# A Fuzzy Inference System for Unsupervised Deblurring of Motion Blur in Electron Beam Calibration
## 1 Introduction  
The main goal of fuzzy systems is to define and control sophisticated processes by incorporating and taking advantage of human knowledge and experience. Nowadays, fuzzy logics are widely used in industry for various applications ranging from cameras to cement kilns, trains, and vacuum cleaners [1]. Furthermore, deblurring techniques have versatile applications and they are either performed in spatial [2] or frequency domains [3; 4; 5]. [PERSON] [6] modeled the electron beam (EB) measurement process with a linear motion blur and evaluated three of the well-established deblurring techniques for EB restoration. In this study [6], [PERSON] used a Weiner inverse filter and blind Richardson-Lucy deconvolutions to restore the EB distribution and correct the measurements through deblurring. A simple motion blur is formulated in Equation (1). \[g(x)=\int f(x)h(x)+n(x), \tag{1}\]  
where in the spatial domain, \(f\), \(g\), \(h\), and \(n\) are the ground truth signal (EB distribution) of length \(L_{f}\), degraded signal (measurement from probe), point spread function (PSF) of length \(L_{h}\), and noise respectively. Their frequency domains are represented by uppercase letters \(F\), \(G\), and \(H\). In the case of electron beam measurements, the ground truth signal is the distribution of EB and the degraded signal is the measurement acquired from the probe. The electron absorption of a slit or wire probe of size \(L_{h}\) is modeled with a PSF kernel [6]. Linear motion blur point spread function has two distinct characteristics of motion direction and length (\(L\)) [7]. The PSF is known for having harmonically spaced vanishing magnitudes in the frequency domain due to its limited length in the spatial domain [8]. There are several approaches to estimate \(L_{h}\) such as log power spectrum, cepstrum, bispectrum, and pitch detection algorithms. In image deblurring jargon, it is assumed that the frequency spectrum of \(F\) is smooth and does not contain vanishing frequencies, hence any vanishing frequencies in \(G\) are associated to \(H\)[9; 10]. However, this assumption usually does not hold for EB measurements, especially where the \(L_{f}\) is in the same order of \(L_{h}\). This similarity makes it complicated to distinguish between \(L_{f}\) and \(L_{h}\) and therefore compromises the deblurring process by an incorrect detection of null frequencies. Such an erroneous deblurring process is likely to produce an incorrect but convincing result, notably when \(f\) and \(h\) have remarkable cross-correlation. This ambiguity is likely to happen in EB measurements, because: (a) \(f\) and \(h\) are usually in the same order of magnitude and they have relatively high cross-correlation; and (b) the \(L_{f}\) can be inconsistent. In Reference [6], a prior knowledge of \(L_{h}\) is used to estimate the position of null frequency of \(h\) from the spectrum analysis of \(G\). [PERSON] limited the spectrum of \(G\) to \(\pm 15\%\) of the nominal \(L_{h}\) by applying a window to its log-power spectrum, thereby ignoring vanishing frequencies outside of this interval. This algorithm is available in Reference [11]. This strategy relies on knowing the \(L_{h}\). Therefore, it is a good approach when it is known accurately. There are a few limitations with this method due to the varying nature of \(L_{f}\) during the calibration and measurement process. As a result, the beam's vanishing frequency (or its harmonics) can be located within the applied window and cause a false detection. Furthermore, if the inaccuracy of \(L_{h}\) is more than \(15\%\), the null frequency of \(h\) is ignored by the window resulting in an erroneous restoration. In addition, any inaccuracy of more than \(\pm 15\%\) cannot be compensated. One solution to effectively address this uncertainty is to use fuzzy systems. Fuzzy inference systems are widely used to address instrumental uncertainties. A comprehensive review and explanation of fuzzy inference systems are provided in Reference [12].