Computational Intelligence and Neuroscience
Volume 2020 (2020), Article ID 6524919, 19 pages
https://doi.org/10.1155/2020/6524919

Research Article

A Complex Chained P System Based on Evolutionary Mechanism for Image Segmentation

Xiyu Liu,1,2 Lin Wang,1,2 Jianhua Qu,1,2 and Ning Wang1,2

¹Institute of Management Science, Shandong Normal University, Jinan 250000, China
²College of Business, Shandong Normal University, Jinan, China

Correspondence should be addressed to Xiyu Liu; xyliu@sdnu.edu.cn

Received 1 December 2019; Revised 6 February 2020; Accepted 25 February 2020; Published 7 August 2020

Academic Editor: Rodolfo E. Haber

Copyright © 2020 Xiyu Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A new clustering membrane system using a complex chained P system (CCP) based on evolutionary mechanism is designed, developed, implemented, and tested. The purpose of CCP is to solve clustering problems. In CCP, two kinds of evolution rules in different chained membranes are used to enhance the global search ability. The first kind of evolution rules using traditional and modified particle swarm optimization (PSO) clustering techniques are used to evolve the objects. Another based on differential evolution (DE) is introduced to further improve the global search ability. The communication rules are adopted to accelerate the convergence and avoid prematurity. Under the control of evolution-communication mechanism, the CCP can effectively search for the optimal partitioning and improve the clustering performance with the help of the distributed parallel computing model. This proposed CCP is compared with four existing PSO clustering approaches on eight real-life datasets to verify the validity. The computational results on tested images also clearly show the effectiveness of CCP in solving image segmentation problems.

1. Introduction

Membrane computing, also known as membrane systems or P systems, is a novel research of bioinspired computing initiated by Păun [1]. It seeks to discover novel biological computing models from the structure of biological cells as well as the cooperation of cells in tissues, organs, and populations of cells. Over the past years, there are three investigated P systems, cell-like P systems, tissue-like P systems, and neural-like P systems, including spiking neural P systems.

P systems have several characteristics: nondeterminism, programmability, extensibility, and readability [2]. Research shows that the some models of P systems present the same computing power as Turing machines and is more efficient to some extent [3]. Therefore, the analysis of computing power and computational efficiency of P systems is one of the important basic studies [4, 5]. Other studies are focused on the variation of P systems to solve optimization problems, including the variant of rules and structures [6, 7]. In addition, some intelligence techniques, such as evolutionary computation and fuzzy theory, is also introduced to the variant P systems in some specific optimization problems [8].

Because the parallel computation in membrane systems can avoid the increase in time consumption with the increase in the number of data points, the membrane systems are suitable for solving clustering problems [9]. There are a lot of interesting researches in variant P systems to solve clustering problems. Liu and Xue [10] proposed a new cluster splitting technique based on Hopfield networks and P systems. Liu et al. [11] presented an improved Apriori algorithm, named ECTPPT-Apriori, based on evolution-communication tissue-like P system with promoters and inhibitors. Peng et al. [12] developed an extended P system with active membranes, in which a modification differential evolution mechanism is used to find the optimal clustering centers in clustering problems. Peng et al. [13] introduced a multiobjective clustering framework using a tissue-like P system for solve fuzzy clustering problems. Wang et al. [14] proposed a new cell-like P system using a modified genetic algorithm to evolve the objects and using communication rules in the cell-like P system.

Image segmentation is an important part of image processing; it also has critical impact on the final quality of image analysis and subsequent tasks [15]. In the previous studies, the segmentation technique can be divided into region-based methods, edge-based methods, cluster-based methods, and threshold-based methods [16], in which the threshold methods can be classified into bilevel and multilevel threshold methods based on the number of clusters [17]. The region-based methods can obtain high segmentation quality but are sensitive to the parameters. The edge-based methods have high segmentation quality in different regions or targets and are also more sensitive to noise. The cluster-based methods are simple and easily implemented, but the clustering results rely on the number of clusters and feature selection in the colour space. The threshold-based methods are simple to computation, requiring no prior knowledge, but the continuity of the regions is not guaranteed due to the lack of space information [18]. So far, image segmentation has been used with wide recognized significance in machine vision, computer-aided diagnosis of the medical imaging, feature extraction, and analysis [19]. In this paper, the experiment on tested images is simple and easily implemented, and the application of image segmentation is not used in our works, so more details are not described in the following.

Because these segmentation methods mentioned above have their respective advantages and limitations, a lot of works have been done to find robust and optimum segmentation techniques [20]. The threshold technique is one of the most popular segmentation techniques which are based on the gray level of images. It is simplicity and easy implementation, which has lower computational complexity [21]. Li et al. [22] presented a novel thresholding extraction method based on variational mode decomposition (VMD). Zhao et al. [23] introduced a gradient-based adaptive particle swarm optimization (PSO) combined with improved external optimization (EO) to overcome being trapped into local optima in high-dimensional tasks. Wang et al. [24] designed a new P system with related interaction rules; the PSO mechanism is used to maximize entropy threshold. Tan et al. [25] proposed a hybrid clustering models using ensemble deep neural networks for skin lesion segmentation.

P systems are a class of distributed parallel computing models that can be used to improve the global search ability of PSO [26]. The commutation rules of chained P systems can be used to accelerate convergence of PSO. Compared with the crossover and mutation operation of DE, the partitioning information only can be used in the velocity updating of particles [27, 28]. Besides, although the neural networks have high quality for solving the optimization problems, the space and time consumption is too much, and the parallel computing model of P systems is also not executed on the neural networks. Therefore, a new variant of P systems based on PSO mechanism is proposed, which is named complex chained P (CCP) systems. And, the concepts of membrane, objects, and rules based on a special chained structure are introduced in the literature [29]. In this CCP system, two kinds of evolution rules in different chained membranes are introduced to enhance the global search ability. One of evolution rules is using the traditional and modified PSO mechanism to evolve the objects, and the partitioning information is introduced as the environmental factor to improve the clustering performance. Another is based on differential evolution (DE) to evolve the global chained objects in order to enhance the global search ability. Compared with genetic algorithm (GA) [30], the DE is simple and easily implemented, which has less predefined parameters and quick convergence speed. The communication rules for global objects between chained membranes are used to accelerate the convergence speed and avoid prematurity [31, 32]. At last, the CCP system is evaluated on eight benchmark clustering problems and eight tested images with the compared clustering and image segmentation techniques; the experimental results verify the validity and performance of proposed CCP.

