Position and radius of spheres from single off-axis catadioptric images

In this paper we address the problem of sphere localization from a single image taken with a noncentral catadioptric camera. We propose a method for determining both the radius and the position of an unknown sphere from a single, catadioptric image. The method can find its application in the field of robotic vision, especially in mobile robots playing soccer in RoboCup contests, in order to improve robot capabilities related to playing with a flying ball. Recently, a method for sphere reconstruction from single image taken with a noncentral, axial-symmetric, catadioptric camera has been proposed. In an axial symmetric catadioptric cameras, the pinhole of the camera is placed on the mirror axis. Though axial symmetric cameras help to simplify the geometrical treatment of the problem, they are difficult to set up since they require a precise alignment, usually hard to check. In this paper we deal with the general case of off-axis catadioptric cameras, with the camera pinhole placed in a general position w.r.t. the mirror. We devise a simple geometrical method by which we determine both the position of a sphere and its radius from its apparent image contour. Since our approach is based on coplanar viewing rays, it has a wider applicability w.r.t. the previous method, as it relaxes the constraint on camera position, i.e. it does not require a precise alignment, and the constraint on mirror axial symmetry, i.e. it can be applied to a wider class of mirrors. Some preliminary experiments both on simulated and real image are also presented.