Mountainous SAR Image Registration Using Image Simulation and an L₂E Robust Estimator

Abstract

Synthetic Aperture Radar (SAR) is one of the most widely utilized methods to extract elevation information and identify large-scale deformations in mountainous areas. Homologous points in stereo SAR image pairs are difficult to identify due to complex geometric and radiometric distortions. In this paper, a new approach for mountainous area images is suggested. Firstly, a simulated SAR image and a look-up table based on DEM data are generated by a range-Doppler model and an empirical formula. Then, a point matching RPM-L₂E 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

Mountainous areas are geomorphic terrains that accounts for 24% of the global land area [1]. Due to their large topographic relief and complex geological structure, human activities [2] (mining and large areas of steep slope reclamation, overgrazing, etc. [3]) and other factors can cause frequent disasters [4,5,6]. Therefore, terrain monitoring in mountainous areas is necessary.

It is very complicated to identify geological hazards in a large mountainous area. Generally, surface deformation is considered an important index of potential geological hazards. The use of Global Navigation Satellite System (GNSS) for 3D positioning in mountainous areas is rapidly growing [7]. Chwedczuk et al. measured and validated altitudes from existing sources with direct GNSS measurements and airborne Lidar data [8]. Langbein et al. estimated the background noise of a Global Positioning System-derived time series for positions at 740 sites in the western United States [9]. Even though the GNSS measurements provide high-accuracy subsidence measurements at discrete locations, they have limitations in providing more detailed and comprehensive information on the ground settlement [10,11,12]. In addition, operating these measurement networks in the field is time consuming and costly.

SAR [13,14] is a well-grounded imaging tool that can generate high resolution images, function around the clock, and is not affected by any interferences, such as weather conditions. In contrast to the GNSS surveying, the satellite SAR technique has the ability of measuring deformation with a high accuracy over a large area [15,16,17,18]. Thus, it is widely implemented for monitoring surface changes [19] and identifying disasters [20] in mountainous areas. Registering SAR images, which is a major step necessary in its implementation, directly affects the accuracy of subsequent data processing [21]. However, an area’s image with subsequent topographic changes ostensibly contains a great deal more distortions than a flat area’s image obtained under the same imaging conditions, since SAR sensors have a mode of imagery described as side looking. The distortions mainly include geometric deformations (translation effects, rotation, scale distortion, occlusion, and viewpoint differences), and foldover and shadow effects [22]. Therefore, traditional SAR image registration alone cannot meet the requirements of mountainous terrain registration, and mountain SAR image registration remains a challenge.

The methods utilized to register images are generally split into two groups: intensity-based and feature-based approaches [23,24]. When images suffer from apparent geometric distortions, feature-based approaches are more robust and dependable than intensity-based approaches, as suggested by several kinds of research [25]. Salient attributes are firstly extracted by feature-based approaches. Afterward, similarity measures are employed to match them so that the geometric correspondence between two images is constructed. The underlined characteristics of these approaches are described as fast, and resilient to both the noise distortions and to significant radiometric errors that are common in registering images of areas with complex geometries. Nevertheless, their performances merely rely on suitably extracted attributes (points, edges, contours, and regions) and dependable algorithms to match them. Besides, invariant descriptors, spatial relations, and relaxation approaches are contained by the renowned feature matching approaches [23].

The performance capabilities of the descriptors were conducted by Mikolajczyk et al. [26], which suggested that the scale-invariant feature transform (SIFT) [27] outperforms most of them. Also, the extraction of disparate invariant features in images is achieved by the SIFT. Thus, performing dependable matching across a substantial range of affine distortions could be realized when there are changes in 3-D viewpoints, when noise is added, and when there is an alteration in exposition. Even though SIFT provides practical advantages, some issues occur once images of the remote sensing are directly employed, i.e., the number of the detected featured matches could be small, and the complicated characteristics of remote sensing images are distributed unevenly. Moreover, feature pairs could have several outliers when the overlaid regions of remote sensing images have significant differences concerning image intensity. Consequently, a singly implemented SIFT could not generate optimal outcomes, as another recent algorithm implementing SIFT utilizing remote sensing images suggested [25]. In other words, most of these methods fail to consider the effects of the side-view imaging mode and topographic relief when a geometric transformation is conducted, and they fail to harness the advantages of rich texture features in mountainous regions.

