An Extended Tissue ‐ like P System Based on Membrane Systems and Quantum ‐ Behaved Particle Swarm Optimization for Image Segmentation

: An extended membrane system using a tissue ‐ like P system with evolutional symport/antiport rules and a promoter/inhibitor, which is based on the evolutionary mechanism of quantum ‐ behaved particle swarm optimization (QPSO) and improved QPSO, named CQPSO ‐ ETP, is designed and developed in this paper. The purpose of CQPSO ‐ ETP is to enhance the optimization performance of statistical network structure ‐ based membrane ‐ inspired evolutionary algorithms (SNS ‐ based MIEAs) and the QPSO technique. In CQPSO ‐ ETP, evolution rules with a promoter based on a standard QPSO mechanism are introduced to evolve objects, and evolution rules with an inhibitor based on an improved QPSO mechanism using self ‐ adaptive selection, and cooperative evolutionary and logistic chaotic mapping methods, are adopted to avoid prematurity. The communication rules with a promoter/inhibitor for objects are introduced to achieve the exchange and sharing of information between different membranes. Under the control of the evolution and communication mechanism, the CQPSO ‐ ETP can effectively improve the performance with the help of a distributed parallel computing model. The proposed CQPSO ‐ ETP is compared with PSO, QPSO and two existing improved QPSO approaches which are conducted on eight classic numerical benchmark functions to verify the effectiveness. Furthermore, computational experiments which are made on eight tested images with three comparative clustering approaches are adopted, and the experimental results demonstrate the clustering validity of the proposed CQPSO ‐ ETP.


Introduction
Membrane computing (MC) is an important branch of bio-inspired computing, which is initiated by Păun [1], and the computing models of membrane computing are also called membrane systems or P systems. P systems focus on abstracting some fundamental concept from the structure and functioning of the living cells, cell tissues or colonies of cells. Research shows that some P systems have the same computing power as Turing machines, or are more efficient, to some extent [2]. There are three classic computing models of P systems according to the structure of the membrane or cell arrangement in previous studies and researches, including cell-like P systems, tissue-like P systems and neural-like P systems [3]. Many variants of P systems based on biological facts, mathematical biological cells, theoretical computer science or application motivations have been presented for solving difficult optimization problems in real life [4,5]. comparison experiments on eight tested images from image segmentation problems are performed to validate the clustering efficiency of this proposed CQPSO-ETP.
The rest of this paper is organized as follows: the basic framework of the tissue-like P system with evolutional symport/antiport rules and a promoter/inhibitor are described in Section 2. More details about the evolutionary mechanism of QPSO and improved QPSO are given in Section 3. The extended tissue-like P system based on the tissue-like P system with evolutional symport/antiport rules and a promoter/inhibitor, and QPSO and improved QPSO are proposed in Section 4, and evolution and communication rules with a promoter/inhibitor for objects are described in more details in this section. Experimental results and analysis on eight classic numerical benchmark functions with four comparative approaches are reported in Section 5. Section 6 gives the experimental results and discussion which are made on eight tested images with three classic existing clustering approaches to evaluate the clustering efficiency of this proposed extended tissue-like P system. Section 7 provides some conclusions and outlines future research directions.

