# Using an Artificial Physarum polycephalum Colony for Threshold Image Segmentation

^{*}

## Abstract

**:**

## Featured Application

**Image segmentation can be applied to image recognition and computer vision as the most important preprocess, and the**

**artificial Physarum polycephalum colony algorithm proposed in this article can also be used to solve**

**all kinds of complex problems in image processing.**

## Abstract

## 1. Introduction

- To overcome the shortcomings of an algorithm search, an APPCA is designed, which can search for the optimal solutions by the expansion and contraction of a lot of artificial hyphae. Different Physarum polycephalum hyphae can learn from each other and produce more hyphae in expansion to enhance the global search capability. In contraction, the artificial Physarum polycephalum colony (APPC) can select the best hyphae with the highest fitness values through a quick sort algorithm to improve the convergence performance. Different from traditional artificial intelligence algorithms, the population sizes of APPCA in expansion and contraction are different, and the proposed algorithm uses a variable population size to search for optimal segmented images.
- To explore the average information content of different pixels in image segmentation, a fitness function is modeled based on Kapur’s entropy for the proposed artificial Physarum polycephalum colony algorithm to search for the optimal threshold segmentation solutions. The fitness function can help the APPCA solve the image segmentation problem with high accuracy and short computational time.
- To test the proposed artificial Physarum polycephalum colony algorithm, a series of benchmark experiments are implemented, and some state-of-the-art approaches are compared. The experimental results verified the effectiveness of the proposed APPCA. The APPCA can obtain better accuracy and convergence speed, and is not easier to fall into the local optimal solution too early compared with traditional artificial intelligence algorithms.

## 2. Related Work

## 3. Methodology

#### 3.1. Symbol Definitions

#### 3.2. Artificial Physarum polycephalum Colony

#### 3.3. A Fitness Function for Entropy Threshold

_{0}, w

_{1}, …, w

_{n}indicate the possibility for each segmentation cluster, and p

_{i}indicates the possibility of the occurrence of pixels with a gray value i. Now, Kapur’s entropy can be obtained for each cluster and instruct the artificial Physarum polycephalum colony to evaluate the characteristics of clusters. There is

#### 3.4. Performance Measurement

_{ij}is a pixel on the segmented image A with a pair of coordinates (i, j), and b

_{ij}is a pixel on the benchmark image B with a pair of coordinates (i, j). The drawback of MSE is that the value often has the same weight in different grayscale. For example, a high MSE can be obtained from a noisy background on a well-recognizable image.

_{ij}is a pixel on the segmented image A with a pair of coordinates (i, j).

#### 3.5. Algorithm Flow of APPCA

_{th}, the number of iterations Ite_max, or a maximum calculation time limit can be used as an end judgment before the optimal solutions are generated.

## 4. Algorithm Steps of APPCA

#### 4.1. Step 1: Initialization

_{s}of social learning, and the possibility p

_{r}of free learning should be set first. The artificial Physarum polycephalum colony algorithm can search for the optimal solutions in the whole solution space and the artificial hyphae can move from one pixel to another to connect them into a segmented image. The segmented image for hyphae $k$ can be seen as a feasible solution. The fitness in Equation (5) can be set to zero at the start.

_{max}, maximum iteration steps Ite_max, and minimum iteration error threshold e

_{th}. If any of these conditions are met, the iterative calculation can be ended.

#### 4.2. Step 2: Expansion

_{s}, while the other part different from the parent comes from other hyphae with a probability of 1 − p

_{s}. The larger the p

_{s}, the more experience is retained for self-learning, the stronger the local search ability, and the faster the convergence speed; on the contrary, the smaller the p

_{s}, the more social-learning experience it has, and the stronger its global search ability, but the slower its convergence speed. In Equation (16), only one child is randomly selected for the next iteration.

_{f}of free learning, that is, a rand-segmented image of the free-learning population ${x}_{free}^{m}\left(Ite\right)$.

