## 3 Cyst Extraction with FCM Algorithm  
The next step is FCM-based quantization [28]. The FCM algorithm is an unsupervised clustering method that has been widely used for ultrasound image analysis whereby pixels with the same features are grouped into the same cluster. The FCM-based quantization algorithm used in this paper is as follows:  
**Step 1:** Initialize the number of cluster \(c\) (\(2\leq c<n\)), exponential weight m (\(1\leq m<\infty\)), the membership degree \(u(0)\), and the error threshold (\(\epsilon\)). **Step 2:** Compute the central vector \(V_{ij}\) as Equation (8) for \(\{v_{i}\ |\ i=1,\,2,\,\ldots,\,c\}\). \[V_{ij}\ =\ \frac{\sum_{k\ =\ 1}^{n}{(U_{ik})^{m}X_{kj}}}{\sum_{k\ =\ 1}^{n}{(U_{ik})^{m}}} \tag{8}\]  
where \(X\) is the input pattern, \(i\) is the cluster index, and \(j\) is the pattern node index. \(k\) is the pattern index, \(n\) is the number of patterns, and \(U\) is the membership function. **Step 3:** Define the FCM cost function \(J\) as Equation (9) where \(d_{ik}\) is the distance between the \(k\)-th pattern \(x_{k}\) and the central vector of the \(i\)-th cluster, and \(u_{ik}\) is the membership degree of \(x_{k}\) among patterns in the \(i\)-th cluster. \[J(U_{ik},v_{i})\ =\ \sum_{i\ =\ 1}^{c}{\sum_{k\ =\ 1}^{n}{(U_{ik})^{m}(d_{ik})^{2}}} \tag{9}\]  
Figure 4: Effect of fuzzy stretching. (**a**) Input image, (**b**) fuzzy stretched. Figure 5: Effect of cubic spline interpolation. (**a**) After stretching, (**b**) after cubic spline. To minimize \(J\), \(d_{ik}\) and membership function U are defined as Equations (10) and (11), respectively. \[d_{ik}\ =\ \sqrt{\sum_{i\ =\ 1}^{l}\left(x_{kj}-v_{ij}\right)^{2}} \tag{10}\]  
\[U_{ik}\ =\ \frac{1}{\sum_{i\ =\ 1}^{C}\left(\frac{d_{ik}}{d_{ik}}\right)^{\frac{ 2}{m-1}}} \tag{11}\]  
where \(l\) is the number of pattern nodes and \(C\) is the number of clusters. **Step 4:** Compute the difference between the new and previous membership degrees (\(U_{ik}\)\((r+1)-U_{ik}(r)\)). If the difference is larger than the error threshold (\(\varepsilon\)), then go to Step 2; otherwise the algorithm stops. The effect of FCM-based quantization is shown in Figure 6.  
To remove noises and to clarify the target region effectively, we first apply an \"expansion\" operation to the image. The \"expansion\" operator expands the \"white pixel\" in size so that the bright area can be more emphasized. The brighter area is then expanded by filling adjacent neutral pixels as white and connecting these pixels to form the candidate object. The 8-directional contour tracking algorithm that is explained in detail in [28] is applied to remove such noises using masks and directional search. The object found in the search process that is less than 10% of the given image or pixels surrounded by zeros are removed as noise. The effect of the 8-directional search is shown in Figure 7.  
Connected component labeling [29] is a simple and efficient algorithm. It is applied after the image has been segmented. We call it a connected component if the pixels have a similar color and are adjacent to each other. Every connected component in the image is labeled uniquely. The ultrasound image is represented by intensity levels from 0 to 255 in grayscale. From this image, we can differentiate bones, tissues, and fluid. Fluid is dark, tissue is gray, and bone is bright. Because of the segmentation, the ultrasound image, which originally  
Figure 6: Effect of FCM-based quantization. (**a**) Region of interest, (**b**) FCM quantization. Figure 7: Effect of 8-directional search. (**a**) Quantized result, (**b**) candidate cyst. has a 0-255 range of grayscale, turns into an image that has \(k\) groups of grayscales (\(k=4\) in this paper). Since the ganglion cyst contains a fluid that is darker in the image, the darkest intensity group is set as foreground color while others are determined as background color. Labeled objects that are too small or too big are removed as noise.