# Optimization of Phase-Only Computer-Generated Holograms Based on the Gradient Descent Method
## 4 Verification of Proposed Technique  
This section presents a verification of the proposed technique. Figure 3 shows three target images used for the verification, including one grayscale image (\"mandrill\") and two binary images (\"hikari\" and \"circle\"). The improvements in the reconstructed images obtained via the optimization process are shown in Figure 4. Here, the reconstructed images for \"hikari\" obtained by the proposed technique after 1, 2, 5, 10, 50, and 100 iterations are shown. For comparison, the reconstructed images obtained by the GS algorithm are shown in Figure 5. These results demonstrate that the proposed technique required few iterations than the GS algorithm for the phase optimization task. Figure 2: Calculation process of the proposed technique  
Figure 4: Reconstructed images obtained by the proposed algorithm after (**a**) 1, (**b**) 2, (**c**) 5, (**d**) 10, (**e**) 50, and (**f**) 100 iterations. Figure 3: Target images: (**a**) mandrill (resolution 512 \(\times\) 512), (**b**) hikari (resolution 256 \(\times\) 256), and (**c**) circle (resolution 512 \(\times\) 512). Figure 6 shows the reconstructed images for the three target images using the proposed technique and the GS algorithm after 100 iterations. The changes in the MSE defined for the intensity distribution of the reconstructed image during the optimization processes are shown in Figure 7. These results show that lower MSE values were obtained with the proposed algorithm than with the GS algorithm. The SNRs of the reconstructed images obtained after 100 iterations were about 30 dB, when the proposed technique was adopted. For 100 iterations, 3D intensity profiles of the reconstructed images for the \"circle\" target image obtained by the proposed algorithm and the GS algorithm are shown in Figure 8a,b, respectively. As shown by the results, the proposed technique more precisely controlled the intensity distribution of the reconstructed image than the GS algorithm. The intensity variation corresponding to the white region is much smaller for the proposed algorithm than for the GS algorithm, while that corresponding to the black region is larger for the proposed algorithm than for the GS algorithm. The GS algorithm reduces evenly the MSE based on the amplitude distribution of the reconstructed image, while the proposed algorithm reduces the MSE defined for the intensity distribution. Therefore, the GS algorithm can uniformly control the amplitude distribution, whereas, the proposed technique can evenly influence the intensity distribution. Because the GS algorithm cannot impact the intensity distribution uniformly, it can be used for the more precise control of low intensities but not for high intensities. Therefore, the GS algorithm could control the black region more strictly, while the proposed technique could manage the white region more accurately. Figure 5: Reconstructed images obtained by the GS algorithm after (**a**) 1, (**b**) 2, (**c**) 5, (**d**) 10, (**e**) 50, and (**f**) 100 iterations. Figure 6: Reconstructed images obtained with (**a–c**) the proposed technique and (**d–f**) the GS algorithm; (**a,d**) mandrill, (**b,e**) hikari, and (**c,f**) circle. Figure 7: Change in mean square error (MSE) during the optimization process for three target images: (**a**) mandrill, (**b**) hikari, and (**c**) circle. The effectiveness of the proposed approach for determining the step length coefficient was verified. Figure 9a-c show changes in the MSE during the optimization processes when constant step length coefficients were used (\(\gamma\) = 0.20, 0.25, and 0.30) and when an optimized step length coefficient was used. When constant step length coefficients were used, the decrease in the MSE was dependent on the test image. When an optimized step length coefficient was used, the MSE was minimized for all test images.