# Improvement of Fresnel Diffraction Convolution Algorithm
## 3 Experiments and Results
A similar situation occurs in the SSIM curves, where the SSIM value of the D-FFT algorithm is 0.9777 and that of the T-FFT algorithm is 0.9882, whereas the D-FFT (Tukey) can take a maximum SSIM value of 0.9979 at \(\beta=0.91\). That is to say, the quality of the reconstructed amplitudes obtained using the D-FFT (Tukey) algorithm is significantly improved and outperforms the T-FFT algorithm's results. From the curves, the maximum values of PSNR and SSIM both appear with the change in \(\beta\). The maximum values of PSNR and SSIM appear at \(\beta=0.89\) and \(\beta=0.91\), respectively. Considering PSNR and SSIM together, \(\beta=0.9\) finally becomes the optimal value for diffraction calculation. The results obtained by the D-FFT (Tukey) and T-FFT algorithms are shown in Figure 5.  
As shown in Figure 5, the amplitude obtained by the T-FFT algorithm is heavily contaminated by a ringing noise, with a large number of oscillations in the background part, while the background part of the amplitude obtained by the D-FFT (Tukey) algorithm is very homogeneous and the ringing noise is well suppressed. Not only the amplitude but also the phase obtained by the D-FFT (Tukey) algorithm are superior to the T-FFT. As can be seen from Figure 5, the phase obtained by the T-FFT algorithm has a pseudo-boundary at USAF1951 group 6, element 2, and at the same location, the fringe boundary obtained by D-FFT (Tukey) matches the real situation. Similarly, at the black rectangular hole in the red box, the phase reconstructed by T-FFT has remarkable oscillations inside the hole, while  
Figure 4: Different algorithms for USAF1951’s reconstruction results of the PSNR curves and SSIM curves: (**A**) PSNR curves; (**B**) SSIM curves. the phase reconstructed by the D-FFT (Tukey) has clear and smooth boundaries. Most importantly, some fringes of group 7, element 2 calculated by T-FFT are missing, while the corresponding bars calculated by DFFT (Tukey) are intact, as shown in the blue box of Figure 5. The above is shown more visually in Figure 6.  
As shown in Figure 6A, in the phase reconstructed by the D-FFT algorithm, all three fringes in group 7, element 2 of USAF1951 can be resolved, whereas in the phase recon  
Figure 5: Intensity and phase of reconstruction images obtained by (**A**) D-FFT (Tukey) (\(\beta=0.9\)) and (**B**) T-FFT. Figure 3:0 Profiles of normalized phase obtained by D-FFT (Tukey) (\(\beta=0.9\)) and T-FFT: (**A**) located in group 7, element 2 (in the blue box in Figure 5); (**B**) located in the rectangular window of group 7 (in the red box in Figure 5). structed by the T-FFT algorithm, there is a severe distortion at this location and only the centrally located fringe can be resolved. According to the profile line shown in Figure 6B, the phase distribution inside the rectangular hole obtained by the T-FFT algorithm has serious oscillations, which cannot distinguish the rectangular hole from the fringe effectively, while the D-FFT (Tukey) algorithm can distinguish the fringe from the rectangular hole in the same situation. Although the phase inside the rectangular hole obtained by the D-FFT (Tukey) algorithm still has oscillations, it does not affect the recognition of the rectangular hole. That is to say, DEFT (Tukey) is capable of obtaining more accurate and higher-resolution phase distributions than T-FFT. To verify the effectiveness of D-FFT (Tukey) in practical applications, we discuss a Fresnel digital holographic diffraction reconstruction issue of a lung cancer slice. The relevant parameters are shown in Figure 3:1 The experimental light path is shown in Figure 3:2  
As shown in Figure 7, a laser with a wavelength of 671 nm is passed through a beam-splitter prism (BS1) and divided into object light and reference light.