The Tissue-Like P System with Evolutioanl Symport/Antiport Rules
In the computing model of a basic tissue-like P system with symport/antiport rules and its variants, different membranes or regions exchange and share information or objects according to the standard symport/antiport rules, but the objects are not modified during this communication process. In fact, this communication method only changed the place of the objects in the system, but did not change the chemical properties of the objects. Chemical substances are modified in the cell biology when the substances enter or exit in different membranes, and this biological phenomenon is called the evolution of substances. Therefore, the evolution idea for objects during the communication process is inspired from the evolution of substances which can be considered in the tissue-like P system, then the objects can be modified through some evolution rules. A tissue-like P system with an evolutional symport/antiport is described as a tuple in the following [24]: where (1)  is a non-empty finite alphabet of objects, each alphabet represents an object; (2)  is a set of initial objects located in the environment; (3)  is the membrane structure of the system that consisting m membranes; (4) 1 , , m    are finite multisets of the initial objects over  ; (5) R is a finite set of evolution rules of the following forms: <1> Evolutional symport rules:       ' <2> Evolutional antiport rules:     u v   , ' ' * , u v  ; (6) out  is the output region or membrane in the tissue-like P system with the evolutional symport/antiport rules, u v v u          , provides a new kind of communication way from membrane i to membrane j . It only can be executed on a moment if there is a membrane in a configuration which contains a multiset of objects u , and another membrane which contains a multiset of objects that are denoted by v ( v  ). When the evolutional antiport rule associated with membrane i and membrane j is applied, the objects u in membrane i are evolved to the new objects ' u and are sent to membrane j . At the same time, the objects v in membrane j are evolved to the new objects that are denoted by ' v ( ' v   ) and are sent to membrane i . Note that objects u in membrane i and object v in membrane j are consumed during this evolutional communication process. Especially, if the objects are not modified but have just changed their place during the communication process in the tissue-like P system, then it is a particular case of the tissue-like P system with evolutional symport/antiport rules, and simply called the tissuelike P system with symport/antiport rules.

The Tissue-Like P System with Evolutional Symport/Antiport Rules and Promoter/Inhibitor
In general, the evolutional rules in the traditional tissue-like P system with evolutional symport/antiport rules are adopted in this work, which are applied in a maximal and spontaneous parallel manner. The evolutional symport/antiport rules for objects are only executed on this moment when the objects exist in the membranes or environment. Therefore, the promoter and inhibitor are introduced to dynamically change the evolution-communication process of this system in order to adjust the execution sequence rather than the uncertain working manner of the traditional P system. Therefore, a recognizer tissue-like P system with evolutional symport/antiport rules and a promoter/inhibitor is defined, and simply called an ETP system. An ETP system is a tuple which can be described in the following [15]: where (1)  is a finite non-empty alphabet of objects; (2)  is a set of initial objects located in the environment; (3)  is the membrane structure consisting of m membranes; are finite multisets of initial objects over  ; (5) R is a finite set of rules that contained two kinds of evolution rules, i.e., evolutional symport rules with a promoter/inhibitor and evolutional antiport rules with a promoter/inhibitor, which are described in following forms: <1> Evolutional symport rules: <2> Evolutional antiport rules: out  is the output region or membrane in the ETP system, where In ETP system, an evolutional symport rule with a promoter/inhibitor,   provides a new kind of communication pathway between membrane i and membrane j . It can be executed on a moment if there is a membrane in a configuration which not only contains a multiset of objects, u ( u   ), but also contains the promoter/inhibitor objects that are denoted by p . Specially, promoter/inhibitor objects p in the subscript are the promoter or inhibitor of the form * p p  and * p p   , where * p represents the promoter and * p  represents the inhibitor. When the evolutional antiport rule with the promoter/inhibitor associated with membrane i and membrane j is applied, the objects u with the promoter/inhibitor objects p in membrane i are evolved to the new objects ' u and are sent to membrane j . Note that the objects u and promoter/inhibitor objects p in membrane i are consumed during this evolutional communication process. An evolutional antiport rule with a promoter/inhibitor, provides a new kind of communication way between membrane i to membrane j . It can be executed on a moment if there is a membrane in a configuration which contains a multiset of objects u and promoter/inhibitor objects p , and another membrane in the same configuration which contains a multiset of objects v ( v  ). When the evolutional antiport rule with the promoter/inhibitor associated with membrane i and membrane j is applied, the objects u with the promoter/inhibitor objects p in membrane i are evolved to the new objects ' u and are sent to membrane j . And the objects v in membrane j are evolved to the new objects ' v and are sent to membrane i at the same time. Note that the objects u and the promoter/inhibitor objects p in membrane i , and the objects v in membrane j are consumed during this evolutional communication process.
An ETP system with the degree m ( 1 m  ) can be regarded as a set of m cells or membranes which are labeled from 1 to m , and the environment is usually labeled by 0 . Traditionally, the mathematics structure of the ETP system can be viewed as a graph in topology. A configuration of the ETP system at any time can be described by all multisets of objects over  with corresponding cells or membranes in the system, and the multisets of objects over   with the environment at the same time. All objects can be evolved through a maximally parallel manner at each step. The ETP system started from the initial configuration and evolved to the execution of the evolutional symport/antiport rules with a promoter/inhibitor, as described above, then a sequence of consecutive configurations will be generated and changed during this evolution process. If no rules can be executed in the system, the computation of the ETP system will be stopped. The end of the configuration is called the halting configuration. Note that only the halting configuration will gives the final computational results obtained from the system, which are usually encoded to the number of the objects placed in the output region or membrane out  .

