SIViP DOI 10.1007/s11760-016-0992-4 ORIGINAL PAPER C-means clustering fuzzified by two membership relative entropy functions approach incorporating local data information for noisy image segmentation R. R. Gharieb 1 · G. Gendy 2 · A. Abdelfattah 1 Received: 21 September 2015 / Revised: 22 September 2016 / Accepted: 26 September 2016 © Springer-Verlag London 2016 Abstract In this paper, C-means algorithm is fuzzified and regularized by incorporating both local data and membership information. The local membership information is incorpo- rated via two membership relative entropy (MRE) functions. These MRE functions measure the information proximity of the membership function of each pixel to the member- ship average in the immediate spatial neighborhood. Then minimizing these MRE functions pushes the membership function of a pixel toward its average in the pixel vicin- ity. The resulting algorithm is called the Local Membership Relative Entropy based FCM (LMREFCM). The local data information is incorporated into the LMREFCM algorithm by adding to the standard distance a weighted distance com- puted from the locally smoothed data. The final resulting algorithm, called the Local Data and Membership Rela- tive Entropy based FCM (LDMREFCM), assigns a pixel to the cluster more likely existing in its immediate neighbor- hoods. This provides noise immunity and results in clustered images with piecewise homogeneous regions. Simulation results of segmentation of synthetic and real-world noisy images are presented to compare the performance of the proposed LMREFCM and LDMREFCM algorithms with several FCM-related algorithms. Keywords Noisy image segmentation · FCM algorithm · Local spatial information · Membership relative entropy B R. R. Gharieb rrgharieb@gmail.com 1 Faculty of Engineering, Assiut University, Assiut 71516, Egypt 2 Al Rajhy Liver Hospital, Assiut University, Assiut, Egypt 1 Introduction Image segmentation is a principle process in many image, video, scene analysis and computer vision applications [1]. It is the process of assigning a label to every pixel in an image such that pixels with the same label share certain char- acteristics [2]. Several image segmentation methods have been developed but still no satisfactory performance attained especially in noisy images [2–7], which makes development of segmentation algorithms to handle noise an active area of research. The existing segmentation algorithms can be categorized into threshold-based, region-based and edge- based, probabilistic-based, artificial neural-network-based and clustering-based methods [2–5]. Metaheuristic algo- rithms such as artificial bee colony (ABC) and genetic-based fuzzy ones have been used for segmentation to add diversity to the algorithms [6, 7]. Clustering and fuzzy-based clustering techniques have been widely adopted by many researchers since clustering needs no training examples [8–12]. C-means clustering algorithm is unsupervised approach in which data are basically partitioned based on locations and distance between various data points. Partitioning the data into C-clusters is carried out by compacting data in the same clusters and separating data in different ones. C-means clus- tering provides crisp segmentation which does not take into account fine details of infrastructure such as hybridization or mixing of data [13]. Fuzzy C-means (FCM) is one of the methods widely used for image segmentation. FCM’s success is chiefly attributed to the introduction of fuzzy sets and membership of belong- ing [14, 15]. Compared with the C-means algorithm which yields hard or crisp segmentation, the FCM one is able to provide soft one by incorporating membership degree of belonging [16]. However, one disadvantage of the standard FCM is not considering any spatial or local information 123