# Hybrid Grasshopper Optimization Algorithm and Differential Evolution for Multilevel Satellite Image Segmentation

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## Abstract

**:**

## 1. Introduction

- Propose an efficient satellite image segmentation method.
- Apply the hybrid algorithm of GOA and DE to the multilevel thresholding domain.
- Introduce an alternative hybrid model for a meta-heuristic algorithm.

## 2. Grasshopper Optimization Algorithm

Algorithm 1 Pseudocode of grasshopper optimization algorithm for optimization problem |

1. Begin2. Initialize a randomly distributed population in the search space; 3. Initialize the best search agent $\widehat{{T}_{d}}$; 4. while $t<{t}_{\mathrm{max}}$5. Evaluate $c$ using Equation (3); 6. for $i=1:n$7. Calculate the objective value of each grasshopper ${f}_{i}$; 8. Update the best search agent $\widehat{{T}_{d}}$; 9. Normalize the distance between grasshoppers in [1,4]; 10. Update the position of grasshopper ${x}_{i}$ using Equation (2); 11. Correct the position of the current grasshopper if it is beyond the border; 12. end for13. end while14. return $\widehat{{T}_{d}}$, which represents the optimal position of optimization; 15. End |

## 3. Multilevel Thresholding

## 4. Proposed Methodology

#### 4.1. Differential Evolution

#### 4.1.1. Mutation

#### 4.1.2. Crossover

#### 4.1.3. Selection

#### 4.2. Self-Adapting Differential Evolution (jDE)

Algorithm 2 Pseudocode of jDE algorithm for an optimization problem |

1. Begin2. Initialize a randomly distributed population in the search space; 3. Initialize the best search agent ${x}_{best}$; 4. while $t<{t}_{\mathrm{max}}$5. for $i=1:n$6. Calculate the objective value of each search agent ${f}_{i}$; 7. Update the best search agent ${x}_{best}$; 8. Evaluate the control parameters $SF$ and $CR$ of each search agent using Equations (8)–(9); 9. Mutation: Generate a mutant individual using Equation (5), and then check the position;10. Crossover: Choose the trial individual from current individual and mutant individual using Equation (6);11. Selection: Select the better individual that will be preserved for the next generation using Equation (7);12. end for13. end while14. return ${x}_{best}$, which represents the optimal position of optimization; 15. End |

#### 4.3. Hybrid Algorithm of GOA and jDE (GOA–jDE)

Algorithm 3 Pseudocode of GOA–jDE-based multilevel satellite image thresholding |

Input: The given satellite image.Output: Segmentation thresholds.
$$**********************************\phantom{\rule{0ex}{0ex}}**\mathrm{Get}\mathrm{information}\mathrm{about}\mathrm{the}\mathrm{image}**\phantom{\rule{0ex}{0ex}}**********************************$$
1. Read the given color satellite image; 2. Extract the histogram of each color component (R, G, and B);
$$**********************************\phantom{\rule{0ex}{0ex}}****\mathrm{GOA}-\mathrm{jDE}****\phantom{\rule{0ex}{0ex}}**********************************$$
3. Initialize a randomly distributed population in the search space; 4. Initialize the best search agent $\widehat{{T}_{d}}$; 5. Initialize the fitness values of the grasshoppers ${f}_{i}$; 6. Set population size $N$ and maximum number of iterations ${t}_{\mathrm{max}}$; 7. Set the dimensions of the optimization problem $dim$, namely the number of thresholds; 8. while (termination condition is not met $(t<{t}_{\mathrm{max}})$)9. Check the boundary and evaluate the fitness value of each grasshopper ${f}_{i}$ using Equation (4); 10. Update the location $\widehat{{T}_{d}}$ and fitness value ${f}_{\mathrm{best}}$ of best search agent if there is a better one; 11. Evaluate the parameter $c$ using Equation (3); 12. Calculate the average fitness value $\overline{f}$ of the population; 13. for (each grasshopper $(i=1:n$))14. if (${f}_{i}>\overline{f}$) % GOA Algorithm15. Update the position of grasshopper using Equation (2); 16. else % jDE Operator17. Evaluate $SF$ and $CR$ of each search agent using Equations (8)–(9); 18. Mutation, Crossover, and Selection using Equations (5)–(7). 19. end if20. end for21. end while |