_{f}, the total population size is (2 + p

_{f})S after the expansion operation. There is

Algorithm 1 Expansion with a triple-loop | |

1: | Input: the original image, H, V, grayscale L |

2: | Define: the fitness function in Equation (5) |

3: | Initialization: the APPC, M, S, p_{s}, p_{f} |

4: | Calculate: randomly generate hyphae $\left\{{x}_{}^{k}\left(1\right)\right\}$ |

5: | Initialization: T_{max}, Ite_max, e_{th} |

6: | For Ite = 1:Ite_max |

7: | For Physarum polycephalum $m=1:M$ |

8: | For hyphae $k=1:S$ |

9: | Random select a pixel |

10: | Self-learning |

11: | Social learning by p_{s} |

12: | Free learning by p_{f} |

13: | To integrate the three parts of the expansion population |

14: | End for |

15: | To update the Physarum polycephalum colony by Equation (18) |

16: | End for |

17: | To update the Physarum polycephalum colony by Equation (19) |

18: | Return the solutions to threshold image segmentation |

19: | //contraction algorithm |

20: | End for |

21: | //Output the optimal segmented image |

Algorithm 2 Expansion with a dual-loop | |

1: | Input: the original image, H, V, grayscale L |

2: | Define: the fitness function in Equation (5) |

3: | Initialization: the APPC, M, S, p_{s}, p_{f} |

4: | Calculate: randomly generate hyphae $\left\{{x}_{}^{k}\left(1\right)\right\}$ |

5: | Initialization: T_{max}, Ite_max, e_{th} |

6: | For Ite = 1:Ite_max |

7: | For hyphae $k=1:S$ |

8: | Random select a pixel |

9: | Self-learning |

10: | Social learning by p_{s} |

11: | Free learning by p_{f} |

12: | To integrate the three parts of the expansion population |

13: | To update the Physarum polycephalum colony by Equation (18) |

14: | End for |

15: | To update the Physarum polycephalum colony by Equation (19) |

16: | Return the solutions to threshold image segmentation |

17: | //contraction algorithm |

18: | End for |

19: | //Output the optimal segmented image |

#### 4.3. Step 3: Contraction

_{f})S after expansion, but not all hyphae can survive. Only the hyphae with the highest fitness of population size S can survive after contraction, the other population (1 + p

_{f})S with low fitness will disappear. There is

_{f})MS in total. The pseudo-code of contraction is shown in Algorithm 3.

Algorithm 3 Contraction algorithm with a triple-loop | |

1: | For Ite = 1:Ite_max |

2: | //expansion algorithm |

3: | For hyphae k = 1:(2 + p_{f})MS |

4: | For hyphae s = 1:(2 + p_{f})MS-1 |

5: | If fitness (s) > fitness(s + 1) |

6: | temp = fitness (s) |

7: | fitness (s) = fitness(s + 1) |

8: | fitness(s + 1) = temp |

9: | End if |

10: | End for |

11: | End for |

12: | To store the optimal solution Temp_s = 1 |

13: | To store the optimal fitness Temp_fitness = fitness(1) |

14: | To calculate the iterative error e=|fitness(1)(Ite)- fitness(1)(Ite-1)| |

15: | To judge the error with end condition: T_{max}, Ite_max, e_{th} |

16: | If the end condition does not fit |

17: | Select the top S best hyphae |

18: | Return to the expansion algorithm |

19: | Else |

20: | Exit |

21: | End for |

22: | Exit: output the optimal segmented image ${x}_{}^{1\ast}\left(Ite\right)$ |

23: | Exit: output the optimal fitness $fitness\left({x}_{}^{1\ast}\left(Ite\right)\right)$ |

_{th}, the maximum iterations Ite_max, and the maximum simulation time T

_{max}, the optimal solution(rank No.1) and the corresponding fitness (fitness of No.1) will be output. Otherwise, the top S best solutions will be selected from all solutions and returned to the expansion algorithm. The APPC contraction algorithm seems to have good sorting capabilities, but the computation is still too time-consuming.

