# Hyperspectral Unmixing Using Robust Deep Nonnegative Matrix Factorization

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Linear Mixing Model

#### 2.2. DNMF

#### 2.3. Proposed ${\mathsf{\ell}}_{2,1}$-RDNMF

Algorithm 1 Proposed ${\mathsf{\ell}}_{2,1}$-RDNMF |

Input: HSI data $\mathbf{X}$; number of layers L; inner ranks ${M}_{l}$. |

Output: Endmembers $\mathbf{W}$; abundances $\mathbf{H}$. |

Pretraining stage: |

1: for
$l=1,\cdots ,L$
do |

2: Obtain initial ${\mathbf{V}}_{l}$ and ${\mathbf{H}}_{l}$ by VCA-FCLS. |

3: repeat |

4: Compute ${\mathbf{G}}_{l}$ using Equation (21). |

5: Update ${\mathbf{V}}_{l}$ by Equation (22). |

6: Update ${\mathbf{H}}_{l}$ by Equation (23). |

7: until convergence |

8: end for |

Fine-tuning stage: |

9: repeat |

10: for $l=1,\cdots ,L$ do |

11: Compute ${\mathbf{C}}_{l}$ using Equation (8). |

12: Compute ${\mathbf{D}}_{l}$ using Equation (9). |

13: Compute ${\mathbf{Q}}_{l}$ using Equation (26). |

14: Update ${\mathbf{V}}_{l}$ by Equation (28). |

15: end for |

16: Update ${\mathbf{H}}_{L}$ by Equation (29). |

17: until the stopping criterion is satisfied. |

18: Compute $\mathbf{W}$ using Equation (14). |

19: Compute $\mathbf{H}$ using Equation (15). |

20: return $\mathbf{W}$ and $\mathbf{H}$ |

#### 2.4. Implementation Issues

## 3. Experiments

#### 3.1. Experiments on Synthetic Data

#### 3.1.1. Experiment 1 (Investigation of the Number of Layers)

#### 3.1.2. Experiment 2 (Investigation of Noise Intensities)

#### 3.1.3. Experiment 3 (Investigation of Noise Types)

#### 3.2. Experiments on Real Data

#### 3.2.1. Samson Dataset

#### 3.2.2. Cuprite Dataset

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**SAD results of proposed ${\mathsf{\ell}}_{2,1}$-RDNMF with different settings of the number of layers.

**Figure 3.**RMSE results of proposed ${\mathsf{\ell}}_{2,1}$-RDNMF with different settings of the number of layers.

**Figure 4.**SAD results of different methods on data simulated with various intensities of Gaussian noise.

**Figure 5.**RMSE results of different methods on data simulated with various intensities of Gaussian noise.

**Figure 9.**Abundance maps estimated by ${\mathsf{\ell}}_{2,1}$-RDNMF on the Cuprite dataset. (

**a**) Alunite. (

**b**) Andradite. (

**c**) Buddingtonite. (

**d**) Dumortierite. (

**e**) Kaolinite #1. (

**f**) Kaolinite #2. (

**g**) Muscovite. (

**h**) Montmorillonite. (

**i**) Nontronite. (

**j**) Pyrope. (

**k**) Sphene. (

**l**) Chalcedony.

VCA-FCLS | ${\mathsf{\ell}}_{2,1}$-NMF | DNMF | SDNMF-TV | CSsRS-NMF | MLNMF | CANMF-TV | ${\mathsf{\ell}}_{2,1}$-RDNMF |
---|---|---|---|---|---|---|---|

1.30 s | 9.51 s | 20.72 s | 141.60 s | 123.80 s | 21.82 s | 149.63 s | 57.06 s |

**Table 2.**SADs of different methods on synthetic data simulated with various types of noise. The best results are marked in bold.

Noise Type | Methods | |||||||
---|---|---|---|---|---|---|---|---|

VCA-FCLS | ${\mathsf{\ell}}_{2,1}$-NMF | DNMF | SDNMF-TV | CSsRS-NMF | MLNMF | CANMF-TV | ${\mathsf{\ell}}_{2,1}$-RDNMF | |

Gaussian noise | 0.0190 | 0.0159 | 0.0168 | 0.0187 | 0.0113 | 0.0172 | 0.0209 | 0.0080 |

Gaussian noise and Impulse noise | 0.2006 | 0.2348 | 0.1635 | 0.1919 | 0.1466 | 0.3062 | 0.2078 | 0.0190 |

Gaussian noise and Dead pixels | 0.2017 | 0.3170 | 0.2810 | 0.4668 | 0.5244 | 0.4296 | 0.5716 | 0.0233 |

Gaussian noise, Impulse noise, and Dead pixels | 0.4721 | 0.5641 | 0.2865 | 0.4507 | 0.3660 | 0.5692 | 0.6282 | 0.0617 |

**Table 3.**RMSEs of different methods on synthetic data simulated with various types of noise. The best results are marked in bold.