The rest of this paper is organized as follows: the framework of the chained P systems is described in Section 2. Section 3 gives more details about the complex chained P system for clustering problems, and the evolution rules and communication rules are described in this section. In order to verify the validity of CCP, some experiments which are conducted on benchmark clustering problems, are taken in Section 4.2. Furthermore, some tested images are used to evaluate the competitive performance of CCP in Section 4.3. Section 5 provides some conclusions and outlines future research directions.

2. Chained P Systems

Some concepts can be briefly defined as follows. A special membrane is called chained membrane, which contains many objects based on a chained structure, and these objects are called chained object. The formal descriptions of the chained P system can be expressed as the following tuple [29].

where

(1) is a finite nonempty alphabet, and the symbols are called chained objects.(2) is the membrane structure consisting of membranes, which is composed of two parts: the structure of chained membranes and the structure of the whole chained P system.(3) are the initial finite sets of chained objects; represents the chained objects in membrane , for , .(4) are the finite sets of chained rules, and represents a finite set of chained rules in membrane , for , which consists of subrules that are executed on a special order. There are many chained rules on objects, for example, object addition rule, object subtraction rule, object crossover rule, and object variation rule.(5) is the input region or membrane in the chained P systems, which contains the initial objects in the systems.(6) is the output region or membrane in the chained P systems. If a certain chained rule cannot be executed in the chained P system, the computation process will be stopped and the computation results or objects will be transported in the output region or membrane.

3. Proposed Complex Chained P System for Clustering

The proposed chained P system based on evolutionary mechanism with a complex chained structure for clustering problems is presented in this section. First, the general framework of this complex chained P system (CCP) is described, and the basic membrane structure is given. Next, the evolution and communication mechanisms of chained P system are introduced in the CCP system. Finally, the computational process of the CCP system is described in the last parts.

3.1. General Framework of CCP System

The general framework of CCP is similar to that of the chained P systems, but the main differences are the membrane structure of whole P system and the evolution rules for chained objects. In complex chained P system, there are two kinds of chained membranes, chained membranes to and membranes , which contains different evolution rules. These chained membranes are labelled from 1 to . Respectively, the formal descriptions of the CCP system are as follows :

where

(1) is a finite set of alphabets, which includes all chained objects or strings in the CCP systems.(2) is the membrane structure of CCP system consisting of membranes.(3) are the multisets of the initial chained objects, with , for , and represents the chained objects.(4) represent finite sets of evolution rules in chained membranes to , represent a set of evolution subrules in chained membrane , and the number of the subrules are denoted by . The subevolution rules are of the form: , with , which means that chained object will be evolved to chained object .(5) represent the communication rules in chained membranes to . The communication rules are of the form: , with , for , which means the chained object in membrane will be changed in object and transported into membrane .(6) represents finite sets of evolution rules in chained membrane . For two arbitrary chained objects and , , there are two kinds of evolution rules existing in the membranes can be executed: crossover rules and variation rules, .(7) is the input region or membrane in the CCP system, which contains the initial objects of the whole system.(8) is the output region or membrane in the CCP systems. Once the computation is completed, the computation results or objects will be transported to the output region or membranes. Figure 1 gives the membrane structure of the proposed CCP system.

Figure 1: The membrane structure of CCP system.

In this proposed CCP, the chained objects are considered to be feasible solutions in the search space, and the partition obtained by the clustering technique is represented by , for . Thus, the -th chained object can be defined as , where represents the -th cluster center, is the number of the clusters, and the dimension of cluster center is denoted by . At last, the best chained object that represents the best partitioning results in the system will be output to the when the computation is completed.

3.2. Evolution Rules on Chained Objects

There are two kinds of evolution rules on chained objects existing in the chained membranes, the evolution rules in chained membranes to and another evolution rules in chained membrane .

3.2.1. Evolution Rules in Chained Membranes to

There are two kinds of evolution subrules in chained membranes to , which are based on different velocity updating strategies for chained objects. Firstly, the particle swarm optimization (PSO) mechanism is used to evolve the chained objects, and the traditional velocity model of PSO is introduced to update the velocity of chained object. At time, the velocity formula of the -th object in -th chained membrane is determined by (3) in the following equation:

where the inertia weight is denoted by and and represent the learning factor, which is usually restricted to 2. The iteration counter is denoted by . and are two independent uniform random numbers. The local best of in history is denoted by , specially, the global best in is denoted by .

Another velocity updating strategy for objects is based on environmental factors. At time, a modified velocity formula of EPSO is determined by (4) in the following equation:

where represents a positive constant, is a uniform random number, and represents the environmental factor around to the object , which is based on the information of clusters. The partitioning information of cluster is changed dynamically through the evolution of objects, and the geometric center of the data point belonging to the corresponding cluster in the object is used as the environment factor, where , for , which is given by (5) at time in the following equation:

where represents the -th cluster centroid of the environment factor , represents the -th data point in the dataset, for , is the number of the data point in the datasets, and number of data points belonging to the corresponding cluster is denoted by .

