## 3 Noise Characteristics in the Reconstructed Images  
The perceived image quality could depend on the type of distortions in the image. In this work, the test image pattern shown in Figure 1 is used to investigate the noise characteristics in the reconstructed images from the holograms calculated by IFTAs. The pattern had \(512\times 512\) pixels and consisted of three areas from the top to the bottom. The top and the bottom areas had the exact same dimensions of \(128\times 512\) pixels and the corresponding grey values were 64 and 192, respectively. The middle area had \(256\times 512\) pixels with a grey value of 128. First, the standard GS algorithm was used to calculate the phase-only holograms that were able to reconstruct this target image pattern in Figure 1. Figure 2 shows the reconstructed images after 50, 100, 150 and 200 GS iterations. The differences between them are small but still can be perceived by human eyes. It can be seen that the quality of the reconstructed images is improved as the number of iterations increases. Figure 3 shows the histograms of grey values in these images. It can be seen that the distributions of the grey values are a superposition of three Gaussian distributions. Fittings for these grey value distributions were carried out based on the following equation:  
\[f(x)=\ a_{1}\mathrm{e}^{-(\frac{x-p_{1}}{\beta_{1}})^{2}}+a_{2}\mathrm{e}^{-( \frac{x-p_{2}}{\beta_{2}})^{2}}+a_{3}\mathrm{e}^{-(\frac{x-p_{2}}{\beta_{3}})^ {2}}. \tag{8}\]  
Figure 1: The image pattern used to analyze the noise characteristics in the image reconstructed from holograms calculated by iterative Fourier transform algorithms (IFTAs). Figure 2: Reconstructed images after (**a**) 50; (**b**) 100; (**c**) 150; and (**d**) 200 Gerchberg–Saxton (GS) iterations. The fitting results are listed in Table 1. The coefficient of determination, i.e., \(r^{2}\), is >0.999 for all of the fittings. This indicates that Equation (8) is able to reflect the grey value distribution in the reconstructed images accurately. Therefore, the noise in the reconstructed images has a Gaussian distribution. The improvement in the quality of the reconstructed image as the iteration progresses can be reflected by the decrease in the Gaussian variance, i.e., values of \(\delta_{1}\), \(\delta_{2}\) and \(\delta_{3}\). In other words, the reconstructed images are less affected by white Gaussian noise with increased number of GS iterations. Fienup and FiDOC algorithms were also used to calculate the phase-only holograms. In the FiDOC algorithm, the don't-care area accounts for 25% of the original image in this dimension. Therefore, the total dimension became \(640\times 640\) pixels. Figure 4 compares the reconstructed images from the calculated holograms after 200 GS, Fienup and FiDOC iterations. It can be seen that both the Fienup and FiDOC algorithms outperformed the GS algorithm. The distributions of the grey values in these three reconstructed images are plotted in Figure 5.  
The corresponding fitting results based on Equation (8) are listed in Table 2. It should be noted that the results for Figure 4c only include the target area. Again, the coefficient of determination, i.e., \(r^{2}\), exceeds 0.9999 in all these three fittings. This further confirms the main source of distortion in the images reconstructed from the holograms calculated by IFTAs is the white Gaussian noise. Different types of IFTAs will only lead to different degrees of white Gaussian noise in the reconstructed images. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Coefficients**} & \multicolumn{4}{c}{**Number of Iterations**} \\ \cline{2-5}  & **50** & **100** & **150** & **200** \\ \hline a\({}_{1}\) & 6087.9 & 9317.1 & 12,103.0 & 14,522.3 \\ a\({}_{2}\) & 11,365.6 & 17  
that the algorithm can be terminated once the quality meets the system requirement. Therefore, the image quality metrics used in IFTAs need to be able to accurately predict the perceived image quality so that the algorithm can stop at the right time.