Fitness function (Minimum Cross entropy) |

Input: Histogram of a color component, and segmentation thresholds ${x}_{i}$.Output: Fitness function value ${f}_{i}$.1. The histogram is divided into $n+1$ parts by $n$ thresholds; 2. Calculate the proportion of pixels in each gray level $\left({p}_{j},j\in \left[0,255\right]\right)$ to the total based on the histogram; 3. Compute the zero-moment ${m}_{i}^{0}$ and first-moment ${m}_{i}^{1}$ on partial range of the image histogram; 4. Calculate the minimum cross entropy of each part $-{m}_{k}^{1}\mathrm{ln}\left({m}_{k}^{1}/{m}_{k}^{0}\right)$ $\left(k\in \left[0,n\right]\right)$; 5. The sum of the entropies of all parts represents the fitness function value; 6. ${f}_{i}=-{m}_{0}^{1}\mathrm{ln}\left({m}_{0}^{1}/{m}_{0}^{0}\right)-{m}_{1}^{1}\mathrm{ln}\left({m}_{1}^{1}/{m}_{1}^{0}\right)-\cdots -{m}_{n}^{1}\mathrm{ln}\left({m}_{n}^{1}/{m}_{n}^{0}\right)$; |

#### 4.4. Computational Complexity

## 5. Experimental Results and Discussion

#### 5.1. Experimental Setup

**boldface**. The experimental environment is given as follows: MATLAB 2017 and Microsoft Windows 10 operating system.

#### 5.2. Performance Measures

#### 5.3. Experimental Series 1: Comparison of Satellite Image Thresholding Methods Based on MCE

#### 5.3.1. Results and Discussions

**bolded**results are best). Note that a lower function value indicates better performance. It can be seen from the table that the proposed method presents the lowest value in most cases. For example, under the conditions of “Image3” (for K = 12), the fitness values are −646.3107, −646.2853, −646.3092, −646.3039, −646.3077, −646.3098, −646.2148, and −646.2988 for GOA–jDE, GOA, DE, MGOA, hjDE, BDE, BA, and PSO, respectively. It is obvious that GOA–jDE outperforms the compared algorithms, indicating its remarkable optimization ability and balanced exploration–exploitation. At the same time, this convincingly demonstrates that the segmented image obtained by the proposed approach is detailed, informative, and of high quality because the entropy of a given image indicates its average information content [65]. More specifically, the reason for the high precision is the use of a powerful hybrid model. In GOA–jDE, the whole population is divided into two parts according to the average fitness value. Compared to the standard GOA and DE, the exploration and exploitation stages of optimization are well balanced, which increases the probability of obtaining the global optimum. Compared to other algorithms, the GOA–jDE algorithm can maintain the population diversity in the late iteration, thus enhancing the ability to avoid local optimization. For example, the main drawback of PSO is the parameter $\omega $. If the starting and ending value of parameter $\omega $ for the current problem are not determined in the best form, the PSO may converge prematurely and fall into a local optimum. For the BA algorithm, more parameters need to be adjusted for different problems, such as ${r}_{i}$ (rate of pulse emission), ${A}_{i}$ (loudness value), and ${v}_{\mathrm{max}}$ (maximum velocity). The determination of these parameters greatly reduces the universality of BA algorithm, which affects the performance to some extent.

#### 5.3.2. Statistical Tests

#### 5.4. Experimental Series 2: Performance on Other Objective Functions

**bolded**results are best). In the comparison with MABC–Tsallis, the GOA–jDE algorithm exhibits a remarkable performance based on the Tsallis entropy thresholding technique, which is reflected in all the obtained values. Moreover, considering the experiment on natural images, GOA–jDE–Tsallis presents better results in most cases, indicating the robustness and universality of the algorithm. For the comparison with the other two methods, it can be seen that the proposed algorithm-based methods are, again, superior to other methods. In general, the compared method based on MCE is relatively more competitive than the compared method based on Tsallis entropy and Otsu’s method, because the first one gives better results in more cases.

