# Comparison of the Noise Robustness of FVC Retrieval Algorithms Based on Linear Mixture Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. LMM-Based Algorithms

#### 2.1. Algorithm-1: Reflectance-Based LMM

#### 2.2. Algorithm-2: VI-Based LMM

**ρ**, and the VI coefficients are

#### 2.3. Algorithm-3: Isoline-Based LMM

## 3. Error Propagation in FVC

#### 3.1. Measurement Errors in the Reflectance Spectra and Propagated Errors in the FVC

#### 3.2. Relationships Among the Errors Propagated in the FVC

## 4. Comparison of the Propagated Errors

#### 4.1. Derivation of the Angle

## 5. Comparison between Algorithms-2 and -3 under Identical VI Conditions

## 6. Numerical Demonstrations

## 7. Conclusions

## Acknowledgements

## Appendix

## References

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**Figure 1.**Error model in the measured spectrum (

**a**) and the errors propagated in the FVCs by algorithm-1 and -3, (

**b**) in the red-NIR reflectance space. In (

**a**), the blue dot indicates the target spectrum and the red dot indicates the band-correlated noise, a distance ${\sigma}_{t}$ from the target spectrum. The circle around the blue dot indicates the choices of band-correlated noise. In (

**b**), the FVCs and propagated errors for the two algorithms are indicated by empty circles on the line spanned by the vegetation and non-vegetation endmember spectra.

**Figure 2.**An example of the relationship between ${\u03f5}_{1}$, and ${\u03f5}_{2}$, determined by numerical simulations. (

**a**) shows the target spectrum (0.1,0.2), the vegetation spectrum (0.05,0.4), and the non-vegetation endmember spectrum (0.2,0.2), indicated by the blue dot, the filled and empty squares, respectively. The magnitude of the input error is set to 0.01. NDVI is used as the endmember model in algorithm-2. In (

**b**), the ellipse is obtained by varying θ in Equations (7) and (8).

**Figure 3.**Schematic diagram illustrating the relationships among the errors propagated by the FVC calculated according to each of the three algorithms. The relationships are indicated by bidirectional arrows of different colors. The figure and section numbers in the illustration indicate the results of numerical validation calculations and the derivations described in a previous study [29].

**Figure 4.**Example of the relationship between ${\u03f5}_{2}$ and ${\u03f5}_{3}$. (

**a**) The target spectrum, vegetation spectrum, and non-vegetation endmember spectrum denoted by the blue dot, filled squares and empty squares, respectively. The values are the same as those shown in Figure 2 except non-vegetation endmember spectrum; (

**b**) The error relationship between algorithm-2 and -3.

**Figure 5.**Relationship between ${\u03f5}_{1}$ and ${\u03f5}_{2}$. The segments of the ellipse, on which algorithm-1 is more robust than algorithm-2, are indicated by the red dotted lines. The values of θ are shown at the boundaries of the segments.

**Figure 6.**Two examples of an asymmetric ellipse describing the relationship between ${\u03f5}_{1}$ and ${\u03f5}_{2}$. In (

**a**), the slope of the major axis exceeds unity, meaning that the average value of $|{\u03f5}_{1}|$ is less than that of $|{\u03f5}_{2}|$. This suggests that algorithm-1 is more robust than algorithm-2 in terms of the propagated error; (

**b**) The opposite case as is shown in (

**a**), the conditions under which algorithm-1 is less robust than algorithm-2.

**Figure 7.**Example of an asymmetric ellipse obtained by setting ${p}_{1,1}=0$ in Equation (14) (

**a**), and the definitions of the angle ${\theta}_{0}$ and the new coordinate system (${x}^{\prime},{y}^{\prime}$) (

**b**).

**Figure 9.**Results of the numerical simulation describing the distribution of $tan{\theta}_{0}$ on a log scale over the red-NIR reflectance space, for comparing the performance of algorithm-1 and -2, or algorithm-1 and -3 (comparison of the inter-algorithm relationship) using EM1 as the endmember spectra. The results indicate the influences in the target spectrum and the choice of VI on $tan{\theta}_{0}$. (

