# A Supervised Method for Nonlinear Hyperspectral Unmixing

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Generating Linearly Mixed Training Spectra

#### 2.2. Mapping

#### 2.2.1. Mapping Using NN

#### 2.2.2. Mapping Using KRR

#### 2.2.3. Mapping Using GP

#### 2.3. Linear Unmixing

## 3. Eperimental Data

#### 3.1. Dataset 1: Simulated Mineral Dataset

#### 3.2. Dataset 2: Ray Tracing Vegetation Dataset

#### 3.3. Dataset 3: Drill Core Hyperspectral Dataset

## 4. Experiments

#### 4.1. Experimental Set-Up

#### 4.1.1. Comparison to Unsupervised Unmixing Models

#### 4.1.2. Comparison to Other Mapping Methods

#### 4.1.3. Performance Measures

#### 4.2. Experiments Using the Simulated Mineral Dataset

#### 4.2.1. Comparison with Direct Mapping to the Fractional Abundances

#### 4.2.2. Robustness to Noise and the Number of Endmembers

#### 4.2.3. Accuracy of the Mapping

#### 4.3. Experiments on the Ray Tracing Vegetation Dataset

#### 4.4. Experiments on the Drill Core Hyperspectral Dataset

## 5. Discussion

- The supervised methods all outperform the use of nonlinear spectral mixture models. This is of course partially due the fact that these methods make use of training data. However, the fact that they do not rely on a specific mixture model and the generic nature of these methods allows them to better account for nonlinearities and spectral variability in the data. Results show that the supervised methods can take nonlinearities of different nature simultaneously into account.
- The strategy of mapping onto the linear mixture model outperforms methods that directly map onto the fractional abundances. The main difference is that the proposed methodololgy inherently meets the nonnegativity and sum-to-one constraints. As opposed to the direct unconstrained mapping of the spectra to abundances, the estimated fractional abundances have a clear physical meaning and consequently the estimates are more accurate. Another difference with direct mapping is that our approach requires endmembers, while the direct mapping methods do not. A clear advantage is that in case no pure pixels are available in the data, endmember spectra from a spectral library can be applied. The mapping accounts for the spectral variability between these and the actual endmembers of the data.
- The proposed methodology is generic in the sense that a mapping can be learned to any model. However, learning a mapping to the linear model is favorable over learning mappings to nonlinear models. The higher the nonlinearity of the model, the higher the errors seem to be. The reason for this is that it is just easier to project onto a linear manifold, since a linear manifold can more easily be characterized by a limited number of training samples.
- The proposed methodology requires high quality ground truth data. Learning the mapping based on pure or linearly mixed spectra, as is done in the state of the art literature, will not improve results over the linear mixture model. High-quality ground truth data for spectral unmixing is hard to obtain, in particular in remote sensing applications. One way is to make use of a high-resolution multispectral image of the same scene (if available) to generate a groundcover classification map that can be used to generate ground truth fractional abundances for a low-resolution hyperspectral image. However, in the domain of mineralogy, it is more common to generate validation data with other characterization techniques, such as the MLA technique in the drill core example.
- An advantage of the proposed methodology is that any nonlinear regression method can be applied for learning the mapping. In this work, we compared three different ones. Gaussian processes generally seems to outperform kernel ridge regression and feedforward neural networks. Compared to kernel ridge regression, Gaussian processes contains more hyperparameters for a band-by-band adaptation to the nonlinearities. However, Gaussian processes can overfit the data in case the ground truth fractional abundance values are not very trustworthy. This can be observed in the obtained abundance maps of the entire drill core sample (Figure 13 and Figure 14). On the other hand, a neural network has better generalization properties, but its training can be computationally expensive.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**(

**a**) RGB image of the drill core sample. The red rectangle represents the area where the ground truth fractional abundance maps were obtained by the scanning electron microscope (SEM)—mineral liberation analysis (MLA) analysis; (

**b**) MLA image.

**Figure 6.**Root mean squared error (RMSE) (20 runs) for the methods using the proposed strategy (Gaussian processes (GP)_LM, kernel ridge regression (KRR)_LM, and neural networks (NN)_LM) and neural networks method with the softmax activation function in the last layer of the network (SM) as a function of signal-to-noise ratio (SNR).

**Figure 7.**RMSE (20 runs) for the methods using the proposed strategy (GP_LM, KRR_LM, and NN_LM) and SM as a function of the number of endmembers.

**Figure 8.**RMSE (100 runs) obtained by the four studied supervised methods as a function of the applied number of training samples in the ray tracing vegetation dataset.

**Figure 9.**Ground truth (GT) and estimated abundance maps and absolute difference with the GT for the four proposed supervised methods on the ray tracing dataset. All images are normalized by the largest abundance value in the GT map.

**Figure 10.**RMSE (100 runs) obtained by the four studied supervised methods as a function of the applied number of training samples in the drill core dataset.

**Figure 11.**Estimated abundance maps and absolute differences with the ground truth for the four supervised methods on the drill core dataset. All images are normalized by the largest abundance value in the GT map.

