Size sensitivity is one of the primary disadvantages of fuzzy c-means (FCM). Existing improvements assume that clusters with more samples are larger than clusters with less samples in terms of the count of the cluster samples. An obvious counter-example is that samples with the same spatial coordinates do not increase territory of the cluster they belong to. Given a spatial transformation, we first define the territory size of each cluster in the space. We then propose a distance normalized FCM (DFCM) where distances of samples to each cluster centre are self-adaptively adjusted based on the territory size of the cluster to prevent cluster centres deviating from smaller clusters to adjacent larger clusters. In addition, FCM is taken as a balance term, which is gradually weakening during the iteration, to the objective function of DFCM to guide DFCM from being trapped by a local optimal solution. We specialize the spatial transformation as kernel functions to address the non-linear separability problem in the Hilbert space. As kernel selection is still an open problem, we propose a homogeneous and an inhomogeneous sample partitions to construct an undirected graph and specialize DFCM in the graph space. We finally evaluate and compare DFCM and its specializations with 12 FCM based methods on 6 datasets in terms of 8 metrics. The performance generally improves by 2%–5%. Initialization sensitivity, parameter effects and settings, robustness to noise and bias field, limitations, and future works are also discussed. Results show that the proposed method is more robust than competitors.
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