Algorithm 4 Contraction algorithm with a dual-loop | |

1: | For Ite = 1:Ite_max |

2: | //expansion algorithm |

3: | For hyphae k = 1:(2 + p_{f}) S |

4: | QuickSort(s, 1, (2 + p_{f}) S) |

5: | To store the optimal solution Temp_s = 1 |

6: | To store the optimal fitness Temp_fitness = fitness(1) |

7: | End for |

8: | To calculate the iterative error e=|fitness(1)(Ite)- fitness(1)(Ite-1)| |

9: | To judge the error with end condition: T_{max}, Ite_max, e_{th} |

10: | If the end condition is not fit |

11: | Select the top S best hyphae |

12: | Return to the expansion algorithm |

13: | Else |

14: | Exit |

15: | End for |

16: | Exit: output the optimal segmented image ${x}_{}^{1\ast}\left(Ite\right)$ |

17: | Exit: output the optimal fitness $fitness\left({x}_{}^{1\ast}\left(Ite\right)\right)$ |

#### 4.4. Step 4: End Judgment

_{th}or the number of iterations Ite_max. If the end conditions are not satisfied, the top S best solutions in the contraction algorithm will be returned to the expansion algorithm, and the process of expansion and contraction will be repeated until the end conditions are fit. Finally, an optimal segmented image ${x}_{}^{1\ast}\left(Ite\right)$ will be output as a solution to the threshold image segmentation.

## 5. Benchmark Tests and Comparison

#### 5.1. Experiment Design

- It is assumed that the ethics, privacy, politics, race, topic, history, ecology, and complexity factors of the original images are not taken into account.
- It is assumed that the differences between the binarization methods, the filters, and the color differences of the pixels are not considered.
- It is assumed that the performance difference of different algorithms running on different software and hardware of personal computers is ignored.
- It is assumed that the optimal parameter adjustment on determined datasets is not discussed. Although each algorithm has a lot of improved editions to obtain better performance, this is not considered here.

_{s}= 0.9, and the free-learning probability is set as p

_{s}= 0.2.

_{in}= 8), and eight output channels (c

_{out}= 8). The learning rate of the offset item was twice that of the weight. The extension edge was set to 0, the weight was initialized to Gaussian, and the value of the constant offset item was 0.

#### 5.2. Benchmark Test on Pascal VOC 2012 Dataset

#### 5.3. Benchmark Test on Stanford Background Dataset

#### 5.4. Experiment Summary

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Convergence curves on Pascal VOC 2012 dataset. (

**a**) image ID 2012_001155; (

**b**) image ID 2012_003403; (

**c**) image ID 2012_003438; (

**d**) image ID 2012_004019; (

**e**) image ID 2012_004134; (

**f**) image ID 2012_004248.

**Figure 6.**Convergence curves on Stanford background dataset. (

**a**) image ID 0105146; (

**b**) image ID 5,000,226; (

**c**) image ID 6,000,141; (

**d**) image ID 6,000,321; (

**e**) image ID 9,000,002; (

**f**) image ID 9,005,294.

Symbol | Definition |
---|---|

O | An original image |

A | A segmented image with additive noise |

B | A benchmark image |

H | The image size in the horizontal direction |

V | The image size in the vertical direction |

N | The total number of pixels on the original image |

L | The grayscale level of an image |

${a}_{ij}$ | A pixel with a pair of coordinates (i, j) on a gray image A |

${\mu}_{A}$ | The average gray level of a gray image A |

${\sigma}_{A}$ | The standard deviation of a gray image A |

${A}_{ij}$ | The intensity value on each pixel with a pair of coordinates (i, j) |

K | Kapur’s entropy |

${K}_{i}$ | Kapur’s entropy on a segmentation area |

${w}_{i}$ | The possibility for each segmentation cluster |

${p}_{i}$ | The possibility of the occurrence of pixels with a gray value i |

t | A set of thresholds t = {${t}_{1},{t}_{2},{t}_{3},\dots ,{t}_{n}$} |

M | The population size of Physarum polycephalum |

m | The number of Physarum polycephalum |

S | The population size of hyphae |

k | The hyphae number |

p_{s} | The possibility of social learning |

p_{f} | The possibility of free learning |

${x}^{k}\left(Ite\right)$ | A solution of the hyphae k at the iteration of Ite |

${x}_{self}^{k}\left(Ite\right)$ | The self-learning hyphae k at the iteration of Ite |

