# Urban Area Tomography Using a Sparse Representation Based Two-Dimensional Spectral Analysis Technique

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. One-Dimensional Tomographic SAR Imaging Model

#### 2.2. Two-Dimensional Tomographic SAR Imaging Model

#### 2.3. Compressive Sensing-Based 2D SAR Tomography Method

_{1}minimization:

## 3. Datasets

## 4. Numerical Experiment

#### 4.1. Simulated Experiments

- (1)
- In first of all, we investigate the super-resolution power of the approach along range direction. We assume the two scatterers are in the same elevation but difference range position, i.e., the range distance between the two scatterers $\mathsf{\Delta}r>0$ m and the elevation distance between the two scatterers $\mathsf{\Delta}s=0$ m. Based on the CS 2D TomoSAR method, 500 independent numerical experiments were implemented. The detection rate of two scatterers with difference range distance $\mathsf{\Delta}r$ is shown in Figure 8a. We can see that the closest range distance of two scatterers such that they can be discriminated with a detection rate more than 90% is nearly 1.8 m. In other words, the range resolution of the CS 2D TomoSAR method in this case is 1.8 m, which is much smaller than the range resolution of the SAR data.
- (2)
- Then, we investigate the super-resolution power of the approach along elevation direction. We assume the two scatterers are in the same range but at different elevation positions, i.e., the range distance between the two scatterers $\mathsf{\Delta}r=0$ m and the elevation distance between the two scatterers $\mathsf{\Delta}s>0$ m. Based on the CS 2D TomoSAR method, 500 independent numerical experiments also were implemented. The detection rate of two scatterers with difference elevation distance $\mathsf{\Delta}s$ is shown in Figure 8b. We can see that the closest elevation distance of two scatterers such that they can be discriminated with a detection rate more than 90% is nearly 10 m. In the other word, the elevation resolution of the CS 2D TomoSAR method in this case is 10 m, which is also much smaller than the Rayleigh resolution in elevation. Moreover, we compared the elevation resolution of the CS 1D TomoSAR method with the one from our proposed method. Under the same experiments parameters and number of independent numerical experiments, the elevation resolution of the CS 1D TomoSAR method is 18 m, which is nearly twice that of the CS 2D TomoSAR method.

- (1)
- We first investigated the distance estimation accuracy of the two scatterers by fixing the elevation distance between the two scatterers. We analyzed the estimation accuracy under four different conditions, where the elevation distances between two scatterers $\mathsf{\Delta}s\text{}\mathrm{were}\text{}0$, 6, 10, and 30 m. Figure 9a shows that when $\mathsf{\Delta}s\text{}\mathrm{is}\text{}30$ m, the CS 2D TomoSAR method provided good estimation accuracy with a decreasing $\mathsf{\Delta}r$. This is because the elevation distance between the two scatterers was larger than the elevation resolution of the method. When $\mathsf{\Delta}s\text{}\mathrm{was}\text{}10$, which was equal to the elevation resolution of the CS 2D TomoSAR method, the estimation accuracy of the CS 2D TomoSAR method converged to a higher value with decreasing $\mathsf{\Delta}r$, due to the limited elevation resolution. When $\mathsf{\Delta}s\text{}\mathrm{was}\text{}0$ m, the estimation accuracy of the CS 2D TomoSAR method converged to a higher value with decreasing $\mathsf{\Delta}r$ until $\mathsf{\Delta}r\text{}\mathrm{was}\text{}1.8$ m. This was due to the limitation in the range resolution. Notably, the worst estimation accuracy of the CS 2D TomoSAR method occurred when $\mathsf{\Delta}s$ was 0 m, which was smaller than when $\mathsf{\Delta}s$ was 10 m. Moreover, when $\mathsf{\Delta}s\text{}\mathrm{was}\text{}6$ m, the worst estimation accuracy of the CS 2D TomoSAR method was between $\mathsf{\Delta}s=0$ m and $\mathsf{\Delta}s=10$ m, and the resolution power was also between $\mathsf{\Delta}s=0$ m and $\mathsf{\Delta}s=10$ m.
- (2)
- Then, we investigated the distance estimation accuracy of the two scatterers by fixing the range distance between the two scatterers. We also investigated the analysis results of the estimation accuracy under four different conditions, where the range between the two scatterers ($\mathsf{\Delta}r$) were 0, 1, 1.8, and 2.4 m. Figure 9b shows that when $\mathsf{\Delta}r\text{}\mathrm{was}\text{}2.4$ m, the CS 2D TomoSAR method provided good accuracy with decreasing $\mathsf{\Delta}s$. This is because the range distance between the two scatterers was larger than the range resolution of the method. When $\mathsf{\Delta}r\text{}\mathrm{was}\text{}1.8$ m, which was equal to the range resolution of the CS 2D TomoSAR method, the estimation accuracy of the CS 2D TomoSAR method converged to a higher value with decreasing $\mathsf{\Delta}s$, due to a limited elevation resolution. When $\mathsf{\Delta}r\text{}\mathrm{was}\text{}0$ m, the estimation accuracy of the CS 2D TomoSAR method also converged to a higher value with decreasing $\mathsf{\Delta}s$ until $\mathsf{\Delta}s$ was 10 m, because of the limitation in the elevation resolution. Notably, the worst estimation accuracy of the CS 2D TomoSAR method was when $\mathsf{\Delta}r$ at 10 m was larger than with $\mathsf{\Delta}r$ at 1.8 m. Moreover, when $\mathsf{\Delta}r$ was 1 m, the worst estimation accuracy of the CS 2D TomoSAR method was between $\mathsf{\Delta}r=0$ m and $\mathsf{\Delta}r=1.8$ m, and the resolution power was also between $\mathsf{\Delta}r=0$ m and $\mathsf{\Delta}r=1.8$ m. We also compared the estimation accuracy of the CS 1D TomoSAR method with our proposed method. Under the same experimental parameters and number of independent numerical experiments, the estimation accuracy of CS 1D TomoSAR method was always worse than that of the CS 2D TomoSAR method.

