# Convolved Feature Vector Based Adaptive Fuzzy Filter for Image De-Noising
## 4 Proposed Methodology: Convolved Feature Vector Based Fuzzy Filter
### Noise Detection Mechanism
#### 4.1.2 Fuzzy Based Contextual Model  
[PERSON] et al. in [38] used nine overlapping sub-windows as shown in Figure 3 for basic processing of noise estimation. In case of low and medium noise, median values of majority of the sub-windows remain unchanged and do not contribute enough in noise estimation process except the limited number. In case of high noise, the probability of lack of texture problem increases in most of the sub-windows. So, the use of all the nine sub-windows increases the computational cost as well as reduces the noise detection capability of the noise detector. The lack of texture problem usually arises when the number of noisy pixels are in large number than half of the size of the kernel and the median value which is now a noisy one replaces the central pixel value of the kernel, resulting in poor detail preservation. Figure 2: (**a**) Shows a \(5\times 5\) conceptual window formed by padding additional rows and columns to original \(3\times 3\) image window. (**b**) Shows a divide and conquer strategy mechanism: division of a \(5\times 5\) kernel to nine (09) subsets of \(3\times 3\) size kernels with new central locations \(x^{\prime}_{i,j}\). (**c**) Shows a new contextual model for noise detection. Figure 3: A large window of size \(5\times 5\) divided into 9 overlapping sub-windows of size \(3\times 3\) and four edge directions \(d_{1},\ d_{2},d_{3},\ d_{4}\)[38]. The problem described above can be addressed by selecting limited number of sub-windows/kernels which ultimately help the noise detector for better decision making at noise detection level. In this paper, we have used a simple mechanism for kernel selection, i.e., kernels lying at corners as shown in Figure 2b are selected as potential candidates as these kernels have large number of non-overlapping pixels. The selected kernels reduce the computational cost as well as the lack of texture problem at high noise. These kernels act like quadrants of a Cartesian plane as shown in Figure 2c. Let \(W=\left\{x_{i-2j-2},\ldots,\ x_{i+2j+2}\right\}\) be a kernel of size \(5\times 5\) centered at (\(i\),\(j\)) containing twenty five (25) pixel locations. After division and kernel selection, let \(Q_{k}^{W},\ k=1,\ 2,\ 3,\ 4\), be the four subsets of \(W\) with new central location at \(x_{i,j}^{\prime}\). The pixel locations in four subsets/quadrants are as follows,  
\[Q_{1}^{W}=\left\{x_{i+l,j+m}\ :\ 0\leq l\leq 2,\ 0\leq m\leq 2\right\}\]  
\[Q_{2}^{W}=\left\{x_{i+l,j+m}:-2\leq l\leq 0,\ 0\leq m\leq 2\right\}\]  
\[Q_{3}^{W}=\left\{x_{i+l,j+m}:-2\leq l\leq 0,-2\leq m\leq 0\right\}\]  
\[Q_{4}^{W}=\left\{x_{i+l,j+m}\ :\ 0\leq l\leq 2,-2\leq m\leq 0\right\}\]  
Median filter defined in Equation (2) can be used to extract the features from the above four quadrants centered at new locations \(x_{i,j}^{\prime}\) as follows,  
\[M_{k}=MED\ \left\{Q_{k}^{W}\right\},k=1,2,3,4, \tag{5}\]  
where \(M_{k}\) is a feature set of pixel locations in four quadrants namely top right, top left, bottom left and bottom right kernels, respectively. Sort the feature set values \(M_{k},\ k=1,2,3,4\), in descending order as \(m_{1}\leq m_{2}\leq m_{3}\leq m_{4}\), where \(m_{1}\) and \(m_{4}\) are the lowest and highest feature values, respectively. \[\overset{\rightarrow}{V}=\left(m_{k}\right),\ k=1,2,3,4. \tag{6}\]  
\[D^{1}=\sum_{2}^{4}\left(m_{i}-m_{i-1}\right) \tag{7}\]  
\(D^{1}\) is the rank order difference [19] of vector \(\overset{\rightarrow}{V}\) given in (6). An edge pixel can be separated from noisy pixel if its directional rank-ordered absolute difference (DROAD) [25] in any direction is minimum. For example, after division strategy when the central location of small kernel (subset of large kernel) shifts from \(x_{i,j}\) to a new central location \(x_{i,j}^{\prime}\) the new and the previous location lie in a certain direction, which should be minimum in case of edge pixel.