The Standard Quantum-Behaved Particle Swarm Optimization
The Quantum-behaved particle swarm optimization (QPSO) is inspired by the concepts of quantum physics and PSO techniques. As a probabilistic global optimization technique, the basic principle of QPSO is different from the basic principle of the PSO technique. Each particle only depends on the information of the local attractor and local best to adjust the flight trajectory. Compared with traditional PSO, the velocity of a particle is replaced by the local attractor, and the local attractor contains two parts, including self-cognition and social-cognition. Therefore, each particle moves according to the local attractor and the mean position of the local best in the whole population, and will attract to the local attractor based on the position of the local best and global best in the search space. The local attractor is denoted by   i z t of i -th particle at iteration t , which is determined by (1) in the following [98]: where  is a uniform random number which is distributed on the interval from 0 to 1, x of position i x of the i -th particle at iteration 1 t  is defined by (2) in the following [98] where  is a uniform random number which is distributed on the interval from 0 to 1, .  is the adjustment parameter to adjust the search speed of the i -th particle, and the named contraction-expansion coefficient.
  m p t is the position of the mean best based on the average of the local best for all particles in the whole population at iteration t , and it is also used to enhance the collaborative ability and global search ability for particles. The position   m p t of the mean best at iteration t is given by (3) in the following [92], The Equation (3) can be simplified as shown in (4) in the following [92], where N is the total number of particles in the population. The computing procedure of the QPSO technique can be described as follows: (1) Initialization. Initialize the position of all particles with random numbers in the search space; (2) Update. Update the position of the local best and the global best for each particle, and compute the position of the mean best based on the position of the local best for all particles in the population according to Equations (3) or (4); (3) Evaluation. Update the local attractor and position for each particle according to Equations (1) and (2), and evaluate the fitness function which is used to select the local best and global best in the population; (4) Termination. The steps (2)-(3) will be implemented repeatedly with an iterative form until the termination criterion is satisfied, and the termination criterion of the QPSO technique is to evaluate whether the maximum number of iterations is reached.

The Cooperative Evolutionary Self-Adaptive Quantum-Behaved Particle Swarm Optimization
In this section, an improved QPSO is presented to enhance the global search ability and to avoid prematurity, which is simply named CQPSO. The self-adaptive selection method is introduced to dynamically adjust the values of acceleration factors in the updating of the local attractor, and the diversity function is adopted to help particles escape to the local optimum. Then the cooperative evolutionary strategy is used for the updating of the local attractor in the QPSO technique, and to increase the probability of discovering the global optimum in the search space. At last, a logistic chaotic mapping method is introduced to generate the chaotic sequence of random parameters in the position updating of particles. More details about the CQPSO technique are discussed in the following. 3.2.1. Self-Adaptive Selection In the classic QPSO technique as we mentioned above, the local attractor i z of the i -th particle is determined by the position of the local best and global best, according to the Equation (1). Based on this, the regulation parameter i  of the i -th particle is the key factor to balance the influence of the local best and global best. Therefore, the acceleration factors are introduced to generate the regulation parameter in the updating of the local attractor, which are given by (5) in the following [93], where 1 r and 2 r are uniform random numbers which are distributed on the interval from 0 to 1, c and 2 c are two acceleration factors in the updating of the regulation parameter. The acceleration factors have a considerable impact both on the convergence speed and accuracy of the QPSO technique. Therefore, the diversity function, which is denoted by D , is introduced to dynamically adjust the acceleration factors in order to avoid prematurity and to enhance the global search ability. The diversity function is shown in (6) in the following [93], 1 D  , which means the population diversity of particles is good [93]. The position of the particle is far away from the position of the global best in the search space at the early stage. Then the particle adjusts the direction of the flight trajectory though the influence of the local attractor and mean best. Therefore, the position of the particle is gradually close to the position of the global best in the search space. The diversity function D , as we mentioned above, is introduced to dynamically adjust the values of the corresponding acceleration factor 1 c and 2 c , which is based on the proximity of the current position and global best in the population. The corresponding acceleration factors 1 c and 2 c are given by (7) where min c and max c are the minimum and maximum of the acceleration factor. max t is the maximum number of iterations.  is a predefined threshold of acceleration factors.