At time, the position formula of is determined by (6) in the following equation:

and the local best of at time is updated according to (7) in the following equation:

where represents the fitness function of for clustering problems, which can be defined by (8) in the following equation:

The purpose of clustering problems is to find a partitioning result obtained by techniques to minimize the values of fitness function according to equation (8). The global best at time is updated according to (7) in the following equation:

At time, a success rate of chained objects is defined by the following equation:

where represents the success rate of . The success rate in the chained membrane , for , is computed by (11) in the following equation:

where represents the number of the chained object in chained membrane and is the success rate of , , which means the improvement rate at last computation. In this study, a linearly increasing strategy is introduced to adjust the values of the inertia weight dynamically; the possible range is based on the mapping relationship of , which is given by (12) in the following equation:

where and represent the minimum and maximum of inertia weight and .

3.2.2. Evolution Rules in Chained Membrane

The evolution rules in chained membrane contain objects crossover rules and object variation rules, differential evolutionary (DE) approach is further used as the variant of evolution rules in , and the mutation and crossover mechanism is also introduced to help the objects escape the local optima. At time, the -th chained object in is denoted by , where represents the dimension of the objects and and are two randomly chained objects in ; the mutation operation is defined by (13) in the following equation:

where represents the created donor, represents the best object in , the scaling factor is denoted by , and the value is given by , . The mutant comes from the created donor through crossover mechanism, which can be defined by

where the crossover rate is denoted by , it is a predefined constant within the range from 0 to 1, is an independent uniform random number, and is the random dimension from 1 to . At time, the position formula of is determined by

3.3. Communication Rules in Chained Membranes

The communication rules of chained P system are used to enhance the cooperation between the chained membranes, which provide good foundation for exchange and sharing of the information. There are two kinds of communication rules in the CCP system:

(1), for . At time, the global best in is sent to the chained membrane and transformed into the -th object . At each iterative computation, the chained membrane only contains objects that come from the chained membranes to .(2), for . At time, the global best in is transported to the chained membranes to , which is considered to be the global best in chained membrane . Meanwhile, the global best in membrane is sent to the output membrane , which is viewed as the best object or the final computation results of the CCP at time.

3.4. Computation Process of CCP

3.4.1. Initialization

The input membrane contains all initial objects in the P system, denoted by . The position of chained object is randomly initialized in the search space. After initialization, the chained objects in the membrane will be transported to the chained membrane to , and each chained membrane contains objects.

3.4.2. Evolution in Chained Membranes to

The evolution rules on chained objects are used to completed the evolution process according to the equations (3), (4),and (6) in chained membrane . And, the selection for velocity formula is based on a random strategy. The local best and global best in are updated through the equations (7) and (9).

3.4.3. Communication between Chained Membranes

The first kinds of communication rules are used to transport the global best in the to the chained membrane .

3.4.4. Evolution Rules in Chained Membranes

The evolution rules on chained objects are used to completed the evolution process according to the equations (13) and (14) in chained membrane , and global best in is updated by the equation (15).

3.4.5. Communication between Chained Membranes

The second kinds of communication rules are used to transport the global best in to each chained membrane .

3.4.6. Halting and Output

The evolution and communication will be implemented repeatedly with an iterative form during the computation process. The termination criterion of the system is stopped to whether the maximum number of iterations or computation is reached. When the system halts, the output membrane will send the global best to the environment, and this object is regarded as the final computing results of CCP system.

4. Experimental Analysis of Clustering Problems

In this section, the feasibility and effectiveness of the proposed CCP will be demonstrated through the experimental analysis. The datasets of the experiment are introduced first, and the artificial dataset from the previous studies [33] is used to tune the parameters in CCP. Eight real-life datasets from UCI machine leaning repository [34] is used to compare the performance with currently existing clustering approaches. All clustering approaches, including CCP, are implemented on MATLAB 2016b, and all the experiments are conducted on a DELL desktop computer with an Intel 4.00 GHz i7-8550U processor and 8 GB of RAM in a Windows 10 environment.

4.1. Parameter Setting

The numbers of chained membranes have important influences on the performance of CCP. Therefore, four artificial datasets [33], Data_5_2, Data_9_2, Size5, and Square4, are used to tune this parameter in order to ensure the equity of the experiment. The details of the four artificial datasets are given in Table 1.

Table 1: Properties of the artificial datasets.

Datasets
Data
Feature
Class

Data_5_2
250
2
5

Data_9_2
900
2
9

Size_5
1000
2
4

Square4
1000
2
4

Different CCP systems with different degrees of , , [12] are used to evaluate the effects of the number of the chained membranes. The maximum of iterations is set to , and the positive constant is a random number which distributed to 0.6 and 3. The lower and upper limits of the inertia weight are set to and . The mutation probability is randomly generated from 0.2 to 0.8. Other parameters which are not tested in this experiment will be maintained the same values for the fairness of the comparison experiments. And, the number of independent running is set to 30 to eliminate the effect of the random factors. The values of mean and SD of the fitness function obtained by CCP system with different degrees are reported in Table 2.

Table 2: Performance of CCP with different degrees.

Datasets

Mean
SD
Mean
SD
Mean
SD

Data_5_2
326.5071
0.0301
326.4696
0.0123
326.4641
0.0110

Data_9_2
590.8587
0.0624
590.7589
0.0463
590.7413
0.0287

Size_5
2493.1450
4.8048
2491.9680
0.0789
2491.9020
0.0565

Square4
2367.6443
0.0317
2367.6052
0.0260
2367.5917
0.0141

Table 2 reports the clustering results of the CCP system with different degrees on four artificial datasets. The best values of mean and SD for each dataset are given in bold. These results show that the mean and the SD values obtained by the CCP system when are the best among these different systems.

4.2. Clustering Problems

The performance of the CCP is compared with four clustering approaches that have been reported in the literature to further evaluate the effectiveness, such as standard particle swarm optimization (PSO) [35], differential evolution (DE) [36], environment particle swarm optimization (EPSO) [37], and adaptive particle swarm optimization (APSO) [38]. The comparison experiments are conducted on eight real-life datasets from the UCI machine learning repository, Iris, Newthyroid, Seeds, Diabetes, Yeast, Glass, CMC, and Lung Cancer. More details about these datasets are presented in Table 3.

