# Underlying Topography Inversion Using TomoSAR Based on Non-Local Means for an L-Band Airborne Dataset

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## Abstract

**:**

## 1. Introduction

## 2. Non-Local Means (NLM) Tomographic Synthetic Aperture Radar (TomoSAR) Method

#### 2.1. SAR Tomography Model

#### 2.1.1. Beamforming

#### 2.1.2. Adaptive Beamforming (Capon)

#### 2.1.3. Multiple Signal Classification (MUSIC)

#### 2.2. The NLM Algorithm

#### 2.3. The Traditional Spectral Estimation Methods Based on NLM

- (1)
- Solve the SCM $\mathit{R}$ of all the pixels in the area of interest.
- (2)
- Specify the size of the search window $W$ and matching window $P$.
- (3)
- Calculate the spatial similarity ${f}_{s}\left({x}_{0},{x}_{i}\right)$ and radiometric similarity ${f}_{r}\left({x}_{0},{x}_{i}\right)$ of the matching window between the central pixel ${x}_{0}$ and neighboring pixel ${x}_{i}$. (A pixel $\left(m,n\right)$ in the research region is located at ${x}_{0}$ within its search window $W$).
- (4)
- Calculate the weight of the neighborhood pixel ${x}_{i}$ based on the spatial and radiometric similarity.
- (5)
- Calculate the optimal weighted CM of the center pixel by the using of the SCM of all the neighborhood pixels (except for the center pixel) in the search window and their corresponding weights.
- (6)
- Substitute the estimated CM into the spectrum estimation formula to estimate the pixel’s spectrum.
- (7)
- Traverse the whole study area, and repeat steps (3) to (6) to obtain the spectra over the whole area.

## 3. Experiments and Results

#### 3.1. Study Area and Dataset

#### 3.2. Experimental Results and Analysis

#### 3.2.1. Comparison of the Tomograms

#### 3.2.2. Inversion of the Underlying Topography

## 4. Discussion

#### 4.1. Optimal Covariance Matrix Estimation

#### 4.2. Tomograms in the HV and VV Channels

#### 4.3. Forest Height Estimation

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Geometry and spectral estimation of the tomographic synthetic aperture radar (TomoSAR) configuration.

**Figure 2.**The principle of optimal CM estimation based on NLM ($p$ represents the size of the matching window, and $w$ denotes the size of the search window).

**Figure 4.**Tomograms of the HH polarization channel at the selected profile estimated by the different TomoSAR methods: (

**a**), (

**c**), (

**e**) are, respectively, the results of local means (LM)-based beamforming, adaptive beamforming (Capon), and multiple signal classification (MUSIC); and (

**b**), (

**d**), (

**f**) are the results of NLM-based beamforming, Capon, and MUSIC, respectively. The solid black line is the LiDAR DTM, and the dashed white line is the estimated values.

**Figure 5.**The underlying topography at the selected profile: (

**a**) the results of the LM-based TomoSAR method; and (

**b**) the results of the NLM-based TomoSAR method.

**Figure 7.**Underlying topography from LiDAR and TomoSAR: (

**a**,

**b**) are the LiDAR DTM; (

**c**), (

**e**), (

**g**) are, respectively, the beamforming, Capon, and MUSIC results based on the LM method; and (

**d**), (

**f**), (

**h**) are, respectively, the beamforming, Capon, and MUSIC results based on the NLM method.

**Figure 8.**The 2D joint distribution between the LiDAR and TomoSAR DTM of different methods: (

**a**) LiDAR DTM and DM by LM beamforming; (

**b**) LiDAR DTM and DTM by LM Capon; (

**c**) LiDAR DTM and DTM by LM MUSIC; (

**d**) LiDAR DTM and DTM by NLM beamforming; (

**e**) LiDAR DTM and DTM by NLM Capon; (

**f**) LiDAR DTM and DTM by NLM MUSIC.

**Figure 9.**Comparison of the amplitude and phase of the CM for the different methods: (

**a**), (

**c**) are, respectively, the amplitude and phase of the CM of the LM method; and (

**b**), (

**d**) are, respectively, the amplitude and phase of the CM of the NLM method.

**Figure 10.**Tomograms of the HV polarization channel at the selected profile estimated by the different TomoSAR methods: (

**a**), (

**c**), and (

**e**) are, respectively, the results of LM-based beamforming, Capon, and MUSIC; and (

**b**), (

**d**), (

**f**) are, respectively, the results of NLM-based beamforming, Capon, and MUSIC. The solid black line is the LiDAR DTM, and the dashed white line is the estimated values.