Cooperative Evolutionary
In the traditional QPSO technique, the local attractor of each particle is determined by the position of the local best and global best in the population. Furthermore, the population diversity of particles decreases under the guidance of the local attractor during the optimization process [99]. The position of the particles will be limited in a rectangle, the vertices of which are the position of the global best and local best, with the decreasing of the local attractor in the possible distribution space. Then particles will be easily trapped into the local optimum and appear prematurely. Therefore, the modification of Equation (1) is given in (8) in the following [99], is a sequence number that is generated by Equation (5) at iteration t .
  1 r p t is the position of the local best for particle 1 r , it can be randomly selected from the population of particles at iteration t .
is a perturbation vector at iteration t , which is given by (9) in the following [99], where position of the local best for particle 2 r , which is randomly selected from the population of particles at iteration t ; note that

Logistic Chaotic Mapping
A logistic map, which was developed by May [100], is a classic kind of chaotic mapping method. And the chaotic variant   1 ch t  at iteration 1 t  , based on the logistic mapping method, is given by (10) in the following [101]  , and the Lyapunov exponent of logistic map is greater than 0 , it means that the chaotic system is in a stable state at iteration t . The values of the chaotic variant based on varying values of the chaotic coefficient are shown in Figure 1a. Thus, a random sequence of the chaotic variant, which is generated by Equation (10), is distributed on the interval from 0 and 1 with some features, such as ergodicity, nonlinear, random similarity. A logistic chaotic map sequence of a chaotic variant after 100 iterations is given in Figure  1b  In the position updating of particles in the QPSO technique,  is a uniform random number which is distributed on the interval from 0 to 1 through Equation (2), . Therefore, the logistic mapping method, as we mentioned above, is adopted to generate the random parameter in Equation (2), it can help particles to explore the vicinity region of a potential solution by oscillating in the search space compared with the random generation method. The modification of the random parameter  at iteration +1 t is given in (11) in the following is the values of parameter  for the i -th particle at iteration t through Usually, in order to ensure the stability of the chaotic system, the value of the chaotic coefficient  is set to 4.

The Proposed CQPSO-ETP
In this section, an extended tissue-like P system based on the evolutioncommunication mechanism of ETP and the evolution mechanism of QPSO and improved QPSO is proposed, and simply named CQPSO-ETP. The evolutionary mechanism for objects and communication mechanism for global objects are introduced in this extended P system. The rest of this section is organized in the following. Firstly, the general framework of this extended tissue-like P system is described, and the basic membrane structure is given in more details. Secondly, the evolution mechanism of QPSO and CQPSO and the evolution-communication mechanism of a tissue-like P system with evolutional symport/antiport rules and a promoter/inhibitor are introduced in the CQPSO-ETP to improve the performance of the extended P system. The computation of the proposed CQPSO-ETP is given in the following content. The complexity analysis of the CQPSO-ETP is described in the last parts.

