## 2 Appendix Segmentation with Double-Layered Outlier Rejection FCM
### Double-Layered Outlier Rejection FCM (DORFCM)
#### 2.3.2 Layer-2 Learning (Step 5 to Step 7)  
**Step 5:** With output of the first layer ORFCM as the input of the second layer, compute the value of central vector \(w_{si}\) as shown in Equation (6) for \(\{w_{si}\ \ \text{$s$}\ =1,\,2,\,\ldots,\,cc\}\) where \(cc\) is the number of clusters for the second layer. \[w_{si}^{(t)}=\frac{\sum_{s=1}^{cc}(q_{si})^{m}u_{ik}}{\sum_{i=1}^{cc}(q_{si})^{ m}} \tag{6}\]  
**Step 6:** Compute the membership degree of the clusters \(q_{si}\) in the second layer with distance function \(d_{sk}\) as Equations (7) and (8) respectively. \(p_{s}\) is the intensity in a cluster of layer-2. \[{q_{si}}^{(t+1)}=\frac{1}{\sum_{i=1}^{cc}\left(\frac{\beta\|q_{s}-u_{ik}(t)\| }{\beta\|q_{i}-u_{ik}(t)\|}\right)^{\frac{2}{(m-1)}}} \tag{7}\]  
\[\beta=\frac{p_{s}+0.01}{256} \tag{8}\]  
**Step 7:** Compute the difference between the new membership and the previous membership degree. If the difference is less than the final error threshold (\(\varepsilon^{\prime}\)), then the algorithm terminates otherwise go to Step 5. During the second layer learning, the value of \(\beta\) is computed differently from the first layer learning since values of numerator and denominator are the results of first layer clustering, they should not exceed 1. The flow of the proposed method can be illustrated as shown in Figure 2 and the Pascal-like pseudo code is given as Algorithm 1. Figure 2: DORFCM flow chart. ``` ProcedureDORFCM(x) returns prototypes and partition matrix Input data x = {x}, x2,..., x3  Local fuzzification parameter: m threshold: \(\varepsilon\), norm: \(\parallel\parallel\)  INITALZE-PARTITION-MATRIX \(\text{g }\leftarrow\text{0,t }\gets\text{0}\) repeat fori = 1 ccdo \(v_{ij}(g)\leftarrow\frac{\sum_{k=1}^{n}(U_{k})^{n}u_{kj}}{\sum_{k=1}^{n}(U_{k} )^{n}}\) compute prototypes fori = 1



FCM is the weakest among four methods and Figure 4 shows the case that only FCM failed to locate inflamed appendix. FCM tends to underestimate the target area. Double-layered learning structure with FCM (Figure 4c) shows better quality than single FCM learning. DFCM showed better performance than FCM to avoid local maximum in clustering. However, the outlier effect can mislead other methods, but DORFCM as one is verified in Figure 5. Two methods without outlier treatment failed to locate the right region and DFCM looks like it suffered by false positive and tends to overestimate when it fails. The standard ORFCM showed better result than FCM but worse than double-layered algorithms in inflamed appendix detection accuracy as shown in Table 1, but it searches different areas from FCM with outlier rejection strategy. However, our DORFCM showed better quality than single ORFCM and standard DFCM in cluster quality as shown in Figure 6.

Figure 4: FCM failed to locate the right area. (**a**) Input, (**b**) FCM, (**c**) DFCM, and (**d**) DORFCM. Figure 5: Only DORFCM succeeds. (**a**) Input, (**b**) FCM, (**c**) DFCM, and (**d**) DORFCM. Figure 3: Extraction of inflamed appendix by methods. (**a**) Input, (**b**) FCM, (**c**) DFCM, and (**d**) DORFCM. Thus, we provide the second performance index is to compare the qualitative performance between DFCM, ORFCM, and DORFCM with respect to the human expert's area marking as the ground truth. True positive (TP) is the set of pixels that both human and the software agrees that they belong to the inflamed appendix, false positive (FP) is the set of pixels that only the software thinks they belong to the inflamed appendix, and false negative (FN) is the set of pixels that only human expert classifies as the inflamed appendix. The performance index is as following. Precision = TP/(TP + FP)

Recall or Sensitivity = TP/(TP + FN)

Accuracy = (TP + TN)/all area

Table 2 summarizes the performance comparison between DFCM and DORFCM among 91 cases that both methods locate the target object successfully. As one can see, the outlier rejection effect is significant.