# Firefly-Based Approaches of Image Recognition

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. 2M-ES Algorithm for Image Alignment and Recognition

Algorithm 1. 2M-ES Algorithm for Image Registration |

1. Inputs:${\text{}\mathsf{\sigma}}_{\mathrm{ini}},\vartheta ,\mathsf{\nu},\mathsf{\tau},\mathrm{Max},\text{}\mathrm{I},\text{}\mathrm{T}$ |

2. $\mathrm{t}\text{}\leftarrow \text{}0$ |

3. Randomly generate an initial point $\mathrm{x}$ |

4. ${\mathrm{x}}_{\mathrm{c}}\leftarrow \mathrm{x}$; $\mathsf{\sigma}\text{}\leftarrow {\text{}\mathsf{\sigma}}_{\mathrm{ini}}$; $\mathrm{it}\text{}\leftarrow \text{}0$; $\mathrm{p}\leftarrow 0$ |

5. while $\mathrm{t}<\mathrm{Max}\mathrm{and}\mathrm{fitness}({\mathrm{x}}_{c})<\mathsf{\tau}$ do |

6. Compute ${\mathrm{y}=\mathrm{x}}_{\mathrm{c}}+\mathrm{z}$, $z$ randomly generated $N\left(0,\sigma \right)$ |

7. if $\mathrm{fitness}({\mathrm{x}}_{c})<\mathrm{fitness}\left(\mathrm{y}\right)$ |

8. ${\mathrm{x}}_{\mathrm{c}}\leftarrow \mathrm{y}$ |

9. $\mathrm{p}\leftarrow \mathrm{p}+1$ |

10. end if |

11. $\mathrm{it}\leftarrow \mathrm{it}+1$ |

12. if $\mathrm{it}=\mathsf{\nu}$ |

13. Apply the Rechenberg rule $\sigma \leftarrow \{\begin{array}{c}\frac{\mathsf{\sigma}}{\vartheta},\raisebox{1ex}{$\mathrm{p}$}\!\left/ \!\raisebox{-1ex}{$\mathsf{\nu}$}\right.0.2\\ \mathsf{\sigma}\xb7\vartheta ,\raisebox{1ex}{$\mathrm{p}$}\!\left/ \!\raisebox{-1ex}{$\mathsf{\nu}$}\right.0.2\\ \mathsf{\sigma},\raisebox{1ex}{$\mathrm{p}$}\!\left/ \!\raisebox{-1ex}{$\mathsf{\nu}$}\right.=0.2\end{array}$ |

14. $\mathrm{it}\leftarrow 0$; $\mathrm{p}\leftarrow 0;$ |

15. end if |

16. $\mathrm{t}\leftarrow \mathrm{t}+1$ |

17. end while |

18. Output:
${\mathrm{I}}_{{\mathrm{x}}_{\mathrm{c}}}$ |

## 4. Image Registration Using the Firefly Algorithm

## 5. Two-Stage Hybrid Algorithm for Image Recognition

_{c}.

Algorithm 2. Two-Stage Hybrid Firefly Algorithm |

1. Inputs: ${\mathrm{n},\mathrm{NMax},\mathsf{\beta}}_{0}{,\mathsf{\gamma},\mathsf{\tau},\mathsf{\sigma}1}_{\mathrm{ini}},\vartheta ,\mathsf{\nu},\mathsf{\tau}1,\mathrm{Max},\mathrm{I},\mathrm{T}$ |

2. APPLY FA |

3. $\mathrm{t}\leftarrow 0$ |

4. Randomly generate an initial population ${\mathrm{P}}_{\mathrm{t}}{=\{\mathrm{x}}_{1}{,\mathrm{x}}_{2}{,\mathrm{x}}_{3}{,\dots \mathrm{x}}_{\mathrm{n}}\}$ |

5. Compute $\mathrm{fitness}\left({\mathrm{x}}_{\mathrm{i}}\right)$, the light intensity of each ${\mathrm{x}}_{\mathrm{i}}\in {\mathrm{P}}_{\mathrm{t}}$ |

6. Compute $\mathrm{best}:\mathrm{fitness}\left(\mathrm{best}\right)=\underset{\mathrm{x}\in {\mathrm{P}}_{\mathrm{t}}}{\mathrm{max}}\mathrm{fitness}\left(\mathrm{x}\right)$ |

7. while $\mathrm{t}<\mathrm{NMax}\mathrm{and}\mathrm{fitness}\left(\mathrm{best}\right)<\mathsf{\tau}\mathrm{do}$ |

