## 1 Introduction  
Synthetic Aperture Radar (SAR) is an advanced microwave equipment of earth observation and has obtained much more interests for it working at all-weather and all-time, strong permeability, multi-bands and polarization information [1]. Unfortunately, the images obtained by SAR are generally degenerated by speckle noise because of its distinct and special imaging mechanism. The existence of speckle usually leads to the degradation of image quality and has a directly impact on the SAR image recognition and interpretation. Furthermore, another important issue in SAR image understanding is how to efficiently distinguish different land covers in this type of images. That is to say, the fast and correctly partition or recognition of massive overlapped pixels into different local regions. In generally, the above two issues are two main crucial points in SAR image segmentation. Consequently, an efficient SAR image segmentation methodology should consider these two issues simultaneously. The theory of fuzzy set is very suitable to SAR image segmentation because the pixel is the returning signal of radar waves reflected from many elementary scatterers within a resolution cell. Nevertheless, the resolution cell usually is denoted by an area of the land space, which may has regions of different characteristics. Thus, one could try to discover the uncertainty by the fuzzy set theory. Among the fuzzy clustering methods, fuzzy c-means (FCM) and its variants have been developed very popular in the recent years. To compensate the shortcomings of FCM, two important aspects are usually incorporated to the original FCM, which are local spatial constraint and local gray-level relationship. [PERSON] _et al._[2] introduced a neighboring item into the objective function to bias the solution toward piecewise-homogeneous regions. The neighboring item is calculated by the neighboring pixels variation to its cluster center. [PERSON] and [PERSON] [3] used the mean or median of neighboring pixels lying within local window to replace the pixels in the local window to simply its computation. [PERSON] _et al._[4] proposed an efficient local similarity measure by fusing local spatial and gray information for medical image segmentation. Besides, the segmenting time is only dependent on the number of the gray-levels by histogram statistics. Another two recent important literatures of FCM are [PERSON] and [PERSON] [5], and [PERSON] _et al._[6]. In [5,6], they present two fuzzy local neighborhood factors incorporating the spatial and gray level relationship. Additionally, the kernel distance measure is introduced into the objective function to improve its robustness to noise or outliers. Overall, considering above variants of FCM, some common features in these algorithms can be found. They use only one clustering index to guide their searching process, which may be very appropriate for particular data distribution of clusters. Hence, it is natural to consider multiple objective functions and optimize them simultaneously. Besides, the SAR images with multiplicative speckle noises are seldom involved in these algorithms. Another important issue for SAR image segmentation lies in its vast pixels with overlapping characteristic. Taking the typical small SAR image (512 \(\times\) 512) as an example, it consists of 262,144 pixels, which is great amount of samples for the traditional clustering methodologies. To this end,the watershed transformation (WT) is utilized to preliminarily partition the image into many local non-overlapping patches or super-pixels. We can see WT merges great amount of local pixels with similar intensity. Here, a gradient-based water-shed transformation (GWT) and morphological operator [7] are employed here. Currently, many similar algorithms can be used for initial segmentation or segmentation, such as mean shift based gradient vector flow [8, 9], anisotropic mean shift based fuzzy c-means segmentation [10]. Here, GWT is used for merge the local similar pixels though it may create many over-segmented results.