# Genetic Algorithm for Sparse Optimization of Mills Cross Array Used in Underwater Acoustic Imaging

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mills Cross Multiplicative Array

_{m}and b

_{n}are the weights of two arrays, k is the wave number, c is the speed of sound in water, $\omega $ is the angular frequency, u = sinθcosφ and v = sinθsinφ.

## 3. Improved Genetic Algorithm

- (1)
- Population coding

**I**= [I

_{1}, I

_{2}, …, I

_{N}], I

_{n}= 0 or 1

**w**= [w

_{1}, w

_{2}, …, w

_{N}]

_{n}is also binary coded. If

**I**and

**w**are combined, a long binary-encoded chromosome will be obtained, expressed as [

**I**,

**w**]. Such a chromosome includes a certain quantity of genes, which indicates whether an individual element of the array survives or not and shows the weight of each element. The initialization of all the genes is random, and an initial population will comprise of some different chromosomes.

- (2)
- Fitness function

_{max}is the peak value of the main lobe, w

_{n}is the weight of the n-th array element, $\theta ,\phi \in S$ specify a reasonable sidelobe angular range. The lower the maximum sidelobe level is, the higher the above fitness value is, the more probably the corresponding chromosome survives.

- (3)
- Selecting operation

- (4)
- Two-point Orthogonal Crossover Operation

^{3}= 8. Considering the representative set of combinations, L

_{8}(2

^{7}) orthogonal table, illustrated in Table 1, is employed to present the layout of two levels per factor and eight combinations of levels. In the above orthogonal design, seven factors have been simplified to three factors [31]. It is important that all the selected combinations are good representatives for all of the possible combinations. Finally, two offspring with higher fitness value after the crossover will be retained.

- (5)
- Mutation operator

_{mut}.

- (6)
- Elitist Preservation

## 4. Sparse Optimization of Mills Cross Array

## 5. Conclusions

- (1)
- Compared with the rectangular planar array, Mills cross multiplicative array has distinct advantage in reducing elements of acoustic imaging array, without losing imaging resolution. It is an available choice to be used in UAI application.
- (2)
- Sparse optimization of Mills cross multiplicative array can be implemented through improved genetic algorithm. The sparse cross array decreases the number of elements by 8.25% compared with MCMA, while it still keeps its advantages of beamwidth and maximum sidelobe level.
- (3)
- The improved genetic algorithm is effective to obtain a sparse solution of cross array. The fitness function based on pattern indices is applicable. Two-point orthogonal crossover operator can retain elitist individuals with a high probability. The sparse solution is an evolved result of mutual restraint between array elements’ survival and their weights.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Schematic configuration of a rectangular planar array with $(2M+1)\times (2M+1)$ elements.

**Figure 4.**Flow of improved genetic algorithm with two-point orthogonal crossover and the elitism preservation operation.

**Figure 5.**Beam patterns (where M = 12). (

**a**) RPA with (25 × 25) elements; (

**b**) MCMA with (49 + 49) elements.

**Figure 6.**Element distribution of Mills cross multiplicative array. (

**a**) before optimization; (

**b**) after optimization.

**Figure 7.**Element weights of Mills cross multiplicative array. (

**a**) before optimization; (

**b**) after optimization.

Individual Substring Number Individual after | I | II | III |
---|---|---|---|

Crossover Operation | |||

1 | a | a | a |

2 | a | a | b |

3 | a | b | a |

4 | a | b | b |

5 | b | a | a |

6 | b | a | b |

7 | b | b | a |

8 | b | b | b |

Cross Array Beamforming Method | Beamwidth | The Maximum Sidelobe Level |
---|---|---|

cross array based on conventional beamforming | 3.15° | −4.97 dB |

Mills cross multiplicative array before optimization | 4.08° | −13.5 dB |

sparse cross array after optimization | 4.76° | −14.27 dB |

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**MDPI and ACS Style**

Teng, D.; Li, Y.; Yang, H.; Wei, Z.; Li, Y. Genetic Algorithm for Sparse Optimization of Mills Cross Array Used in Underwater Acoustic Imaging. *J. Mar. Sci. Eng.* **2022**, *10*, 155.
https://doi.org/10.3390/jmse10020155

**AMA Style**

Teng D, Li Y, Yang H, Wei Z, Li Y. Genetic Algorithm for Sparse Optimization of Mills Cross Array Used in Underwater Acoustic Imaging. *Journal of Marine Science and Engineering*. 2022; 10(2):155.
https://doi.org/10.3390/jmse10020155

**Chicago/Turabian Style**

Teng, Duo, Yatian Li, Hu Yang, Zhiqiang Wei, and Yaan Li. 2022. "Genetic Algorithm for Sparse Optimization of Mills Cross Array Used in Underwater Acoustic Imaging" *Journal of Marine Science and Engineering* 10, no. 2: 155.
https://doi.org/10.3390/jmse10020155