# High Accuracy Interpolation of DEM Using Generative Adversarial Network

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Generative Adversarial Networks

_{g}and D maximizing θ

_{d}. An objective function of this game can be denoted as:

_{g}to map vector z’s prior distribution P

_{z(z)}to vector x’s distribution P

_{data(x)}and generates G(z). D maximizes θ

_{d}to best distinguish the real data from fake samples (G(z)).

#### 2.2. CEDGAN

#### 2.3. Model Architecture

#### 2.3.1. Gated Convolution

#### 2.3.2. Dilated Convolution Structure

#### 2.3.3. SN-PatchGAN and its Discriminative Loss

^{sn}represents the spectral-normalized discriminator, and G is the image interpolation network that takes the masked image z as the input. G(z) is the generated image x is the ground truth image.

#### 2.3.4. The Proposed Generator’s Structure and its Loss

#### 2.4. Data Description

#### 2.5. Evaluation Metrics

#### 2.5.1. Quantitative Evaluation

_{elevatio}

_{n,}Equation (11)) across tiles, the average slope error (E

_{slope}, Equation (12)) was also computed since DEM errors affect terrain factors. All the generated values can be mapped back to their original elevation range by Equation (6). E

_{elevatio}

_{n}is calculated using the mapped elevations, and so is the slope (Equations (8)– (10)).

_{max}and H

_{min}are the true maximum and minimum elevation values in each DEM tile, H represents the true or mapped elevation from the generator’s output, and H

_{G}is the normalized elevation for the generator’s input or the elevation generated by the generator. Figure 2 shows the relative locations of the elements in Equations (8)–(9). Slope is calculated by Equation (10), where the variable Slope

_{we}(Equation (8)) denotes the slope in the horizontal direction and the Slope

_{sn}(Equation (9)) denotes slope in the vertical direction slope [34]. Slope is measured in degrees. Cellsize represents DEM’s resolution (in meters). Equations (11) and (12) compute the RMSE of the elevation and slope, respectively, where n represents the tile’s pixels, “true” represents the ground true DEM image, and “fake” represents the interpolated DEM image.

#### 2.5.2. Visual Evaluation

## 3. Results

#### 3.1. Training and Validation

^{−4}, where β1 and β2 were 0.5 and 0.999, respectively. The batch size was 16. The learning rates were the default rates in DeepFill v2, as we found that a lower learning rate made little contribution to improving the accuracy but slowed down the convergence. Intuitively, we set λ as 0.05, as we found during the training that the backpropagation gradient of the adversarial loss was about 20 times more than that of the L1 distance loss.

^{−4}. For the CEDGAN training, to make sure that the structure of the discriminator did not change, we had to crop the 256 × 256 tiles to 32 × 32 for training, the same size as the input’s for CEDGAN in [26]. As in Section 2.2, we analyzed that the discriminator only accepted 32 × 32 for training. For the validation, we also cropped the 256 × 256 tiles to 32 × 32, with an 8 × 8 pixel overlap, and concatenated them to 256 × 256 for a fairer comparison, since we found that inputting the 256 × 256 size image tiles led to extremely high errors. The reason for this is that the limited convolutional layers of the generator limited its receptive field for bigger images. As can be seen in Table 1, all the deep learning-based methods outperformed the traditional IDW method in terms of elevation and slope errors. Although the proposed methods obtained a similar elevation interpolation accuracy to CEDGAN, they did achieve a higher slope accuracy. Compared to the GAN-based methods, it was concluded that EDSR took an overwhelming advantage in quantitative quality. The main reason was that GAN including a discriminator can always produce some new textures and, thus, increase the interpolation uncertainty at the pixel level.

#### 3.2. Potentials

_{elevation}and average E

_{slope}values were obtained for comparing the different interpolation methods (Big DR values are bold)

## 4. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Generator’s structure. In this figure, “c64k7s2” means there are 64 channels, kernel size with 7, and stride with 2 in this convolutional layer. The ”deconv” is a deconvolution layer. Number in “2-4-8-16/2-4-8-8-4-2” means different dilation rate.

**Figure 2.**Moving window for slope and roughness computation. In this figure, e

_{1}~e

_{9}represent the elevation values on the Digital Elevation Model (DEM) image.

**Figure 3.**Training set’s average L1 loss among epochs of different combinations (convolutions (vanilla or gated) and dilation convolution structure (2-4-8-8-4-2 or 2-4-8-16)) during the training procedure.

**Figure 4.**Visualization of different methods’ results. In this figure, (

**A**) Ground Truth, (

**B**) inverse distance weighted (IDW), (

**C**) enhanced deep super-resolution network (EDSR), (

**D**) conditional encoder-decoder GAN (CEDGAN), (

**E**) VSUGAN (Vanilla conv + Increase-Down dilated conv), (

**F**) GIUGAN (Gated conv + Increase dilated conv), (

**G**) GSUGAN (Gated conv + Increase-Down dilated conv). (

**A**–

**G**) are elevation maps with 256 × 256 cells and (

**a**–

**g**) in (1)–(3) are enlarged regions marked by rectangles above. Warmer colors label higher elevation values. It can been seen that instead of small elevation errors, closer-to-zero DR values reveal better visual quality.

