An Exact Robust Method to Localize a Known Sphere by Means of One Image

In this article we provide a very robust algorithm to compute the position of the center of a sphere with known radius from one image by a calibrated camera. To our knowledge it is the first time that an exact sphere localization formula is published that only uses the (pixel) area and the ellipse center of the sphere image. Other authors either derived an approximation formula or followed the less robust and more time consuming procedure of fitting an ellipse through the detected edge pixels. Our method is analytic and deterministic, making use of the unique positive real tool of a cubic equation. We observe that the proposed area method is significantly more accurate and precise than an ellipse fitting method. Furthermore, we investigate in what conditions for sphere images the proposed exact method is preferable to the robust approximation method. These observations are validated by virtual, synthetic and real experiments.