In addition, one of the most common disadvantages of SAR image registration based on SIFT algorithms is that only image information is used. Under the condition of the same pass and the same side-looking mode, these methods can meet the requirements of SAR image registration despite the different resolutions and incident angles. However, in different pass or different side-looking mode configurations, the layover and foreshortening will completely change, which would lead to serious geometric distortions of SAR images. Furthermore, due to the side-looking imagery mode of SAR sensors, the image covering an area of significant topographic changes evidently has larger distortions than the one covering a flat area in the same imaging conditions. Consequently, many false feature detections and false matches occur, few features will be correctly matched, and it may be impossible to register SAR images with large terrain fluctuations.

Image simulation can obtain simulated SAR images with rich textures and clear features in areas with large terrain fluctuations, and image simulation also has a high similarity of texture features with real SAR images, which is very conducive to the interpretation of SAR images. Therefore, this paper proposes a new SAR image registration approach for mountainous areas. Our approach is based on DEM and SAR image satellite orbit data [28], the R-D model [29,30], and uses the image simulation method to simulate two SAR images and generate the corresponding lookup table (which stores the correspondence between auxiliary DEM and simulated SAR images). Firstly, we select some control points from the DEM uniformly and use the look-up table to obtain the corresponding control points on the two simulated images. Second, a shape context, called descriptors of features, is employed to establish similarity relation [31]. The transformation is estimated by employing the robust L₂-minimizing estimate (L₂E) 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 L₂E: RPM-L₂E

In this study, the transformations concerning correspondences are estimated utilizing the robust L₂E 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 L₂E: Robust Estimation

Assume $\mathrm{S}=\left\{\left(x_{\mathrm{n}},y_n\right):n\in IN\right\}$ is called the initial point set of the SAR image with outliers that are generated by perturbation of both noise and outlier data. The noises in each component have Gaussian distribution with zero average value, and a uniform standard deviation is assumed. Therefore, the correspondence of an inlier datum satisfies $y_n-f\left(x_{\mathrm{n}}\right)~N\left(0,{\sigma }^{2}I\right),$ where $I$ represents a $\mathrm{D}\times \mathrm{D}$ identity matrix where the point’s dimension is denoted by $\mathrm{D}$. By applying the L₂E [31,34,35] criterion to the problem dealing with point set matching, the following criterion is obtained:

$${\mathrm{L}}_2\mathrm{E}\left(f,{\sigma }^2\right)=\frac{1}{2^{\mathrm{D}}{\left(\pi \sigma \right)}^{\mathrm{D}/2}}-\frac{2}{N}{{\displaystyle \sum }}_{\mathrm{n}=1}^{\mathrm{N}}\Phi \left(y_{\mathrm{n}}-f\left(x_{\mathrm{n}}\right)|0,{\sigma }^{2}I\right)$$

(1)

A reproducing kernel Hilbert space (RKHS) $H$ is defined [36,37,38] by a positive definite matrix-valued kernel $\mathsf{\Gamma }\left(x,x^{\prime }\right):I{R}^{\mathrm{D}}\times I{R}^{\mathrm{D}}\to I{R}^{\mathrm{D}\times \mathrm{D}}$. The optimal transformation f that minimizes the L₂E 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:

$$f\left(x_{\mathrm{n}}\right)={{\displaystyle \sum }}_{\mathrm{m}=1}^{\mathrm{M}}\mathsf{\Gamma }\left(x,{\tilde{x}}_{\mathrm{m}}\right)c_{\mathrm{m}}$$

(2)