**Figure 5.**Visualization of terrains for different methods. In this figure, four different methods’ performances on four kinds of terrains are visualized by enlarging some regions. Generative adversarial networks (GANs’) results show more detail information, especially in complicated terrains.

**Table 1.**Elevation’s and Slope’s root-mean-square error (RMSEs) and DRs of Different Methods on Validation Set.

Methods | E_{elevation} | E_{slope} | DR |
---|---|---|---|

IDW | 0.186 | 3.389 | −0.0025 |

EDSR [19] | 0.088 | 1.842 | −0.0032 |

CEDGAN [26] | 0.168 | 3.345 | 0.0058 |

VSUGAN (Vanilla conv + Increase-Down dilated conv) | 0.176 | 2.825 | 0.0030 |

GIUGAN (Gated conv + Increase dilated conv) | 0.165 | 2.877 | 0.0018 |

GSUGAN (Gated conv + Increase-Down dilated conv) | 0.153 | 2.710 | 0.0005 |

Terrain | Resolution | Meanh | E_{elevation} | Slope | E_{slope} | ${\overline{R}}_{truth}$ | DR | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

IDW | ESDR | CEDGAN | GSUGAN | IDW | ESDR | CEDGAN | GSUGAN | IDW | ESDR | CEDGAN | GSUGAN | |||||

Plain | 1 m | 2.283 | 0.055 | 0.055 | 0.070 | 0.052 | 3.1844 | 1.888 | 1.567 | 1.795 | 1.598 | 0.0342 | −0.0160 | −0.0068 | 0.0015 | −0.0105 |

1/3 s | 9.260 | 0.135 | 0.125 | 0.160 | 0.134 | 0.954 | 0.436 | 0.386 | 0.466 | 0.421 | 0.0812 | −0.0234 | −0.0094 | 0.0025 | −0.0111 | |

1 s | 15.034 | 0.298 | 0.268 | 0.338 | 0.275 | 0.563 | 0.328 | 0.279 | 0.328 | 0.295 | 0.1607 | −0.0507 | −0.0176 | 0.0044 | −0.0260 | |

Plateau | 1 m | 6.469 | 0.021 | 0.023 | 0.028 | 0.024 | 3.031 | 0.634 | 0.641 | 0.763 | 0.675 | 0.0199 | −0.0049 | −0.0039 | −0.0009 | −0.0038 |

1/3 s | 49.056 | 0.286 | 0.150 | 0.253 | 0.220 | 3.488 | 0.797 | 0.487 | 0.762 | 0.671 | 0.2290 | −0.0152 | −0.0051 | 0.0107 | −0.0020 | |

1sec | 92.474 | 0.986 | 0.599 | 0.892 | 0.783 | 3.234 | 0.933 | 0.630 | 0.867 | 0.786 | 0.6289 | −0.0838 | −0.0122 | 0.0383 | −0.0020 | |

Basin | 1 m | 139.351 | 0.202 | 0.093 | 0.196 | 0.174 | 25.626 | 2.910 | 1.460 | 3.174 | 2.472 | 0.2816 | 0.0009 | −0.0017 | 0.0086 | 0.0031 |

1/3 s | 224.816 | 0.722 | 0.305 | 0.607 | 0.519 | 8.571 | 1.614 | 0.813 | 1.562 | 1.265 | 0.8115 | −0.0330 | −0.0096 | 0.0237 | 0.0029 | |

1 s | 520.604 | 2.964 | 1.660 | 2.783 | 2.466 | 8.407 | 2.379 | 1.464 | 2.347 | 2.057 | 2.4317 | −0.3054 | −0.0895 | 0.0794 | −0.0168 | |

Hill | 1 m | 39.995 | 0.092 | 0.054 | 0.093 | 0.075 | 11.071 | 2.420 | 1.631 | 2.571 | 1.694 | 0.1031 | −0.0061 | −0.0044 | 0.0028 | −0.0025 |

1/3 s | 178.745 | 0.737 | 0.446 | 0.719 | 0.6000 | 7.655 | 2.017 | 1.356 | 2.044 | 1.694 | 0.7182 | −0.0609 | −0.0322 | 0.0181 | −0.0192 | |

1 s | 344.718 | 2.358 | 1.581 | 2.291 | 2.000 | 6.946 | 2.123 | 1.568 | 2.155 | 1.879 | 2.0041 | −0.2610 | −0.1228 | 0.0363 | −0.0820 |

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Yan, L.; Tang, X.; Zhang, Y.
High Accuracy Interpolation of DEM Using Generative Adversarial Network. *Remote Sens.* **2021**, *13*, 676.
https://doi.org/10.3390/rs13040676

**AMA Style**

Yan L, Tang X, Zhang Y.
High Accuracy Interpolation of DEM Using Generative Adversarial Network. *Remote Sensing*. 2021; 13(4):676.
https://doi.org/10.3390/rs13040676

**Chicago/Turabian Style**

Yan, Li, Xingfen Tang, and Yi Zhang.
2021. "High Accuracy Interpolation of DEM Using Generative Adversarial Network" *Remote Sensing* 13, no. 4: 676.
https://doi.org/10.3390/rs13040676