Optimal consensus set and preimage of 4-connected circles in a noisy environment

This paper exploits the problem of fitting special forms of annuli that correspond to 4-connected digital circles to a given set of points in 2D images in the presence of noise by maximizing the number of inliers, namely the consensus set. We prove that the optimal solutions can be described by solutions with three points on the annulus boundary. These solutions correspond to vertices of the preimage of the annulus in the parameter space thus allowing us to build the preimage and to enumerate all the optimal solutions.