**Figure 11.**Profile line spectrum estimation results for the VV polarization channel: the solid black line is the LiDAR DTM, and the dashed white line is the estimated values; (

**a**), (

**c**), and (

**e**) are, respectively, the estimation results of LM-based beamforming, Capon, and MUSIC; and (

**b**), (

**d**), (

**f**) are the estimation results of NLM-based beamforming, Capon, and MUSIC, respectively.

**Figure 12.**Profile line spectrum estimation results for the HV polarization channel (terrain removed): the solid white line is the LiDAR canopy height model; (

**a**), (

**c**), and (

**e**) are, respectively, the estimation results of LM-based beamforming, Capon, and MUSIC; and (

**b**), (

**d**), and (

**f**) are the estimation results of NLM-based beamforming, Capon, and MUSIC.

Initialization | $\mathit{R}=\frac{1}{L}{\displaystyle \sum}_{l=1}^{L}\mathit{g}\left(l\right)\mathit{g}{\left(l\right)}^{H}$ |

Traverse | |

repeat | |

${f}_{s}\left({x}_{0},{x}_{i}\right)={\delta}_{{\gamma}_{s}}\left(\left|\left|{x}_{0}-{x}_{i}\right|\right|\right)$ | |

${f}_{r}\left({x}_{0},{x}_{i}\right)={\delta}_{{\gamma}_{r}}{\left[\frac{1}{{P}^{2}}{\displaystyle \sum _{p\in P}}{d}^{2}\left({\mathit{R}}_{{x}_{i}+p},{\mathit{R}}_{{x}_{0}+p}\right)\right]}^{1/2}$ | |

${w}_{{x}_{0},{x}_{i}}={f}_{s}\left({x}_{0},{x}_{i}\right){f}_{r}\left({x}_{0},{x}_{i}\right)$ | |

$\widehat{\mathit{R}}\left({x}_{0}\right)=\frac{{{\displaystyle \sum}}_{{x}_{i}\in W}{w}_{{x}_{0},{x}_{i}}{\mathit{R}}_{{x}_{i}}}{{{\displaystyle \sum}}_{{x}_{i}\in W}{w}_{{x}_{0},{x}_{i}}}$ | |

${\widehat{\mathit{P}}}_{\mathrm{BF}}=\frac{a{\left({z}_{d}\right)}^{H}\widehat{\mathit{R}}a\left({z}_{d}\right)}{{N}^{2}}$ ${\widehat{\mathit{P}}}_{\mathrm{CP}}=\frac{1}{a{\left({z}_{d}\right)}^{H}{\widehat{\mathit{R}}}^{-1}a\left({z}_{d}\right)}$ ${\widehat{\mathit{P}}}_{\mathrm{MUSIC}}=\frac{1}{a{\left({z}_{d}\right)}^{\mathrm{H}}\mathit{U}{\mathit{U}}^{\mathrm{H}}a\left({z}_{d}\right)}$ | |

Until (finish) |

**Table 2.**The parameters of the E-SAR airborne system [40].

Items | Parameters |
---|---|

Wavelength | 0.23 m (L-band) |

Polarimetric channel | HH + HV + VV |

Incidence angle | 25–55° |

Center slant range | 3900 m |

Range resolution | 2.12 m |

Azimuth resolution | 1.20 m |

**Table 3.**The baseline information for the InSAR pairs [40].

Identifier | Acquisition Date | Baseline (m) |
---|---|---|

08biosar0201 × 1 | 15 October 2008 | 0 |

08biosar0203 × 1 | −6 | |

08biosar0205 × 1 | −12 | |

08biosar0207 × 1 | −18 | |

08biosar0209 × 1 | −24 | |

08biosar0211 × 1 | −30 |

Item | Beamforming (m) | Capon (m) | MUSIC (m) |
---|---|---|---|

LM | 3.24 | 2.87 | 1.55 |

NLM | 2.11 | 1.77 | 1.06 |

Improvement | 34.87% | 38.28% | 31.61% |

Item | Beamforming (s) | Capon (s) | MUSIC (s) |
---|---|---|---|

LM | 26 | 25 | 24 |

NLM | 369 | 371 | 374 |

Beamforming (m) | Capon (m) | MUSIC (m) | |
---|---|---|---|

LM | 2.85 | 2.56 | 1.61 |

NLM | 1.83 | 1.67 | 1.12 |

Improvement | 35.78% | 34.76% | 30.43% |

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## Share and Cite

**MDPI and ACS Style**

Peng, X.; Wang, Y.; Long, S.; Pan, X.; Xie, Q.; Du, Y.; Fu, H.; Zhu, J.; Li, X.
Underlying Topography Inversion Using TomoSAR Based on Non-Local Means for an L-Band Airborne Dataset. *Remote Sens.* **2021**, *13*, 2926.
https://doi.org/10.3390/rs13152926

**AMA Style**

Peng X, Wang Y, Long S, Pan X, Xie Q, Du Y, Fu H, Zhu J, Li X.
Underlying Topography Inversion Using TomoSAR Based on Non-Local Means for an L-Band Airborne Dataset. *Remote Sensing*. 2021; 13(15):2926.
https://doi.org/10.3390/rs13152926

**Chicago/Turabian Style**

Peng, Xing, Youjun Wang, Shilin Long, Xiong Pan, Qinghua Xie, Yanan Du, Haiqiang Fu, Jianjun Zhu, and Xinwu Li.
2021. "Underlying Topography Inversion Using TomoSAR Based on Non-Local Means for an L-Band Airborne Dataset" *Remote Sensing* 13, no. 15: 2926.
https://doi.org/10.3390/rs13152926