# Hyperspectral Unmixing from Incomplete and Noisy Data

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## Abstract

**:**

## 1. Introduction

#### 1.1. Hyperspectral Unmixing

#### 1.2. Contribution

#### 1.3. Related Work

## 2. Novel Mathematical Model

#### 2.1. Unmixing Model with Spatial Regularization and Proposed Model

**Remark 1.**

**Lemma 1.**

**Proof.**

#### 2.2. Model Reformulation

## 3. Algorithm

**Remark 2.**

## 4. Numerical Results

#### 4.1. Hyperspectral Line Camera

**Figure 1.**Hyperspectral line camera (

**left**) and principle of measuring a line simultaneously at all wavelengths

**(right**).

**Figure 2.**Measured object region (

**left**) and sensor frames $y={y}_{j}$ of the hypercube measured each in one sensor snapshot (

**right**); they correspond to one line of the object.

#### 4.2. Numerical Results for Real Data

**Figure 3.**(

**a**,

**b**) Two sensor frames $y={y}_{0}$ with the spectral direction along the z-axis and (

**c**) the mask of working sensor pixels.

**Figure 4.**Sensor frames of masked noisy original input and after inpainting by Navier–Stokes (NS) and our Model (6), respectively; (

**a**–

**c**): sensor frame $y=30$, and (

**d**–

**f**): $y=60$.

**Figure 5.**Channels $10,70,90$ of: (

**a**) the noisy original hypercube; (

**b**) the masked original known to the algorithm; (

**c**) the restoration by Navier–Stokes; and (

**d**) the restoration by our method.

^{cube}by [20].

**Figure 6.**(

**a**) The four endmember spectra ($p=4$) corresponding to each of the plastics blends in Figure 2 (left); (

**b**) the eight endmember spectra ($p=8$) used in Section 4.5.

#### 4.3. Numerical Results for Artificial Data (Pure Regions)

**Figure 7.**The artificial image is constructed by filling region j of the image on the left with the j-th endmember spectrum on the right ($p=4$).

**Figure 8.**(

**a**) Upper left corner of the $240\times 256$ mask for 3% working sensor pixels; (

**b**) label maps obtained from the unmixing by assigning to pixel $(i,\phantom{\rule{0.166667em}{0ex}}j)$ the index r of the largest coefficient ${X}_{ijr}^{\mathrm{low}}\in \{{X}_{ij1}^{\mathrm{low}},\cdots ,{X}_{ij4}^{\mathrm{low}}\}$ at that location.

**Table 1.**Percentage of image pixels, for which the largest of the material abundances found by the unmixing, correctly identifies the material at that pixel.

sensor pixels known, in % | 30 | 10 | 3 | 1 | 0.3 | 0.1 |

correctly assigned image pixels using TV, in % | 100 | 100 | 99.5 | 96.3 | 83.9 | 54.1 |

correctly assigned image pixels using ${TV}_{\mathrm{aniso}}$, in % | 100 | 100 | 99.4 | 95.5 | 82.8 | 52.7 |

#### 4.4. Numerical Results for Artificial Data (Mixed Regions)

**Figure 9.**Ground truth in false colors (

**left**); Channel 40 of the noisy hypercube (

**middle**); Channel 40 noisy and masked (

**right**); here, 10% are known.

**Figure 10.**Unmixing result, knowing 10% of the noisy hypercube. The images show the estimated abundances ${X}_{ij1}^{\mathrm{cube}},\cdots ,{X}_{ij4}^{\mathrm{cube}}$ corresponding to the four pure spectra.

#### 4.5. Numerical Results for Non-Occurring Endmembers

**Figure 11.**Unmixing result with eight endmembers of which four occur in the image; with 10% of the noisy hypercube known. The images show the estimated abundances ${X}_{ij1}^{\mathrm{cube}},\cdots ,{X}_{ij8}^{\mathrm{cube}}$ corresponding to the eight pure spectra.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Montag, M.J.; Stephani, H.
Hyperspectral Unmixing from Incomplete and Noisy Data. *J. Imaging* **2016**, *2*, 7.
https://doi.org/10.3390/jimaging2010007

**AMA Style**

Montag MJ, Stephani H.
Hyperspectral Unmixing from Incomplete and Noisy Data. *Journal of Imaging*. 2016; 2(1):7.
https://doi.org/10.3390/jimaging2010007

**Chicago/Turabian Style**

Montag, Martin J., and Henrike Stephani.
2016. "Hyperspectral Unmixing from Incomplete and Noisy Data" *Journal of Imaging* 2, no. 1: 7.
https://doi.org/10.3390/jimaging2010007