Residual-Sparse Fuzzy <i>C</i>-Means Clustering Incorporating Morphological Reconstruction and Wavelet Frame

Instead of directly utilizing an observed image including some outliers,\nnoise or intensity inhomogeneity, the use of its ideal value (e.g. noise-free\nimage) has a favorable impact on clustering. Hence, the accurate estimation of\nthe residual (e.g. unknown noise) between the observed image and its ideal\nvalue is an important task. To do so, we propose an $\\ell_0$\nregularization-based Fuzzy $C$-Means (FCM) algorithm incorporating a\nmorphological reconstruction operation and a tight wavelet frame transform. To\nachieve a sound trade-off between detail preservation and noise suppression,\nmorphological reconstruction is used to filter an observed image. By combining\nthe observed and filtered images, a weighted sum image is generated. Since a\ntight wavelet frame system has sparse representations of an image, it is\nemployed to decompose the weighted sum image, thus forming its corresponding\nfeature set. Taking it as data for clustering, we present an improved FCM\nalgorithm by imposing an $\\ell_0$ regularization term on the residual between\nthe feature set and its ideal value, which implies that the favorable\nestimation of the residual is obtained and the ideal value participates in\nclustering. Spatial information is also introduced into clustering since it is\nnaturally encountered in image segmentation. Furthermore, it makes the\nestimation of the residual more reliable. To further enhance the segmentation\neffects of the improved FCM algorithm, we also employ the morphological\nreconstruction to smoothen the labels generated by clustering. Finally, based\non the prototypes and smoothed labels, the segmented image is reconstructed by\nusing a tight wavelet frame reconstruction operation. Experimental results\nreported for synthetic, medical, and color images show that the proposed\nalgorithm is effective and efficient, and outperforms other algorithms.\n