Table 3: Properties of the UCI datasets.

Datasets
Data
Feature
Class

Iris
150
4
3

Newthyroid
215
5
3

Seeds
210
7
3

Diabetes
768
8
2

Yeast
1484
8
10

Glass
214
9
6

CMC
1473
9
3

Lung Cancer
32
26
3

In these compared clustering approaches, the crossover probability of the DE approach is randomly generated from 0.2 to 0.8. An environment factor based on the clustering information is embedded in the velocity updating model of EPSO approach, and the learning factor is randomly generated from 0.6 to 3. A nonlinear regressive function of APSO based on the population diversity is used to the adjustment formula of the inertia weight, represents a predefined constant, and is a predefined constant. All adjustable parameters in these clustering approaches are set to the appropriate values which are reported in the respective publications from Table 4.

Table 4: Parameter setting of the comparative clustering approaches used in the experiments.

Parameters
PSO
DE
EPSO
APSO
CCP

Population ()
100
100
100
100
100

200
200
200
200
200

,
2,2
N
2,2
N
2,2

N
N
(0.6, 3)
N
(0.6, 3)

,
(0,1)
N
(0, 1)
N
(0, 1)

N
N
(0, 1)
N
N

(,)
1
N
(0.4, 0.6)
N
(0.4, 1.2)

N
(0.2, 0.8)
N
N
(0.2, 0.8)

N
N
N
2.1
N

N
N
N
2
N

N
N
N
N
10

Each clustering approach, including CCP, ran for 50 times for each dataset to eliminate the effects of random factors. Simple statistics including worst values (Worst), best value (Best), Mean, and SD of fitness function according to (8) are used in the experiments as the evaluation criteria of clustering results. The experimental environment is the same for all comparative clustering approaches.

Figure 2 shows the convergence of these clustering approaches on the eight test datasets for typical runs of these approaches. The fitness value obtained by CCP declines faster at the beginning of the evolution process and then obtains fine convergence for each dataset. The values of the fitness function of PSO and DE decrease slowly at the beginning of the evolution process and do not apparently have better convergence performance than other approaches. Although EPSO and APSO show better performance than the above clustering approaches, they are also easily trapped into local optima, as shown in Figures 2(e), 2(f), and 2(h). Therefore, CCP has better convergence speed and higher clustering quality than the comparative approaches for all these datasets, as shown in Figure 2.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 2: Convergence of the approaches on each of the eight real-life datasets. (a) Iris, (b) Newthyroid, (c) Seeds, (d) Diabetes, (e) Yeast, (f) Glass, (g) CMC, and (h) Lung Cancer.

Simple statistics of the fitness function values of these clustering approaches on these datasets are reported in Table 5. Results in Table 5 show that CCP has the overall best performance on these eight test datasets. Due to the characteristics of the test datasets, some clustering approaches performed better on some specific datasets with smaller SD, but the performance of CCP on these ten datasets is all considered comparable. Table 6 provides the average computation time in seconds taken by each of the five clustering techniques when running 50 times on each of the datasets. It can be seen from Table 6 that the proposed CCP has a larger average computation time as compared with PSO, DE, EPSO, and APSO. The evolution process in the chained membranes is time-consuming so that CCP takes more time than other techniques does.

Table 5: Performance of the comparative clustering approaches on the eight real-life datasets.

Datasets
Statistics
Clustering approaches

PSO
DE
EPSO
APSO
CCP

Iris
Worst
153.8256
109.9969
96.8183
96.7113
96.6555

Best
114.6361
101.9888
96.7063
96.6672
96.6555

Mean
133.4267
105.2555
96.7431
96.6863
96.6555

S.D.
8.7150
2.3362
0.0374
0.0093
1.23E − 13

Newthyroid
Worst
2400.0152
1977.3729
1943.6091
1908.2618
1895.9965

Best
2040.0058
1924.2365
1887.7578
1885.4950
1866.5183

Mean
2271.0529
1950.8237
1903.7699
1897.2114
1888.1277

S.D.
91.1317
14.0217
11.1225
5.0863
7.5021

Seeds
Worst
452.8039
353.1875
312.1602
311.9663
311.7978

Best
384.2174
333.9361
311.8824
311.8493
311.7978

Mean
422.1335
342.6432
312.0224
311.9148
311.7978

S.D.
23.0106
5.4078
0.0715
0.0257
4.27E − 07

Diabetes
Worst
59885.0835
47911.2714
47629.8621
47590.4655
47561.1362

Best
51628.1083
47693.6403
47585.4012
47561.6124
47561.1262

Mean
54535.4335
47779.1879
47607.6176
47573.1668
47561.1262

S.D.
2147.4319
73.2248
11.8570
6.9363
3.79E − 08

Yeast
Worst
430.0923
378.2874
248.9934
287.4479
245.0999

Best
391.0852
348.2327
236.9439
256.7699
235.3785

Mean
407.6461
367.3743
242.7818
271.6039
239.6706

S.D.
10.9776
7.2050
3.3082
8.0848
2.4396

Glass
Worst
451.0029
334.2308
226.0186
253.7243
213.4793

Best
328.2842
292.1006
212.6117
210.8631
212.1629

Mean
418.7840
314.2182
216.2731
230.8009
212.9308

S.D.
31.2743
8.9158
3.5290
14.8939
0.3110

CMC
Worst
6852.7697
5720.0002
5537.8142
5535.5675
5532.3096

Best
6335.6713
5628.6990
5535.5853
5533.1403
5532.1857

Mean
6551.6558
5683.0978
5536.8242
5534.1784
5532.1913

S.D.
174.0443
28.1480
0.5543
0.5962
0.0278

Lung Cancer
Worst
170.2152
153.9227
127.4346
125.6464
125.6485

Best
184.9632
159.9095
138.4711
126.3331
125.6685

Mean
178.8476
157.0802
133.1676
125.7308
125.6609

S.D.
4.1542
1.6051
3.3466
0.1595
0.0052

Table 6: Average computation time in seconds over 50 runs.

