# Satellite Attitude Determination and Map Projection Based on Robust Image Matching
## 3 Method
### Finding Matched Feature Points from Satellite and Base Images
#### 3.2.3 Robust Feature Matching  
To make the attitude determination more robust, we developed a screening procedure based on a RANSAC-based algorithm [8]. As mentioned in the last paragraph of Section 3.2.2, this procedure is essential to the success of the method. We will examine four variants from the RANSAC family: baseline RANSAC, M-estimator SAC (MSAC) [27], maximum likelihood estimation SAC (MLESAC) [27], and progressive SAC (PROSAC) [28]. The following is the baseline RANSAC procedure for attitude parameter estimation, which will form the basis for the other three algorithms:  
1. Randomly choose three feature pairs from the rough matching result.
2. Estimate \(\hat{\mathbf{M}}\) using three chosen feature pairs.  
Calculate the vector angles between \(\mathbf{V}_{\text{C,i}}\) and \(\hat{\mathbf{\mathbf{M}}}^{\prime}\mathbf{V}_{\text{E,i}}\) \[\theta_{i}=\cos^{-1}\bigl{(}\mathbf{V}_{\text{C,i}}\cdot\hat{\mathbf{\mathbf{M}}}^{ \prime}\mathbf{V}_{\text{E,i}}\bigr{)}\frac{180^{\circ}}{\pi}\,\] (9) and find the largest angle \(\theta_{max}\) in the three pairs (ideally \(\theta_{i}=0\)).
4. If \(\theta_{max}\) is less than a specified threshold, estimate the angles of \(\mathbf{V}_{\text{C,i}}\) and \(\hat{\mathbf{\mathbf{M}}}^{\prime}\mathbf{V}_{\text{E,i}}\) for all other feature pairs using the obtained \(\hat{\mathbf{\mathbf{M}}}^{\prime}\), and find the inliers whose vector angles are less than the threshold.
5. Repeat Steps 1-4 for a pre-specified number of times \(N_{\text{iter}}\), and find a condition that maximizes the number of inliers \(L\).
6. Recalculate \(\hat{\mathbf{\mathbf{M}}}^{\prime}\) using all inliers in the maximized \(L\) found in Step 5.  
In this study, we set the \(\theta_{\text{max}}\) threshold as \(0.2^{\circ}\), which is reasonable because the vector angles of incorrect pairs are several degrees, much larger than the empirical vector angle of correct pairs (0.02-0.05\({}^{\circ}\)). At Step 5 of the procedure, \(N_{\text{iter}}=2000\) is empirically found to be enough for our cases. An early stopping technique will be discussed in Section 5.1. The RANSAC approach should properly eliminate incorrect feature pairs, because such pairs are unrelated within the geometrical constraints; in particular, they will yield larger vector angles \(\theta\) between \(\mathbf{V}_{\text{C}}\) and \(\hat{\mathbf{\mathbf{M}}}^{\prime}\mathbf{V}_{\text{E}}\) than correct pairs. Although a threshold of \(0.2^{\circ}\) gave satisfactory results for all our experiments, an optimal threshold may vary according to the violation of the geometrical constraints (e.g., by strong camera distortions and significant land appearance changes), and it is worth investigating in future to find a safe and stable threshold.  
Among robust estimation methods, we have employed the RANSAC family because of the low inlier ratio in our problem. As shown in Section 4.1, the inlier ratio in this study can be as small as 20%. Since M-estimator approaches have their breakdown point at the 50% inlier ratio (cf. [29]), they have difficulties to find correct matched pairs and are out of choice. In contrast, RANSAC and its derivatives can find correct pairs even when the inlier ratio is less than 50%, though many iterations can be required if the inlier ratio is small. Details of the other three algorithms adopted in this paper, MSAC, MLESAC, and PROSAC, are available in Appendix A.