The General Framework of CQPSO-ETP
The general framework of CQPSO-ETP is similar to the tissue-like P system with evolutional symport/antiport rules and a promoter/inhibitor. However, the rules for objects in the CQPSO-ETP are divided into two kinds of rules for objects, including the evolution rules for objects and communication rules for global objects. The membranes in the CQPSO-ETP system, are labeled from 1 to m , and simply denoted by 1 2 , , , m Respectively, CQPSO-ETP is a tuple which can be formally described in the following, where (1)  is a finite non-empty alphabet of objects; (2)  is the membrane structure of CQPSO-ETP consisting of m membranes; (4) R represents finite sets of evolution rules in the CQPSO-ETP, represents a finite set of commutation rules for global objects associated with membrane i to membrane j . The form of the communication rules is also the same as the form of the evolutional symport/antiport rules, which can be described in the form: q represents the promoter, and * q  represents the inhibitor; note that p q  .
Once the computation is completed, the computational results or objects in the output membrane will be transported to the environment. The membrane structure of CQPSO-ETP is especially graphically depicted in Figure 2. In Figure 2, the CQPSO-ETP contains m membranes, which are simply denoted by 1 2 , , , m     represents the output region out  , which is used to store computational results or objects of the system for each iteration.

Evolution Rules
There are two kinds of evolution rules based on different updating mechanisms of QPSO and improved QPSO for objects in the proposed CQPSO-ETP, including the evolution rules with the promoter * p and the evolution rules with the inhibitor The evolution rules with the promoter/inhibitor are normally executed only on the elementary membrane in order to generate the position of objects.

The Evolution Rules with Promoter
In this work, the mechanism of QPSO is adopted to generate the position of the local attractor and objects in the elementary membrane  (12) in the following, which is based on the logistic chaotic mapping method according to Equation (11).
according to Equation (2). The values of contraction-expansion coefficient are dynamically updated according to a linear decreasing method for each iteration, and are given by (13) in the following, where max t is the maximum number of iterations. In Equation (12) (3), and it is given by (14) in the following, where

The Evolution Rules with Inhibitor
The mechanism of CQPSO is adopted to generate the position of the local attractor and objects in the elementary membrane . Updating the position for objects in the evolution rules with the inhibitor is determined by Equation (12). A perturbation vector is introduced to the updating method of the local attractor for the objects, which is different from the updating method based on the QPSO technique in the evolution rules with a promoter through Equation (15). The local attractor  at iteration t is given by (16) in the following, is a perturbation vector, which is determined by (17) according to Equation (9) in the following, where i  is the adjustment parameter of the local attractor i z ,

Communication Rules
In the proposed CQPSO-ETP, the evolutional antiport rules, as we mentioned above, are introduced to the communication rules to improve the convergence speed and accuracy. The exchange and sharing of information for different membranes or regions is achieved by the execution of the evolutional antiport rules for the objects. Two kinds of communication rules with a promoter/inhibitor are adopted to the CQPSO-ETP, including the communication rules with promoter The commutation relationship is established from the execution of communication rules with a promoter/inhibitor between the elementary membrane and its adjacent membrane, which is depicted by a loop topology structure in mathematics, as shown in Figure 3a. Besides this, the communication relationship based on the communication rules with a promoter/inhibitor also constructed the neighborhood structure of the elementary membranes. The exchange and sharing of information only performed on the elementary membrane and its neighboring membranes through communication rules with a promoter/inhibitor, as shown in Figure 3b.   Step2: Position initialization

Compuatation of CQPSO-ETP
The position of the objects is randomly initialized in the search space. The membrane structure of this extended P system is shown in Figure 2. In general, the optimization problem is considered as the minimum optimization problem; Step3: Update the position of the local best and global best Update the position of the local best and global best for all objects in each elementary membrane. Note that the promoter and inhibitor will not simultaneously exist in the same elementary membrane. The evolution and communication rules with the promoter/inhibitor are only executed on a configuration at a moment when the promoter/inhibitor objects have appeared in the elementary membrane.
(2) Evolution rules for objects The promoter * p and inhibitor * p  in the evolution rules are described as some restricted conditions of the system, including the comparison condition and stagnation condition. The comparison condition is adopted to compare the fitness values of the global best in the elementary membrane and its neighboring membranes, and the stagnation condition is adopted to evaluate whether the position of the global best for all objects in the elementary membrane cannot be further improved for limit iterations.  The evolution and communication rules for objects in the proposed CQPSO-ETP will be implemented repeatedly with an iterative form until the termination criterion is satisfied. The termination criterion of CQPSO-ETP is stopped when the maximum number of iterations is reached. When the system halts, the position of the last global best, which is stored in the environment, is regarded as the final computational results for the system. Algorithm 1 depicted the main pseudo code of the computation for proposed CQPSO-ETP.