${x}_{social}^{k}\left(Ite\right)$ | The social-learning hyphae k at the iteration of Ite |

${x}_{free}^{m}\left(Ite\right)$ | The free-learning hyphae k at the iteration of Ite |

fit | A fitness function |

Ite | An iterative computation counter |

Ite_max | The maximization of the iterative computation counter |

e_{th} | The error threshold of iterative computation |

$MSE$ | The mean squared error |

$PSNR$ | The peak signal-to-noise ratio |

$CorCoe$ | The correlation coefficient between two images |

$Jaccard$ | Jaccard’s index, also called Intersection-Over-Union, IoU |

$Dice$ | The Sørensen Dice coefficient |

$SSIM$ | The structural similarity index |

Algorithms | Metrics | 2012_001155 | 2012_003403 | 2012_003438 | 2012_004019 | 2012_004134 | 2012_004248 |
---|---|---|---|---|---|---|---|

APPCA | MSE | 395 | 356 | 379 | 387 | 443 | 398 |

PSNR | 22.16 | 22.62 | 22.34 | 22.25 | 21.67 | 22.13 | |

SSIM | 0.63 | 0.65 | 0.65 | 0.63 | 0.59 | 0.62 | |

DL | MSE | 376 | 1214 | 1629 | 391 | 405 | 461 |

PSNR | 22.38 | 17.29 | 16.01 | 22.21 | 22.06 | 21.49 | |

SSIM | 0.64 | 0.46 | 0.41 | 0.61 | 0.61 | 0.58 | |

GWO | MSE | 1602 | 2213 | 1007 | 2166 | 1211 | 856 |

PSNR | 16.08 | 14.68 | 18.10 | 14.77 | 17.30 | 18.81 | |

SSIM | 0.42 | 0.36 | 0.48 | 0.36 | 0.48 | 0.48 | |

PSO | MSE | 739 | 383 | 476 | 502 | 722 | 2353 |

PSNR | 19.44 | 22.30 | 21.35 | 21.12 | 19.55 | 14.41 | |

SSIM | 0.52 | 0.63 | 0.59 | 0.58 | 0.52 | 0.35 | |

DEA | MSE | 584 | 391 | 726 | 718 | 507 | 1829 |

PSNR | 20.47 | 22.21 | 19.52 | 19.57 | 21.08 | 15.51 | |

SSIM | 0.56 | 0.61 | 0.53 | 0.54 | 0.57 | 0.41 | |

ACO | MSE | 827 | 745 | 636 | 2029 | 744 | 661 |

PSNR | 18.96 | 19.41 | 20.10 | 15.06 | 19.42 | 19.93 | |

SSIM | 0.51 | 0.50 | 0.55 | 0.38 | 0.51 | 0.53 | |

GA | MSE | 549 | 574 | 373 | 712 | 588 | 2012 |

PSNR | 20.74 | 20.54 | 22.41 | 19.61 | 20.44 | 15.09 | |

SSIM | 0.57 | 0.55 | 0.65 | 0.55 | 0.54 | 0.39 | |

ABC | MSE | 3216 | 4637 | 1405 | 2683 | 2556 | 409 |

PSNR | 13.06 | 11.47 | 16.65 | 13.84 | 14.06 | 22.01 | |

SSIM | 0.32 | 0.28 | 0.45 | 0.33 | 0.41 | 0.61 | |

AFSA | MSE | 2134 | 3936 | 1982 | 2351 | 425 | 2439 |

PSNR | 14.84 | 12.18 | 15.16 | 14.42 | 21.85 | 14.26 | |

SSIM | 0.35 | 0.33 | 0.39 | 0.34 | 0.60 | 0.35 | |

IS | MSE | 1482 | 4335 | 347 | 417 | 3875 | 746 |

PSNR | 16.42 | 11.76 | 22.73 | 21.93 | 12.25 | 19.40 | |

SSIM | 0.43 | 0.30 | 0.67 | 0.60 | 0.31 | 0.51 |

Algorithms | Metrics | 0105146 | 5000226 | 6000141 | 6000321 | 9000002 | 9005294 |
---|---|---|---|---|---|---|---|

