# Automatic Matching of Multi-Source Satellite Images: A Case Study on ZY-1-02C and ETM+
## 2 Methods
### Initial Matching Point Detection
#### 2.1.1 Estimation of Variation in Rotation and Scale  
To address the problem of rotation and scale variances between images, establishing a uniform space reference is crucial. Image point pairs that correspond to the same ground tie point can be determined with RPC in the ZY-1-02C imagery and geographic information in the DOMs of ETM+ and SRTM, which were first detected by Harris detectors. As shown in Figure 2, a uniform image coordinate system was adopted. The coordinate system of one image was selected as the standard so variances in rotation, scale, and translation would be reflected in the coordinates of the corresponding points.  
Figure 3 shows the model for transforming one image space to another image space. There are four steps in this process. Step 1: Detect feature points on the DOM of ETM+ by Harris detector. Step 2: Use geographic information to find the ground tie point (X, Y, Z). The X and Y are found in the DOM, and Z is found in the SRTM. Step 3: Use the RPC to find the ground tie point (X, Y, Z) on the ZY1-02C imagery with Equation (1). Step 4: The feature points create a line, where DETM+ and DZY-102C represent the length of the line in ETM+ and ZY-1-02C imagery. Contrast the length of the line and the angle between the lines by using Equation (2).  
\[\left\{\begin{array}{l}r_{n}=\frac{P_{1}(X_{n},Y_{n},Z_{n})}{P_{2}(X_{n},Y_ {n},Z_{n})}\\ c_{n}=\frac{P_{1}(X_{n},Y_{n},Z_{n})}{P_{2}(X_{n},Y_{n},Z_{n})}\end{array}\right. \tag{1}\]  
Figure 2: Transformation between images. Here, \((x_{i},y_{1})\) and \((x_{r},y_{r})\) are the normalized coordinates of the two images, \(A\) is the rotation angle, and \(k\) is a scale parameter.  
where \((r_{n},c_{n})\) and \((X_{n},Y_{n},Z_{n})\) are the normalized coordinates of the image space and the ground space points, respectively, and \(P_{i}(i=1,2,3,4)\) are the cubic polynomials that represent the rigorous geometric sensor model of satellite imagery.  
The rotation angles of the lines were used to obtain the value of A, and the lengths of the lines were used to obtain the value of k, as shown in Figure 2. The following equation represents a simple and effective model for determining the relationship between the two images. To calculate the model in Equation (2), the RPCs and the geographic information of the DOMs play important roles. As shown in Figures 2 and 3, two or more pairs of image points should be used to connect the lines between two image spaces. These statistical points can calculate the value of \(A\) and \(k\), which is the preparation work for image matching.  
\[\left\{\begin{array}{ll}A=&\frac{\sum(A_{1}-A_{r})_{n}}{\eta}\\ k=&\frac{\sum(D_{1}/D)_{n}}{n}\end{array}\right. \tag{2}\]  
where \(A_{1}\) and \(A_{r}\) are the rotation angles of the lines in the two images, \(D_{1}\) and \(D_{r}\) are the lengths of the lines in the two images, \(A\) is the rotation angle, \(k\) is a scale parameter, and \(n\) is the number of statistical samples.  
Owing to the inconsistent positioning precision, these point pairs may be non-correspondent or far away. However, the relative relationship in ZY-1-02C imagery space between the point pairs tends to be consistent, regardless of whether the geographical positioning is accurate or not. For instance, the distances and angles of lines between two groups of point pairs could highlight the rotation and scale variances between images. In this study, the lengths and angles of lines, consisting of image point pairs with identical geographic coordinates, were calculated to determine the relationship between images in terms of rotation and scale.  
Figure 3: Model of tie point between ground space and image space.