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Mountainous SAR Image Registration Using Image Simulation and an L_{2}E Robust Estimator

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## Abstract

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_{2}E algorithm is used to match images obtained by the simulation and in real-time to indirectly obtain the feature points of the real SAR images. Finally, the accurate registration of mountainous areas in the SAR images is achieved by a polynomial transform. Experimental verification is performed by using the data of mountainous SAR images from the same sensor and different sensors. When the registration accuracy of the method is compared with that of two state-of-the-art image registration algorithms, better outcomes are experimentally shown. The suggested approach can effectively solve the registration problem of SAR images of mountainous areas, and can overcome the disadvantages of poor adaptability and low accuracy of traditional SAR image registration methods for mountainous areas.

## 1. Introduction

_{2}-minimizing estimate (L

_{2}E) estimator. This enables practitioners to cope with both outliers and noises in the correspondences. Thirdly, the control points on the simulated images are converted to points on the real SAR images by polynomial transformation, so that the corresponding control points on the real image can be obtained. Finally, we determine the transformation model between the reference and the sensed images. The precise SAR registration in mountainous areas is completed by running a resampling procedure.

## 2. Robust Point Matching Algorithm Based on L_{2}E: RPM-L_{2}E

_{2}E estimator [32,33], which enables us to obtain reliable results even if a great quantity of outliers exists in the sample. Then, the robust algorithm is applied to match features between two sets of SAR images, which are called simulated and real.

#### 2.1. Formulation of the Problem Based on L_{2}E: Robust Estimation

_{2}E [31,34,35] criterion to the problem dealing with point set matching, the following criterion is obtained:

_{2}E functional in (2) then takes the form $f\left({x}_{\mathrm{n}}\right)={\displaystyle \sum}_{\mathrm{n}=1}^{\mathrm{N}}\mathsf{\Gamma}\left(x,{x}_{\mathrm{n}}\right){c}_{\mathrm{n}}$ [38,39], where ${c}_{\mathrm{m}}$ denotes a $\mathrm{D}\times 1$ coefficient vector (that will be determined). To improve the computational efficiency, the sparse approximate optimal solution is used for approximation as follows:

_{2}E functional in (1), which is defined by

_{2}E functional in (3) may be conveniently denoted by

#### 2.2. Estimation of the Transformation

_{2}E is required to estimate the transformation. Equation (4) should be checked concerning matrix C, which is denoted by

#### 2.3. Nonrigid Point Set Registration

Algorithm 1: Nonrigid Point Set Registration |

Input: $\left\{{x}_{\mathrm{n}}:\mathrm{n}\in I{N}_{\mathrm{N}}\right\},\left\{{y}_{\mathrm{l}}:l\in I{N}_{\mathrm{L}}\right\}$ are two-point sets, respectively, correspondence set $\mathrm{S}=\left\{\left({x}_{\mathrm{n}},{y}_{\mathrm{n}}\right):\mathrm{n}I{N}_{\mathrm{N}}\right\}$, parameters $\gamma $, $\beta $, $\lambda $Output: Aligned model point set $\left\{{\widehat{x}}_{\mathrm{n}}:\mathrm{n}\in I{N}_{\mathrm{N}}\right\}$, the optimal transformation $f$1 Calculate feature descriptors for the target point set $\left\{{y}_{\mathrm{l}}:l\in I{N}_{\mathrm{L}}\right\}$; 2 Repeat3 Calculate feature descriptor for the model point set $\text{}\left\{{x}_{\mathrm{n}}:\mathrm{n}\in I{N}_{\mathrm{N}}\right\}$; 4 Predict the initial correspondences utilizing the feature descriptors of the two-point set; 5 Determine Gram $\mathsf{\Gamma}$ and U matrices.6 Assign random values to parameters ${\sigma}^{2}$ and $\mathrm{C}$; 7 Deterministic annealing:8 Employ Equation (5), the objective function (4) is optimized by a numerical method (e.g., the quasi-Newton algorithm $\mathrm{C}$ based on the previous value); 9 Update the parameters $\mathrm{C}\leftarrow argmi{n}_{\mathrm{C}}{\mathrm{L}}_{2}\mathrm{E}\left(\mathrm{C},{\sigma}^{2}\right)$;10 Anneal ${\sigma}^{2}\to \gamma {\sigma}^{2}$; 11 The transformation $\mathrm{f}$ is found by Equation (5); 12 Update model point set $\left\{{x}_{\mathrm{n}}:\mathrm{n}\in I{N}_{\mathrm{N}}\right\}\leftarrow \left\{f\left({x}_{\mathrm{n}}\right):\mathrm{n}\in I{N}_{\mathrm{N}}\right\}$;13 until reaching the maximum iteration numbers;14 The aligned model point set $\left\{{\widehat{x}}_{\mathrm{n}}:\mathrm{n}\in I{N}_{\mathrm{N}}\right\}$ is given by $\left\{f\left({x}_{\mathrm{n}}\right):\mathrm{n}\in I{N}_{\mathrm{N}}\right\}$. |

**Parameter Settings**

_{2}E: γ, β and λ. The parameter γ denotes the annealing rate. The parameter β and λ determine the influence of the smoothness constraint on the transformation f. RPM-L

_{2}E is robust and parameter transformation has little effect on the algorithm. Thus, we set γ = 0.5, β = 1 and λ = 0.1 throughout this paper.