APPCA | MSE | 374 | 377 | 351 | 396 | 349 | 382 |

PSNR | 22.40 | 22.37 | 22.68 | 22.15 | 22.70 | 22.31 | |

SSIM | 0.61 | 0.64 | 0.65 | 0.61 | 0.63 | 0.62 | |

DL | MSE | 411 | 1616 | 370 | 354 | 368 | 345 |

PSNR | 21.99 | 16.05 | 22.45 | 22.64 | 22.47 | 22.75 | |

SSIM | 0.59 | 0.46 | 0.62 | 0.62 | 0.61 | 0.64 | |

GWO | MSE | 477 | 615 | 1116 | 1252 | 720 | 768 |

PSNR | 21.35 | 20.24 | 17.65 | 17.15 | 19.56 | 19.28 | |

SSIM | 0.57 | 0.51 | 0.46 | 0.47 | 0.52 | 0.51 | |

PSO | MSE | 1954 | 342 | 751 | 716 | 731 | 1462 |

PSNR | 15.22 | 22.79 | 19.37 | 19.58 | 19.49 | 16.48 | |

SSIM | 0.39 | 0.67 | 0.53 | 0.53 | 0.53 | 0.42 | |

DEA | MSE | 1095 | 568 | 637 | 459 | 694 | 726 |

PSNR | 17.74 | 20.59 | 20.09 | 21.51 | 19.72 | 19.52 | |

SSIM | 0.43 | 0.54 | 0.55 | 0.57 | 0.54 | 0.53 | |

ACO | MSE | 614 | 539 | 990 | 836 | 555 | 657 |

PSNR | 20.25 | 20.81 | 18.17 | 18.91 | 20.69 | 19.96 | |

SSIM | 0.54 | 0.52 | 0.49 | 0.51 | 0.56 | 0.54 | |

GA | MSE | 927 | 425 | 429 | 702 | 572 | 1121 |

PSNR | 18.46 | 21.85 | 21.81 | 19.67 | 20.56 | 17.63 | |

SSIM | 0.47 | 0.57 | 0.58 | 0.54 | 0.55 | 0.45 | |

ABC | MSE | 2628 | 2664 | 1711 | 1692 | 3272 | 1635 |

PSNR | 13.93 | 13.88 | 15.80 | 15.85 | 12.98 | 16.00 | |

SSIM | 0.38 | 0.37 | 0.39 | 0.40 | 0.35 | 0.41 | |

AFSA | MSE | 3063 | 2941 | 1624 | 1570 | 4647 | 3343 |

PSNR | 13.27 | 13.45 | 16.02 | 16.17 | 11.46 | 12.89 | |

SSIM | 0.36 | 0.33 | 0.44 | 0.43 | 0.28 | 0.34 | |

IS | MSE | 3590 | 946 | 865 | 955 | 673 | 3619 |

PSNR | 12.58 | 18.37 | 18.76 | 18.33 | 19.85 | 12.54 | |

SSIM | 0.31 | 0.49 | 0.50 | 0.49 | 0.54 | 0.29 |

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## Share and Cite

**MDPI and ACS Style**

Cai, Z.; Li, G.; Zhang, J.; Xiong, S.
Using an Artificial *Physarum polycephalum* Colony for Threshold Image Segmentation. *Appl. Sci.* **2023**, *13*, 11976.
https://doi.org/10.3390/app132111976

**AMA Style**

Cai Z, Li G, Zhang J, Xiong S.
Using an Artificial *Physarum polycephalum* Colony for Threshold Image Segmentation. *Applied Sciences*. 2023; 13(21):11976.
https://doi.org/10.3390/app132111976

**Chicago/Turabian Style**

Cai, Zhengying, Gengze Li, Jinming Zhang, and Shasha Xiong.
2023. "Using an Artificial *Physarum polycephalum* Colony for Threshold Image Segmentation" *Applied Sciences* 13, no. 21: 11976.
https://doi.org/10.3390/app132111976