# Improvement of Fresnel Diffraction Convolution Algorithm
## 3 Experiments and Results
When \(\beta\) is greater than 0.2, the number of fringes in [PERSON]'s double-hole diffraction field obtained by the D-FFT (Tukey) algorithm decreases. When \(\beta=0.3\), the amplitude of the fringes at the edges of the picture is significantly reduced, although there are still 15 fringes in Figure 3B. When \(\beta\) takes 0.5 and 0.7, there are only 13 fringes in Figure 3C,D. In other words, the degree of edge smoothing of the Tukey window directly affects the accuracy of the calculated amplitude distribution, and the value of the edge smoothing factor \(\beta\) needs to be selected according to the actual situation when using the D-FFT (Tukey) algorithm to deal with different diffraction problems. To compare the reconstructed amplitudes and phases of the D-FFT (Tukey) and T-FFT at diffraction distances more than \(\mathrm{d_{t}}\), an extra simulation experiment of Fresnel holographic reconstruction of USAF1951 was conducted. In the simulation, limited by the set pixel size, the pixel widths of the horizontal lines were one row of pixels and two rows of pixels in group 7, element 2. The physical widths of these lines were 1.5 \(\mathrm{\SIUnitSymbolMicro m}\) and 3 \(\mathrm{\SIUnitSymbolMicro m}\), respectively. The relevant parameters are shown in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Variable** & **Value** \\ \hline Wavelength & \(\lambda\) & 0.6328 \(\mathrm{\SIUnitSymbolMicro m}\) \\ Pixel size & Pix & 10 \(\mathrm{\SIUnitSymbolMicro m}\) \\ Pixel number of hologram & N & 1024 \(\times\) 1024 \\ Diffraction distance & \(\mathrm{d^{\prime}}\) & 235.5 \(\mathrm{\SIUnitSymbolMicro m}\) \\ Reconstructed distance & d & 353.3 \(\mathrm{\SIUnitSymbolMicro m}\) \\ Demarcation distance & \(\mathrm{d_{t}}\) & 161.8 \(\mathrm{\SIUnitSymbolMicro m}\) \\ Radius of reconstructed light & R & 706.5 \(\mathrm{\SIUnitSymbolMicro m}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of simulation of Fresnel hologram. Figure 3: Intensity profiles through center of [PERSON]’s double-hole experiment fringes obtained by D-FFT (Tukey): (**A**) \(\beta=0.1\); (**B**) \(\beta=0.3\); (**C**) \(\beta=0.5\); (**D**) \(\beta=0.7\). As mentioned earlier, the two simulations should have had different values of \(\beta\) due to the different complexities of the objects. That is to say, in practice, D-FFT (Tukey) must choose an appropriate \(\beta\) value according to the diffractive plane, to obtain the best results. \(\beta\) can be determined with the help of the peak signal-to-noise ratio (PSNR) and the structural similarity (SSIM). \[\text{PSNR}(\text{x},\text{y})=10\,\log_{10}\left(\text{Max}^{2}/\text{MSE}( \text{x},\text{y})\right), \tag{8}\]  
\[\text{SSIM}(\text{x},\text{y})=\frac{(2\mu_{\text{x}}\mu_{\text{y}}+\text{C} _{1})(2\sigma_{\text{xy}}+\text{C}_{2})}{(\mu_{\text{x}}^{2}+\mu_{\text{y}}^{2 }+\text{C}_{1})(\sigma_{\text{x}}^{2}+\sigma_{\text{y}}^{2}+\text{C}_{2})} \tag{9}\]  
where Max is the maximum gray value of the image, \(\text{MSE}(\text{x},\text{y})\) is the mean square error between image x and image y, \(\mu\) is the grayscale mean of all pixel points of the image, \(\sigma\) is the variance of all pixel points of the image, and \(\sigma_{\text{xy}}\) is the covariance between image x and image y. A larger PSNR and SSIM mean a better reconstruction quality. In the simulation experiment of USAF1951, the object on the diffraction screen was used as the reference image to obtain two curves of \(\beta\) values versus PSNR and SSIM. The PSNR and SSIM curves are shown in Figure 4.  
In Figure 4, The PSNR of the reconstructed amplitude obtained by the D-FFT algorithm is only 6.335, while the PSNR of the reconstructed amplitude obtained by the T-FFT algorithm is 10.89, and the PSNR of the reconstructed amplitude obtained by the D-FFT (Tukey) algorithm can reach 19.22 when \(\beta=0.89\).