#### 5.5. Experimental Series 3: Further Evaluation on SIPI Image Database

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Variation of function s at different values of f and l. (

**a**) $l=1.5$ and $f$ in $\left[0,1\right]$, (

**b**) $f=0.5$ and $l$ in $\left[1,2\right]$.

**Figure 2.**(

**a**) The average value of crossover rate (

**CR**) and (

**b**) scaling factor (

**F**) for 30 runs using jDE algorithm.

**Figure 4.**Original test images named ‘Image1’, ‘Image2’, ‘Image3’, ‘Image4’, ‘Image5’, ‘Image6’, ‘Image7’, and ‘Image8’, and the corresponding histograms for each of the color channels (red, green, and blue). (

**a**) 1800 × 1200, (

**b**) 2796 × 1864, (

**c**) 5339 × 3559, (

**d**) 2712 × 1808, (

**e**) 4310 × 4019, (

**f**) 2856 × 1904, (

**g**) 3467 × 2311, (

**h**) 1512 × 1008; (

**left figure**) Original test images, (

**right figure**) Histogram of each frame.

**Figure 5.**The segmented results and local zoom maps of “Image4” at 4, 8, and 12 threshold levels. (

**a**) K = 4, (

**b**) K = 8, (

**c**) K = 12.

**Figure 6.**The segmented results and local zoom maps of “Image7” at 4, 8, and 12 threshold levels. (

**a**) K = 4, (

**b**) K = 8, (

**c**) K = 12.

**Figure 7.**The convergence curves for fitness function using MCE method at 12 threshold levels. (

**a**) Image2, (

**b**) Image4, (

**c**) Image6, (

**d**) Image8.

**Figure 8.**Comparison of PSNR values for different algorithms using MCE method at 4, 6, 8, 10, and 12 levels.

**Figure 10.**Comparison of FSIM values over all images using MCE method at 4, 6, 8, 10, and 12 levels.

**Figure 11.**The boxplot for fitness function using MCE method at 12 threshold levels. (

**a**) Image5, (

**b**) Image6, (

**c**) Image7, (

**d**) Image8.

**Figure 12.**Two natural images from the Berkeley segmentation dataset, which are named ‘Elephant’, and ‘Plane’, respectively, and the corresponding histograms for each of the color channels (red, green, and blue). (

**a**) Elephant (481 × 321), (

**b**) Plane (481 × 321).

$\mathit{S}\mathit{F}=0.1$ | $\mathit{S}\mathit{F}=0.3$ | $\mathit{S}\mathit{F}=0.5$ | $\mathit{S}\mathit{F}=0.7$ | $\mathit{S}\mathit{F}=0.9$ | |
---|---|---|---|---|---|

$CR=0.5$ | 14.0625 | 13.75 | 16.6875 | 9.75 | 11.75 |

$CR=0.6$ | 7.0625 | 13.4375 | 9.8125 | 13 | 12.6875 |

$CR=0.7$ | 15.6875 | 16.6875 | 14.3125 | 15.1875 | 17.0625 |

$CR=0.8$ | 13.8125 | 13.8125 | 14.5 | 12.6875 | 9.5 |

$CR=0.9$ | 15.875 | 9 | 3 | 14.875 | 17 |

Image Number | Explanation |
---|---|

1. | The Aïr Mountains dispersed across the Sahara Desert in northern Niger. |

2. | Glacier cover in the mountainous region of northwestern Venezuela. |

3. | Dukan Lake in the Zagros Mountains, the largest lake in Iraqi Kurdistan. |

4. | Candeleros rock containing quite a menagerie of fossilized fauna. |

5. | The waters of Foxe Basin, which have been choked with sea ice for most of the year. |

6. | The Port of Busan at the southeastern tip of the Korean Peninsula, which has been a trading hub since at least the 15th century. |

7. | A fire in Northern California during the summer of 2018. |

8. | The Ebro Delta, located more than 200 kilometers (120 miles) southwest of Barcelona. |

No. | Algorithm | Parameter Setting | Year | Reference |
---|---|---|---|---|

1. | GOA–jDE | $CR=0.9,\text{}SF=0.5,\text{}S{F}^{low}=0.1,\text{}S{F}^{up}=0.9,\text{}{\tau}_{1}={\tau}_{2}=0.1$ | — | — |