Datasets
Clustering approaches

PSO
DE
EPSO
APSO
KEPSO-CCP

Iris
2.4630
2.7315
3.0608
2.6708
3.3833

Newthyroid
2.4785
2.8251
3.3350
2.8360
3.6576

Seeds
2.4263
2.7072
3.2817
2.8446
3.5799

Diabetes
2.7341
3.1116
4.9156
3.2315
5.2833

Yeast
5.1674
5.2355
9.8255
5.6831
10.3316

Glass
2.6526
3.0069
3.6378
2.9946
4.0859

CMC
3.7550
4.0391
8.0583
3.9940
8.1726

Lung Cancer
2.3451
2.7382
2.8267
2.7655
3.2006

In order to evaluate the clustering performance of these compared clustering approaches, clustering accuracy (CA) is used to evaluate the quality of clustering results obtained by the clustering techniques; the overall accuracy of partitioning results is defined by

where represents the number of the data points in both belonging to actual cluster and partitioning cluster , for . The simple statistics of clustering approaches on eight datasets are reported in Table 7. It can be seen that CCP has overall better performance on these datasets. Although some approaches show a better performance on some specific datasets, the performance of CCP on these specific datasets is also considered compared from Table 7.

Table 7: Performance of the comparative clustering approaches on the eight real-life datasets.

Datasets
Statistics
Clustering approaches

PSO
DE
EPSO
APSO
CCP

Iris
Worst
0.6667
0.8933
0.8933
0.9000
0.8667

Best
0.9267
0.9000
0.9000
0.9000
0.9600

Mean
0.8077
0.8983
0.8997
0.9000
0.9013

SD
0.0904
0.0027
0.0015
1.03E − 17
0.0248

Newthyroid
Worst
0.6977
0.7488
0.7721
0.7721
0.7767

Best
0.8372
0.8093
0.8047
0.8047
0.8047

Mean
0.7391
0.7777
0.8000
0.8000
0.8014

SD
0.0337
0.0187
0.0114
0.0114
0.0080

Seeds
Worst
0.6476
0.8477
0.8952
0.8952
0.8952

Best
0.8953
0.9143
0.8952
0.8952
0.8953

Mean
0.8367
0.8814
0.8952
0.8952
0.8952

SD
0.0850
0.0207
1.14E − 16
1.14E − 16
1.14E − 16

Diabetes
Worst
0.6510
0.6510
0.6510
0.6510
0.6510

Best
0.6510
0.6510
0.6510
0.6510
0.6654

Mean
0.6510
0.6510
0.6510
0.6510
0.6518

SD
1.14E − 16
1.14E – 16
1.14E − 16
1.14E − 16
0.0032

Yeast
Worst
0.3120
0.3309
0.4508
0.3942
0.4683

Best
0.3376
0.4016
0.5370
0.4569
0.5418

Mean
0.3171
0.3640
0.4965
0.4377
0.5200

SD
0.0057
0.0248
0.0275
0.0134
0.0152

Glass
Worst
0.3551
0.4720
0.5607
0.5093
0.5841

Best
0.4813
0.5234
0.5981
0.5841
0.5841

Mean
0.4140
0.5021
0.5834
0.5486
0.5841

SD
0.0444
0.0144
0.0122
0.0267
1.14E − 16

CMC
Worst
0.4332
0.4358
0.4515
0.4562
0.4515

Best
0.4630
0.4582
0.4569
0.4562
0.4569

Mean
0.4510
0.4492
0.4556
0.4562
0.4562

SD
0.0087
0.0074
0.0016
5.66E − 17
0.0012

Lung Cancer
Worst
0.4062
0.4063
0.5313
0.4063
0.5625

Best
0.5625
0.5000
0.5625
0.6250
0.5625

Mean
0.4641
0.4281
0.5578
0.5188
0.5625

SD
0.0433
0.0250
0.0114
0.0644
1.28E − 16

4.3. Proposed CCP for Image Segmentation

In this section, some typical experiment and analysis for tested image are proposed to evaluate the segmentation performance of this proposed CCP. These tested images are used in the previous studies and researches, which are provided from the Berkeley segmentation dataset and benchmark [39]. The size of the tested image is 481 × 321; Figure 3 gives the original image of these tested images.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3: Benchmark tested images. (a) Church. (b) Train. (c) Roman. (d) Starfish. (e) Surfer II. (f) Cow. (g) Crocodile. (h) Elephant.

OSTU proposed by Qtsu [40] is one of most popular segmentation methods, and it has been used to determine whether the optimal threshold method can give a satisfactory segmentation results. The following discriminant criterion measure of OSTU can be described as follows: An image contains pixels from 0 to , usually is set to 255, need to segmented in clusters. Thus, thresholds are needed that divided the original image. The number of -th gray level pixels or frequencies is denoted by , represents the probability of -th gray level pixel in the image,, where . The optimal thresholds are determined by the following equations:

where , , for .

