## 3 Proposed Algorithm
\[\begin{array}{ll}\text{V}_{1}\big{(}\text{x}_{i},\text{y}_{j}\big{)}&=[\text {P}\big{(}\text{x}_{i+1},\text{y}_{j-1}\big{)}+\text{P}\big{(}\text{x}_{i}, \text{y}_{j-1}\big{)}+\text{P}\big{(}\text{x}_{i-1},\text{y}_{j-1}\big{)}+ \text{P}\big{(}\text{x}_{i-1},\text{y}_{j}\big{)}+\big{\{}\text{P}\big{(} \text{x}_{i-2},\text{y}_{j-2}\big{)}\] \[+M, the linear equation \(\rm E_{R}(x)\) is represented by Equation (13). In addition, taking into account only the X-axis direction for the pixel values between R and N, the linear equation \(\rm E_{LD}(x)\) is represented by Equation (14). If all gradients of Equations (12) to (14) are greater than zero, the surface characteristic is estimated as a simple inclined surface where the pixel value on the right is greater than that on the left. If all gradients of Equations (12) to (14) are less than zero, the surface characteristic is estimated as a simple inclined surface whose left pixel value is greater than the right pixel value. If all gradients of Equations (12) to (14) are zero, the surface characteristic is estimated as a flat surface.  
If the surface characteristic in the X-axis direction at location 1 is a simple inclined surface or a flat surface, the predicted value \(\rm V_{2}(x_{i},y_{i})\) at location 1 is calculated using Equation (15). Thereafter, the rounded value of \(\rm V_{2}\) is used as the predicted pixel value at location 1. If location 1 is not a simple inclined surface or flat surface, Step 1.7 is executed to check the presence of curved surface characteristics at location 1.  
\[\rm y=\left\{(y_{2}-y_{1})/(x_{2}-x_{1})\right\}(x-x_{1})+y_{1} \tag{11}\]  
\[\rm E_{L}(x)=\left\{(N-O)/(x_{i}-x_{i-1})\right\}(x-x_{i-1})+O \tag{12}\]  
\[\rm E_{R}(x)=\left\{(M-N)/(x_{i+1}-x_{i})\right\}(x-x_{i})+N \tag{13}\]  
\[\rm E_{LD}(x)=\left\{(N-R)/(x_{i}-x_{i-1})\right\}(x-x_{i-1})+R \tag{14}\]  
\[\rm V_{2}\left(x_{i},y_{j}\right)=(M+N+O+R)/4 \tag{15}\]  
1.7: The \(\rm avg(x_{i},y_{j})\), which is the average of the neighboring 11 pixels, is calculated using Equation (16). The \(\rm dev2(x_{i},y_{j})\), which is the deviation between \(\rm avg(x_{i},y_{j})\) and the neighboring pixels, is calculated using Equation (17). If \(\rm dev2(x_{i},y_{j})\) is equal to or less than the reference value (\(\varepsilon\)), Step 1.8 is executed because location 1 has a gently changing curved surface characteristic. If \(\rm dev2(x_{i},y_{j})\) is larger than \(\varepsilon\), the rounded value of \(\rm V_{1}(x_{i},y_{j})\) is used as the predicted pixel value. \[\rm avg\left(x_{i},y_{j}\right) =\left\{P\left(x_{i-2},y_{j-2}\right)+P\left(x_{i-1},y_{j-2} \right)+P\left(x_{i},y_{j-2}\right)+P\left(x_{i+1},y_{j-2}\right)+P\left(x_{i +2},y_{j-2}\right)\right. \tag{16}\] Figure 6:0 Figure 6:1  
Figure 6:2 Figure 6:3 Figure 6:4 Figure 6:5  
1.8: If the values of the neighboring pixels satisfy Equations (18) to (21), it is assumed that location 1 has simple convex surface characteristic. The predicted value \(\rm V_{3V}(x_{i},y_{j})\) at location 1 is calculated as shown in Equation (26) using Equations (22) to (25). Taking into account only the X-axis direction for the pixel values between Q and R, A and O, and E and M, the linear equations \(\rm con0[x]\), \(\rm con1[x]\), and \(\rm con3[x]\) are represented by Equations (22), (23), and (25), respectively. Similarly, taking into account only the Y-axis direction for the pixel values between C and N, the linear equation is represented by Equation (24). The rounded value of \(\rm V_{3V}(x_{i},y_{i})\) is used as the predicted pixel value. If all Equations (18) to (21) are not satisfied, Step 1.9 is executed to check the presence of simple concave surface characteristic at location 1  
Figure 6:6  
Figure 6:7  
\[\text{edge (x) }=\left\{ \begin{array}{ll}\text{strong \_edge}&\text{if (diff [x] = 0)}\\ \text{normal\_edge}&\text{if (diff [x] = 1)}\\ \text{weak\_edge}&\text{if (diff [x] = 2)}\\ \text{very\_weak\_edge}&\text{if (diff [x] = 3)}\\ 0&\text{otherwise}\end{array} \right.