The chosen point set $\left\{x_{\mathrm{m}}:\mathrm{m}\in I{N}_{\mathrm{M}}\right\}$ is more or less similar to the “control points” [40]. Imposing a smooth constraint on the transformation by including a regularization term leads to the L₂E functional in (1), which is defined by

$${\mathrm{L}}_{2}E(f,{\sigma }^2)=\frac{1}{2^{\mathrm{D}}{\left(\pi \sigma \right)}^{\mathrm{D}/2}}-\frac{2}{N}{{\displaystyle \sum }}_{\mathrm{n}=1}^{\mathrm{N}}\frac{1}{{\left(2\pi {\sigma }^2\right)}^{\mathrm{D}/2}}{e}^{-\frac{\Vert y_{\mathrm{n}}-{{\displaystyle \sum }}_{\mathrm{m}=1}^{\mathrm{M}}\mathsf{\Gamma }\left(x,{\tilde{x}}_{\mathrm{m}}\right)c_{\mathrm{m}}{\Vert }^2}{2{\sigma }^2}}+\lambda \Vert f{\Vert }_{\mathsf{\Gamma }}^2$$

(3)

where the strength of regularization is controlled by $\mathsf{\lambda }>0$, an inner kernel defines the stabilization denoted by $\Vert f{\Vert }_{\mathsf{\Gamma }}^2$ [39]: $\mathsf{\Gamma }\left(x_{\mathrm{i}},x_{\mathrm{j}}\right)=k\left(x_{\mathrm{i}},x_{\mathrm{j}}\right)·I=exp\left\{-\frac{\Vert x_{\mathrm{i}},x_{\mathrm{j}}{\Vert }^2}{{\beta }^2}\right\}·I$, where $\beta$ determines the width of the range of interaction between samples. The L₂E functional in (3) may be conveniently denoted by

$${\mathrm{L}}_2\mathrm{E}(\mathrm{C},{\sigma }^2)=\frac{1}{2^{\mathrm{D}}{\left(\pi \sigma \right)}^{\mathrm{D}/2}}-\frac{2}{N}{{\displaystyle \sum }}_{\mathrm{n}=1}^{\mathrm{N}}\frac{1}{2^{\mathrm{D}}{\left(\pi \sigma \right)}^{\mathrm{D}/2}}{e}^{-\frac{\Vert y_{\mathrm{n}}{}^T-U_{\mathrm{n}}\mathrm{C}{\Vert }^2}{2{\sigma }^2}}+\lambda tr({\mathrm{C}}^T\mathsf{\Gamma }\mathrm{C})$$

(4)

where the kernel matrix $\mathsf{\Gamma }\in I{R}^{\mathrm{M}\times \mathrm{M}}$ is called the Gram matrix with ${\mathsf{\Gamma }}_{\mathrm{ij}}=k\left(\widetilde{x_{\mathrm{i}}},\widetilde{x_{\mathrm{j}}}\right)=exp\left\{-\frac{\Vert \widetilde{x_{\mathrm{i}}}-{\widetilde{x_{\mathrm{j}}}\Vert }^2}{{\beta }^2}\right\}$;$U\in I{R}^{\mathrm{N}\times \mathrm{M}}$ with $U_{\mathrm{ij}}=k\left(x_{\mathrm{i}},\widetilde{x_{\mathrm{j}}}\right)=exp\left\{-\frac{\Vert x_{\mathrm{i}}-{\widetilde{x_{\mathrm{j}}}\Vert }^2}{{\beta }^2}\right\}$; $U_{\mathrm{n},}$ denotes the $\mathrm{ith}$ row in $U$; $\mathrm{C}={\left(c_1,\cdots ,c_{\mathrm{m}}\right)}^T$ denotes $\mathrm{M}\times \mathrm{D}$ coefficient matrix, and $tr(·)$ gives race.

