# A Robust RANSAC-Based Planet Radius Estimation for Onboard Visual Based Navigation
## 4 Results  
To validate our method, we performed a set of experiments on a dataset of 50 synthetic 1024 \(\times\) 1024 16-bit grayscale images of celestial bodies in different lighting conditions and positions, including partial frame occlusions (see Figure 5). The dataset has been generated with opensource software Celestia v1.6.1. We generated images of celestial bodies taken at various random distances, with the radius of the celestial body ranging from a minimum of 5 pixels to a maximum of 1024 pixels. We evaluated the accuracy of our method on the original dataset by performing a repeated radius estimation for each image. We reported the mean value and standard deviation for the target center and radius. For brevity, we present only a subset of relevant samples in Table 1. In Figure 6 we display qualitative results for some image samples. Figure 4: Complete image processing pipeline. **(1)** Input image corrupted with additive Gaussian noise; **(2)** binary edge mask resulting from Equation (3); **(3)** binary edge mask applied to result of edge detection, compared to (2) the resulting image contains the magnitude of the image gradient as floating point values. **(4)** Point cloud step. Points without sector binning are in black, dataset with sector binning is in blue and the set of inliers returned by RANSAC is in orange. **(5)** Final radius estimation with inliers. We compared the radius and center estimation obtained by our method and Least Squares Estimation for circle identification. The effect of scarce illumination is particularly relevant for the Least Squares Estimator (LSE), whereas our RANSAC-based approach is able to provide better results for the same lighting conditions. For each image, we repeated the estimation 1000 times and reported mean value and standard deviation for the radius estimate and center coordinates. We performed the same experiment using LSE. It can be seen that LSE has a lower standard deviation than RANSAC for noise-free images; however, when we introduce noise LSE performance rapidly degrades. We performed a qualitative analysis to estimate the sensitivity of different sources of noise in the input image. We considered additive Gaussian noise with an increasing level of intensity, to simulate channel disturbances. We also reproduced frame errors and data packet drop by altering and removing random image scanlines, see Figures 7 and 8. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Filename** & \(\mathbf{err_{x}}\) & \(\mathbf{err_{y}}\) & \(\mathbf{err_{r}}\) \\ \hline
0001 & 0.074 \(\pm\) 0.050 & 0.08  
We altered the original image with an additive Gaussian noise as shown in Equation (3), where \(\mathcal{N}(0,\sigma^{2})\) is a zero-mean Gaussian distribution with \(\sigma^{2}\) variance (see Figure 9). We set the \(\sigma^{2}\) variance to be equal to \(SNR^{-1}\). \[I_{ij}=J_{ij}+\mathcal{N}(0,\sigma^{2}) \tag{3}\]  
Figure 8: Output estimation for images corrupted by various sources of noise. Figure 7: Images altered by various sources of noise. In Figure 9, we reported the effect of additive Gaussian noise on the model parameter for increasing value of SNR. We compared the LSE model fitting to RANSAC. For \(SNR>2\) (i.e., Gaussian noise with variance 0.5), RANSAC outperforms LSE significantly. We performed a similar analysis introducing local frame distortions (see Figure 10). The distortion value is the local translation of a random scanline by a certain amount of pixels. We reported the effect of frame distortion errors in Figure 10. In general, RANSAC provides better results than LSE. However, it can be noted that the estimation error of the \(x\) coordinate is higher for RANSAC. The higher error in the x coordinate is related to the horizontal scanline translation. We simulated transmission errors by randomly dropping scanlines (see Figure 11). In Figure 11, we evaluated the effect of random scanline drop by comparing the estimation errors for RANSAC and LSE. Figure 10: (a) Effect on parameter estimation for input image altered by local frame distortion.