Fuzzy One-Mean Algorithm: Formulation,Convergence Analysis,and Applications

From the clustering point of view, the fuzzy one-mean (FOM) algorithm can be considered to be an extension of the fuzzy c-means (FCM) algorithm to one-class clustering. Given N scalar input values, the FOM algorithm yields the one-mean value bounded by the arithmetic-mean and the median values. The optimal one-mean value is determined by the exponent value in the FOM algorithm. Fuzzy one-mean filtering (FOMF) is the convolution of the FOM algorithm with signal samples. According to signal processing theory, arithmetic-mean filters are optimal in smoothing white gaussian noise, while median filters are optimal in rejecting outliers. A FOMF can therefore be taken as a soft outlier-rejection filter. Indeed, a FOMF can to a certain degree both smooth white gaussian noise and reject outliers. An analysis of the convergence behavior of the FOM algorithm is presented. The soft outlier-rejection feature of FOM algorithm is utilized to formulate a special filter referred to as the FOM derivative filter (FOM-DF) which functions as an edge detector and simultaneously as an outlier-rejection filter.