Fuzzy ellipsoidal-shell clustering algorithm and detection of elliptical shapes

Fuzzy c-Ellipsoidal Shell (FCES) algorithm that utilizes hyper-ellipsoidal-shells as cluster prototypes is proposed. FCES is a generalization of the Fuzzy Shell Clustering (FSC) algorithm. The generalization is achieved by allowing the distances measured through a norm inducing matrix that is symmetric, positive definite. In case of fixed, known norms, the extension of FCS to FCES is straightforward. Two different strategies are recommended when the norm is unknown. The first strategy considers use of non-linear least-squared fit approach with fuzzy memberships as weights. The second approach considers norm inducing matrix as a variable of optimization, thus making FCES an adaptive norm type algorithm. An adaptive norm theorem is presented. The results of first approach is used to detect ellipses having unequal sizes and orientations in two-dimensional data-sets. Non-linear equations of the FCES algorithm are more complex than those of the FSC algorithm. Numerical issues related to both the FCES algorithm and the FSC algorithm are discussed.