# Modeling and Unsupervised Unmixing Based on Spectral Variability for Hyperspectral Oceanic Remote Sensing Data with Adjacency Effects

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## Abstract

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## 1. Problem Statement

- (1)
- In the field of remote sensing with hyperspectral imaging, when only simple situations are considered, data may be represented by the standard mixing model (i.e., the linear model without spectral variability). In contrast, we here consider much more challenging configurations, related to sea bottom analysis, where the so-called adjacency effect occurs. In [50], we showed that such configurations must be addressed by using the much more complex data model that we developed in that paper [50]. The present paper is thus the second paper wherein this cutting-edge data model is used to improve data analysis performance. This model is detailed in Section 2.1, especially in (1). As explained in that section, it differs from the standard mixing model as follows:
- (a)
- It first includes a term that is a modified version of the standard mixing model: the latter model, namely $S\phantom{\rule{3.33333pt}{0ex}}{a}_{i}$, is here altered by the vectors ${k}_{1,i}$ defined in Section 2. These vectors ${k}_{1,i}$ depend on the considered scene (with one vector per pixel) and are initially unknown, which is the first challenge of the considered configurations.
- (b)
- Moreover, the data model (1) includes a second term, namely ${k}_{2,i}\odot \left(S\phantom{\rule{3.33333pt}{0ex}}A{p}_{i}\right)$, which accounts for the adjacency effect and thus does not appear at all in the simple configurations addressed in the literature. This term depends on the vectors ${k}_{2,i}$ defined in Section 2. These vectors ${k}_{2,i}$ also depend on the considered scene (with one vector per pixel) and are initially unknown, which is the second challenge of the considered configurations.

- (2)
- The class of data analysis methods that is considered in this paper is hyperspectral unmixing. The unmixing methods proposed in the literature for the standard mixing model cannot be used here, because we consider a more complex mixing model, as stated above. Therefore, in [50], we developed the first method intended for our new data model. However, this method is restricted because it requires one to know (i.e., to previously estimate) the values of all parameter vectors ${k}_{1,i}$ and ${k}_{2,i}$ of the data model mentioned above, i.e., the method is supervised. Therefore, our main theoretical contribution in the present paper consists of a new approach to handling our recent data model in an unsupervised way, hence without knowing all pixel-dependent parameter vectors ${k}_{1,i}$ and ${k}_{2,i}$. To this end, we first provide an analysis of that model, which then allows us to show how to handle the considered data with an unsupervised unmixing method, which we propose for data that have spectral variability. Thus, our approach is promising thanks to our ability to handle spectral variability, whereas standard unmixing methods (such as VCA) cannot handle it and provide low performance if applied to data that have such variability.
- (3)
- The above expectation about unmixing performance is actually confirmed in this paper, because we also provide an experimental contribution, which proves the attractiveness of our approach. More precisely, we show that our algorithm yields better performance than the VCA unmixing method (intended for the standard mixing model) when applied to the considered realistic data.

## 2. Mixing Model for Oceanic Remote Sensing: Original and New Versions

#### 2.1. Original Mixing Model

#### 2.2. Analysis of the Mixing Model

## 3. Blind Unmixing Method for Spectral Variability

## 4. Data and Performance Criteria

#### 4.1. Considered Data

#### 4.2. Performance Criteria

## 5. Test Results and Discussion

#### 5.1. Results for Clear Shallow Water

#### 5.2. Results for Standard-Quality Shallow Water

#### 5.3. Results for Clear Deep Water

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Real pure reflectance spectra: Caulerpa Taxifolia, substratum, sand and Posidonia. These spectra contain 31 spectral bands over the [400 nm, 700 nm] range.

**Figure 2.**Maps of coefficients ${a}_{j,i}$ for all classes of pure materials (Caulerpa Taxifolia, substratum, sand and Posidonia). We numerically generated the coefficients ${a}_{j,i}$ and ${p}_{n,i}$ of (6), so as to simulate relevant pure material “abundances”. To this end, we used the same protocol as in [50]. The considered images contain $100\times 24$ pixels.

