−13
	8	Mean	4028.9410	4027.7892	4028.8474	4027.7210	4028.9619
		S.D.	1.83 × 10−2	9.47 × 10−1	6.11 × 10−1	1.69 × 10−0	1.18 × 10−2
	10	Mean	4043.8213	4042.9309	4043.7499	4043.5949	4043.3274
		S.D.	6.24 × 10−1	1.17 × 10−0	7.05 × 10−1	7.29 × 10−1	8.42 × 10−1

.

It can be seen that results obtained by the proposed DSPSO-ECP are the best results on most of the test benchmark images compared to WOA, GWO, WOA-TH, and GWO-TH as shown in Table 9.

Table 10. Best fitness values of comparative approaches on 5, 8 and 10 number of threshold values.

Images	$\mathit{K}$	Statistics	Comparative Approaches
			WOA	GWO	WOA-TH	GWO-TH	DSPSO-ECP
Stones	5	Best	5340.7385	5340.7385	5340.7385	5340.7385	5340.7385
	8	Best	5380.5433	5380.5037	5380.5433	5380.5433	5380.5433
	10	Best	5389.3200	5389.2840	5389.3200	5389.3200	5389.3200
Island	5	Best	1664.6061	1664.6061	1664.6061	1664.6061	1664.6061
	8	Best	1707.4168	1707.3259	1707.4168	1707.4168	1707.4168
	10	Best	1719.3673	1719.2667	1719.3673	1719.3673	1719.3673
Tiger	5	Best	1554.2736	1554.2736	1554.2736	1554.2736	1558.4539
	8	Best	1598.2471	1598.1950	1598.2471	1598.2471	1602.5735
	10	Best	1609.8391	1609.7846	1609.8391	1609.8391	1614.1956
Worker	5	Best	3616.8847	3616.8712	3616.8847	3616.8847	3616.8847
	8	Best	3677.4483	3677.4143	3677.4483	3677.4483	3677.4483
	10	Best	3692.7307	3692.6752	3692.7307	3692.7307	3692.7307
Surfer I	5	Best	1713.3169	1713.3169	1713.3169	1713.3169	1713.3169
	8	Best	1764.2431	1764.2386	1764.2431	1764.2431	1764.2431
	10	Best	1779.3398	1779.3106	1779.3398	1779.3398	1779.3398
Teste4	5	Best	5355.8555	5355.8555	5355.8555	5355.8555	5428.3717
	8	Best	5399.6582	5399.6372	5399.6582	5399.6582	5473.9078
	10	Best	5411.1063	5411.0714	5411.1063	5411.1063	5485.8333
Church	5	Best	3376.1738	3376.1738	3376.1738	3376.1738	3381.9246
	8	Best	3417.7417	3417.6603	3417.7417	3417.7417	3423.6059
	10	Best	3430.2596	3430.0797	3430.2787	3430.2787	3435.9076
Cow	5	Best	3965.2932	3965.2932	3965.2932	3965.2932	3965.2932
	8	Best	4028.9668	4028.8155	4028.9668	4028.9668	4028.9668
	10	Best	4044.1498	4043.9150	4044.1498	4044.1498	4044.1498

reported the best fitness values achieved by all comparative approaches for 100 independent times.

6.4. The Wilcoxon Signed Ranks Test

As a non-parametric statistical hypothesis, the Wilcoxon signed ranks test, which is reported in previous research [83], is introduced to validate the difference between the proposed DSPSO-ECP and other segmentation approaches. The mean fitness values obtained by WOA, GWO, WOA-TH, GWO-TH, and DSPSO-ECP are used as comparison criterion in this statistical test of the Wilcoxon signed ranks. The null hypothesis of the Wilcoxon signed ranks test is described as follows. There is no difference between the proposed DSPSO-ECP and other existing segmentation approaches.

Wilcoxon’s test is defined as follows. Let $d_i$ is the difference between the performance scores of two compared approaches on $i$-th out of $n$ problems, where $n=24$. $‖W_R^{+}‖$ is the sum of ranks for the problems that DSPSO-ECP outperformed compared approach, $‖W_R^{-}‖$ is the sum of ranks for the problems that compared approach outperformed DSPSO-ECP, which are determined by (16) and (17) in the following

$$‖W_R^{+}‖={\displaystyle \sum _{d_i>0}\mathrm{rank}\left(d_i\right)}+\frac{1}{2}{\displaystyle \sum _{d_i=0}\mathrm{rank}\left(d_i\right)}$$

(16)

$$‖W_R^{-}‖={\displaystyle \sum _{d_i<0}\mathrm{rank}\left(d_i\right)}+\frac{1}{2}{\displaystyle \sum _{d_i=0}\mathrm{rank}\left(d_i\right)}$$

(17)

where $T_R$ is the smaller of the sums $‖W_R^{+}‖$ and $‖W_R^{-}‖$, where $T_R=\min \left(‖W_R^{+}‖,\hspace{0.33em}‖W_R^{-}‖\right)$. The results obtained by all pairwise comparison concerning DSPSO-ECP with a level of significance $\alpha =5\%$ are shown in

Table 11. Wilcoxon signed ranks test results with a level of significance $\alpha =5\%$.