2. | GOA | $c\in \left[0.00001,1\right]$ | 2017 | [33] |

3. | DE | $CR=0.9,\text{}SF=0.5$ | 1997 | [16] |

4. | MGOA | $\beta =0.8\left(\mathrm{L}\xe9\mathrm{vy}\mathrm{flight}\mathrm{parameter}\right)$ | 2019 | [55] |

5. | hjDE | ${p}_{a}=0.25\left(\mathrm{switching}\mathrm{probability}\right)$ | 2016 | [21] |

6. | BDE | $a\in \left[0,1\right]\left(\mathrm{beta}\mathrm{distribution}\mathrm{parameter}\right)$ | 2018 | [56] |

7. | BA | ${r}_{i}\in \left[0,1\right]\left(\mathrm{rate}\mathrm{of}\mathrm{pulse}\mathrm{emission}\right),{A}_{i}\in \left[1,2\right]\left(\mathrm{loudness}\mathrm{value}\right)$ | 2010 | [17] |

8. | PSO | ${c}_{1}={c}_{2}=2,w\in \left[0.4,0.9\right],{v}_{max}=25.5$ | 1995 | [15] |

9. | MABC | $K=300\left(\mathrm{chaotic}\mathrm{iteration}\right)$ | 2015 | [57] |

10. | IDSA | — | 2018 | [58] |

11. | CS | ${p}_{a}=0.25\left(\mathrm{mutation}\mathrm{probability}\right),\beta =1.5\left(\mathrm{scalingfactor}\right)$ | 2017 | [13] |

No. | Measures | Formulation | Remark | Reference |
---|---|---|---|---|

1. | Average Fitness Function Value | $\mathrm{Average}=\frac{{\sum}_{i=1}^{N}{f}_{i}}{N}$ | Indicates the center value of sample data. | [46] |

2. | Standard Deviation (STD) | $\mathrm{Std}=\sqrt{\frac{1}{N-1}{{\displaystyle \sum}}_{i=1}^{N}{\left({f}_{i}-\mathrm{Average}\right)}^{2}}$ | Reflects the degree of dispersion in a dataset. A lower value shows better performance. | [46] |

3. | Peak Signal to Noise Ratio (PSNR) | $\mathrm{PSNR}=10\text{}{\mathrm{log}}_{10}\left(\frac{{255}^{2}}{\mathrm{MSE}}\right)$ | The ratio of the maximum possible power of the signal to the destructive noise power. | [60] |

4. | Mean Squared Error (MSE) | $\mathrm{MSE}=\frac{1}{MN}{{\displaystyle \sum}}_{i=1}^{M}{{\displaystyle \sum}}_{j=1}^{N}{\left[I\left(i,j\right)-K\left(i,j\right)\right]}^{2}$ | Computes the difference between the predicted value. | [60] |

5. | Structural Similarity Index (SSIM) | $\mathrm{SSIM}=\frac{\left(2{\mu}_{x}{\mu}_{y}+{c}_{1}\right)\left(2{\sigma}_{xy}+{c}_{2}\right)}{\left({\mu}_{x}^{2}+{\mu}_{y}^{2}+{c}_{1}\right)\left({\sigma}_{x}^{2}+{\sigma}_{y}^{2}+{c}_{2}\right)}$ | Defines the similarity between the original image and the segmented image. | [61] |

6. | Feature Similarity Index (FSIM) | $\mathrm{FSIM}=\frac{{\sum}_{x\in \Omega}{S}_{L}\left(x\right)\xb7P{C}_{m}\left(x\right)}{{\sum}_{x\in \Omega}P{C}_{m}\left(x\right)}$ | Reflects the similarity of feature structure, the maximum value is 1. | [62] |

7. | Average Computation Time | $Time=\frac{{\sum}_{i=1}^{N}tim{e}_{i}}{N}$ | Indicates the operating efficiency of each method. | [14] |