Algorithms 1 CQPSO-ETP:
Step 2: Update the global best in the system end end

Complexity Analysis
The complexity of the proposed CQPSO-ETP is analyzed in this subsection. As defined earlier, N is the total number of objects in the system.

Experimental Results and Analysis
Computational experiments which are made on some classic numerical benchmark functions are conducted to verify the optimization performance of the proposed CQPSO-ETP. More details about eight classic numerical benchmark functions which are used in this computational experiment are given in this section. Furthermore, the optimized efficiency of CQPSO-ETP is compared with classic PSO, QPSO and two existing improved QPSO approaches in the comparison experiment. All optimized techniques, including CQPSO-ETP, 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 10 Environment.

Numerical Benchmark Functions
In this work, eight classic numerical benchmark functions from the previous work, which are reported in [102], are adopted to the computational experiments, including unimodal and multimodal functions. Unimodal and multimodal functions are usually used to evaluate the exploitation and exploration efficiency of optimized approaches.
The domains and minimums of eight classic numerical benchmark functions are depicted in Table 1, including the Rastrigin function, the Sum power function, the Alpine function, the Schwefel 1.2 function, the Rosenbrock function, the Sum Squares function, the Quartic function, and the Schwefel function. Additionally, the dimension of the eight benchmark functions which are used in the computational experiment is set to 2 and 10 in order to get more meaningful results. The shape and range of eight classic numerical benchmark functions for 2 D  are depicted in Figure 4.

Comparision with Other Optimized Approaches
The optimized efficiency of the proposed CQPSO-ETP is compared with the classic PSO, QPSO and two improved QPSO approaches, i.e., sequential synchronous quantumbehaved particle swarm optimization (SAQPSO) [103] and improved quantum-behaved particle swarm optimization based on adaptive behavior selection (AQPSO) [93] as described above, which have been reported in the previous literature. The population of particles in the SAQPSO are divided into multiple sub-populations, and the global best in each sub-population is adopted to the commutation information with the others. A diversity function is introduced to the update of the acceleration factors which are based on the proximity between the current position and the global best for each particle in the AQPSO. The values of the adjustable parameters in the comparative techniques are the best ones which have been reported in the respective references, as listed in Table 2.
The proposed CQPSO-ETP and other compared approaches were also run 50 independent times 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 fitness function are adopted to the computational experiment as the evaluation criterion to measure the effectiveness of these optimized techniques. Figure 5 shows the convergence results of these comparative techniques on eight numerical benchmark functions from  Compared with classic PSO, QPSO and AQPSO, the fitness values of the proposed CQPSO-ETP declined quickly at the beginning of the evolutionary process from Figure 5a-d,f, and the slope of the convergence curve obtained by the proposed CQPSO-ETP is the maximum, compared with other optimization techniques. It means that the proposed CQPSO-ETP has small fitness values among comparative techniques on the eight numerical benchmark functions, as shown in Figure 5.
To be more clear, the experimental results of the proposed CQPSO-ETP and other compared approaches, which are made on eight classic numerical benchmark functions, are reported in Table  3.  The difference of comparison results obtained by optimized techniques is obvious, as indicated by Table 3, and the statistical results obtained by CQPSO−ETP, including Mean and S.D., which are the minimum among the comparison results. Therefore, the proposed CQPSO−ETP has a better performance than classic PSO, QPSO, SAQPSO and AQPSO on most of the classic numerical benchmark functions with 2 D  . The mean of the computational time which is taken by the comparative techniques is given in Table 4, thus the computational time of CQPSO−ETP is acceptable compared with other optimized techniques. Furthermore, Figure 6 gives the convergence of these optimized techniques on eight numerical benchmark functions from  