2.2. Estimation of the Transformation

Computing the derivative of the cost function of the L₂E is required to estimate the transformation. Equation (4) should be checked concerning matrix C, which is denoted by

$$\frac{\partial {\mathrm{L}}_2\mathrm{E}}{\partial \mathrm{C}}=\frac{2U^{\mathrm{T}}\left[P·\left(Q\otimes 1_{1\times \mathrm{D}}\right)\right]}{N{\sigma }^2{\left(2\pi {\sigma }^2\right)}^{\mathrm{D}/2}}+2\lambda \mathsf{\Gamma }\mathrm{C}$$

(5)

where $P=UC-Y$ and $Y={\left(y_1,\cdots ,y_{\mathrm{N}}\right)}^T$ represent $\mathrm{N}\times \mathrm{D}$ matrices, $Q=exp\left\{\frac{diag\left(P{P}^{\mathrm{T}}\right)}{2{\sigma }^2}\right\}$ denotes an $\mathrm{N}\times 1$ vector, $diag(·)$ represents the diagonal elements of the matrix, $1_{1\times D}$ represents a $1\times \mathrm{D}$ vector whose components are 1s, a product called the Hadamard is represented by ⊙, and a product called the Kronecker is denoted by $\otimes$.

Gradient-based numerical optimization methods, which are called the quasi-Newton methods, and the nonlinear conjugate gradient method are employed to resolve the problem by utilizing Equation (5). However, the convexity is satisfied only when the optimal solution is in the neighborhood set for the cost function given in (4). Then, a coarse-to-fine approach is employed to implement the deterministic annealing on the inlier noise parameter ${\sigma }^2$ to enhance convergence. So, a large initial ${\sigma }^2$ value is needed. The process denoted by ${\sigma }^2\to \gamma {\sigma }^2$ is targeted, which $\gamma$ is called the annealing rate.

2.3. Nonrigid Point Set Registration

Note that both the transformation and point correspondence are not resolved by the proposed method. An initial correspondence is needed to resolve the transformation between two-point sets. In this paper, shape context is utilized as the feature descriptor [35] and the Hungarian approach with the ${\chi }^2$ test statistic is utilized to measure the cost for matching. An iteration algorithm is conducted to estimate both correspondences and transformations so that a dependable outcome is attained. In this paper, the number of iterations is fixed to 10, which is a typical choice. However, greater than 10 could be used when the amount of noise and the percentage of outliers appear large in the original point sets. Algorithm 1 dealing with point set registration is summarized as follows:

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 Repeat  3    Calculate feature descriptor for the model point set $\left\{x_{\mathrm{n}}:\mathrm{n}\in I{N}_{\mathrm{N}}\right\}$;  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

There are four main parameters in RPM-L₂E: 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-L₂E 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

It is difficult to select homologous points in the process of SAR stereo image matching in mountainous areas. To solve this problem, this paper uses auxiliary DEM data to convert direct registration between SAR images into indirect registration between simulated images and SAR images. The flowchart is shown in Figure 1.

3.1. SAR Image Simulation

The target’s spatial coordinate determines the location of any pixel in a digital image. Then, the model of the geometric imaging calculates its position [29,30]. The widely implemented R-D model defines the imaging geometry of the SAR system intuitively and accurately to a great extent. SAR simulated images include geometric simulation and grayscale simulation components.

3.1.1. SAR Image Geometric Simulation

To simulate the SAR image, the grid points of each DEM were first calculated according to the principle of imaging radar, that is, the DEM grid coordinates were transformed into the corresponding coordinates of the pixel points of the SAR image. SAR sensors are slant-range imaging sensors based on the Doppler principle, therefore, we use the range-Doppler model to build the relationship between DEM grid points $\left(X, Y, Z\right)$ and the coordinates of imaging position points (x, y). The spatial geometry of a side-looking SAR image is shown in Figure 2.

