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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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The low spatial resolution of hyperspectral images leads to the coexistence of multiple ground objects in a single pixel (called mixed pixels). A large number of mixed pixels in a hyperspectral image hinders the subsequent analysis and application of the image. In order to solve this problem, a novel sparse unmixing method, which considers highly similar patches in nonlocal regions of a hyperspectral image, is proposed in this article. This method exploits spectral correlation by using collaborative sparsity regularization and spatial information by employing total variation and weighted nonlocal low-rank tensor regularization. To effectively utilize the tensor decomposition, nonlocal similar patches are first grouped together. Then, these nonlocal patches are stacked to form a patch group tensor. Finally, weighted low-rank tensor regularization is enforced to constrain the patch group to obtain an estimated low-rank abundance image. Experiments on simulated and real hyperspectral datasets validated the superiority of the proposed method in better maintaining fine details and obtaining better unmixing results.