## 7 Edge Detection  
As another case study we are presenting image edge detection with IT2 fuzzy systems. Edge detection is a recent area of application with great results for IT2 fuzzy logic, and for example the works in [26; 27; 28]. The IT2 FIS can be used to find if a pixel is or not an edge, and is necessary to consider the pixel's neighborhood as shown in Figure 9.  
This is because the pixel's gradient is obtained as shown in Equation (28), this strategy, using the gradients is used for example in [6] and produces a good result. Figure 8: Average error comparison of the type-reduction algorithms. \begin{table}
\begin{tabular}{l l l l} \hline \hline  & KM Type Reduction & Approximate CEKM Type Reduction \\ \hline  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Performance comparison. \[ \begin{array}{l}D1=|I2-I8|\\ D2=|I4-I6|\\ D3=|I3-I7|\\ D4=|I1-I9|\end{array}  \tag{28}\]  
The gradients are approximated in this case as the differences in absolute values and these are the inputs to the IT2 FIS, the values are from 0 to 255, and the fuzzy sets can be found in Figure 10.  
The MFs distribution for the fuzzy sets are the same for each input, and are parameterized as proposed in [6] allowing adaptation to different contexts. The proposed fuzzy sets are formed by two type-2 fuzzy membership functions as is shown in Figure 11.  
Figure 11: Output Fuzzy set. Figure 9: Neighborhood of a particular pixel. As can be observed, the output fuzzy set is a composed function. * Step 1 Equations (29) and (30) define the partial knowledge of the system. \[\overline{\mu}B(z)=\left\{ \begin{array}{rl}&2\overline{\theta}_{N}(127-z),\;0< z<127\\ &2\overline{\theta}_{E}(z-127),\;127<z<255\\ &0,\;otherwise\end{array} \right.\] (29) \[\underline{\mu}B(z)=\left\{ \begin{array}{rl}&2\underline{\theta}_{N}(117-z), \;0<z<117\\ &2\underline{\theta}_{E}(z-117),\;137<z<255\\ &0,\;otherwise\end{array} \right.\] (30)
* Step 2 Defines the output membership functions intervals, and these domains are reported in Table 10.
* Step 3 In this case, study, step 3 is not necessary because the MFs do not have intersections. * Step 4 Tables 11 and 12 show the integral of each interval for the output MFs. \begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Upper MFs**} \\ \cline{2-5}  & **Interval** & **Partial Knowledge** & **Interval** & **Partial Knowledge** \\ \hline
1 & \(0<z<127\) & \(2\overline{\theta}_{N}(127-z)\) & \(0<z<117\) & \(2\underline{\theta}_{N}(117-z)\) \\
2 & \(127<z<255\) & \(2\overline{\theta}_{E}(z-127)\) & \(137<z<255\) & \(2\underline{\theta}_{E}(z-117)\) \\ \hline \hline \end{tabular} 
* Step 3 In this case, study, step 3 is not necessary because the MFs do not have intersections. * Step 4 Tables 11 and 12 show the integral of each interval for the output MFs. \begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Upper MFs**} \\ \cline{2-5}  & **Interval** & **Partial Knowledge** & \(\overline{\Delta}\) & \(\overline{\psi}\) \\ \hline
1 & \(0<z<127\) & \(2\overline{\theta}_{N}(127-z)\) & \(-(\xi_{0}-z)\overline{\theta}_{N}(z-127)^{2}\) & \(\frac{\overline{\theta}_{N}(127)^{3}}{3}\) \\
2 & \(127<z<255\) & \(2\overline{\theta}_{E}(z-127)\) & \((\xi_{0}-z)\overline{\theta  
The proposed approximation is faster than other approximations, such as the NT Method. This is because it provides a continuous approximation of the output, which means that it not only eliminates the type-reduction process, but also reduces the defuzzification method to a polynomial, and on the other hand presents slightly higher error with respect to the theoretical model and other approximations. Based on the presented comparison we can state that the proposed algorithm is not recommended for:  
* Applications where the error is a critical requirement. * Applications that require adapting the IT2 FIS online. We recommend the implementation of this approximate CEKM type-reduction algorithm in:
* Applications that require a high processing rate.