Images	$\mathit{K}$	WOA	GWO	WOA-TH	GWO-TH
Stones	5	0.0177 (4)	0.7888 (8)	0.0000 (0)	0.0000 (0)
	8	0.9723 (12)	2.1260 (12)	0.0069 (4)	0.8039 (13)
	10	1.1945 (15)	2.2051 (13)	0.0077 (5)	0.1623 (7)
Island	5	0.0109 (2)	0.1075 (1)	0.0000 (0)	0.0009 (1)
	8	0.0259 (7)	0.4134 (6)	−0.0088 (7)	0.1333 (6)
	10	0.0241 (6)	0.9304 (9)	−0.0046 (2)	0.2454 (9)
Tiger	5	4.1805 (16)	4.2927 (16)	4.1803 (12)	4.2129 (15)
	8	4.8776 (19)	5.2928 (17)	4.3273 (14)	4.3822 (16)
	10	4.7274 (18)	6.0928 (19)	4.3666 (15)	4.4960 (17)
Worker	5	0.4901(10)	0.1499 (4)	0.0062 (3)	0.0206 (3)
	8	1.1433 (13)	3.6313 (15)	0.0007 (1)	0.1192 (5)
	10	1.1483 (14)	3.4633 (14)	0.0613 (9)	0.1108 (4)
Surfer I	5	0.0080 (1)	0.1078 (2)	0.0000 (0)	0.0191 (2)
	8	0.0997 (8)	0.6099 (7)	0.0561 (8)	0.2023 (8)
	10	0.1567 (9)	1.1668 (10)	0.0084 (6)	0.5999 (11)
4Teste4	5	72.5197 (22)	72.5984 (22)	72.5162 (18)	72.5162 (21)
	8	74.2514 (23)	74.8380 (23)	74.2513 (19)	74.3440 (22)
	10	74.7325 (24)	75.4768 (24)	74.7293 (20)	74.9797 (23)
Church	5	6.0525 (21)	7.7049 (20)	6.0128 (17)	9.9433 (20)
	8	4.6762 (17)	5.9162 (18)	4.2701 (13)	5.7019 (18)
	10	5.3819 (20)	8.2702 (21)	5.1481 (16)	6.6783 (19)
Cow	5	0.0137 (3)	0.1193 (3)	0.0000 (0)	0.6836 (12)
	8	0.0209 (5)	1.1727 (11)	0.1145 (10)	1.2409 (14)
	10	−0.4939 (11)	0.3965 (5)	−0.4225 (11)	−0.2675 (10)
$‖W_R^{+}‖$		289	300	190	266
$‖W_R^{-}‖$		11	0	20	10
$T_R$		11	0	20	10
$H$		1	1	1	1
$p$-value		7.14 × 10−5	1.82 × 10−5	1.51 × 10−3	9.90 × 10−5

. The critical values of distribution $T$ of Wilcoxon for $n$ degrees of freedom is 81 [84]. The values of Wilcoxon test statistic are 11, 0, 20, and 10 from Table 11, where $T_R=\left\{11,0,20,10\right\}$ and $T=81$. Therefore, the null hypothesis of Wilcoxon signed ranks test is rejected, and the proposed DSPSO-ECP outperforms other existing segmentation approach with the $p$-value (the $p$-value have been computed by using MATLAB) associated.

7. Conclusions

An extended membrane system combining a cell-like P system with evolutional symport/antiport rules and active membranes, and an improved PSO mechanism, called DSPSO-ECP, is proposed for solving optimization problems. This extended membrane system under the framework of membrane computing uses a cell-like P system with a dynamic membrane structure, and integrates evolution rules for objects and membranes. There are two kinds of evolution rules for objects in the proposed DSPSO-ECP, including basic evolution rules based on PSO in elementary membranes and local evolution rules based on improved SPSO in sub-membranes. In basic evolution rules, the updated model of velocity and position in PSO is adopted to evolve objects in elementary membranes. The modified update model of velocity in improved SPSO is used in local evolution rules to evolve objects in sub-membranes. A group of sub-membranes for elementary membrane are specially designed to avoid prematurity through membrane creation and dissolution rules with promoter/inhibitor. The exchange and sharing of information between different membranes are achieved by communication rules for objects, which are based on the evolutional symport rules of ECP. The proposed DSPSO-ECP is evaluated on eight numerical benchmark functions, and is compared with classic PSO and three improved PSO approaches. Computational results clearly validate the effectiveness of this extended membrane system. Furthermore, comparison experiments with segmentation approaches, which are made on eight test images, are conducted to verify the performance, and results show the validity of the proposed DSPSO-ECP.

P systems, under the framework of distributed parallel computing model, have highly effective efficiency in solving optimization problems with linear or polynomial complexity. The evolution-communication mechanism of P systems processes data in parallel and also provides a new way to solve complex problems. However, the application of P systems has been limited by the incompleteness of fundamental computation and the complicacy of implementation. Therefore, extended membrane systems, integrated classic P systems, and evolutionary computation are proposed to try to solve these problems as we mentioned above. In the proposed DSPSO-ECP, the unilateral connection of parent-child membranes based on cell-like P systems is too simple and single, and more multilateral connections based on complicated P systems or biochemical reactions may be introduced in future studies. Comparison experiments are only made on low-dimensional benchmark functions, and DSPSO-ECP may have some limitations on high-dimensionality functions or large datasets. More work is needed to apply this extended membrane system to solving more complex optimization problems.