## 3. Methodology

#### 3.1. SAR Image Simulation

#### 3.1.1. SAR Image Geometric Simulation

#### 3.1.2. SAR Image Grayscale Simulation

#### 3.1.3. DEM Interpolation

#### 3.2. Transformation of the Image Coordinate Relation

_{2}E algorithm is employed to match simulated and real SAR images and the corresponding association between them is established; namely,

## 4. The Results of the Experiments and Their Analysis

#### 4.1. Experimental Data

#### 4.2. The Results of the Experiments

_{2}E algorithm uses the L

_{2}E estimator to make a robust estimation of the transformation from the corresponding relationship so that it can remove the outliers in the sample set well and obtain more correct matching points.

- Registration results between SAR images

_{2}E algorithm, which fully considers the terrain characteristics of mountainous areas, and the homologous points are obtained indirectly through the simulated images, thus further improving the registration accuracy.

## 5. Conclusions

_{2}E point set matching algorithm. The feature points are obtained indirectly, which solves the problem of finding feature points directly in mountainous areas.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Matching results between simulated images and real images in Experiment 1: (

**a**) is the matching result of reference image point sets obtained by the SIFT approach; (

**b**) is the matching outcome of sensed image point set obtained by the SIFT method; (

**c**) is the matching outcome of reference image point sets obtained by RPM-L

_{2}E algorithm; (

**d**) is the matching outcome of sensed image point set obtained by using RPM-L

_{2}E algorithm. Red lines show false matches, blue lines show true matches.

**Figure 5.**Matching outcomes between simulated images and real images in Experiment 2: (

**a**) is the matching result of reference image point sets obtained by the SIFT approach; (

**b**) is the matching outcome of sensed image point set obtained by the SIFT approach; (

**c**) is the matching outcome of reference image point set obtained by RPM-L

_{2}E algorithm; (

**d**) is the matching outcome of sensed image point set obtained by using RPM-L

_{2}E algorithm. Red lines show false matches, blue lines show true matches.

**Figure 6.**The qualitatively illustrated registration results of the proposed, RANSAC and SC methods, utilizing Experiment 1.

**Figure 7.**The qualitatively illustrated registration results of the proposed, RANSAC and SC methods, utilizing Experiment 2.

SAR Images | Sensor | Spatial Resolution | Incident Angle | Elevation Range | Size | Date of Acquisition | Location of Acqusition | |
---|---|---|---|---|---|---|---|---|

Experiment 1 | Ref. image | TerraSAR | 3 m | 30.99° | 600 m ~800 m | 600 × 600 | 11 October 2011 | Yuncheng |

Sen. image | TerraSAR | 3 m | 30.99° | 600 m ~800 m | 600 × 600 | 2 November 2011 | Yuncheng | |

Experiment 2 | Ref. image | TerraSAR | 1.5 m | 41.8° | 1800 m ~2100 m | 300 × 300 | 24 January 2016 | Heifangtai |

Sen. image | Sentinel | 15 m | 33.84° | 1800 m ~2100 m | 300 × 300 | 13 April 2017 | Heifangtai |

SAR Images | Method | Number of Matching Points | |||
---|---|---|---|---|---|

Reference Image | Correct Match | Reference Image | Correct Match | ||

Experiment 1 | SIFT | 15 | 10 | 16 | 9 |

RPM-L_{2}E | 20 | 20 | 16 | 16 | |

Experiment 2 | SIFT | 26 | 13 | 26 | 15 |

RPM-L_{2}E | 26 | 26 | 23 | 23 |

**Table 3.**The quantitatively compared outcomes of the suggested approach, and the RANSAC and SC methods.

SAR Images | Methods | RMSE/Pixel | MI |
---|---|---|---|

Experiment 1 | RANSAC | 0.028 | 0.48 |

SC | 0.047 | 0.097 | |

Our approach | 0.024 | 0.54 | |

Experiment 2 | RANSAC | 0.036 | 0.64 |

SC | 0.071 | 0.020 | |

Our approach | 0.035 | 0.66 |

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**MDPI and ACS Style**

Zhang, S.; Sui, L.; Zhou, R.; Xun, Z.; Du, C.; Guo, X.
Mountainous SAR Image Registration Using Image Simulation and an L_{2}E Robust Estimator. *Sustainability* **2022**, *14*, 9315.
https://doi.org/10.3390/su14159315

**AMA Style**

Zhang S, Sui L, Zhou R, Xun Z, Du C, Guo X.
Mountainous SAR Image Registration Using Image Simulation and an L_{2}E Robust Estimator. *Sustainability*. 2022; 14(15):9315.
https://doi.org/10.3390/su14159315

**Chicago/Turabian Style**

Zhang, Shuang, Lichun Sui, Rongrong Zhou, Zhangyuan Xun, Chengyan Du, and Xiao Guo.
2022. "Mountainous SAR Image Registration Using Image Simulation and an L_{2}E Robust Estimator" *Sustainability* 14, no. 15: 9315.
https://doi.org/10.3390/su14159315