8. | Wilcoxon’s Rank Sum Test | ${R}^{+}={\displaystyle {\displaystyle \sum}_{{d}_{i}>0}}rank\left({d}_{i}\right)+\frac{1}{2}{\displaystyle {\displaystyle \sum}_{{d}_{i}=0}}rank\left({d}_{i}\right)\phantom{\rule{0ex}{0ex}}{R}^{-}={\displaystyle {\displaystyle \sum}_{{d}_{i}<0}}rank\left({d}_{i}\right)+\frac{1}{2}{\displaystyle {\displaystyle \sum}_{{d}_{i}=0}}rank\left({d}_{i}\right)$ | Whether there is a significant difference between two algorithms. | [63] |

9. | Friedman Test | ${F}_{f}=\frac{12n}{k\left(k+1\right)}\left[{{\displaystyle \sum}}_{j}{R}_{j}^{2}-\frac{k{\left(k+1\right)}^{2}}{4}\right]$ | Detects significant differences between the behaviors of two or more algorithms. | [64] |

Images | K | GOA–jDE | GOA | DE | MGOA | hjDE | BDE | BA | PSO |
---|---|---|---|---|---|---|---|---|---|

Image1 | 4 | −701.0136 | −700.9701 | −701.0136 | −701.0136 | −701.0136 | −701.0136 | −701.0132 | −701.0136 |

6 | −701.2706 | −701.2705 | −701.2705 | −701.2705 | −701.2706 | −701.2704 | −701.2267 | −701.2702 | |

8 | −701.3808 | −701.3829 | −701.3803 | −701.3806 | −701.3807 | −701.3807 | −701.3152 | −701.3782 | |

10 | −701.4401 | −701.4371 | −701.4395 | −701.4389 | −701.4392 | −701.439 | −701.3746 | −701.4042 | |

12 | −701.4753 | −701.473 | −701.4737 | −701.4736 | −701.4751 | −701.4749 | −701.3678 | −701.4573 | |

Image2 | 4 | −370.8833 | −370.8833 | −370.8833 | −370.8833 | −370.8833 | −370.8833 | −370.7795 | −370.8833 |

6 | −371.1958 | −371.1958 | −371.1959 | −371.1958 | −371.1959 | −371.1959 | −371.045 | −371.1898 | |

8 | −371.3386 | −371.3108 | −371.3372 | −371.3209 | −371.3385 | −371.3381 | −371.2998 | −371.2486 | |

10 | −371.4126 | −371.3998 | −371.4082 | −371.3839 | −371.4124 | −371.4115 | −371.2348 | −371.3697 | |

12 | −371.457 | −371.4345 | −371.4507 | −371.4395 | −371.4569 | −371.4558 | −371.3314 | −371.3988 | |

Image3 | 4 | −645.6498 | −645.6498 | −645.6498 | −645.6498 | −645.6498 | −645.6498 | −645.646 | −645.6498 |

6 | −646.008 | −645.9872 | −646.008 | −646.008 | −646.008 | −646.0079 | −645.98 | −646.0059 | |

8 | −646.1744 | −646.1662 | −646.1735 | −646.1743 | −646.1742 | −646.1741 | −646.066 | −646.1298 | |

10 | −646.261 | −646.2592 | −646.2597 | −646.2605 | −646.2603 | −646.2582 | −646.1845 | −646.235 | |

12 | −646.3107 | −646.2853 | −646.3092 | −646.3039 | −646.3077 | −646.3098 | −646.2148 | −646.2988 | |

Image4 | 4 | −474.0258 | −474.0258 | −474.0257 | −474.0258 | −474.0258 | −474.0254 | −474.0241 | −474.0257 |

6 | −474.3694 | −474.3683 | −474.3691 | −474.3694 | −474.3694 | −474.3693 | −474.3543 | −474.3685 | |

8 | −474.526 | −474.5238 | −474.5229 | −474.5259 | −474.5251 | −474.5257 | −474.3738 | −474.5036 | |

10 | −474.6101 | −474.6052 | −474.6082 | −474.6068 | −474.6088 | −474.6082 | −474.4214 | −474.5932 | |

12 | −474.6598 | −474.6508 | −474.6569 | −474.6563 | −474.6596 | −474.6585 | −474.5078 | −474.6356 | |

Image5 | 4 | −498.1325 | −498.1325 | −498.1325 | −498.1325 | −498.1325 | −498.1325 | −498.1271 | −498.1325 |

