# A Fuzzy Inference System for Unsupervised Deblurring of Motion Blur in Electron Beam Calibration
## 3 Practical Result  
### Membership Function Parameters  
Membership function parameters were investigated pragmatically by testing the explained algorithm for various degraded EB measurement samples. In all degraded measurements, \(h\) and \(f\) had approximately similar sizes as a result of which \(\hat{L}_{1}\cong\hat{L}_{2}\). The membership functions were designed with smooth transitions to provide a general solution and more flexibility, except for the attenuation. To further discriminate between \(E_{1}\) and \(E_{2}\), the attenuation membership function parameters were adjusted to have more emphasis between the interval of 0 to 0.3 dB. This intuitive definition was done by observing the magnitude of null frequencies in several degraded signals where the attenuation of the null frequencies was always under 0.3 dB. The membership function parameters are presented in Table 2.  
The membership functions of attenuation (\(B_{high}\) & \(B_{Low}\)) and residue (\(C_{low}\) & \(C_{high}\)) fuzzy sets are depicted in Figure 4, according to their values in Table 2. The fuzzy sets of PSF deviation and restoration quality were also defined with the similar membership functions to that of residues.  
\begin{table}
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c} \hline \hline \multicolumn{2}{c|}{**PSF Deviation**} & \multicolumn{3}{c|}{**Attenuation**} & \multicolumn{3}{c|}{**Residue**} & \multicolumn{3}{c}{**Restoration Quality**} \\ \hline \(\mu_{m}\) & \(\mu_{m}^{\prime}\) & \(\mu_{o}\) & \(\mu_{o}^{\prime}\) & \(\mu_{r}\) & \(\mu_{r}^{\prime}\) & \(\mu_{q}\) & \(\mu_{q}^{\prime}\) \\ \hline \(a_{m}\) & \(c_{m}\) & \(a_{m}\) & \(c_{m}\) & \(a_{0}\) & \(c_{0}\) & \(a_{0}\) & \(c_{0}\) & \(a_{r}\) & \(c_{r}\) & \(a_{q}\) & \(c_{q}\) & \(a_{q}\) & \(c_{q}\) \\The analysis of a few of the samples are shown in Figures 5 and 6. For a few of the EB measurements, the \(L_{h}\) (probe sizes) were known to be 1.00, 0.20, and 0.40 mm respectively. The crisp fuzzy inputs and deconvolution grades \(E_{i}\) were also provided for every sample. The restoration that resulted in the higher \(E_{i}\) was selected by the system as the correct solution and its corresponding \(\hat{L}_{i}\) therefore represents the probe size (\(\hat{L}_{h}\leftarrow\hat{L}_{i}\)). To validate the proposed system with the ground truth signal (\(f\)) [6], both restorations (\(\hat{f}_{1,2}\)) were compared against their ground truth signal using cross-correlation. For the \(\hat{f}_{i}\) with the higher \(E_{i}\), the cross-correlation of \(\hat{f}_{i}\) and \(f\) also produced greater coefficients, supporting the accuracy and reliability of the system. As another benchmark, full width at half maximum (FWHM) analysis was used, as it is a popular measure in the EB calibration jargon. The FWHM of \(f\) and the \(\hat{f}_{i}\) that had the higher \(E_{i}\) produced a similar result, further confirming that the FIS had successfully identified the correct restoration process.  
Figure 4: Attenuation and deconvolution residue membership functions.  
Figure 5: Null frequencies in the spectrum of the degraded pulse. Result of restoration with detected null frequencies, expected PSF length of 1 mm on the left and 0.2 mm on the right.