Within a range direction scan line, any point on the slant-range image should satisfy the following range equation and Doppler equation:

$$\{\begin{array}{c}{R}_{\mathrm{p}}=\sqrt{{\left(X-X_{\mathrm{S}}\right)}^2+{\left(Y-Y_{\mathrm{S}}\right)}^2+{\left(Z-Z_{\mathrm{S}}\right)}^2}=R_0+K{r}_x\\ f=\frac{-2\left[V_{\mathrm{X}}\left(X-X_{\mathrm{S}}\right)+V_{\mathrm{Y}}\left(Y-Y_{\mathrm{S}}\right)+V_{\mathrm{Z}}\left(Z-Z_{\mathrm{S}}\right)\right]}{\lambda R_{\mathrm{p}}}\end{array}$$

(6)

where $S\left(X_{\mathrm{S}},Y_{\mathrm{S}},Z_{\mathrm{S}}\right)$ and $P\left(X, Y, Z\right)$ denote positions of the SAR sensor and target in vector representations; $R_0$ is the initial slant distance; $R_{\mathrm{P}}$ denotes the slant sensor-to-target distance; K denotes the coordinates of P in the slant-range direction; $r_{\mathrm{x}}$ represents the pixel size of the slant range direction; $V_{\mathrm{X}}$, $V_{\mathrm{Y}}$, and $V_{\mathrm{Z}}$ denote the instantaneous velocity of $S$; $f$ represents the Doppler frequency associated with the radar echo and $\lambda$ denotes the wavelength of the radar.

Satellite position vector $\left(X_{\mathrm{S}},Y_{\mathrm{S}},Z_{\mathrm{S}}\right)$ and speed vector $\left(V_{\mathrm{X}}, V_{\mathrm{Y}}, V_{\mathrm{Z}}\right)$ are functions of time t, namely:

$$\begin{array}{c}\begin{array}{c}{X}_{\mathrm{S}}=X_0+V_{\mathrm{X}0}t+\frac{1}{2}{a}_{\mathrm{X}}{t}^2\\ Y_{\mathrm{S}}=Y_0+V_{Y0}t+\frac{1}{2}{a}_{\mathrm{Y}}{t}^2\\ Z_{\mathrm{S}}=Z_0+V_{\mathrm{Z}0}t+\frac{1}{2}{a}_{\mathrm{Z}}{t}^2\end{array}\\ V_{\mathrm{X}}=V_{X0}+a_{\mathrm{X}}t\\ V_{\mathrm{Y}}=V_{\mathrm{Y}0}+a_{\mathrm{Y}}t\\ V_Z=V_{Z0}+a_{\mathrm{Z}}t\end{array}\}$$

(7)

where $\left(X_0, Y_0, Z_0\right)$ is the position of the starting row,$\left(V_{\mathrm{X}0}, V_{\mathrm{Y}0}, V_{\mathrm{Z}0}\right)$ is the velocity of the starting row, and $\left(a_{\mathrm{X}},a_{\mathrm{Y}},a_{\mathrm{Z}}\right)$ is acceleration. This information can be calculated from satellite orbit data, and time t is closely related to the scan line of the SAR image, i.e.,$t=y/P_{\mathrm{RF}}$, and $P_{\mathrm{RF}}$ is the pulse repetition rate of SAR, which is a fixed known quantity. Thus, Equations (7) and (8) establish the corresponding relationship between image coordinates (x, y) and ground coordinates $\left(X_{\mathrm{S}},Y_{\mathrm{S}},Z_{\mathrm{S}}\right)$. Only x and y are unknowns. Each DEM grid point can accurately calculate the corresponding image coordinates (x, y) according to this relationship. The iterative method of Newton’s nonlinear equations is used to solve the problem.

3.1.2. SAR Image Grayscale Simulation

An SAR image is a power and amplitude image of a radio echo. For a fixed radar system, the radar backscattering coefficient ${\sigma }_0$ is the key to determining the value of the radar image, which is also related to the wavelength, polarization mode, local incident angle, complex dielectric constant of ground objects, vegetation cover, surface roughness, and other factors. At present, it is difficult to obtain an accurate radar backscattering coefficient in the imaging area. Therefore, some physical models are usually obtained based on theoretical analysis, or an empirical formula is obtained based on the ground measurement results to simulate the number of radar backscattering systems. The current theoretical analysis of physical models includes the specular reflection model, the physical optics model, the Bragg surface scattering model, the geometric optics model, and the Lambert surface scattering model. The semiempirical formula simplifies the radar backscattering coefficient into a function model of the local incident angle on the ground, as proposed by D.O. Muhleman [41]:

$$\sigma =\frac{0.0133\cos I}{{\left[\sin I+0.1\cos I\right]}^3}$$

(8)

