A simple least squares method for fitting of ellipses and circles depends on border points of a two-tone image and their 3-D extensions

Fitting circles and ellipses of an object is a problem that arises in many application areas, e.g. target detection, shape analysis and biomedical image analysis. In the past, algorithms have been proposed, which fit circles and ellipses in some least squares sense without minimizing the geometric distance to the given points. In this paper, the problem of fitting circle or ellipse to an object in 2-D as well as the problem of fitting sphere, spheroid or ellipsoid to an object in 3-D have been considered. The proposed algorithm depends on the border points of the object. Here, assume that the center of the ellipse or circle coincides with the centroid of all border points of the object. The major and minor axes of the ellipse are presented by least sum perpendicular distance of all border points of the object. The main concept is that the border points satisfy the equation of conic. On the basis of this concept, all the border points of the object will generate an error function (algebraic function) and the other parameters of the conic are estimated by minimizing this error function. The extension of this idea in 3-D for fitting sphere, spheroid and ellipsoid are proposed.