The compared experiments are performed on the test images with a different number of thresholds, , , and [41], to evaluate the performance of the CCP in both low- and high-dimensional multilevel thresholding problems. And, the proposed CCP is compared with PSO, DE, EPSO, and APSO approaches as mentation above. The purpose of image segmentation is to find a set of thresholds to maximize the values of Ostu’s function according to equation (17). Figures 4–7 give the segmented results on tested images with a different number of thresholds.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

Figure 4: Segmented images of church at different thresholds calculated by all approaches: (a) PSO (Th = 3), (b) DE (Th = 3), (c) EPSO (Th = 3), (d) APSO (Th = 3), (e) CCP (Th = 3), (f) PSO (Th = 5), (g) DE (Th = 5), (h) EPSO (Th = 5), (i) APSO (Th = 5), (j) CCP (Th = 5), (k) PSO (Th = 8), (l) DE (Th = 8), (m) EPSO (Th = 8), (n) APSO (Th = 8), (o) CCP (Th = 8).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

Figure 5: Segmented images of starfish at different thresholds calculated by all approaches: (a) PSO (Th = 3), (b) DE (Th = 3), (c) EPSO (Th = 3), (d) APSO (Th = 3), (e) CCP (Th = 3), (f) PSO (Th = 5), (g) DE (Th = 5), (h) EPSO (Th = 5), (i) APSO (Th = 5), (j) CCP (Th = 5), (k) PSO (Th = 8), (l) DE (Th = 8), (m) EPSO (Th = 8), (n) APSO (Th = 8), (o) CCP (Th = 8).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

Figure 6: Segmented images of Surfer II at different thresholds calculated by all approaches: (a) PSO (Th = 3), (b) DE (Th = 3), (c) EPSO (Th = 3), (d) APSO (Th = 3), (e) CCP (Th = 3), (f) PSO (Th = 5), (g) DE (Th = 5), (h) EPSO (Th = 5), (i) APSO (Th = 5), (j) CCP (Th = 5), (k) PSO (Th = 8), (l) DE (Th = 8), (m) EPSO (Th = 8), (n) APSO (Th = 8), (o) CCP (Th = 8).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

Figure 7: Segmented images of elephant at different thresholds calculated by all approaches: (a) PSO (Th = 3), (b) DE (Th = 3), (c) EPSO (Th = 3), (d) APSO (Th = 3), (e) CCP (Th = 3), (f) PSO (Th = 5), (g) DE (Th = 5), (h) EPSO (Th = 5), (i) APSO (Th = 5), (j) CCP (Th = 5), (k) PSO (Th = 8), (l) DE (Th = 8), (m) EPSO (Th = 8), (n) APSO (Th = 8), (o) CCP (Th = 8).

Figures 4–7 provide the segmented images on church, starfish, surfer II, and elephants obtained by the compared techniques. It can be observed that the segmented quality has been improved with the increasing number of thresholds. And, CCP has a better performance than others on these tested images. Respectively, the segmented images obtained by APSO and CCP achieve better consistency than those by PSO, DE, and EPSO when . The gray-level histogram is often regarded as a kind of distributions to determine the thresholds for the image segmentation [42]. And, the peak value of the histogram is one of the important factors that affect the segmentation accuracy. Therefore, the thresholds of the compared approaches on church and elephant images are shown in gray-level histogram as follows.

Figures 8 and 9 show the segmented results of church and elephant at 8 thresholds level obtained by CCP and compared approaches. The optimal thresholds values of CCP with {40, 79, 114, 130, 158, 191, 213, 235} is similar to those of the PSO with {40, 77, 114, 130, 157, 190, 213, 235} on church images and also similar to those of the elephant image. From these figures, it can be seen that the optimal thresholds values by DE and EPSO are very different from those by PSO, APSO, and CCP in most cases. Because the segmentation results depend on the information of classes according to the thresholds level, the optimal thresholds of EPSO have statistical difference with others. Furthermore, it is not difficult to find that the selected optimal thresholds heavily depend on the objective function that is chosen. Besides EPSO, other threshold selection approaches, including the CCP approach, can segment the test images more reasonably, as shown in Figures 4–7. Table 8 provides the mean and S of Ostu’s function with 3, 5, and 8 thresholds achieved by all compared approaches in 50 run times.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8: Segmented images of church at 8 thresholds level calculated by all approaches and their respective gray histogram. (a) Church, (b) PSO, (c) DE, (d) EPSO, (e) APSO, and (f) CCP.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9: Segmented images of elephant at 8 thresholds level calculated by all approaches and their respective gray histogram. (a) Elephant, (b) PSO, (c) DE, (d) EPSO, (e) APSO, and (f) CCP.

Table 8: The values of Ostu’s function with a different number of thresholds calculated by all approaches.