Noise Type | Methods | |||||||
---|---|---|---|---|---|---|---|---|

VCA-FCLS | ${\mathsf{\ell}}_{2,1}$-NMF | DNMF | SDNMF-TV | CSsRS-NMF | MLNMF | CANMF-TV | ${\mathsf{\ell}}_{2,1}$-RDNMF | |

Gaussian noise | 0.0251 | 0.0298 | 0.0371 | 0.0307 | 0.0265 | 0.0319 | 0.0345 | 0.0225 |

Gaussian noise and Impulse noise | 0.1719 | 0.1401 | 0.1060 | 0.1332 | 0.1780 | 0.1793 | 0.1256 | 0.0339 |

Gaussian noise and Dead pixels | 0.3326 | 0.3330 | 0.2338 | 0.3165 | 0.4166 | 0.3804 | 0.3164 | 0.0441 |

Gaussian noise and Impulse noise and Dead pixels | 0.4216 | 0.3737 | 0.2740 | 0.3411 | 0.4680 | 0.3222 | 0.3801 | 0.0522 |

Endmember | Methods | |||||||
---|---|---|---|---|---|---|---|---|

VCA-FCLS | ${\mathsf{\ell}}_{2,1}$-NMF | DNMF | SDNMF-TV | CSsRS-NMF | MLNMF | CANMF-TV | ${\mathsf{\ell}}_{2,1}$-RDNMF | |

Soil | 0.2680 | 0.1817 | 0.0568 | 0.1584 | 0.1675 | 0.1250 | 0.0903 | 0.0274 |

Tree | 0.0518 | 0.0868 | 0.0524 | 0.1284 | 0.0870 | 0.0811 | 0.0625 | 0.0505 |

Water | 0.1281 | 0.1235 | 0.1373 | 0.2076 | 0.2635 | 0.2492 | 0.1440 | 0.1467 |

Mean | 0.1493 | 0.1307 | 0.0822 | 0.1648 | 0.1726 | 0.1518 | 0.0989 | 0.0748 |

Endmember | Methods | |||||||
---|---|---|---|---|---|---|---|---|

VCA-FCLS | ${\mathsf{\ell}}_{2,1}$-NMF | DNMF | SDNMF-TV | CSsRS-NMF | MLNMF | CANMF-TV | ${\mathsf{\ell}}_{2,1}$-RDNMF | |

Alunite | 0.0847 | 0.0968 | 0.0914 | 0.0917 | 0.0946 | 0.1567 | 0.1351 | 0.2765 |

Andradite | 0.0778 | 0.0826 | 0.1084 | 0.0897 | 0.0831 | 0.1146 | 0.0781 | 0.0731 |

Buddingtonite | 0.1215 | 0.1430 | 0.1115 | 0.1264 | 0.1224 | 0.1142 | 0.0601 | 0.1132 |

Dumortierite | 0.1000 | 0.1329 | 0.1172 | 0.1323 | 0.1046 | 0.1648 | 0.1095 | 0.0821 |

Kaolinite #1 | 0.0855 | 0.0688 | 0.0629 | 0.0652 | 0.0738 | 0.0846 | 0.1504 | 0.0785 |

Kaolinite #2 | 0.1133 | 0.0888 | 0.0640 | 0.0613 | 0.0859 | 0.0941 | 0.1225 | 0.0599 |

Muscovite | 0.1976 | 0.1349 | 0.2191 | 0.1532 | 0.1924 | 0.4113 | 0.1662 | 0.1216 |

Montmorillonite | 0.1110 | 0.0632 | 0.0619 | 0.1276 | 0.0634 | 0.0665 | 0.0620 | 0.0637 |

Nontronite | 0.0996 | 0.0982 | 0.1135 | 0.1227 | 0.0934 | 0.1382 | 0.1007 | 0.0833 |

Pyrope | 0.1118 | 0.0583 | 0.1908 | 0.0573 | 0.0567 | 0.0787 | 0.0574 | 0.0524 |

Sphene | 0.0544 | 0.2495 | 0.0545 | 0.2320 | 0.2506 | 0.0960 | 0.0898 | 0.0703 |

Chalcedony | 0.1902 | 0.1242 | 0.1288 | 0.1283 | 0.1154 | 0.1525 | 0.1326 | 0.1464 |

Mean | 0.1123 | 0.1118 | 0.1103 | 0.1156 | 0.1114 | 0.1393 | 0.1054 | 0.1017 |

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**MDPI and ACS Style**

Huang, R.; Jiao, H.; Li, X.; Chen, S.; Xia, C.
Hyperspectral Unmixing Using Robust Deep Nonnegative Matrix Factorization. *Remote Sens.* **2023**, *15*, 2900.
https://doi.org/10.3390/rs15112900

**AMA Style**

Huang R, Jiao H, Li X, Chen S, Xia C.
Hyperspectral Unmixing Using Robust Deep Nonnegative Matrix Factorization. *Remote Sensing*. 2023; 15(11):2900.
https://doi.org/10.3390/rs15112900

**Chicago/Turabian Style**

Huang, Risheng, Huiyun Jiao, Xiaorun Li, Shuhan Chen, and Chaoqun Xia.
2023. "Hyperspectral Unmixing Using Robust Deep Nonnegative Matrix Factorization" *Remote Sensing* 15, no. 11: 2900.
https://doi.org/10.3390/rs15112900