Fuzzy c-means (FCM) has been considered as an effective algorithm for image segmentation. However, it still suffers from two problems: one is insufficient robustness to image noise, and the other is the Euclidean distance in FCM, which is sensitive to outliers. In this paper, we propose two new algorithms, generalized FCM (GFCM) and hierarchical FCM (HFCM), to solve these two problems. Traditional FCM can be considered as a linear combination of membership and distance from the expression of its mathematical formula. GFCM is generated by applying generalized mean on these two items. We impose generalized mean on membership to incorporate local spatial information and cluster information, and on distance function to incorporate local spatial information and image intensity value. Thus, our GFCM is more robust to image noise with the spatial constraints: the generalized mean. To solve the second problem caused by Euclidean distance (l2 norm), we introduce a more flexibility function which considers the distance function itself as a sub-FCM. Furthermore, the sub-FCM distance function in HFCM is general and flexible enough to deal with non-Euclidean data. Finally, we combine these two algorithms to introduce a new generalized hierarchical FCM (GHFCM). Experimental results demonstrate the improved robustness and effectiveness of the proposed algorithm.

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