Firedman Test Statistic
The Friedman test is introduced in the comparison experiment to investigate the difference of these optimized techniques, and the average value of the fitness function obtained by the comparative techniques is used to the evaluation criterion of Friedman test. The null hypothesis of the Friedman statistic test is that all optimized techniques in this experiment have the same performance for any one benchmark function with 2 D  and 10 D  . Mathematically, the Friedman test works as follows [104]. In the Friedman test, the average of the fitness values of the comparative techniques of the eight benchmark functions are ranked from the smallest to largest [105]. Moreover, the rank of comparative technique j on benchmark function i is denoted by ij r , for , where j and i are the labels of the comparative approach and benchmark function. The average of these ranks is denoted by   where n is the total number of the benchmark functions in the experiment, for

The Proposed CQPSO−ETP for Image Segmentation Problems
In this section, experiments which are made on image segmentation problems with different tested images are performed, and the comparison results obtained by different classic clustering approaches, including CQPSO−ETP, are reported and discussed in order to validate the clustering efficiency of the proposed CQPSO−ETP. Eight tested images from the classic image segmentation datasets are used in the comparison experiments. All comparative clustering 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 16 GB of RAM in a Windows 10 Environment.

Tested Images
Eight tested images are used in the comparison experiment, including a Swan, Aircraft, Eagle, Goshawk, Plane, White Bear, Daisy and Parrot, from the previous studies and researches about the Berkeley Segmentation Dataset and Benchmark [106]. The size of the tested image is set to 481*321. More details about these tested images are given in Figure 7, and the label information about these tested images using the grey region are given in Figure 8. As is graphically depicted in Figure 8

Evaluation Funcion
As we mentioned above, the position of the i −th object in the proposed CQPSO−ETP can be viewed as the potential solution in the search space for the optimization problems.
Therefore, in the image segmentation problems, the position of the i −th object i u can be regarded as a partitioning of an image, which is represented by a set of cluster centers, The classification of a pixel is correct or accurate if it is clustered into the right cluster or class [107]. Therefore, the classification rate, also called clustering accuracy, is denoted by A . The clustering accuracy of an image is defined to the proportional correctly classified pixels in an image, as shown in (19) in the following, where   ) represents the j −th real cluster.
is the total number of correctly classified pixels, and G is the total number of pixels in an image. In this work, the clustering accuracy A is also used to evaluate the clustering performance of these comparative clustering techniques.

Comparision Results
Superpixel segmentation is an important image preprocessing technique, reported in the previous work, which groups pixels into many perceptually meaningful regions on the atomic level, rather than the traditional rigid structure of the pixel in the image. A lot of researches and works about superpixel segmentation approaches have been reported, and both of them have their own particular application. In this work, a classic superpixel technique, named simply as linear iterative clustering (SLIC) [108], is introduced to the preprocessing of the tested images in order to simplify the clustering data for the comparative clustering techniques. The SLIC based on a K−means clustering technique is used to generate superpixels based on the similarity of pixels through a linear iterative manner. The number of superpixels is usually set to 200 for the tested images. The achievable segmentation results based on the SLIC technique on eight tested images are shown in Figure 9. The clustering performance of the proposed CQPSO−ETP is compared with three classic clustering techniques, including K−means, Spectral clustering (SC) and PSO, and the intra−cluster compactness as the fitness function of the PSO and the proposed CQPSO−ETP. The comparison experiments which are made on the segmentation results based on the SLIC technique are conducted to verify the clustering efficiency of the proposed CQPSO−ETP. These comparative clustering approaches were run for 50 independent times for each tested image so as to get some meaningful clustering results and eliminate the effects of random factors.
The segmentation results obtained by these comparative clustering approaches on the eight tested images are shown in Figure 10. The boundary and regions of the segmentation image obtained by the proposed CQPSO−ETP are more clear than others.
Furthermore, Table 8 gives the statistic results of clustering accuracy A from these comparative clustering approaches on the tested images, including Worst, Best, Mean and S.D. The computing results on the eight tested images are shown in Table 8. The proposed CQPSO−ETP has better a clustering performance than the other three classic clustering techniques on the tested images, especially in the mean results of the fitness function.         The mean of the computational time obtained by these comparative clustering techniques is given in Table 9, and the computational time of the proposed CQPSO−ETP is acceptably compared with the other clustering techniques. Lastly, the Friedman test is also applied to the Means of the clustering accuracy in this comparison experiment in order to prove the difference between the proposed CQPSO−ETP and other comparative techniques. The ranks of the clustering accuracy obtained by the comparative clustering approaches for each tested image are presented in Table 10 . Therefore, these comparative clustering approaches obtained significantly different clustering accuracies in this comparison experiment [105].