**Figure 3.**Variations in ${k}_{1,i}$ vs. wavelength, for all image pixels (one plot per pixel). Conditions: reference depth ${d}_{ref}=1.5$ m, depth variations over considered scene defined by $\mathsf{\Delta}=0.05$ m, clear water. The ${k}_{1,i}$ and ${k}_{2,i}$ vectors of (6) were derived from the physical model of the OSOAA radiative transfer simulator [57]. These vectors depend on the considered water quality (clear or standard) and on the depth ${d}_{i}$ of the considered pixel i.

**Figure 4.**Variations in ${k}_{2,i}$ vs. wavelength, for all image pixels (one plot per pixel). Conditions: reference depth ${d}_{ref}=1.5$ m, depth variations over considered scene defined by $\mathsf{\Delta}=0.05$ m, clear water. The ${k}_{1,i}$ and ${k}_{2,i}$ vectors of (6) were derived from the physical model of the OSOAA radiative transfer simulator [57]. These vectors depend on the considered water quality (clear or standard) and on the depth ${d}_{i}$ of the considered pixel i.

**Figure 5.**Contributions ${\rho}_{i,j}$ of Caulerpa taxifolia in the mixed reflectance spectra corresponding to all pixels (see (8), one plot per pixel). Conditions: clear water, reference depth ${d}_{ref}=1.5$ m and depth variations over considered scene defined by $\mathsf{\Delta}=0.25$ m.

**Figure 6.**Contributions ${\rho}_{i,j}$ of substratum in the mixed reflectance spectra corresponding to all pixels (see (8), one plot per pixel). Conditions: clear water, reference depth ${d}_{ref}=1.5$ m and depth variations over considered scene defined by $\mathsf{\Delta}=0.25$ m.

**Figure 7.**Contributions ${\rho}_{i,j}$ of sand in the mixed reflectance spectra corresponding to all pixels (see (8), one plot per pixel). Conditions: clear water, reference depth ${d}_{ref}=1.5$ m and depth variations over considered scene defined by $\mathsf{\Delta}=0.25$ m.

**Figure 8.**Contributions ${\rho}_{i,j}$ of Posidonia in the mixed reflectance spectra corresponding to all pixels (see (8), one plot per pixel). Conditions: clear water, reference depth ${d}_{ref}=1.5$ m and depth variations over considered scene defined by $\mathsf{\Delta}=0.25$ m.

**Figure 9.**Pure material spectra estimates ${e}_{i,j}$ for Caulerpa taxifolia and all pixels (one plot per pixel). Conditions: clear water, reference depth ${d}_{ref}=1.5$ m (i.e., shallow water) and depth variations over considered scene defined by $\mathsf{\Delta}=0.25$ m (highest considered value). The parameter $\mu $ of the IP-NMF method was set to 150 and we performed 400 iterations with this method.

**Figure 10.**Pure material spectra estimates ${e}_{i,j}$ for substratum and all pixels (one plot per pixel). Conditions: clear water, reference depth ${d}_{ref}=1.5$ m (i.e., shallow water) and depth variations over considered scene defined by $\mathsf{\Delta}=0.25$ m (highest considered value).The parameter $\mu $ of the IP-NMF method was set to 150 and we performed 400 iterations with this method.

**Figure 11.**Pure material spectra estimates ${e}_{i,j}$ for sand and all pixels (one plot per pixel). Conditions: clear water, reference depth ${d}_{ref}=1.5$ m (i.e., shallow water) and depth variations over considered scene defined by $\mathsf{\Delta}=0.25$ m (highest considered value). The parameter $\mu $ of the IP-NMF method was set to 150 and we performed 400 iterations with this method.

**Figure 12.**Pure material spectra estimates ${e}_{i,j}$ for Posidonia and all pixels (one plot per pixel). Conditions: clear water, reference depth ${d}_{ref}=1.5$ m (i.e., shallow water) and depth variations over considered scene defined by $\mathsf{\Delta}=0.25$ m (highest considered value). The parameter $\mu $ of the IP-NMF method was set to 150 and we performed 400 iterations with this method.

**Table 1.**Overall SAM, in degrees, of our IP-NMF method, versus its parameter $\mu $. Conditions: clear water, reference depth ${d}_{ref}=1.5$ m (i.e., shallow water), depth variations over considered scene defined by $\mathsf{\Delta}=0.25$ m.

