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Top1. Introduction
Because of the imperfection of medical information, the field of medicine has become a very attractive domain for the application of fuzzy set theory. This is due to the large imprecision caused by the Partial Volume Effect (PVE) and the uncertainty due to noise (Behroozi & Daliri, 2012). Recently, there has been considerable interest in the use of fuzzy clustering methods for medical image segmentation, which retain more information from the original image than hard clustering methods (Vasuda & Satheesh, 2010). In hard clustering, each voxel of the image belongs to exactly one cluster or specific tissue, and a membership value of zero or one is assigned to it; in fuzzy clustering, each voxel belongs to every cluster, with its membership to each one varying between 0 and 1 (El-Melegy, Zanaty, Abd-Elhafiez & Farag, 2007). In medical images the complexity of tissue boundaries causes many voxels to contain a mixture of tissues (Links, Beach, L. Subramana et al., 1998). Fuzzy methods are better suited for this kind of situation.
In the literature on fuzzy clustering, the Fuzzy C-Means (FCM) clustering algorithm, proposed by Dunn (Dunn, 1974) and extended by Bezdek (Bezdek, J.C. 1981) is the best known and most used method. Although FCM is a useful clustering method, its membership results do not always correspond accurately to the degrees of belonging of the data, and it may be inaccurate in a noisy environment (Krishnapuram & Keller, 1993). To help address those weaknesses of FCM, methods stemming from possibilistic logic appeared.
Possibilistic logic was introduced by Zadeh (Zadeh, 1978) following its former works in fuzzy logic (Zadeh, 1965) in order to simultaneously represent imprecise and uncertain knowledge. Possibilistic logic is a weighted logic developed in the realm of artificial intelligence, with a view to develop a simple and rigorous approach to automated reasoning from uncertain or prioritized incomplete information. It enjoys properties that keep it close to classical fuzzy logic, although the introduction of weights (of different kinds) substantially increases its representation capabilities, especially for inconsistency handling (Dubois & Prade, 2004). Particularly, it has gained popularity recently in modeling and propagating uncertainty and precision in imaging applications (Pal, Pal, Keller & Bezdek, 2005).
To overcome the weakness of FCM against noise and PVE, Krishnapuram and Keller (Krishnapuram & Keller, 1993) proposed to relax the constraint of fuzziness and produce memberships that have a good explanation of the degrees of belonging for the data. They established the first Possibilistic C-Means algorithm (PCM) which used a possibilistic type of membership function to describe the degree of belonging. They showed that algorithms with possibilistic memberships are more robust to noise and outliers than FCM and its variants (Krishnapuram & Keller, 1993). However, Barni et al. (Barni, Cappellini & Mecocci, 1996) showed that the PCM algorithm is sensitive to initialization and generates coincident clusters. Timm et al. (Timm, Borgelt, Doring & Kruse, 2004) proposed then the possibilistic fuzzy clustering, and Pal et al. (Pal, Pal, Keller & Bezdek, 2005) proposed another Fuzzy Possibilistic C-Means (FPCM) that can avoid the coincident clusters of PCM while being less sensitive to noise than FCM.