Conclusions
An extended tissue−like P system combining the evolutionary mechanism of QPSO and improved QPSO and the evolution-communication mechanism of tissue−like P systems is proposed, called the CQPSO−ETP, to solve optimization problems and image segmentation problems. This extended tissue−like P system under the framework of membrane computing using the tissue−like P system with evolutional symport/antiport rules and a promoter/inhibitor for objects, and the distributed parallel computing model of tissue−like P systems is adopted to enhance the global search ability of the CQPSO−ETP. Different from the existing SNS−based MIEAs, the CQPSO−ETP has two kinds of evolution rules with a promoter and inhibitor for objects. The evolution rules with a promoter are based on the basic position updating strategy for particles in a standard QPSO. Other evolution rules with an inhibitor are based on the position updating strategy for particles in an improved QPSO using self−adaptive selection, and cooperative evolutionary and logistic chaotic mapping methods, to accomplish the evolution of the objects in the system. The evolution mechanism for objects based on the QPSO and improved QPSO is used to improve the optimization performance of the CQPSO−ETP. The communication rules with promoter and inhibitor for objects are adopted to transfer the local and global best positions of objects to achieve the exchange and sharing of information between different membranes. The communication mechanism for objects based on communication rules in tissue−like P systems is introduced to improve the convergence speed and accuracy. The promoter and inhibitor are also introduced to control the exchanged direction of information between different membranes. The computational experiments, which are compared with PSO, QPSO and two improved QPSO approaches, are evaluated on eight classic numerical benchmark functions from previous studies and researches, and the results clearly exhibit the optimization effectiveness of this proposed extended tissue−like P system. Furthermore, the comparison experiments which are made on eight tested images from the image segmentation datasets are conducted to verify the clustering performance of the CQPSO−ETP as compared with three classic clustering techniques, including K−means, SC, and PSO, and the computational results show the validity of this extended tissue−like P system.
The computing model of tissue−like P systems is the parallel computing model, which is highly effective and more efficient in solving optimization problems with linear or polynomial complexity. However, the application of tissue−like P systems has been limited by the incomplete fundamental operation and implementation difficulties. The evolution mechanism based on the evolution computing techniques provides a new way to achieve the evolutionary process for objects in P systems. Then, the computation of tissue−like P systems in biology, which contain the execution of the evolution and communication rules for objects, is converted to the update and exchange of potential solutions in mathematics. The proposed CQPSO−ETP takes the tissue−like P system with evolutional symport/antiport rules and a promoter/inhibitor as the basic computational structure, and the communication rules for objects will establish the communication relationship between different membranes and regions in P systems. These relation links between different membranes and regions are bidirectional, which are simple and easy to implement. However, these static relation links in future studies are going to be replaced by dynamic relation links in order to further accelerate convergence and improve the population diversity. Besides, a more complicated membrane computing structure in P systems may be introduced in future studies to improve the optimization performance of the SNS−based MIEAs. Furthermore, the computing experiments were only executed on a low dimensional search space for classic numerical benchmark functions, and the proposed CQPSO−ETP may have some limitations in a high dimensional space. Therefore, future studies may test the effectiveness of CQPSO−ETP using high dimensional benchmark functions, and may also focus on MIEAs based on the extended neural−like P systems and other bio−inspired computing models. More works are also needed to balance the local and global search abilities of this proposed extended P system.