8. for $\mathrm{i}=1\dots \mathrm{n}$ |

9. for $\mathrm{j}=1\dots \mathrm{n}$ |

10. if $\mathrm{fitness}({\mathrm{x}}_{\mathrm{i}})<\mathrm{fitness}\left({\mathrm{x}}_{\mathrm{j}}\right)$ |

11. Compute ${\mathrm{x}}_{\mathrm{new}}$ by moving firefly ${\mathrm{x}}_{\mathrm{i}}$ toward firefly ${\mathrm{x}}_{\mathrm{j}}$ (12) |

12. For ${\mathrm{x}}_{\mathrm{new}}$ use the border reflection mechanism (14) |

13. Compute the brightness of ${\mathrm{x}}_{\mathrm{new}}$, $\mathrm{fitness}\left({\mathrm{x}}_{\mathrm{new}}\right)$ |

14. Replace ${\mathrm{x}}_{\mathrm{i}}$ with ${\mathrm{x}}_{\mathrm{new}}$ in ${\mathrm{P}}_{\mathrm{t}}$ |

15. end if |

16. end for |

17. end for |

18. Compute ${\mathrm{best}}_{\mathrm{c}}$: $\mathrm{fitness}\left({\mathrm{best}}_{\mathrm{c}}\right)=\underset{\mathrm{x}\in {\mathrm{P}}_{\mathrm{t}}}{\mathrm{max}}\mathrm{fitness}\left(\mathrm{x}\right)$ |

19. if $\mathrm{fitness}({\mathrm{best}}_{\mathrm{c}})>\mathrm{fitness}(\mathrm{best})$ |

20. ${\mathrm{best}=\mathrm{best}}_{\mathrm{c}}$ |

21. end if |

22. $\mathrm{t}\leftarrow \mathrm{t}+1$ |

23. end while |

24. APPLY 2M-ES PROCEDURE |

25. Compute ${\mathrm{best}}_{\mathrm{c}}:$ ${2\mathrm{M-ES}(\mathrm{best},\mathsf{\sigma}1}_{\mathrm{ini}},\vartheta ,\mathsf{\nu},\mathsf{\tau}1,\mathrm{Max},\mathrm{I},\mathrm{T})$ |

26. Output:${\mathrm{I}}_{{\mathrm{best}}_{\mathrm{c}}}$ |

## 6. The Memetic Approaches of Image Registration

Algorithm 3. Generic Memetic Algorithm |

1. Initialize population |

2. Evaluate each candidate |

3. while Not Termination Condition do |

4. Select parents |

5. Recombine parents to obtain offspring |

6. Mutate offspring |

7. Evaluate offspring |

8. Improve offspring via local search |

9. Select the next generation |

10. end while |

Algorithm 4. Memetic Firefly Algorithm |

1. Inputs: ${\mathrm{n},\mathrm{NMax},\mathsf{\beta}}_{0}{,\mathsf{\gamma},\mathsf{\tau},\mathrm{s},\mathsf{\sigma}1}_{\mathrm{ini}},\vartheta ,\mathsf{\nu},\mathsf{\tau}1,\mathrm{Max},\mathrm{I},\mathrm{T}$ |

2. $\mathrm{t}\leftarrow 0$ |

3. Randomly generate an initial population ${\mathrm{P}}_{\mathrm{t}}{=\{\mathrm{x}}_{1}{,\mathrm{x}}_{2}{,\mathrm{x}}_{3}{,\dots \mathrm{x}}_{\mathrm{n}}\}$ |

4. for $\mathrm{i}=1\dots \mathrm{n}/\mathrm{s}$ |

5. Randomly select $\mathrm{x}\in {\mathrm{P}}_{\mathrm{t}}$ |

6. Compute ${\mathrm{x}}_{\mathrm{new}}$: ${2\mathrm{M}-\mathrm{ES}(\mathrm{x},\mathsf{\sigma}1}_{\mathrm{ini}},\vartheta ,\mathsf{\nu},\mathsf{\tau}1,\mathrm{Max},\mathrm{I},\mathrm{T})$ |

7. Replace $\mathrm{x}$ with ${\mathrm{x}}_{\mathrm{new}}$ in ${\mathrm{P}}_{\mathrm{t}}$ |

8. end for |

9. Evaluate each ${\mathrm{x}}_{\mathrm{i}}\in {\mathrm{P}}_{\mathrm{t}}$ |

10. Compute $\mathrm{best}:\mathrm{fitness}\left(\mathrm{best}\right)=\underset{\mathrm{x}\in {\mathrm{P}}_{\mathrm{t}}}{\mathrm{max}}\mathrm{fitness}\left(\mathrm{x}\right)$ |

