# Approximating Empirical Surface Reflectance Data through Emulation: Opportunities for Synthetic Scene Generation

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## Abstract

**:**

## 1. Introduction

## 2. Interpolation and Emulation

#### 2.1. Interpolation

- Nearest-neighbor: This is the simplest method for interpolation, which is based on finding the closest node ${\mathbf{x}}_{i}$ to a query point ${\mathbf{x}}_{q}$ (e.g., by minimizing their Euclidean distance) and associating their output variables, i.e., $\hat{\mathbf{f}}({\mathbf{x}}_{q})=\mathbf{f}({\mathbf{x}}_{i})$. This fast method is valid for both gridded and scattered datasets. However, it produces discontinuities of the underlying model being interpolated.
- Piece-wise linear: This method is commonly used in remote sensing applications due to its balance between computation time and interpolation error. The implementation of linear interpolation is based on the Quickhull algorithm [21] for triangulations in multi-dimensional input spaces. For the scattered input data, the piece-wise linear interpolation method is reduced to finding the corresponding Delaunay simplex [22] (e.g., a triangle when $D=2$) that encloses a query D-dimensional point ${\mathbf{x}}_{q}$ (see Equation (1)):$${\hat{\mathbf{f}}}_{i}({\mathbf{x}}_{q})=\sum _{j=1}^{D+1}{\omega}_{j}\mathbf{f}({\mathbf{x}}_{j}),$$However, linear interpolation causes discontinuities on the first derivative of the interpolated model. In addition, in scattered datasets, the underlying Delaunay triangulation is computationally expensive in high dimensional input spaces (typically $D>$ 6) and is also limited by its intensive memory consumption [21,24]. In practice, it implies that it cannot do extrapolation. To predict the missing samples, here linear interpolation is used in combination with the following method:
- Inverse Distance Weighting (IDW) [8]
**:**Also known as Shepard’s method, this method weights the n closest nodes to the query point ${\mathbf{x}}_{q}$ (see Equation (2)) by the inverse of the distance metric $d({\mathbf{x}}_{q},{\mathbf{x}}_{i}):\mathcal{X}\mapsto {\mathbb{R}}^{+}$ (e.g., the Euclidean distance):$$\hat{\mathbf{f}}({\mathbf{x}}_{q})=\frac{{\sum}_{i=1}^{n}{\omega}_{i}\mathbf{f}({\mathbf{x}}_{i})}{{\sum}_{i=1}^{n}{\omega}_{i}},$$$${\omega}_{i}={\left(\frac{R-d({\mathbf{x}}_{q},{\mathbf{x}}_{i})}{R\xb7d({\mathbf{x}}_{q},{\mathbf{x}}_{i})}\right)}^{p},$$

#### 2.2. Emulation

## 3. Description of Used SPARC Dataset and Experimental Setup

#### 3.1. SPARC Dataset

#### 3.2. Experimental Setup

## 4. Results

#### 4.1. Interpolation vs. Emulation

#### 4.2. Emulation of Hyperspectral Imagery

#### 4.2.1. CHRIS-Like Image

#### 4.2.2. HyMap-Like Image

#### 4.2.3. Sentinel-2-Like Hyperspectral Image

## 5. Discussion

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Wavelength-dependent NRMSE (%) results of the two interpolation methods and the three tested emulators, i.e., KRR, GPR, NN for CHRIS (

**left**) and HyMap (

**right**) datasets.

**Figure 2.**Original SPARC 20% validation spectra (

**left**) and GPR-emulated spectra (

**right**) for CHRIS (

**top**) and HyMap (

**bottom**) data, color plotted as a function of LAI.

**Figure 3.**500 GPR-emulated CHRIS-like (

**left**) and HyMap-like (

**right**) surface reflectance spectra, color plotted as a function of LAI.

**Figure 4.**Schematic overview of RGB and emulated synthetic CHRIS image over agricultural site Barrax, Spain (R: 653 nm; G: 553 nm; B: 460 nm).

**Figure 5.**Wavelength-dependent NRMSE (%) comparison of scenes as generated by the three tested emulators (KRR, GPR, NN) against a reference CHRIS scene.

**Figure 6.**Relative difference maps (%) for arbitrarily chosen wavelengths between GPR-emulated scene and reference CHRIS scene.

**Figure 7.**Schematic overview of RGB and emulated synthetic subset of HyMap scene over agricultural site Barrax, Spain (R: 646 nm; G: 555 nm; B: 462 nm).

**Figure 8.**Wavelength-dependent NRMSE (%) comparison of scenes as generated by the three tested emulators (KRR, GPR, and NN) against a reference HyMap scene.

**Figure 9.**Relative difference maps (%) for arbitrarily chosen wavelengths between KRR-emulated scene and reference HyMap scene.

**Figure 10.**Schematic overview of RGB- and KRR-emulated synthetic hyperspectral S2-like image and data cube visualization of emulated subset over agricultural site Valladolid, Spain (R: 646.5 nm; G: 554.9 nm; B: 462.4 nm).

**Table 1.**Interpolation vs. emulation validation results and CPU time for CHRIS (

**top**) and HyMap (

**bottom**) SPARC datasets.

Model | RMSE | NRMSE (%) | CPU (s) |
---|---|---|---|

CHRIS | |||

Interpolation: | |||

- nearest | 653.3 | 20.7 | 0.1881 |

- linear + IDW | 649.4 | 20.5 | 0.3040 |

Emulation: | |||

- KRR | 436.3 | 13.0 | 0.0007 |

- GPR | 420.6 | 13.0 | 0.0096 |

- NN | 432.5 | 13.4 | 0.0070 |

HyMap | |||

Interpolation: | |||

- nearest | 405.4 | 12.5 | 0.1501 |

- linear + IDW | 398.2 | 12.2 | 0.2428 |

Emulation: | |||

- KRR | 269.6 | 8.5 | 0.0006 |

- GPR | 267.2 | 8.4 | 0.0086 |

- NN | 412.0 | 12.6 | 0.0059 |

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**MDPI and ACS Style**

Verrelst, J.; Rivera Caicedo, J.P.; Vicent, J.; Morcillo Pallarés, P.; Moreno, J. Approximating Empirical Surface Reflectance Data through Emulation: Opportunities for Synthetic Scene Generation. *Remote Sens.* **2019**, *11*, 157.
https://doi.org/10.3390/rs11020157

**AMA Style**

Verrelst J, Rivera Caicedo JP, Vicent J, Morcillo Pallarés P, Moreno J. Approximating Empirical Surface Reflectance Data through Emulation: Opportunities for Synthetic Scene Generation. *Remote Sensing*. 2019; 11(2):157.
https://doi.org/10.3390/rs11020157

**Chicago/Turabian Style**

Verrelst, Jochem, Juan Pablo Rivera Caicedo, Jorge Vicent, Pablo Morcillo Pallarés, and José Moreno. 2019. "Approximating Empirical Surface Reflectance Data through Emulation: Opportunities for Synthetic Scene Generation" *Remote Sensing* 11, no. 2: 157.
https://doi.org/10.3390/rs11020157