$\mathit{\mu}$ | SAM of IP-NMF (°) |
---|---|

25 | 4.30 |

50 | 4.26 |

75 | 4.22 |

100 | 4.20 |

125 | 4.17 |

150 | 4.15 |

175 | 4.13 |

200 | 4.13 |

**Table 2.**(a) Overall SAM, in degrees, of our IP-NMF method and of VCA, versus magnitude $\mathsf{\Delta}$ of pixel-depth variations around reference depth ${d}_{ref}$. (b) % of improvement in SAM of IP-NMF, as compared with SAM of VCA. Conditions: clear water, ${d}_{ref}=1.5$ m.

Δ (m) | SAM of IP-NMF | SAM of VCA | % of Improvement |
---|---|---|---|

0.05 | 4.30 | 4.58 | 6.5 |

0.10 | 3.90 | 4.19 | 7.4 |

0.15 | 4.13 | 4.52 | 9.4 |

0.20 | 3.96 | 4.34 | 9.6 |

0.25 | 4.15 | 4.73 | 14.0 |

**Table 3.**(a) Overall SAM, in degrees, of our IP-NMF method and of VCA, versus magnitude $\mathsf{\Delta}$ of pixel-depth variations around reference depth ${d}_{ref}$. (b) % of improvement in SAM of IP-NMF, as compared with SAM of VCA. Conditions: standard-quality water, ${d}_{ref}=1.5$ m.

Δ (m) | SAM of IP-NMF | SAM of VCA | % of Improvement |
---|---|---|---|

0.05 | 3.75 | 3.83 | 2.1 |

0.10 | 3.02 | 3.28 | 8.6 |

0.15 | 3.59 | 4.02 | 12.0 |

0.20 | 3.90 | 4.61 | 18.2 |

0.25 | 5.54 | 6.53 | 17.9 |

**Table 4.**(a) Overall SAM, in degrees, of our IP-NMF method and of VCA, versus magnitude $\mathsf{\Delta}$ of pixel-depth variations around reference depth ${d}_{ref}$. (b) % of improvement in SAM of IP-NMF, as compared with SAM of VCA. Conditions: clear water, ${d}_{ref}=7.5$ m.

Δ (m) | SAM of IP-NMF | SAM of VCA | % of Improvement |
---|---|---|---|

0.05 | 3.97 | 4.03 | 1.5 |

0.10 | 4.75 | 4.82 | 1.5 |

0.15 | 5.00 | 5.18 | 3.6 |

0.20 | 5.32 | 5.75 | 8.1 |

0.25 | 7.00 | 7.44 | 6.3 |

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## Share and Cite

**MDPI and ACS Style**

Deville, Y.; Brezini, S.-E.; Benhalouche, F.Z.; Karoui, M.S.; Guillaume, M.; Lenot, X.; Lafrance, B.; Chami, M.; Jay, S.; Minghelli, A.;
et al. Modeling and Unsupervised Unmixing Based on Spectral Variability for Hyperspectral Oceanic Remote Sensing Data with Adjacency Effects. *Remote Sens.* **2023**, *15*, 4583.
https://doi.org/10.3390/rs15184583

**AMA Style**

Deville Y, Brezini S-E, Benhalouche FZ, Karoui MS, Guillaume M, Lenot X, Lafrance B, Chami M, Jay S, Minghelli A,
et al. Modeling and Unsupervised Unmixing Based on Spectral Variability for Hyperspectral Oceanic Remote Sensing Data with Adjacency Effects. *Remote Sensing*. 2023; 15(18):4583.
https://doi.org/10.3390/rs15184583

**Chicago/Turabian Style**

Deville, Yannick, Salah-Eddine Brezini, Fatima Zohra Benhalouche, Moussa Sofiane Karoui, Mireille Guillaume, Xavier Lenot, Bruno Lafrance, Malik Chami, Sylvain Jay, Audrey Minghelli,
and et al. 2023. "Modeling and Unsupervised Unmixing Based on Spectral Variability for Hyperspectral Oceanic Remote Sensing Data with Adjacency Effects" *Remote Sensing* 15, no. 18: 4583.
https://doi.org/10.3390/rs15184583