6 | −498.5083 | −498.4884 | −498.5083 | −498.5083 | −498.5082 | −498.4757 | −498.4172 | −498.5078 | |

8 | −498.6791 | −498.6698 | −498.6783 | −498.6784 | −498.6791 | −498.679 | −498.5085 | −498.6742 | |

10 | −498.774 | −498.7722 | −498.7672 | −498.7733 | −498.7731 | −498.7736 | −498.6267 | −498.734 | |

12 | −498.8269 | −498.8226 | −498.8201 | −498.826 | −498.8248 | −498.826 | −498.7248 | −498.8073 | |

Image6 | 4 | −306.5464 | −306.5464 | −306.5463 | −306.5464 | −306.5464 | −306.5461 | −306.5389 | −306.5463 |

6 | −306.9244 | −306.9237 | −306.9244 | −306.9244 | −306.9244 | −306.9231 | −306.8676 | −306.8951 | |

8 | −307.0986 | −307.0977 | −307.0981 | −307.0985 | −307.0985 | −307.0985 | −306.8234 | −307.0793 | |

10 | −307.1962 | −307.187 | −307.1699 | −307.1871 | −307.1961 | −307.1912 | −307.0848 | −307.1482 | |

12 | −307.2547 | −307.2322 | −307.2507 | −307.2423 | −307.252 | −307.2517 | −307.0775 | −307.2106 | |

Image7 | 4 | −480.5483 | −480.5483 | −480.5483 | −480.5483 | −480.5483 | −480.5483 | −480.5475 | −480.548 |

6 | −480.8448 | −480.8428 | −480.8441 | −480.8448 | −480.8448 | −480.8448 | −480.7985 | −480.8158 | |

8 | −480.974 | −480.9729 | −480.9725 | −480.9735 | −480.9733 | −480.9725 | −480.83 | −480.9581 | |

10 | −481.041 | −481.0396 | −481.0402 | −481.04 | −481.0408 | −481.0407 | −480.9169 | −481.0313 | |

12 | −481.0802 | −481.0699 | −481.0776 | −481.0723 | −481.0796 | −481.0771 | −480.9164 | −481.0549 | |

Image8 | 4 | −411.1164 | −411.1164 | −411.1164 | −411.1164 | −411.1164 | −411.1164 | −411.1118 | −411.1164 |

6 | −411.4617 | −411.4605 | −411.4616 | −411.4616 | −411.4616 | −411.4611 | −411.308 | −411.4607 | |

8 | −411.6287 | −411.6273 | −411.6285 | −411.6284 | −411.6286 | −411.6231 | −411.5499 | −411.5935 | |

10 | −411.7136 | −411.7007 | −411.7105 | −411.7043 | −411.7119 | −411.7131 | −411.6203 | −411.6984 | |

12 | −411.7664 | −411.7632 | −411.7641 | −411.758 | −411.7662 | −411.7653 | −411.6613 | −411.7395 |

Images | GOA–jDE | GOA | DE | MGOA | hjDE | BDE | BA | PSO |
---|---|---|---|---|---|---|---|---|

1 | 6.67 | 0 | 0 | 0 | 0 | 0 | 3.33 | 0 |

2 | 100 | 0 | 10 | 86.67 | 0 | 0 | 36.67 | 13.33 |

3 | 83.33 | 26.67 | 20 | 40 | 0 | 23.33 | 0 | 10 |

4 | 100 | 100 | 100 | 96.67 | 100 | 93.33 | 100 | 90 |

5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

6 | 100 | 60 | 36.67 | 53.33 | 30 | 70 | 86.67 | 40 |

7 | 73.33 | 0 | 3.33 | 40 | 0 | 6.67 | 13.33 | 0 |

8 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 |

K | GOA–jDE | GOA | DE | MGOA | hjDE | BDE | BA | PSO |
---|---|---|---|---|---|---|---|---|

4 | 1.744575 | 3.339125 | 1.132188 | 3.338625 | 1.632425 | 1.50965 | 1.3265 | 1.081413 |

6 | 1.814575 | 3.407225 | 1.225213 | 3.467038 | 1.734113 | 1.701825 | 1.5425 | 1.1903 |