11. while$\mathrm{t}<\mathrm{NMax}\mathrm{and}\mathrm{fitness}\left(\mathrm{best}\right)<\mathsf{\tau}$ do |

12. for$\mathrm{i}=1\dots \mathrm{n}$ |

13. $\mathrm{for}\mathrm{j}=1\dots \mathrm{n}$ |

14. if $\mathrm{fitness}({\mathrm{x}}_{\mathrm{i}})<\mathrm{fitness}\left({\mathrm{x}}_{\mathrm{j}}\right)$ |

15. Compute ${\mathrm{x}}_{\mathrm{new}}$ by moving firefly ${\mathrm{x}}_{\mathrm{i}}$ toward firefly ${\mathrm{x}}_{\mathrm{j}}$ using (12) |

16. For ${\mathrm{x}}_{\mathrm{new}}$ use the border reflection mechanism (14) |

17. Evaluate ${\mathrm{x}}_{\mathrm{new}}$ |

18. Replace ${\mathrm{x}}_{\mathrm{i}}$ with ${\mathrm{x}}_{\mathrm{new}}$ in ${\mathrm{P}}_{\mathrm{t}}$ |

19. end if |

20. end for |

21. end for |

22. Compute ${\mathrm{best}}_{\mathrm{c}}$: $\mathrm{fitness}\left({\mathrm{best}}_{\mathrm{c}}\right)=\underset{\mathrm{x}\in {\mathrm{P}}_{\mathrm{t}}}{\mathrm{max}}\mathrm{fitness}\left(\mathrm{x}\right)$ |

23. if ${\mathrm{fitness}(\mathrm{best}}_{\mathrm{c}})\le \mathrm{fitness}\left(\mathrm{best}\right)$ |

24. for $\mathrm{i}=1\dots \mathrm{n}/\mathrm{s}$ |

25. Randomly select $\mathrm{x}\in {\mathrm{P}}_{\mathrm{t}}$ |

26. Compute ${\mathrm{x}}_{\mathrm{new}}$: ${2\mathrm{M}-\mathrm{ES}(\mathrm{x},\mathsf{\sigma}1}_{\mathrm{ini}},\vartheta ,\mathsf{\nu},\mathsf{\tau}1,\mathrm{Max},\mathrm{I},\mathrm{T})$ |

27. Replace $\mathrm{x}$ with ${\mathrm{x}}_{\mathrm{new}}$ in ${\mathrm{P}}_{\mathrm{t}}$ |

28. end for |

29. Compute ${\mathrm{best}}_{\mathrm{c}}$: $\mathrm{fitness}\left({\mathrm{best}}_{\mathrm{c}}\right)=\underset{\mathrm{x}\in {\mathrm{P}}_{\mathrm{t}}}{\mathrm{max}}\mathrm{fitness}\left(\mathrm{x}\right)$ |

30. if $\mathrm{fitness}({\mathrm{best}}_{\mathrm{c}})>\mathrm{fitness}(\mathrm{best})$ |

31. ${\mathrm{best}=\mathrm{best}}_{\mathrm{c}}$ |

32. end if |

33. end if |

34. $\mathrm{t}\leftarrow \mathrm{t}+1$ |

35. end while |

36. Output:${\mathrm{I}}_{\mathrm{best}}$ |

## 7. Experimental Results

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Input | SR | Run Time Mean Value | Run Time Min Value | Run Time Max Value |
---|---|---|---|---|

TI1 | 100% | 15.29 | 4.30 | 34.34 |

TI2 | 98% | 26.21 | 5.54 | 95.85 |

Input | SR | FITNESS SNR Mean Value | FITNESS SNR Min Value | FITNESS SNR Max Value |
---|---|---|---|---|

TI1 | 100% | 0.92 26.44 | 0.89 25.02 | 0.969 31.16 |

TI2 | 98% | 0.903 26.21 | 0.054 10.034 | 0.927 27.587 |

Input | SR | Run Time Mean Value | Run Time Min Value | Run Time Max Value |
---|---|---|---|---|

TI1 | 100% | 14.5914 | 1.7948 | 25 |

TI2 | 100% | 16.7172 | 2.1061 | 41.7394 |

Input | SR | FITNESS SNR Mean Value | FITNESS SNR Min Value | FITNESS SNR Max Value |
---|---|---|---|---|

