# A Robust RANSAC-Based Planet Radius Estimation for Onboard Visual Based Navigation
## 2 Previous Works  
The problem of Autonomous Navigation Systems for spacecraft navigation has been long studied and various approaches have been proposed [3; 13]. An emerging set of solutions to this application are based on Image processing techniques to estimate the relative position between the starcraft and the target body. Such techniques can be implemented for onboard computers in modern spacecraft since they require only an image acquisition system. Currently, there is no systematic way to validate these systems, although authors usually test their methods on a simulated dataset or real images acquired by previous missions [11]. The problem of estimating the radius of a celestial body is usually divided into three phases. First, the target is isolated from the deep-space background (segmentation), then the points that lie on the target limb are filtered from the points that are inside the target and those that lie near the terminator (the line separating the illuminated face from the backlit face) [14]. The third phase is a model fitting algorithm on the selected set of points. The output is the set of parameters describing the best fitting model for the set of points. For celestial bodies such as planets and spacecraft, models are generally ellipses; however, if the oblateness (the eccentricity) of the target is negligible, ellipses can be replaced by circles that are computationally simpler to handle. In this paper, we assume a circle model fitting algorithm where the circle is fully described by their center coordinates and their radius. In [7] image differentiation is obtained with a fourth order Richardson extrapolation, then outliers are removed with box-based method and RANSAC. The elliptical model parameters are obtained with Taubin curve fitting [15] algorithm and Circular Sigmoidal Function (CSF) Least Squares model estimation. The Circular Sigmoidal Function fitting is also used in [8] where a camera-only attitude estimation algorithm is described. The algorithm approximates the radius ellipses using a computationally inexpensive Least Squares fit of sigmoidal functions that achieves high accuracy even for smooth limb transition. In [11], a pre-processing phase performs threshold segmentation, background noise filtering, and image sharpening before Canny edge detection. Pseudo-edges (points lying on the terminator) are removed selecting the edges aligned with the direction of the light source. A curve model fitting is then applied to the remaining points. A comparison of Least Squares and Levenberg-Marquardt ellipse-fitting is reported. This approach performs well on targets with different texture and in different lighting conditions; however, it relies on the knowledge of the lighting direction. In [6], the authors developed a full-disk image processing pipeline to estimate the camera-to-moonrelative position assuming a pinhole camera model. The target is segmented from background using thresholding and limb pixels are obtained using first a Canny edge detector. Background noise is removed through a noise removing filter after threshold segmentation. The method requires the direction of the light source that can be retrieved from an external sensor or inferred from further processing. As reported, the procedure has an unpredictable error in case of scarce target illumination. In [9], the authors describe a camera-based three-axis attitude sensor based on the estimation of the spacecraft-to-Moon direction. An image processing pipeline processes a single grayscale image and estimates the Moon center coordinates with a Least Squares estimator minimizing the standard deviation instead of the mean squared. This approach is reported to be less sensitive to spurious points that do not lie on the body edge. In [10], the target is segmented performing thresholding, then a morphological opening operation is applied to remove noise, secondary objects, and background stars. A rectangular Region Of Interest (ROI) is described around the target to reduce the computational demand of the next steps.