# Calculation Model of Radar Terrain Masking Based on Tensor Grid Dilation Operator

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Tensor and Its Application to Spatio-Temporal Data

#### 2.2. Basic Operations on Tensor

- (a)
- Fiber

- (b)
- Slice

## 3. High-Dimensional Electromagnetic Data Tensor Grid Model

#### 3.1. Tensor Grid-Based Modeling of High-Dimensional Electromagnetic Data

#### 3.2. Definition of Three-Dimensional Tensor Grid Dilation Operator

#### 3.3. Definition of Added Points Calculation for Three-Dimensional Tensor Grid Dilation Operator

## 4. Radar Terrain Masking Calculation Algorithm Based on Three-Dimensional Tensor Grid Dilation Operator

#### 4.1. Definition of Data Structure

#### 4.2. Dilation Judgment Factor

#### 4.3. Algorithm Flow

**Step 1**. Calculate the intersection ${S}_{RT}$ of the ideal radar detection range set ${S}_{Rader}$ and the terrain set ${S}_{Terrain}$, which overlap the radar detection range and terrain under ideal conditions. Let the initial set of dilation ${S}_{d}(0)={S}_{RT}$, set the terrain masking set ${S}_{TM}={S}_{Terrain}$, the 3D structuring element $S{E}_{3\mathrm{D}}$, the maximum number of iterations n, and the number of dilation iteration count $i=0$.

**Step 2**. Set the dilation iteration count $i=i+1$. If $i\le n$, turn to

**Step 3**; if $i>n$, turn to

**Step 8**.

**Step 3**. Start the ith dilation from the current dilation set ${S}_{d}(i-1)$, select the uncalculated spatial grid P in the dilation set. If there is no uncalculated grid in the current dilation set, turn to

**Step 7**; otherwise, turn to

**Step 4**.

**Step 4**. Dilate based on the 3D tensor grid dilation operator with 3D structuring element 3D-SE and obtain the added grids, which are defined as to-be-judged grids. Calculate the dilation judging factor ${F}_{Judge\text{\_}P}$ of each grid P, if ${F}_{Judge\text{\_}P}=1$, turn to

**Step 5**; if ${F}_{Judge\text{\_}P}=0$, then turn to

**Step 6**.

**Step 5**. Dilate to this to-be-judged grid and add the current grid to the added dilation set ${S}_{d}(i)$ and turn to

**Step 3**.

**Step 6**. Do not dilate to this to-be-judged grid, and the current grid is not added to the added dilation set, turn to

**Step 3**.

**Step 7**. Record the current added dilation set ${S}_{d}(i)$, find the union set of ${S}_{d}(i)$ and the terrain masking set ${S}_{TM}$, and determine whether ${S}_{d}(i)$ is the empty set. If ${S}_{d}(i)\ne \varnothing $, turn to

**Step 2**, and if ${S}_{d}(i)=\varnothing $, turn to

**Step 8**;

**Step 8**. End the dilation to obtain the terrain masking set ${S}_{TM}$. Then, calculate the actual radar detection range under the influence of terrain, based on the ideal radar detection range.

## 5. Experiments and Results

#### 5.1. Experimental Data

^{n}-Tree”, referred to as GeoSOT [22]. Therefore, we chose to experiment with GeoSOT-3D subdivision framework, which is the three-dimensional extension of GeoSOT. Based on the aforementioned experimental parameters, the ideal radar detection range is calculated, and the spatial subdivision is carried out according to the GeoSOT-3D subdivision framework. The radar detection range under ideal conditions and its representation with subdividing grids are shown in Figure 13.

#### 5.2. Experimental Results and Discussions

#### 5.2.1. Experimental Results of Two Simulated Terrain Datasets

#### 5.2.2. Experimental Results Varying Subdivision Layers and Grid Sizes

#### 5.2.3. Comparison of Computational Efficiency and Accuracy with Existing Algorithms

#### 5.2.4. Experimental Results with Actual Digital Elevation Model (DEM) Data

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 9.**Schematic of the set of added points for two-dimensional successive dilation. (