TI1 | 100% | 0.9063 26.2854 | 0.89 24.6903 | 0.969 31.1685 |

TI2 | 100% | 0.9055 26.1922 | 0.89 21.9707 | 0.963 30.8899 |

Input | SR | Run Time Mean Value | Run Time Min Value | Run Time Max Value |
---|---|---|---|---|

P_{1,1} | 100% | 4.3325 | 0.4764 | 10.2576 |

P_{1,2} | 100% | 5.1893 | 0.625 | 11.0681 |

P_{1,3} | 100% | 3.3141 | 0.4046 | 8.2807 |

P_{1,4} | 100% | 5.8114 | 0.734 | 13.2551 |

P_{1,5} | 100% | 5.7321 | 0.9164 | 12.7348 |

P_{1,6} | 100% | 6.5153 | 0.864 | 14.0022 |

P_{1,7} | 100% | 5.6918 | 1.0151 | 12.5463 |

P_{1,8} | 100% | 5.6389 | 0.5895 | 11.9978 |

Input | SR | Run Time Mean Value | Run Time Min Value | Run Time Max Value |
---|---|---|---|---|

P_{2,1} | 100% | 8.5842 | 0.8736 | 17.5454 |

P_{2,2} | 99.8% | 7.5911 | 1.0809 | 15.0537 |

P_{2,3} | 100% | 5.4495 | 0.6411 | 11.4637 |

P_{2,4} | 99.8% | 4.1837 | 0.7385 | 30.6742 |

P_{2,5} | 100% | 6.2053 | 1.3296 | 11.9818 |

P_{2,6} | 100% | 4.9761 | 0.7192 | 11.4633 |

P_{2,7} | 100% | 4.7902 | 0.8598 | 11.3452 |

P_{2,8} | 100% | 11.3124 | 1.5711 | 24.5833 |

Input | SR | FITNESS SNR Mean Value | FITNESS SNR Min Value | FITNESS SNR Max Value |
---|---|---|---|---|

P_{1,1} | 100% | 0.9113 25.931 | 0.89 | 0.9665 |

P_{1,2} | 100% | 0.9109 26.2857 | 0.89 | 0.9687 |

P_{1,3} | 100% | 0.904 27.5375 | 0.8901 | 0.9304 |

P_{1,4} | 100% | 0.9039 25.8263 | 0.89 | 0.9293 |

P_{1,5} | 100% | 0.9045 25.8614 | 0.89 | 0.9316 |

P_{1,6} | 100% | 0.9044 26.1817 | 0.89 | 0.9366 |

P_{1,7} | 100% | 0.9052 25.6632 | 0.89 | 0.9365 |

P_{1,8} | 100% | 0.9013 26.5873 | 0.89 | 0.9271 |

Input | SR | FITNESS SNR Mean Value | FITNESS SNR Min Value | FITNESS SNR Max Value |
---|---|---|---|---|

P_{2,1} | 100% | 0.9133 26.5621 | 0.89 | 0.9669 |

P_{2,2} | 99.8% | 0.9021 26.06244 | 0.7673 | 0.9289 |

P_{2,3} | 100% | 0.9036 26.22287 | 0.89 | 0.9313 |

P_{2,4} | 99.8% | 0.9048 27.23462 | 0.1789 | 0.943 |

P_{2,5} | 100% | 0.9031 26.26862 | 0.89 | 0.929 |

P_{2,6} | 100% | 0.9051 26.97139 | 0.8901 | 0.9338 |

P_{2,7} | 100% | 0.9043 27.28921 | 0.8901 | 0.9441 |

P_{2,8} | 100% | 0.9011 22.55602 | 0.89 | 0.9236 |

Input | SR | Run Time Mean Value | Run Time Min Value | Run Time Max Value |
---|---|---|---|---|

TI1 | 100% | 5.278 | 0.404 | 14.002 |

TI2 | 99.95% | 6.636 | 0.6411 | 30.674 |

Input | SR | FITNESS SNR Mean Value | FITNESS SNR Min Value | FITNESS SNR Max Value |
---|---|---|---|---|

TI1 | 100% | 0.9057 26.23431 | 0.89 24.444 | 96.8796 31.523 |

TI2 | 99.95% | 0.90472 26.1459 | 0.1789 13.6355 | 96.697 31.042 |

Input | SR | Run Time Mean Value | Run Time Min Value | Run Time Max Value |
---|---|---|---|---|

