# Low-Photon Counts Coherent Modulation Imaging via Generalized Alternating Projection Algorithm
## 4 Verification with Experimental Data  
Optical experiments were performed to verify the CMI-GAP algorithm in a low light flux scenario. A stabilized 632.8 nm He-Ne laser was used to form a quasi-monochromatic and collimated illumination beam. The sample was a positive USAF resolution target stuck on a 1.5 mm hole. Scattered waves from the sample were modulated by a downstream thin modulator. An sCMOS camera (Thorlabs CS2100) with a pixel size of 5.04 \(\upmu\)m was used to collect the diffraction data. A ptychography dataset was recorded with an illumination probe flux of \(10^{8}\) photons. The retrieved modulator will be used in our CMI and CMI-GAP reconstruction. In the near-field experiment, the distance between the sample and the modulator, and the distance between the modulator and the detector, were 22.1 mm and 21.5 mm,  
Figure 6: Error metrics comparison of reconstruction in our simulation for the far-field dataset when \(N=10^{8},\ 10^{7},\ 10^{6},\ 10^{5}\). Bar charts: PSNR comparison of reconstructed amplitudes under two algorithms. Line graphs: RMS comparison of reconstructed amplitudes under two algorithms. Figure 7: Convergence curve of RMS for the two algorithms in far-field geometry under 3 levels of photon flux. The CMI-GAP algorithm has a relatively low RMS value and a faster convergence. respectively. CMI data with four levels of photon flux were acquired. The reconstruction results are shown in Figure 8. The iteration number of the traditional algorithm is about 500 and 50 with CMI-GAP. The overall degradation of quality is believed to be caused by residual reflections and detector noise. From Figure 8, one can see that the amplitude and phase images can be accurately reconstructed when \(N\) equals to \(10^{8}\) or \(10^{7}\). In contrast, the reconstruction from the CMI-GAP algorithm has fewer artifacts and a higher resolution. The quantitative comparison, in terms of RMS, is given in Table 1. The ground truth images were obtained by ptychography. When \(N\) drops to \(5\times 10^{6}\), the traditional CMI algorithm fails to produce a recognizable image, while the proposed algorithm can still reconstruct most of the sample information. Both algorithms failed when the average photon count per pixel dropped to 13. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of Photons** & \(10^{8}\) & \(10^{7}\) & \(5\times 10^{6}\) & \(3\times 10^{6}\) \\ \hline RMS of CMI-GAP & 0.17 & 0.31 & 0.45 & 0.62 \\ \hline RMS of CMI & 0.32 & 0.50 & 0.76 & 0.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RMS corresponding to the near-field CMI. Figure 8: Reconstruction of the USAF given near-field CMI experimental data at varying illumination photons. The reconstruction from the CMI-GAP algorithm has fewer artifacts and a higher resolution. For far-field geometry, we performed optical experiments using setup parameters the same as the simulations. A similar improvement as the near-field scenario was obtained. Here we show the results when applied to the far-field X-ray experiment data [39]. In these experiments, a 6.2 keV X-ray was used to form a coherent beam. The illumination probe, carrying about \(3.6\times 10^{6}\) photons, was formed by a 2.2 \(\upmu\)m pinhole. The modulator was placed at 2.5 mm downstream of the object, while the distance between the modulator and the detector with a 172 mm pixel size was about 7.2 m.  
The photon flux was about \(3.6\times 10^{6}\) for this X-ray dataset. The reconstruction results obtained by CMI and CMI-GAP are shown in Figure 9. With the original CMI, the amplitude recovered is drowned by noise. In contrast, reconstructions with fewer artifacts and higher contrast can be seen clearly using the proposed CMI-GAP algorithm. For the X-ray experiment, the detector had a 172 mm pixel, leading to the insufficient sampling of the diffraction patterns. A detector pixel subdivision was then used to better model the detector sampling process.