Images
Th
P
Segmentation approaches

PSO
DE
EPSO
APSO
CCP

Church
3
Mean
3276.8367
3276.837
3276.7685
3276.8367
3276.8367

SD
1.87E − 12
1.87E − 12
0.155831
1.87E − 12
0.00E + 00

5
Mean
3381.9107
3381.4923
3380.4929
3381.9236
3381.9246

SD
0.0171
0.2245
1.1240
0.0046
9.25E − 13

8
Mean
3422.5621
3420.9578
3416.2082
3420.7834
3423.2006

SD
2.6579
0.9181
3.5574
2.86725
1.2323

Train
3
Mean
2606.5951
2606.5951
2606.5623
2606.5951
2606.5951

SD
4.67E − 13
4.67E − 13
0.0500
4.67E − 13
4.67E − 13

5
Mean
2736.2257
2735.3181
2735.2676
2735.6328
2736.2331

SD
0.0135
0.5241
0.6743
2.6787
9.33E − 13

8
Mean
2790.3435
2788.0338
2786.9829
2790.4068
2790.4461

SD
0.0913
0.5724
2.5511
0.0897
0.0389

Roman
3
Mean
2142.3440
2142.3440
2142.2752
2142.3440
2142.3440

SD
9.33E − 13
9.33E − 13
0.1073
9.33E − 13
1.39E − 13

5
Mean
2221.3026
2220.8465
2220.5998
2221.3051
2221.3056

SD
0.0047
0.2476
0.4636
0.0012
1.67E − 13

8
Mean
2251.7218
2250.5766
2249.7117
2251.7258
2251.7325

SD
0.0592
0.4211
1.2147
0.0673
0.0573

Starfish
3
Mean
2784.2272
2784.2272
2784.0795
2784.2272
2784.2272

SD
1.4E − 12
1.4E − 12
0.1267
1.4E − 12
9.25E − 13

5
Mean
2916.2557
2915.6466
2915.5862
2916.2726
2916.2730

SD
0.0240
0.3693
0.6716
0.0010
0.0009

8
Mean
2973.9976
2971.8186
2971.2207
2974.0391
2974.0829

SD
0.0404
0.7603
1.8786
0.0942
0.0151

Surfer II
3
Mean
7953.4181
7953.4181
7953.3489
7953.4181
7953.4181

SD
2.80E − 12
2.80E − 12
0.0718
2.78E − 12
2.80E − 12

5
Mean
8025.5880
8025.1056
8024.6160
8025.6045
8025.6055

SD
0.0147
0.2639
1.2232
0.0031
0.0019

8
Mean
8055.9754
8054.3387
8052.1159
8056.0443
8056.0530

SD
0.0505
0.5497
2.2741
0.0361
0.0189

Cow
3
Mean
3858.2202
3858.2202
3858.1872
3858.2200
3858.2202

SD
0.00E + 00
0.00E + 00
0.0886
0.0009
0.00E + 00

5
Mean
3965.2806
3964.3722
3964.0770
3965.2925
3965.2932

SD
0.0196
0.4531
1.2480
0.0032
4.67E − 13

8
Mean
4028.8447
4026.0490
4024.4213
4028.9299
4028.9450

SD
0.0758
0.8635
2.6447
0.0548
0.0107

Crocodile
3
Mean
3155.4359
3155.4343
3155.3468
3155.4359
3155.4359

SD
9.33E − 13
0.0069
0.0969
9.33E − 13
9.25E − 13

5
Mean
3291.4296
3290.6065
3290.4743
3291.4263
3291.4407

SD
0.0163
0.4943
1.0594
0.0323
0.00E + 00

8
Mean
3348.6314
3346.4809
3345.8010
3348.6425
3348.7305

SD
0.0719
0.6380
2.0889
0.1033
0.0168

Elephants
3
Mean
1626.7205
1626.7205
1626.6710
1626.7198
1626.7205

SD
0.00E + 00
0.00E + 00
0.1220
0.0035
0.00E + 000

5
Mean
1695.8663
1695.5298
1695.2335
1695.8663
1695.8676

SD
0.0039
0.1415
0.8755
0.0023
0.0005

8
Mean
1728.1706
1726.9276
1726.1534
1728.0408
1728.2503

SD
0.0592
0.3931
1.5228
0.6764
0.0232

The simple statistics of compared approaches on tested images are reported in Table 8. The best values of mean and SD for each image are highlighted, and it is not hard to see that the CCP system is able to find the best values. Because the PSO, DE, EPSO, and APSO are not specially techniques for image segmentation, some segmented approaches based on multilevel threshold, whale optimation algorithm (WOA) [43], gray wolf optimizer (GWO) [44], whale optimization algorithm based on thresholding heuristic (WOA-TH) [41], and gray wolf optimizer based on thresholding heuristic (GWO-TH) [41], are used to evaluate the clustering effectiveness of CCP. The maximum number of iterations is set to 2200, 3000, and 3600 with the number of thresholds being 3, 5, and 8. The mean and SD of Ostu’s function are obtained by the 100 run times to avoid the effects of random factors.

Table 9 provides the values of mean and SD obtained by different segmented approaches on tested images. Obviously, traditional approaches are easily trapped into local optima, and WOA and GWO have worse segmented performance than that of the others. And, the thresholding heuristic has finetuned the best thresholds to enhance the global search ability of WOA and GWO. It also can be observed that the CCP has better and stable performance than compared segmented approaches on the tested images.

Table 9: The values of Ostu’s function with a different number of thresholds calculated by segmentation approaches.