**a**–

**c**) show the $tan{\theta}_{0}$-map for algorithm-1 and -2 using NDVI, SAVI, and EVI2 as the endmember models.(

**d**–

**f**) show the $tan{\theta}_{0}$-map for algorithm-1 and -3 using those VIs as the constraints.

**Figure 10.**Results of numerical simulations obtained by replacing the endmember spectra EM1 with EM2, shown with respect to those used in previous calculations, Figure 9. The results indicate that the relationship is influenced by the choice of VI as well as endmember spectra (non-vegetation class). A comparison with previous results is also shown. (

**a**–

**c**) show $tan{\theta}_{0}$-map for algorithm-1 and -2 using NDVI, SAVI, and EVI2 as the endmember models. (

**d**–

**f**) show $tan{\theta}_{0}$-map for algorithm-1 and -3 using the VIs as constraints.

**Figure 11.**Results of numerical simulations describing the distribution of $tan{\theta}_{0}$ on a log scale over the red-NIR reflectance spectrum. Algorithm-2 or -3 were compared using different VI conditions (comparison of the intra-algorithm relationship) and with EM1 as the endmember spectra. The results indicate the influences on the target spectrum and the choice of VI on $tan{\theta}_{0}$. (

**a**–

**c**) show the $tan{\theta}_{0}$-map for algorithm-2 using NDVI and SAVI, NDVI and EVI2, and SAVI and EVI2 as the endmember models. (

**d**–

**f**) show $tan{\theta}_{0}$-map for algorithm-3 using the same sets of VI used in the upper three panels as constraints.

**Figure 12.**Results of numerical simulations describing a distribution of α on a log scale over the red-NIR reflectance spectrum to compare algorithm-2 and -3 using identical VI conditions (comparison of the inter-algorithm relationship). The results indicate the influence of the target spectrum, endmember spectra, and choice of VI on $tan{\theta}_{0}$. (

**a**–

**c**) show the $tan{\theta}_{0}$-map using NDVI and SAVI, NDVI and EVI2, and SAVI and EVI2 as the conditions for algorithm-2 and -3 based on EM1. (

**d**–

**f**) show the $tan{\theta}_{0}$-map for the VIs assumed in the upper three panels, based on EM2.

Type of algorithm | Endmember model | Constraint |
---|---|---|

Reflectance-based LMM | reflectance spectrum | reflectance spectrum |

VI-based LMM | VI | VI |

Isoline-based LMM | reflectance spectrum | VI |

**Table 2.**Coefficients of the two-band VIs (${p}_{i}$, ${q}_{i}$, and ${r}_{i}$) used as examples in this study.

${\mathit{p}}_{1}$ | ${\mathit{q}}_{1}$ | ${\mathit{r}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{q}}_{2}$ | ${\mathit{r}}_{2}$ | |
---|---|---|---|---|---|---|

NDVI | $-1$ | 1 | 0 | 1 | 1 | 0 |

SAVI | $-(1+L)$ | $1+L$ | 0 | 1 | 1 | L |

EVI2 | $-2.5$ | $2.5$ | 0 | $2.4$ | 1 | 1 |

Class | Vegetation | Non-vegetation | ||

Band | Red | NIR | Red | NIR |

EM1 | 0.05 | 0.4 | 0.2 | 0.2 |

EM2 | 0.05 | 0.4 | 0.1 | 0.1 |

© 2011 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 license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Obata, K.; Yoshioka, H.
Comparison of the Noise Robustness of FVC Retrieval Algorithms Based on Linear Mixture Models. *Remote Sens.* **2011**, *3*, 1344-1364.
https://doi.org/10.3390/rs3071344

**AMA Style**

Obata K, Yoshioka H.
Comparison of the Noise Robustness of FVC Retrieval Algorithms Based on Linear Mixture Models. *Remote Sensing*. 2011; 3(7):1344-1364.
https://doi.org/10.3390/rs3071344

**Chicago/Turabian Style**

Obata, Kenta, and Hiroki Yoshioka.
2011. "Comparison of the Noise Robustness of FVC Retrieval Algorithms Based on Linear Mixture Models" *Remote Sensing* 3, no. 7: 1344-1364.
https://doi.org/10.3390/rs3071344