# Enhanced Phase Retrieval Method Based on Random Phase Modulation
## 4 Experiment and Analysis  
A lens-free imaging system was used to verify our method, which was composed of four parts: a coherent plane wave (532 nm), a sample, an SLM (pixel size = 12.5 \(\upmu\)m), and a CCD camera (pixel size = 3.1 \(\upmu\)m). A calibration target etched with a resolution testing board was used for the sample. The distance from the object to the SLM was 100 mm, and that from the SLM to the CCD was 150 mm. Then, four intensity measurements were sequentially recorded by the CCD as the SLM transformed four times. As the image sharpness is a significant evaluation criterion of an image, we used the sharpness  
Figure 6: The curves of anti-noise performance and reconstruction results with polluted intensity; \(n_{1}\ =\ 5,\ N=80\). (**a**) The relationships between the variances of the added Gaussian noise and MSEs of reconstructed amplitudes from the serial, parallel, and proposed cascaded reconstruction algorithms; (**b–d**) reconstructed amplitudes with polluted intensities of the three methods after 80 iterations, when the noise variance was 30% and mean was 2%; (**e–g**) corresponding recovered phases. functions of retrieved images for a global quantification of the convergence rate and reconstruction quality. The sharpness function is defined as follows [24]:  
\[S_{n}\ =\ [\sum_{\forall x,y}\abs{O_{n}(x,y)}^{2}]/[\sum_{\forall x,y}\abs{O_{n}(x,y )}^{2}, \tag{2}\]  
where \(S_{n}\) is the sharpness of \(O_{n}\), and \(O_{n}\) is the reconstruction result after \(n\) loops. The convergence curves after 600 iterations of the three methods are plotted in Figure 7. The serial process of the proposed cascaded method executed 35 iterations, with \(n_{1}\ =\ 35\) and \(n_{2}\ =\ 565\). Although the convergence speed of the serial method was faster than that of the parallel method, the convergent result of the parallel method was much better than that of the serial method. Obviously, only the cascaded method achieved rapid convergence and quality results at the same time. We used the threshold condition \(\
orm{S_{n+1}}-\abs{S_{n}}\leq 1^{-10}\) to extract the convergent iteration numbers to quantify the convergence rates. The iteration numbers and the corresponding sharpness are exhibited in Table 1. Compared with the parallel method, the proposed cascaded method realized the same quality result with fewer iterations. To clearly show the differences in the reconstruction results with different sharpness, the recovery amplitudes and phases of these three methods are shown in Figure 8a-f. In the reconstructed amplitudes exhibited in Figure 8a-c, the background noise of the serial algorithm was the most severe, compared to the other two methods. Moreover, the data captured from white lines in Figure 8a-c are plotted in Figure 9 to show the detailed distinction of reconstruction quality. Apparently, the contrast of the serial result was reduced by background noise. The parallel and cascaded methods, which held higher sharpness, performed better in terms of reconstruction quality. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Performance** & **Serial** & **Parallel** & **Cascaded** \\ \hline Iteration number & 90 & 425 & 156 \\ Sharpness (\(\times 10^{-5}\)) & 2.0475 & 2.2706 & 2.2740 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reconstructed performance comparison of the three methods. Figure 7: The relationship between iteration number and the sharpness of retrieved amplitudes from the serial, parallel, and the proposed cascaded methods; \(n_{1}\ =\ 35\), \(N\) = 600. The experiments show the availabilities of the proposed cascaded method, which extracted the superiorities of both serial and parallel methods, thereby achieving quality imaging with fewer iterations. Nevertheless, there is still an important question to be addressed in the proposed serial-parallel cascaded reconstruction framework. How does the distribution scheme of iteration numbers affect the reconstruction performances?