Images
Th
P
Segmentation approaches

WOA
GWO
WOA-TH
GWO-TH
CCP

Church
3
Mean
3271.4427
3271.4383
3271.4427
3271.4445
3276.8367

SD
1.24E – 02
9.93E – 03
9.14 E − 13
8.27E − 05
1.87E − 12

5
Mean
3375.8721
3374.2197
3375.9118
3371.9813
3381.9246

SD
2.65E + 00
6.74E + 00
2.62E + 00
9.65E + 00
9.33E − 13

8
Mean
3416.6815
3415.4415
3417.0876
3415.6558
3423.5841

SD
2.40E + 00
1.74E + 00
1.78E + 00
3.51E + 00
2.72E − 02

Train
3
Mean
2606.5951
2611.4987
2611.5081
2611.5081
2606.5951

SD
2.26E − 02
2.53E − 02
4.57E − 12
4.57E − 12
9.52E  − 13

5
Mean
2740.6749
2740.5534
2740.6800
2740.6784
2736.2331

SD
1.57E − 02
1.16E − 01
5.77E − 03
6.18E − 03
2.32E  − 03

8
Mean
2794.6931
2793.9232
2794.7170
2794.2733
2790.4901

SD
5.57E − 02
1.19E + 00
1.90E − 02
5.32E − 01
1.12E − 02

Roman
3
Mean
2138.7893
2138.7837
2138.8057
2138.7992
2142.3440

SD
2.52E − 02
2.62E − 02
0.00E + 00
1.69E − 02
9.33E  − 13

5
Mean
2218.6635
2218.7936
2218.9622
2218.9687
2221.3056

SD
2.53E + 00
7.93E − 01
1.11E − 02
2.35E − 04
9.25E  − 13

8
Mean
2249.3768
2248.6408
2249.6457
2249.6080
2251.7798

SD
1.28E + 00
1.17E + 00
1.37E − 01
2.00E − 01
3.77E  − 02

Starfish
3
Mean
2779.9252
2779.9167
2779.9252
2779.9214
2784.2272

SD
3.20E − 12
6.64E − 03
3.20E − 12
5.91E − 03
1.40E  − 12

5
Mean
2912.8532
2912.7058
2912.8562
2912.8371
2916.2730

SD
1.18E − 02
1.66E − 01
5.48E − 03
4.62E − 02
9.34E-04

8
Mean
2972.2218
2971.3725
2972.3479
2972.2159
2974.0989

SD
1.21E + 00
1.19E + 00
7.36E − 03
1.99E − 01
2.23E − 03

Surfer II
3
Mean
7953.4167
7953.4063
7953.4167
7953.4135
7953.4181

SD
5.07E − 03
1.89E − 02
4.82E − 03
7.80E − 03
6.48E  − 12

5
Mean
8025.0703
8025.4959
8025.6049
8025.6025
8025.6055

SD
7.73E − 03
5.56E − 02
7.11E − 03
3.20E − 03
2.80E  − 12

8
Mean
8055.4215
8054.4135
8056.0617
8055.8463
8056.0646

SD
8.27E − 03
7.36E − 01
2.05E − 03
1.25E − 01
2.01E  − 03

Cow
3
Mean
3858.2202
3858.2188
3858.2202
3858.2202
3858.2202

SD
5.94E − 12
1.03E − 02
5.94E − 12
5.94E − 12
1.85E  − 12

5
Mean
3965.2795
3965.1739
3965.2932
3964.6096
3965.2932

SD
2.01E − 02
9.72E − 02
6.40E − 12
2.93E + 00
1.39E  − 12

8
Mean
4028.9410
4027.7892
4028.8474
4027.7210
4028.9605

SD
1.83E − 02
9.47E − 01
6.11E − 01
1.69E + 00
1.50E  − 02

Crocodile
3
Mean
3155.4359
3155.4337
3155.4359
3155.4359
3155.4359

SD
9.14E − 13
1.39E − 02
9.14E − 13
9.14 E  − 13
9.33E − 13

5
Mean
3291.4340
3291.2945
3291.4366
3291.4287
3291.4407

SD
1.02E − 02
1.22E − 01
5.38E − 03
1.35E − 02
4.67E − 13

8
Mean
3348.0384
3347.4528
3348.7453
3348.6504
3348.7454

SD
2.79E + 00
1.75E + 00
1.94 E  − 03
7.60E − 02
3.27E − 03

Elephants
3
Mean
1626.7183
1626.7192
1626.7205
1626.7196
1626.7205

SD
3.53E − 03
9.10E − 03
1.37E − 12
9.25E − 03
9.25E  − 13

5
Mean
1695.0482
1695.3208
1695.8664
1695.8231
1695.8676

SD
3.99E + 00
2.84E + 00
1.05 E  -   03
8.42E − 02
7.12E − 02

8
Mean
1727.1725
1726.5379
1728.2596
1728.1452
1728.2676

SD
2.86E + 00
1.34E + 00
2.55E − 02
1.35E − 01
1.20E − 02

5. Conclusions

A complex chained P system (CCP) is proposed for solving clustering problems, which combines complex chained P systems and evolution mechanisms, including PSO and DE mechanisms. Two kinds of evolution rules for objects in different chained membranes are introduced to enhance the global search ability of PSO. One of evolution rules contain two subevolution rules, which are based on traditional and modified PSO techniques. The partitioning information as environmental factor is introduced to improve the clustering performance of PSO. Another is based on the DE mechanism to evolve the global chained objects in the chained membrane to enhance the global search ability. In addition, two kinds of communication rules in the chained P systems are defined to enhance the cooperation between chained membranes and avoid prematurity. In order to verify the validity and the performance of CCP, this proposed system is evaluated on eight benchmark clustering problems from the UCI machine learning repository as compared with four developed clustering approaches. Furthermore, eight tested images which from the Berkeley segmentation image databases BSDS300 are used to further evaluate the performance of CCP compared to four existing segmentation techniques. These experimental results verify the validity and performance of this proposed CCP.

P systems, as parallel computing models, are highly effective and efficient in solving optimization problems with linear or polynomial complexity. These parallel computing models based on evolution mechanisms provide new ways for solving clustering problems. The extended clustering chained P system uses the chained P system as the computation structure, and the communication rules between chained membranes are single directional. Although these single directional communication rules are simple and easy to implement, bidirectional communication rules may be introduced in future studies to further accelerate the convergence and improve the diversity of populations. Some more complicated communication structures between different membranes may be used in future studies to improve the performance of the approach. Furthermore, the experiments only used small datasets from the artificial datasets and the UCI Machine Learning Repository, and the proposed approach may have some limitations on high dimensional and large datasets. Future studies may test the effectiveness of CCP using large datasets. Balancing the local and global search abilities is also a hard problem to resolve in the future studies. Future studies may also focus on extended P systems based on tissue-like P systems and other bioinspired computing models. More works are needed to apply these extended membrane systems to solve automatic and multiobjective clustering problems.

Data Availability

The two artificial datasets that were manually generated and often used in the existing literature are from the artificial datasets, available at https://www.isical.ac.in/content/databases (accessed June 2018). The eight real-life datasets are often used in the existing literature from the UCI Machine Learning Repository, available at http://archive.ics.uci.edu/ml/datasets.html (accessed June 2018). The eight tested images are from the Berkeley computer vision group, Berkeley segmentation dataset, and benchmark (BSDS300), available at https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/segbench/(accessed October 2018).

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research project was partially supported by the National Natural Science Foundation of China (61876101, 61802234, and 61806114), the National Natural Science Foundation of Shandong Province, China (ZR2019QF007), the Ministry of Education Humanities and Social Science Research Youth Foundation, China (19YJCZH244), the Social Science Fund Project of Shandong Province, China (16BGLJ06 and 11CGLJ22), the Special Postdoctoral Project of China (2019T120607), and Postdoctoral Project of China (2017M612339 and 2018M642695).

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