# Spatial Resolutions of On-Axis and Off-Axis Transmission Kikuchi Diffraction Methods
## 3 Results and Discussion  
### Quantitative Evaluation of the Spatial Resolution  
In this study, the digital image correlation (DIC) technique [15; 16; 17] was used to investigate the effects of the accelerating voltage and specimen thickness. The DIC method can be used to quantitatively evaluate small variations between patterns by calculating their correlation coefficient. As the electron beam is scanned close to a grain boundary, the interaction volumes of both sides overlap, which leads to overlapping Kikuchi patterns, and thus to a decreased correlation coefficient. Therefore, this method can be used to determine the spatial resolution. It has been extensively used to evaluate the physical resolutions of the EBSD [14; 18] and off-axis TKD methods [19; 20; 21]. The correlation coefficient can be expressed as \(r_{ij}=\frac{\sum{g_{ij}\overline{g}_{ij}}}{\sqrt{\sum{g_{ij}^{2}\sum{ \overline{g}_{ij}^{2}}}}}\), where \(g_{ij}\) and \(\overline{g}_{ij}\) denote the grey scale of the reference and sampling pattern at each position (_i,j_), respectively.  
The middle part of the on-axis and off-axis Kikuchi patterns have good qualities; however, the boundaries of the patterns contain considerable noise. Moreover, a bright spot is generated by the transmitted beam in the center of the on-axis Kikuchi pattern. The size and shape of the bright spot can change during the scanning. Thus, if the whole pattern (\(640\times 480\) pixels) is considered, the correlation coefficient would deviate and the results would be inaccurate. Therefore, regions of interest (ROIs) have to be selected to increase the precision. [PERSON] et al. used this method to determine the resolution of the off-axis TKD [19]. In this study, we have used a similar approach. Circular ROIs were chosen for the off-axis TKD, which contained the overlapping region, as shown in Figure 1C. On the other hand, annular ROIs were selected in the on-axis TKD to ensure that the boundaries and bright spots were eliminated while retaining the overlapping regions (Figure 1B). In addition, the chosen ROIs should be as large as possible to ensure representativity.  
The calculated correlation coefficients at a voltage of 30 kV and specimen thickness of 91 nm are presented in Figure 2. [PERSON] et al. [14] and [PERSON] et al. [20] applied a fast Fourier transform (FFT) filter to reduce the noise, while FFT followed by inverse FFT were applied by [PERSON] et al. [19] to obtain smooth curves. FFT is commonly used for filtering in the frequency domain; if a smooth profile is needed, the peak may be broadened. Therefore, an average filtering algorithm is introduced to avoid the deviation in the correlation coefficient, which can smooth profiles in the spatial domain. As it does not affect the profile width, it is commonly used to filter out Gaussian noise. In Figure 2B, the low-frequency noise was removed and the whole curve became smoother after filtering. The smooth profiles of the correlation coefficients were summed and normalized. The procedure is presented in previous reports [14; 19; 20]. With this approach, we can obtain the sum of the normalized correlation coefficients as a function of the distance (Figure 2C) and the full width at half maximum can be easily calculated, which is defined as the PSR [14].