P_{1,1} | 100% | 7.2073 | 3.0007 | 19.1900 |

P_{1,2} | 100% | 7.7114 | 2.9078 | 24.4584 |

P_{1,3} | 100% | 6.0070 | 2.1924 | 19.0050 |

P_{1,4} | 100% | 8.1369 | 2.9153 | 27.0790 |

P_{1,5} | 100% | 8.3016 | 3.1708 | 25.9697 |

P_{1,6} | 100% | 15.9387 | 2.4286 | 59.3580 |

P_{1,7} | 100% | 9.9297 | 2.7894 | 34.0189 |

P_{1,8} | 100% | 10.7730 | 2.9607 | 33.6675 |

Input | SR | Run Time Mean Value | Run Time Min Value | Run Time Max Value |
---|---|---|---|---|

P_{2,1} | 100% | 10.2538 | 2.7983 | 41.112 |

P_{2,2} | 100% | 20.0726 | 2.9867 | 68.346 |

P_{2,3} | 100% | 10.316 | 2.6688 | 32.1672 |

P_{2,4} | 99.8% | 8.5464 | 2.6749 | 239.634 |

P_{2,5} | 100% | 9.9095 | 2.9316 | 34.5194 |

P_{2,6} | 100% | 8.4755 | 2.8107 | 34.1838 |

P_{2,7} | 100% | 12.579 | 2.5451 | 40.4873 |

P_{2,8} | 100% | 12.4921 | 4.1513 | 43.4075 |

Input | SR | FITNESS SNR Mean Value | FITNESS SNR Min Value | FITNESS SNR Max Value |
---|---|---|---|---|

P_{1,1} | 100% | 0.9114 25.93298 | 0.89 | 0.9631 |

P_{1,2} | 100% | 0.9125 26.395 | 0.8902 | 0.968 |

P_{1,3} | 100% | 0.9042 27.55638 | 0.8901 | 0.9291 |

P_{1,4} | 100% | 0.904 25.838 | 0.89 | 0.9332 |

P_{1,5} | 100% | 0.9035 25.813 | 0.89 | 0.9318 |

P_{1,6} | 100% | 0.9032 26.12029 | 0.89 | 0.9324 |

P_{1,7} | 100% | 0.9051 25.67207 | 0.89 | 0.9395 |

P_{1,8} | 100% | 0.9012 26.58445 | 0.89 | 0.9249 |

Input | SR | FITNESS SNR Mean Value | FITNESS SNR Min Value | FITNESS SNR Max Value |
---|---|---|---|---|

P_{2,1} | 100% | 0.9122 26.48436 | 0.89 | 0.97 |

P_{2,2} | 100% | 0.9015 26.0435 | 0.8902 | 0.9264 |

P_{2,3} | 100% | 0.9039 26.25327 | 0.89 | 0.9296 |

P_{2,4} | 99.8% | 0.9055 27.20916 | 0.8809 | 0.9455 |

P_{2,5} | 100% | 0.903 26.25712 | 0.89 | 0.9296 |

P_{2,6} | 100% | 0.9062 27.02602 | 0.89 | 0.9354 |

P_{2,7} | 100% | 0.9048 27.31797 | 0.89 | 0.9426 |

P_{2,8} | 100% | 0.9 22.54077 | 0.89 | 0.9225 |

Input | SR | Run Time Mean Value | Run Time Min Value | Run Time Max Value |
---|---|---|---|---|

TI1 | 100% | 9.2507 | 2.1924 | 59.358 |

TI2 | 99.9975% | 11.5806 | 2.5451 | 239.634 |

Input | SR | FITNESS SNR Mean Value | FITNESS SNR Min Value | FITNESS SNR Max Value |
---|---|---|---|---|

TI1 | 100% | 0.90569 26.2390 | 0.89 24.513 | 0.9680 31.3535 |

TI2 | 99.9975% | 0.90474 26.14152 | 0.8809 21.87044 | 0.97001 31.3641 |

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## Share and Cite

**MDPI and ACS Style**

Cocianu, C.-L.; Stan, A.D.; Avramescu, M.
Firefly-Based Approaches of Image Recognition. *Symmetry* **2020**, *12*, 881.
https://doi.org/10.3390/sym12060881

**AMA Style**

Cocianu C-L, Stan AD, Avramescu M.
Firefly-Based Approaches of Image Recognition. *Symmetry*. 2020; 12(6):881.
https://doi.org/10.3390/sym12060881

**Chicago/Turabian Style**

Cocianu, Catalina-Lucia, Alexandru Daniel Stan, and Mihai Avramescu.
2020. "Firefly-Based Approaches of Image Recognition" *Symmetry* 12, no. 6: 881.
https://doi.org/10.3390/sym12060881