# Low-Photon Counts Coherent Modulation Imaging via Generalized Alternating Projection Algorithm  
[PERSON]  
1 Hbei Key Laboratory of Power Internet of Things Technology and the Department of Electronics and Communication Engineering, North China Electric Power University, Baoding 071003, China 1  
[PERSON]  
2 Shenzhen Key Laboratory of Robotics Perception and Intelligence and the Department of Electrical and Electronic Engineering, Southern University of Science and Technology, 1088 Xueyuan Avenue, Shenzhen 518055, China 2  
[PERSON]  
3 Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA 34 Singapore-MIT Alliance for Research and Technology (SMART) Centre, 1 Create Way, Singapore 117543, Singapore  
reconstruction quality is achieved by combining the conjugate gradients with preconditioning strategies, as well as regularization accounting for the statistical noise model of real experiments [28; 29].  
For imaging of biological samples, the ultimate limit of resolution is set by the radiation damage, and thus the performance of the imaging method under low illumination flux is essential [30; 31; 32; 33]. Previous researchers also demonstrated that electron ptychography is a promising phase-contrast imaging technique with the advantage of high dose efficiency [34; 30; 35; 36]. According to the dose factorization theorem [37], there is a strong hope that ptychography can keep its high performance under low light conditions. Using two-dimensional MoS2 crystal samples, it was demonstrated that highly quantitative phase can be obtained under low electron dose conditions [34].  
The requirement of multiple measurements in ptychography makes it unsuitable for fast sample dynamics investigation. CMI is an alternative solution in addressing the weakness of CDI, but in a single-shot manner [38; 39]. In CMI, a modulator is placed between the object and the detector to strengthen the coding of the phase of diffracted waves in a diffraction pattern. CMI is shown to work well under illumination flux greater than \(10^{7}\) photons. Nevertheless, collected diffraction patterns can be severely interfered by Poisson noise as the illumination weakens, which affects the stability and accuracy of the algorithm. Therefore, it is worthy of retrieving phase information from contaminative diffraction patterns in biological specimens with efficient algorithm and ideal illumination photons. For lower flux, the current algorithm starts to encounter convergence difficulty and image artifacts in reconstruction. Recently, the PhENN neural network was proposed to improve the low-flux performance of CDI [40], as well as CMI [41]. Their training requirement can be computationally expensive and still there is concern about their suitability on general samples. Untrained learning methods also exist [42], but they are yet to be demonstrated for the low photon scenario.  
In this manuscript, we report a computationally efficient and robust algorithm, based on generalized alternating projections (GAP), that preserves the advantages of CMI to recover the phase information from low signal-to-noise ratios data. GAP consists of two sub-problems, including measurement formation and statistical prior regularization. The paper is organized as follows: Section 2 details the theory and metric of the proposed method. In Section 3, the results and quantitative analysis of the simulation are presented. Experimental results of visible light and X-rays are described in Section 4 before a summary of our method is drawn in Section 5.