Objective The extended ptychographical iterative engine (ePIE) is a powerful phase-retrieval method that provides label-free, high-contrast imaging and robustness. In the ePIE method, all diffraction patterns captured during each iteration are applied to perform batch improvement and iteratively recover the probe and object in sequence. The ePIE can converge reasonably at a proper rate in practical experiments; however, hundreds of time-consuming iterations are required. In our previous work, a technique is proposed to increase the convergence speed of ptychographic imaging based on multiple axial intensity constraints (MAIC-PIE). In this strategy, the diffracted light from the sample is separated axially using a beam splitter and inserting multiple charge-coupled devices (CCDs) into the splitter beams. The multiple CCDs work together independently to obtain multiple axial intensity images. Such multiple axial intensity images function as an additional constraint to strengthen the connection between the intensity and the unknown phase. In MAIC-PIE, each axial intensity is required in the calculation over all iterations; however, with an increasing number of iterations, the multiple axial intensity constraints are not required in the calculation of each iteration. Thus, we modified the MAIC-PIE and propose a fast-convergence ePIE based on nonglobal multiple axial intensity constraints (ngMAIC-PIE) to reduce calculation time.
Methods In the experimental apparatus of ngMAIC-PIE, the light diffracted from the sample is split spatially into two orthogonal beams using a beam splitter. Two CCDs (CCD1 and CCD2) are placed at two axial positions after the beam splitter. They are used to record the diffraction patterns generated by the interaction of the sample with the localized probe. The ngMAIC-PIE reconstruction method involves four stages: multiple axial intensity constraints, ePIE reconstruction, multiple axial intensity constraints, and ePIE reconstruction. In the multiple axial intensity constraint stage, a round-trip iteration between the two CCDs and intensity constraints is added to the ePIE method, where the object function and light field function can be constrained to update along the effective direction and improve the convergence speed. In the ePIE reconstruction stage, the conventional ePIE method is employed, which maintains the rapid convergence in the previous stage and reduces the calculation time of the multiple round trips of the CCDs. The selection rule for the number of iterations for the multiple axial intensity constraint stage is summarized as follows. In the first 20 iterations, two stages of the multiple axial intensity constraints are completed. Here, the number of iterations at the end of the first stage and the beginning of the second stage should vary between 5 and 10. Further, the first and second stages of the multiple axial intensity constraints should be completed in 1--10 and 10--20 iterations, respectively.
Results and Discussion In a simulation evaluation, the root mean square (RMS) value relative to the amplitude and phase error from each iteration is evaluated in terms of reconstruction accuracy and convergence speed. Using the same system parameters, the convergence performance and reconstruction accuracy of the ngMAIC-PIE and MAIC-PIE exceed those of the ePIE and the performance of ngMAIC-PIE is similar to that of MAIC-PIE (Fig. 4). The reconstruction quality can be improved by increasing the number of iterations; however, the convergence rate differs. The ngMAIC-PIE method demonstrates considerably faster convergence than the conventional ePIE method for the target test object (Fig. 4). In an experiment, when reaching the iteration termination condition Et=0.002, the minimum numbers of iterations for ePIE, MAIC-PIE, and ngMAIC-PIE are 155, 70, and 78, respectively. In addition, the computation times achieved by ePIE, MAIC-PIE, and ngMAIC using an I7-6700HQ CPU with 24 GB RAM are 4989.45, 3302.60, and 2720.68 s, respectively (Fig. 7). Compared with MAIC-PIE and ePIE, the computation time of ngMAIC-PIE is reduced by 17.62% and 45.37%, respectively. Experimental results demonstrate that the proposed method can effectively improve the convergence speed and reduce the calculation time. In addition, the proposed method demonstrates higher time efficiency than MAIC-PIE.
Conclusions A ngMAIC-PIE strategy is proposed to improve the convergence speed and time efficiency of ePIE, in which the multiple axial intensity constraints and ePIE reconstruction methods are employed alternatively. The proposed strategy maintains the fast convergence characteristics of MAIC-PIE. Moreover, the round-trip iteration calculation between CCDs can be reduced in the ePIE reconstruction stage, which improves time efficiency and reduces the calculation time. Simulation and experimental results show that the proposed method exhibits fast convergence and high time efficiency. Furthermore, we compared the proposed ngMAIC-PIE with MAIC-PIE and the traditional ePIE. The results demonstrate that MAIC-PIE and ngMAIC-PIE can obtain reasonable solutions quickly, with improved reconstruction quality in previous several iterations. In subsequent iterations, ngMAIC-PIE and MAIC-PIE exhibit nearly similar convergence characteristics and both converge faster than ePIE. The proposed ngMAIC-PIE strategy is both simple and effective and demonstrates fast convergence and high time efficiency.