8 | 2.027638 | 3.493013 | 1.31005 | 3.492675 | 1.987663 | 1.876263 | 1.885988 | 1.321688 |

10 | 2.258513 | 3.567325 | 1.446075 | 3.58 | 2.248538 | 2.023563 | 2.240663 | 1.454863 |

12 | 2.558875 | 3.860313 | 1.5924 | 3.750038 | 2.380238 | 2.263163 | 2.813125 | 1.56585 |

Comparison | p-Value |
---|---|

GOA–jDE vs. GOA | 0.0197 |

GOA–jDE vs. DE | 2.7461 × 10^{−4} |

GOA–jDE vs. MGOA | 2.3012 × 10^{−5} |

GOA–jDE vs. hjDE | 8.7216 × 10^{−4} |

GOA–jDE vs. BDE | 1.6063 × 10^{−9} |

GOA–jDE vs. BA | 3.9766 × 10^{−7} |

GOA–jDE vs. PSO | 5.0219 × 10^{−8} |

K | Average Rank | |||||||
---|---|---|---|---|---|---|---|---|

GOA-jDE | GOA | DE | MGOA | hjDE | BDE | BA | PSO | |

4 | 2 | 5.4625 | 3.6 | 4.55 | 4.075 | 4.9 | 6.4125 | 5 |

6 | 1.475 | 5.425 | 4.1 | 4.3375 | 3.75 | 4.9625 | 6.7625 | 5.1875 |

8 | 1.0625 | 4.6125 | 4.4125 | 4.475 | 4.325 | 4.725 | 6.9625 | 5.425 |

10 | 1.05 | 4.8625 | 4.8125 | 4.6375 | 3.7625 | 4.2125 | 6.9625 | 5.7 |

12 | 1 | 5.0375 | 4.2375 | 4.5875 | 3.725 | 4.3875 | 7.5125 | 5.5125 |

**Table 10.**The chi-square value and p-value (Friedman test) for the experimental results of the MCE method.

K | Chi-Square Value | p-Value |
---|---|---|

4 | 101.901 | 4.36751 × 10^{−19} |

6 | 114.919 | 8.73960 × 10^{−22} |

8 | 126.529 | 3.33534 × 10^{−24} |

10 | 135.448 | 4.56229 × 10^{−26} |

12 | 155.666 | 2.61739 × 10^{−30} |

Images | K | Tsallis | Otsu | MCE | |||
---|---|---|---|---|---|---|---|

GOA–jDE | MABC | GOA–jDE | IDSA | GOA–jDE | CS | ||

Image1 | 20 | 65.6992 | 63.9303 | 2536.0239 | 2533.7508 | −701.5320 | −701.5321 |

25 | 72.0775 | 69.8990 | 2538.4352 | 2537.4078 | −701.5441 | −701.5440 | |

30 | 78.0006 | 74.2769 | 2540.1262 | 2539.0127 | −701.5516 | −701.5513 | |

Image2 | 20 | 65.1949 | 63.3661 | 1380.4683 | 1378.9008 | −371.5264 | −371.5240 |

25 | 71.4121 | 69.0462 | 1382.6497 | 1382.0753 | −371.5386 | −371.5390 | |

30 | 77.1012 | 73.8485 | 1383.9328 | 1383.0637 | −371.5467 | −371.5457 | |

Image3 | 20 | 66.3283 | 64.7228 | 2267.4996 | 2265.5258 | −646.3892 | −646.3875 |

25 | 73.0784 | 70.6841 | 2270.3662 | 2269.1050 | −646.4058 | −646.4052 | |

30 | 78.9913 | 74.9226 | 2272.1397 | 2270.5508 | −646.4160 | −646.4150 | |

Image4 | 20 | 67.2077 | 65.8090 | 1654.2822 | 1652.9518 | −474.7376 | −474.7383 |

25 | 74.0556 | 71.2441 | 1656.8361 | 1655.9701 | −474.7543 | −474.7542 | |

30 | 80.2195 | 75.7998 | 1658.3974 | 1657.8302 | −474.7638 | −474.7638 | |

Image5 | 20 | 67.1305 | 65.6002 | 5516.8189 | 5515.3632 | −498.9233 | −498.9223 |