**Figure 12.**Scatterplot of the whole drill core sample (blue dots) and the MLA region (red circles); without (

**a**) and with (

**b**) normalization.

**Figure 13.**Estimated abundance maps of entire drill core sample (applying a map learned by using 373 training pixels). All images are normalized by the largest abundance value in the GT map.

**Figure 14.**Estimated abundance maps (without rescaling) of the entire drill core sample, for groups of minerals representing the matrix (

**left**) and veins (

**right**) (applying a map learned by using 373 training pixels).

**Table 1.**Linear and nonlinear mixing models and their parameters. The assumptions for all models: $\forall m:{a}_{m}\ge 0,{\sum}_{m}{a}_{m}=1$.

Model | Equation | Parameters |
---|---|---|

Linear | $\mathbf{x}={\sum}_{i=1}^{p}{a}_{i}{\mathbf{e}}_{i}$ | |

FM | $\mathbf{x}=\mathbf{y}+{\sum}_{m=1}^{p-1}{\sum}_{k=m+1}^{p}{b}_{mk}{\mathbf{e}}_{m}\odot {\mathbf{e}}_{k}$ | $\forall m\ge k:{b}_{mk}=0$ |

$\mathbf{y}={\sum}_{i=1}^{p}{a}_{i}{\mathbf{e}}_{i}$ | $\forall m<k:{b}_{mk}={a}_{m}{a}_{k}$ | |

PPNM | $\mathbf{x}=\mathbf{y}+b(\mathbf{y}\odot \mathbf{y})$ | $\forall m,k:{b}_{mk}=b{a}_{m}{a}_{k}$ |

$\mathit{y}={\sum}_{i=1}^{p}{a}_{i}{\mathbf{e}}_{i}$ | $b\in [-0.25,0.25]$ | |

MLM | $\mathbf{x}={\displaystyle \frac{(1-P)\mathbf{y}}{1-P\mathbf{y}}},\mathbf{y}={\sum}_{i=1}^{p}{a}_{i}{\mathbf{e}}_{i}$ | $P\in [0,1]$ |

Hapke | $x={\displaystyle \frac{w}{(1+2\mu \sqrt{1-w})(1+2{\mu}_{0}\sqrt{1-w})}}$ | ${\mu}_{0}$: cosine incident angle |

$\sqrt{1-w}=\frac{\sqrt{{({\mu}_{0}+\mu )}^{2}{x}^{2}+(1+4{\mu}_{0}\mu x)(1-x)}-({\mu}_{0}+\mu )x}{1+4{\mu}_{0}\mu x}$ | $\mu $: cosine reflectance angle |

**Table 2.**Results on 20 runs of 10,000 test pixels generated by the Hapke model. The best performances are denoted in bold.

Method | GP_LM | GP | KRR_LM | KRR | NN_LM | NN | SVR | SM |
---|---|---|---|---|---|---|---|---|

training set 1 | ||||||||

RMSE | 19.88 ± 0.62 | 40.89 ± 0.01 | 31.81 ± 1.71 | 40.89 ± 0.01 | 23.57 ± 1.97 | 36.57 ± 6.40 | 34.91 ± 9.48 | 33.68 ± 0.01 |

NEFA | 0 | 48.32 ± 0.31 | 0 | 25.12 ± 0.27 | 0 | 22.34 ± 0.36 | 24.37 ± 0.62 | 0 |

training set 2 | ||||||||

RMSE | 3.05 ± 1.10 | 5.54 ± 1.31 | 4.05 ± 0.58 | 5.55 ± 0.95 | 4.15 ± 1.17 | 5.15 ± 0.80 | 7.10 ± 0.95 | 15.65 ± 5.88 |

NEFA | 0 | 5.13 ± 1.97 | 0 | 4.66 ± 0.82 | 0 | 8.71 ± 2.40 | 8.96 ± 1.67 | 0 |

**Table 3.**Root mean squared error (RMSE) and reconstruction error (RE) (20 runs) of 250 test pixels of the simulated dataset. The best performances are denoted in bold.

Method | GP_LM | GP_Fan | GP_Hapke | KRR_LM | KRR_Fan | KRR_Hapke | NN_LM | NN_Fan | NN_Hapke |
---|---|---|---|---|---|---|---|---|---|

RMSE | 1.19 ± 0.72 | 1.44 ± 0.93 | 1.46 ± 0.74 | 3.04 ± 0.32 | 3.06 ± 0.35 | 3.61 ± 0.64 | 3.65 ± 0.59 | 4.12 ± 0.52 | 5.20 ± 1.25 |

RE | 2.4 ± 1.98 | 5.8 ± 4.00 | 19 ± 18 | 6.25 ± 3.28 | 12 ± 2.14 | 21 ± 4.47 | 15 ± 7.37 | 29 ± 7.32 | 50 ± 22 |

**Table 4.**RMSE (100 runs) of 360 test pixels of the ray tracing dataset. The best performances are denoted in bold.

Endmember Method | GP_LM | KRR_LM | NN_LM | SM | LMM | Fan | PPNM | MLM | Hapke |
---|---|---|---|---|---|---|---|---|---|

