# Fuzzy Rough C-Mean Based Unsupervised CNN Clustering for Large-Scale Image Data
## 2 Fuzzy Rough C-Mean Based Unsupervised CNN Clustering
### Background of Fuzzy Rough C-Mean (FRCM)  
[PERSON] et al. [69] proposed a FRCM clustering algorithm, which is the development and combination of FCM [70] and RCM [37; 38]. As we know, FCM maps a membership over the range 0-1; each object belongs to some or all of the clusters to some fuzzy degree. RCM classifies the object space into three parts: lower approximation, boundary, and negative region. All the objects with RCM in the lower approximation take the same weight and all the objects in the boundary take another weighting index uniformly. In fact, the objects in the lower approximation definitely belong to a cluster, but the objects in the boundary regions belong to a cluster to some extent and have different influences on the centers and clusters, so different weighting values should be imposed on the boundary objects in computing the new centers. Inspired by the above idea, FRCM [69] integrates the advantage of fuzzyset theory and rough set theory and incorporates fuzzy membership values of each sample to the lower approximation and boundary area of a cluster. Let a set of image data \(I=\{I_{1},I_{2},\ldots,I_{n_{x}}\}\in\mathbb{R}^{d}\), where \(d\) is the dimension of the data points. Each cluster \(C_{j}\)\((j=1,2,\ldots,k)\) is regarded as a rough set. It is categorized by the lower approximations \(\underline{C_{j}}\), the upper approximations \(\overline{C_{j}}\) and the boundary area \(C_{j}^{\text{B}}=\overline{C_{j}}-\underline{C_{j}}\), respectively. Let \(c=\{c_{1},c_{2},\ldots,c_{k}\}\) be a vector composed of \(k\) centers of clusters, where \(c_{j}\in\mathbb{R}^{d}\). The objects in lower approximation belong to a cluster categorically, however the objects in the boundary regions belong to a cluster to some extent and have diverse effect on the centers and clusters, so different weighting values ought to be imposed on the boundary objects in computing the new centers. Let \(u=\{u_{i}(j)\}_{n_{x}\ x\ k}\) be a membership matrix, we can define that membership function as follows:  
\[ \begin{split} u_{ij}&=\left\{\begin{array}{c}1,\\ \frac{1}{\sum_{s=1}^{k}\left(d_{ij}/d_{sj}\right)^{2/(n-1)}}\end{array} \right.,\\ i&=1,2,\ldots,\ n_{x},\\ I_{i}&\in C_{j}\\ I_{i}&\in C_{j}^{\text{B}},\\ j&=1,2,\ldots,k\end{split}  \tag{1}\]  
The exponent \(m>1\) is utilized to change the weighting impact of membership values. The new cluster center is computing as follows  
\[c_{j}^{(l+1)}=\frac{\sum_{i=1}^{n_{x}}\left(u_{ij}^{(l)}\right)^{m}\text{I}_{i }}{\sum_{i=1}^{n_{x}}\left(u_{ij}^{(l)}\right)^{m}},\ j=1,\ 2,\ \ldots,\ k \tag{2}\]  
The objective function of FRCM is  
\[ \begin{split}\text{J}_{m}^{(l)}(u,c)&=\sum_{i=1}^{n _{x}}\sum_{j=1}^{\text{k}}\left(u_{ij}\right)^{m}\left\|I_{i}-c_{j}\right\|^{2 }\end{split}  \tag{3}\]  
The FRCM can be formulated as follows. **Input:** Unlabeled data I, number of cluster \(k\), threshold parameter T, exponent index \(m\), stop criterion \(\varepsilon\). **Output:** membership matrix \(u\), \(k\) cluster centers. **Step 0:** Let \(l=0\), initialization the centers \(\text{c}_{j}^{(l)},\ j=1,\ 2,\ \ldots,\ k\) using random sampling. **Step 1:** Assign the data objects to the approximations  
i. For a given data object \(\text{I}_{i}\) calculate its closest center \(\text{c}_{h}^{(l)}\) and \(\text{A}^{(l)}\) as follows  
\[d^{(l)}=dist\left(I_{i},\ \text{c}_{h}^{(l)}\right)=\min_{1\leq j\leq k}dist \left(I_{i},\ \text{c}_{j}^{(l)}\right) \tag{4}\]  
\[\text{A}=\left\{\forall\ s,\,s=1,2,\ldots,k,\ s\
eq\text{h}:\ \text{d}_{is}^{(l)}- \text{d}_{ih}^{(l)}\leq\text{T}\right\}\]  
ii. If \(\text{A}\
eq\varnothing\), then \(I_{i}\in\overline{\text{c}_{h}}\), \(I_{i}\in\text{C}_{s}\), and \(\text{I}_{i}\
otin\underline{C_{l}}\), \(l=1,2,\ldots,k\). If \(\text{A}\
eq\varnothing\), then \(I_{i}\in\underline{\text{c}_{h}}\), \(I_{i}\in\overline{\overline{\text{C}_{h}}}\). **Step 2:** Compute membership values using Equation (1). **Step 3:** Compute new cluster center by using Equation (2). **Step 4:** Check convergence of the algorithm.