25 | 74.3413 | 71.3003 | 5519.9728 | 5519.1290 | −498.9440 | −498.9433 | |

30 | 79.8769 | 75.9055 | 5521.8659 | 5521.0078 | −498.9556 | −498.9552 | |

Image6 | 20 | 66.7823 | 65.4021 | 3210.8833 | 3209.5405 | −307.3492 | −307.3459 |

25 | 74.1973 | 71.5457 | 3213.7465 | 3212.8257 | −307.3670 | −307.3685 | |

30 | 79.4842 | 75.5905 | 3215.0132 | 3214.2657 | −307.3798 | −307.3795 | |

Image7 | 20 | 65.5192 | 63.8334 | 1803.2208 | 1801.3177 | −481.1434 | −481.1416 |

25 | 72.4605 | 68.99 | 1805.3468 | 1803.8873 | −481.1571 | −481.1556 | |

30 | 77.5298 | 74.1611 | 1806.9075 | 1806.2681 | −481.1634 | −481.1624 | |

Image8 | 20 | 66.8490 | 65.1504 | 2355.6389 | 2354.3242 | −411.8501 | −411.8497 |

25 | 73.6971 | 71.5997 | 2358.1153 | 2356.7401 | −411.8686 | −411.8674 | |

30 | 78.9075 | 75.0792 | 2359.6548 | 2358.5339 | −411.8757 | −411.8768 | |

Elephant | 20 | 62.1191 | 60.7010 | 2009.5697 | 2008.3336 | −457.3469 | −457.3467 |

25 | 67.8767 | 66.4680 | 2011.4073 | 2010.5957 | −457.3577 | −457.3569 | |

30 | 72.9363 | 69.5792 | 2012.2340 | 2011.6498 | −457.3633 | −457.3631 | |

Plane | 20 | 52.1894 | 52.1762 | 706.7254 | 706.6974 | −587.0459 | −587.0450 |

25 | 55.6713 | 55.8603 | 707.6530 | 707.4842 | −587.0525 | −587.0503 | |

30 | 58.7978 | 58.3922 | 707.8326 | 708.1450 | −587.0541 | −587.0538 |

K | Average Rank | ||||||||
---|---|---|---|---|---|---|---|---|---|

GOA–jDE–MCE | MGOA–MCE | hjDE–MCE | BDE–MCE | BA–MCE | PSO–MCE | CS–MCE | IDSA–Otsu | MABC–Tsallis | |

20 | 3.1754 | 4.4781 | 5.5833 | 6.3070 | 5.6974 | 4.9693 | 4.9167 | 3.9430 | 5.9298 |

25 | 2.7368 | 4.9342 | 5.0746 | 6.5219 | 6.4342 | 4.1535 | 4.8640 | 4.2588 | 6.0219 |

30 | 1.3421 | 4.6886 | 4.5526 | 5.9254 | 7.4386 | 5.1798 | 4.6316 | 4.5175 | 6.7237 |

Overall | 2.4181 | 4.7003 | 5.0702 | 6.2515 | 6.5234 | 4.7675 | 4.8041 | 4.2398 | 6.2251 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jia, H.; Lang, C.; Oliva, D.; Song, W.; Peng, X. Hybrid Grasshopper Optimization Algorithm and Differential Evolution for Multilevel Satellite Image Segmentation. *Remote Sens.* **2019**, *11*, 1134.
https://doi.org/10.3390/rs11091134

**AMA Style**

Jia H, Lang C, Oliva D, Song W, Peng X. Hybrid Grasshopper Optimization Algorithm and Differential Evolution for Multilevel Satellite Image Segmentation. *Remote Sensing*. 2019; 11(9):1134.
https://doi.org/10.3390/rs11091134

**Chicago/Turabian Style**

Jia, Heming, Chunbo Lang, Diego Oliva, Wenlong Song, and Xiaoxu Peng. 2019. "Hybrid Grasshopper Optimization Algorithm and Differential Evolution for Multilevel Satellite Image Segmentation" *Remote Sensing* 11, no. 9: 1134.
https://doi.org/10.3390/rs11091134