**a**) Original grid set with structuring element; (

**b**) Result of the first dilation; (

**c**) Result of the second dilation; (

**d**) Result of the third dilation.

**Figure 11.**Schematic diagram of terrain occlusion judgment. (

**a**) The spatial grid is not obscured; (

**b**) The spatial grid is obscured.

**Figure 13.**Radar detection range under ideal conditions. (

**a**) Radar detection range in geographic coordinates; (

**b**) radar detection range displayed by subdividing grids.

**Figure 14.**The spatial location relationship between the simulated terrain, radar detection range, and experimental results. (

**a**) Simulated terrain dataset 1 and experimental results; (

**b**) simulated terrain dataset 2 and experimental results; (

**c**) simulated terrain dataset 3 and experimental results.

**Figure 15.**Experimental results of each layer. (

**a**) Layer 15; (

**b**) Layer 16; (

**c**) Layer 17; (

**d**) Layer 18; (

**e**) Layer 19.

**Figure 16.**The trend of the logarithm base 10 of the total number of grids and number of masking grids and calculation time.

**Figure 17.**Sampling method for line-of-sight visibility method. (

**a**) Sampling method in the azimuthal direction; (

**b**) sampling method in a vertical profile.

Type Flag | Expressed Meaning |
---|---|

0 | The grid has not yet been calculated |

1 | The grid belongs to the calculated actual radar detection range |

2 | The grid is the boundary of the ideal radar detection range |

3 | The grid belongs to the terrain set |

Radar Parameter | Value |
---|---|

Radiation source location | (24°N, 115°E) |

Radiation source altitude | 400 m |

Transmit power | 50 kW |

Antenna gain | 10 |

Radar operating frequency | 1 GHz |

Radar operating wavelength | 0.3 m |

Half-power beamwidth | 30° |

Target radar cross-section | 10 m^{2} |

Minimum output Signal-to-Noise Ratio (SNR) | 20 dB |

Simulated Terrain Datasets | Number of Terrain Grids | Number of Intersecting Grids | Number of Masking Grids | Calculation Time |
---|---|---|---|---|

Dataset 1 | 1382 | 271 | 706 | 0.143 s |

Dataset 2 | 959 | 127 | 361 | 0.093 s |

Dataset 3 | 1153 | 151 | 1092 | 0.230 s |

Subdivision Layer | Grid Size | Total Number of Grids | Number of Intersecting Grids | Number of Masking Grids | Calculation Time |
---|---|---|---|---|---|

15 | 1280 m | 433 | 22 | 63 | 0.006 s |

16 | 640 m | 2558 | 72 | 300 | 0.034 s |

17 | 320 m | 15,987 | 271 | 706 | 0.143 s |

18 | 160 m | 115,223 | 1566 | 4061 | 1.139 s |

19 | 80 m | 873,155 | 10,106 | 20,734 | 9.124 s |

**Table 5.**Comparison results of calculation time and accuracy between current algorithm and existing algorithms.

Dilation Method Calculation Time | Line-of-Sight Visibility Method Calculation Time | Relative Error | |
---|---|---|---|

Dataset 1 | 0.143 s | 0.338 s | 1.29% |

Dataset 2 | 0.093 s | 0.597 s | 0.42% |

Dataset 3 | 0.230 s | 0.406 s | 0.98% |

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**MDPI and ACS Style**

Nie, K.; Fang, S.; Liu, H.; Wei, X.; Zhang, Y.; Yang, J.; Kong, Q.; Chen, B.
Calculation Model of Radar Terrain Masking Based on Tensor Grid Dilation Operator. *Remote Sens.* **2024**, *16*, 1432.
https://doi.org/10.3390/rs16081432

**AMA Style**

Nie K, Fang S, Liu H, Wei X, Zhang Y, Yang J, Kong Q, Chen B.
Calculation Model of Radar Terrain Masking Based on Tensor Grid Dilation Operator. *Remote Sensing*. 2024; 16(8):1432.
https://doi.org/10.3390/rs16081432

**Chicago/Turabian Style**

Nie, Kaiyu, Shengliang Fang, Hao Liu, Xiaofeng Wei, Yamin Zhang, Jianpeng Yang, Qinglei Kong, and Bo Chen.
2024. "Calculation Model of Radar Terrain Masking Based on Tensor Grid Dilation Operator" *Remote Sensing* 16, no. 8: 1432.
https://doi.org/10.3390/rs16081432