Adaptive Fuzzy c-Shells Clustering and Detection of Ellipses

Several generalizations of the fuzzy c-shells (FCS) algorithm are presented for characterizing and detecting clusters that are hyperellipsoidal shells. An earlier generalization, called the adaptive fuzzy c-shells (AFCS) algorithm, is examined in detail, and is found to have global convergence problems when the shapes to be detected are partial. New formulations are considered wherein the norm inducing matrix in the distance metric is unconstrained in contrast to the AFCS algorithm. The resulting algorithm, called the AFCS-U algorithm, is shown to perform better for partial shapes. Another formulation is also considered, which is based on the second-order quadrics equation, resulting in a linear system of equations. These algorithms are shown to be able to detect ellipses and circles in two—dimensional data. Their performance is compared with the Hough transform (HT) based methods for ellipse detection. Existing HT-based methods for ellipse detection are evaluated, and a multistage method incorporating the good features of all the methods is used for comparison. It is demonstrated through numerical examples of real image data that the AFCS algorithm requires less memory than the HT based methods, and it is at least an order of magnitude faster than the HT approach. It is also shown that the use of fuzzy memberships improves the ability to attain global optima compared with the use of hard memberships.