Soil | 1.59 ± 0.16 | 2.39 ± 0.93 | 1.86 ± 0.34 | 3.07 ± 0.46 | 11.69 | 12.40 | 16.15 | 14.06 | 9.79 |

Weed | 1.18 ± 0.18 | 3.77 ± 1.06 | 1.62 ± 0.40 | 3.86 ± 1.42 | 14.84 | 12.72 | 21.40 | 14.42 | 5.94 |

Tree | 2.06 ± 0.21 | 3.95 ± 0.67 | 2.63 ± 0.46 | 3.74 ± 1.20 | 24.36 | 19.13 | 8.28 | 26.06 | 13.39 |

**Table 5.**The mean estimated fractional abundances of 186 test pixels (in %) of the drill core dataset using the supervised methods (100 runs) along with the estimated fractional abundances using the unsupervised methods. The best performances are denoted in bold.

Mineral Method | GT | KRR_LM | GP_LM | NN_LM | SM | LMM | Fan | PPNM | MLM | Hapke |
---|---|---|---|---|---|---|---|---|---|---|

White Mica | 14.14 | 13.40 ± 0.83 | 13.57 ± 0.79 | 13.61 ± 1.22 | 13.56 ± 1.32 | 9.01 | 3.82 | 6.93 | 9.63 | 6.91 |

Biotite | 0.46 | 0.63 ± 0.12 | 0.62 ± 0.12 | 0.59 ± 0.20 | 1.04 ± 0.61 | 0 | 0 | 0 | 0 | 0 |

Chlorite | 4.17 | 4.66 ± 0.35 | 4.56 ± 0.33 | 4.61 ± 0.85 | 3.64 ± 0.60 | 0 | 0 | 0 | 0 | 0 |

Amphiboles | 1.43 | 1.37 ± 0.10 | 1.38 ± 0.09 | 1.43 ± 0.40 | 1.83 ± 0.46 | 0.24 | 0 | 6.46 | 0.81 | 0 |

Gypsum | 10.93 | 11.21 ± 1.24 | 11.11 ± 1.19 | 11.08 ± 1.20 | 11.71 ± 1.61 | 32.32 | 31.31 | 44.26 | 31.66 | 42.54 |

Plagioclase | 33.20 | 32.24 ± 1.58 | 32.74 ± 1.44 | 32.32 ± 2.69 | 33.72 ± 2.69 | 7.30 | 3.66 | 7.62 | 20.43 | 0 |

K-Feldspar | 6.68 | 5.93 ± 0.30 | 5.98 ± 0.32 | 6.27 ± 1.28 | 5.58 ± 0.76 | 4.77 | 6.33 | 4.92 | 7.74 | 41.21 |

Quartz | 26.85 | 28.15 ± 1.26 | 27.62 ± 1.12 | 27.19 ± 2.52 | 25.91 ± 1.70 | 0.43 | 0.14 | 20.64 | 3.79 | 0 |

Barite | 0.33 | 0.46 ± 0.10 | 0.43 ± 0.09 | 0.45 ± 0.24 | 0.98 ± 0.59 | 0 | 0 | 3.40 | 0 | 0 |

Pyrite | 1.80 | 1.95 ± 0.32 | 2.00 ± 0.34 | 2.46 ± 1.14 | 2.02 ± 0.44 | 45.93 | 54.73 | 5.76 | 25.93 | 9.34 |

Method | time${}_{\mathbf{Ray}\mathbf{tracing}}$ (s) | time${}_{\mathbf{Drill}\mathbf{core}}$ (s) |
---|---|---|

KRR_LM | 0.89 | 20.55 |

GP_LM | 43.84 | 840.62 |

NN_LM | 105.15 | 7499.84 |

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## Share and Cite

**MDPI and ACS Style**

Koirala, B.; Khodadadzadeh, M.; Contreras, C.; Zahiri, Z.; Gloaguen, R.; Scheunders, P. A Supervised Method for Nonlinear Hyperspectral Unmixing. *Remote Sens.* **2019**, *11*, 2458.
https://doi.org/10.3390/rs11202458

**AMA Style**

Koirala B, Khodadadzadeh M, Contreras C, Zahiri Z, Gloaguen R, Scheunders P. A Supervised Method for Nonlinear Hyperspectral Unmixing. *Remote Sensing*. 2019; 11(20):2458.
https://doi.org/10.3390/rs11202458

**Chicago/Turabian Style**

Koirala, Bikram, Mahdi Khodadadzadeh, Cecilia Contreras, Zohreh Zahiri, Richard Gloaguen, and Paul Scheunders. 2019. "A Supervised Method for Nonlinear Hyperspectral Unmixing" *Remote Sensing* 11, no. 20: 2458.
https://doi.org/10.3390/rs11202458