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