Considering the large fluctuation of mountainous terrain, the area of the ground scattering unit is defined as the sum of the area of the scattering unit divided diagonally into two triangles, as shown in Equation (9). The size of the lookup table is the same as that of the resampled DEM, and the simulated image coordinates corresponding to geographic coordinates are stored in the following form:

$$D_N=S_{ABC}·\cos \theta +S_{ADC}·\cos \theta$$

(9)

where A, B, C, and D represent the four corners of the DEM pixels, and $\theta$ is the incidence angle of a pixel.

3.1.3. DEM Interpolation

DEM data are usually inconsistent with the resolution of simulated images. Therefore, DEM resampling is required before simulating the image. Each simulated image pixel is assigned at least once, and the following resampling factor should be satisfied:

$$f\gg f_1·f_2,f_1=\frac{{\sigma }_r}{{\sigma }_{rg}},f_2=\sqrt{2}\frac{\mathsf{\Delta }{S}_{dem}}{\mathsf{\Delta }{S}_{out}}.$$

(10)

where ${\sigma }_{rg}$ and ${\sigma }_r$ are the ground range resolution units and pixel spacing slant-range units, respectively; $\mathsf{\Delta }{S}_{dem}$ is the resolution of the DEM data; and $\mathsf{\Delta }{S}_{out}$ is the resolution of the simulated image.

3.2. Transformation of the Image Coordinate Relation

Based on DEM data and the R-D model, the mapping relationship between the coordinate space of the auxiliary DEM and the coordinate space of the simulated image is established. The mapping relationship is stored in a lookup table.

$${\left(X, Y, Z\right)}_{DEM}\leftrightarrow {\left(i,j\right)}_{SIM}$$

(11)

The RPM-L₂E algorithm is employed to match simulated and real SAR images and the corresponding association between them is established; namely,

$${\left(i,j\right)}_{SIM}\leftrightarrow {\left(i_1,j_1\right)}_{SAR}$$

(12)

Equations (12) and (13) are used to determine the relationship between real SAR image coordinates and DEM projections.

$${\left(X, Y, Z\right)}_{DEM}\leftrightarrow {\left(i_1,j_1\right)}_{SAR}$$

(13)

Based on the above ideas, the relationship between the coordinates of two real images and the projection coordinates of the same auxiliary DEM projections can be obtained. By selecting an appropriate number of obvious ground object points from DEM data as control points, the corresponding ground control points on real SAR images can be obtained through (13), and then registration can be carried out. Thus, the difficulty in selecting ground control points in mountainous areas can be solved, and the registration accuracy in mountainous areas can be improved. The mapping process of the DEM projection and real SAR image is shown in Figure 3.

4. The Results of the Experiments and Their Analysis

Two experiments dealing with the registration of SAR remote sensing images taken from the same and different sensors are conducted simultaneously to determine the performance of the suggested approach. Furthermore, the outcomes are also provided to make the comparison between the suggested approach and the others available in the literature. A computer with an Intel Core 2.33-GHz processor and 2.0 GB of physical memory is utilized to conduct the experiments.

4.1. Experimental Data

Here, two pairs of real SAR images are employed to verify the efficacy and accuracy of our approach.

Table 1. Preeminent information of the true SAR images.

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

depicts the corresponding preeminent information of the SAR images. DEM data with a resolution of 30 m are used as auxiliary data in both experiments.

4.2. The Results of the Experiments

In Experiment 2, there was a large difference in the resolution of the data obtained by the two sensors. Therefore, down-sampling of TerraSAR data was required to resolve the two images consistently. Furthermore, numerical values were assigned to parameters that correlate between the simulated and true SAR images.

