## 5 Results  
We tested the proposed method with two real captured datasets with scanning steps of 4.5 and 2 mm, respectively. We used an object placed 2 m away from the pupil plane of the imaging lens. First, a total of 225 LR images with resolution of 231 pixel \(\times\) 231 pixel was obtained with a scanning step of 4.5 mm and a 61% overlap ratio between two adjacent images. Second, by changing the scanning step to 2 mm and with an 84.5% overlap ratio between adjacent images, we obtained 361 LR images with the same resolution. Figure 11 is the dataset under multiple acquisitions showing different results under two different far-field FP experimental setups. Figure 11a, with a scanning step of 4.5 mm and 61% overlap ratio, is an original sequence image composed of 225 LR images with 160 \(\upmu\)m line width and 3.13 lp/mm resolution. Figure 11b, with a scanning step of 2 mm  
Figure 10: The graph of the random distribution function of dark current; through this graph, we generated a new random dark current field. and 84.5% overlap ratio, shows a set of images with 361 LR intensity images with the same resolution. In Figure 12, the images in the red box were recovered from 225 LR images with 231 pixel \(\times\) 231 pixel and a scanning step of 4.5 mm. Using a personal computer (9 th Generation Intel Core i7), we found the single iteration time to be 3 s, and the data converged after 10-15 iterations. The total recovery time was approximately 30-45 s. The reconstructed detailed information of the object is shown in Figure 12(a1,b1). Figure 12(c1) shows the phase of the reconstructed HR image. Figure 12(d1) shows the zoomed-in phase of Group 9 HR image. Meanwhile, the images in the blue box were recovered from 361 LR images with 231 pixel \(\times\) 231 pixel with a scanning step of 2 mm. The single iteration time was 5 s, and the data converged after 15-25 iterations. The total recovery time was approximately 80-130 s. Group 17 of the HR object (Figure 12(a2)) is shown in Figure 12(b2). Figure 12(c2) shows the phase of the reconstructed HR image. Figure 12(d2) shows the zoomed-in phase of Group 17 HR image. The recovery algorithm with a longer scanning step has a shorter running time and higher efficiency. Comparing the experimental results with the two scanning steps (4.5 and 2 mm), we found that when the scanning step was decreased, the increase in the spectrum overlap ratio greatly improved the recovered HR image. The two experiments directly proved the validity of the simulation results along with the correctness, efficiency, and effectiveness of the imaging equations and phase retrieval algorithm. We further processed the images with a scanning step of 2 mm with partition reconstruction, which involved cutting the image into smaller objects for reconstruction. We chose to intercept the central position objects of each LR image for reconstruction. The above algorithm was used to reconstruct the intercepted images again, and the reconstruction results are shown in Figure 13. From the results, we found that the partition reconstruction can effectively reduce the effect of the aberrations and reconstruct smaller feature with higher image quality than HR image reconstructed by full-frame reconstruction. Partition reconstruction yields better results than full-frame reconstruction. For example, full-frame reconstruction can only resolve the feature of Group 17 on the object, as shown in Figure 12, while partition reconstruction can even resolve Group 25 with a 40 um line width of 12.5 lp/mm and a resolution gain of 4\(\times\). Finally, LR images with different off-axis aberrations at different locations were acquired by camera scanning. Although the FP algorithm has a certain correction effect on the aberration, it is more effective to correct the aberration at different positions of the same image by using the partition reconstruction method to obtain more high-frequency image details.