#### 4.2. Real Data Experiments

#### 4.2.1. Tomographic Profiles

#### 4.2.2. 3D View of the Building

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**The spatial resolution in the vertical direction depends on the elevation and range resolution.

**Figure 4.**Flow diagram of compressive sensing-based two-dimensional range-elevation TomoSAR (CS 2D TomoSAR) method.

**Figure 7.**Reconstruction signal in the range-elevation plane: (

**a**) Fourier Beamforming; (

**b**) CS 1D TomoSAR method; and (

**c**) CS-2D-TomoSAR method.

**Figure 8.**Detection rate as a function of (

**a**) $\mathsf{\Delta}r$ at $\mathsf{\Delta}s$ = 0 m and (

**b**) $\mathsf{\Delta}s$ at $\mathsf{\Delta}r$ = 0 m.

**Figure 9.**Estimation accuracy of the distance between two scatterers as function of (

**a**) ∆r at ∆s of 0, 6, 10, and 30 m, and (

**b**) ∆s at $\mathsf{\Delta}r=0$, 1, 1.8, and 2.4 m.

**Figure 10.**Experiments on test area: (

**a**) Google Earth image; (

**b**) SAR image; (

**c**) coherence image; (

**d**) and (

**c**) a zoom of target building in red rectangle in (

**a**) and (

**b**); and (

**e**) target building (oil storage silo with a height of 26 m) on the test line.

**Figure 11.**Elevation-range sections of tomographic reconstruction over the test line, obtained with (

**a**) CS 1D TomoSAR method; (

**b**) CS-2D-TomoSAR with the range resolution enhanced two-fold; and (

**c**) CS-2D-TomoSAR with the range resolution enhanced 1.5-fold; (

**d**) The comparison of the detailed reconstruction results between the CS 1D TomoSAR and CS 2D TomoSAR methods. The color is associated to the estimated scattering magnitude of objects.

**Figure 12.**(

**a**) 3D view of the target building and 3D visualization of the scatterers reconstructed with SAR tomography, after coordinate transformation from range to ground distance; obtained with (

**b**) the CS 1D TomoSAR method, (

**c**) CS-2D-TomoSAR with the range resolution enhanced 1.5-fold and (

**d**) CS-2D-TomoSAR with the range resolution enhanced two-fold. The color is associated to the estimated height.

**Figure 13.**Three-dimensional visualization of the building zone of the scatterers reconstructed with SAR tomography, after coordinate transformation from range to ground distance, obtained with (

**a**) CS 1D TomoSAR method; (

**b**) CS-2D-TomoSAR with the range resolution enhanced 1.5 times; and (

**c**) CS-2D-TomoSAR with the range resolution enhanced two-fold. The color is associated to the estimated height.

Wavelength (m) | Range (km) | Incidence Angle | Total Baseline Span (m) | Azimuth Spacing (m) | Range Spacing (m) |
---|---|---|---|---|---|

0.0555 | 895 | 30° | 405.87 | 5.17 | 4.73 |

Flight Date (2012) | Baseline (m) Flight Date of Master Image: 30 November 2012 |
---|---|

9 July | 141.12 |

2 August | 251.43 |

26 August | −153.12 |

19 September | −138.31 |

11 October | −92.42 |

6 November | −132.73 |

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**MDPI and ACS Style**

Liang, L.; Li, X.; Ferro-Famil, L.; Guo, H.; Zhang, L.; Wu, W. Urban Area Tomography Using a Sparse Representation Based Two-Dimensional Spectral Analysis Technique. *Remote Sens.* **2018**, *10*, 109.
https://doi.org/10.3390/rs10010109

**AMA Style**

Liang L, Li X, Ferro-Famil L, Guo H, Zhang L, Wu W. Urban Area Tomography Using a Sparse Representation Based Two-Dimensional Spectral Analysis Technique. *Remote Sensing*. 2018; 10(1):109.
https://doi.org/10.3390/rs10010109

**Chicago/Turabian Style**

Liang, Lei, Xinwu Li, Laurent Ferro-Famil, Huadong Guo, Lu Zhang, and Wenjin Wu. 2018. "Urban Area Tomography Using a Sparse Representation Based Two-Dimensional Spectral Analysis Technique" *Remote Sensing* 10, no. 1: 109.
https://doi.org/10.3390/rs10010109