When SAR images are simulated, the DEM resampling factors f should satisfy Equation (9). In Experiment 1, resampling factors were set to 4 for both images. In Experiment 2, the resampling factors of the reference and the sensed images were set to 4 and 1, respectively. Figure 4 and Figure 5 show the matching outcomes of the simulated and true SAR images in the two groups of experiments, respectively. The red lines represents false matches and the blue lines represents correct matches.

Table 2. Matching points comparison between SIFT algorithm and RPM-L₂E algorithm.

SAR Images	Method	Number of Matching Points
		Reference Image	Correct Match	Reference Image	Correct Match
Experiment 1	SIFT	15	10	16	9
	RPM-L2E	20	20	16	16
Experiment 2	SIFT	26	13	26	15
	RPM-L2E	26	26	23	23

shows the matching points of the two groups of experiments.

When compared with SIFT, the proposed approach not only attains more initial key points but also achieves a much greater level of accuracy. The reason is that the SIFT algorithm has difficulty obtaining homologous points in large topography relief areas. However, the RPM-L₂E algorithm uses the L₂E 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

The suggested approach is compared to the two other methods to further assess the performance of the algorithm. The RANSAC algorithm [42] is the first method considered for comparison, and SC [35] is the second considered method.

The results are illustrated in Figure 6 and Figure 7. The conducted experiment aims at aligning the reference images to the sensed images by warping the reference ones. The outcomes of the visualized alignment are attained by creating a checkerboard after conducting the image transformation. The displacement deviation of the registration method in this paper is much smaller than that of the other two algorithms, and it is more obvious in regions with large terrain fluctuations. It can be seen that the displacement deviation of our method is much smaller than that of the other two algorithms, especially in the high relief area. This advantage stems from the adoption of the robust RPM-L₂E 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.

A quantitative experiment is manually run on the picked landmarks whose number of them is twenty. They are distributed evenly and located at identifiable locations in the research region. Then, two performance assessment criteria—the root mean square error (RMSE) and the mutual information (MI)—are employed.

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

presents the outcomes of the quantitative assessment regarding each approach.

The proposed method is better than others suggested in the literature. When the measurements of RMSE and MI are utilized, the proposed approach attains superior outcomes, while less robustness and even failure could be observed in other approaches. In Experiment 1, the accuracy is more obviously improved, and the registration accuracy of images from different sensors is lower than that of images from the same sensor. The reason may be that the incident angles of the images in Experiment 2 are quite different, which increases the geometric distortion and radiation difference of the images, bringing certain challenges to registration. However, the performance of the proposed approach attains the best results when compared with the RANSAC and SC approaches, indicating that our approach has great robustness, generality, and ability to handle various matching problems.

5. Conclusions

Aiming at the problem that traditional SAR image registration methods have difficulty obtaining ideal SAR image registration effects in areas with large terrain fluctuations due to complex geometric and radiometric distortions, this paper proposes an approach that registers SAR images for mountainous areas. In this paper, image simulation is carried out through auxiliary DEM data, and the matching relation between simulated SAR images and real SAR images is obtained by the RPM-L₂E point set matching algorithm. The feature points are obtained indirectly, which solves the problem of finding feature points directly in mountainous areas.

The experimental results on SAR images of mountains obtained with the same sensor and different sensors present the advantages of the proposed approach when compared with other state-of-the-art methods, in terms of both qualitative effects and quantitative metrics, thus proving the applicability of the suggested method for multisource SAR image registration. Particularly in high relief areas, the features have comparatively better spatial distribution and position accuracy than those extracted by other methods for SAR images of significant geometric distortions. Moreover, the algorithm overcomes the shortcomings of poor adaptability and low accuracy of traditional SAR image registration methods for mountainous areas to some extent. This method has a certain guiding significance for SAR image registration in mountainous areas.

The suggested approach could be theoretically implemented in the registration of any SAR images. Then, the R-D model can be applied to describe the imaging geometry of the SAR images. Further research could include evaluating the algorithm when multisource time-series SAR images are registered and applying it to surface deformation monitoring and geological hazard identification in complex mountainous areas.