## 3 Experiments and Results
### Results  
#### 3.2.1 Recovery Performance Analysis  
To compare the recovery performance of the conventional imaging correlography with the ptychographic imaging correlography, we perform the experimental simulations using a 512 pixel \(\times\) 512 pixel resolution chart, described in Figure 8a. The resolution chart contains line pairs with varying widths from 20 pixels down to 1 pixel, corresponding to line pairs per pixel in the range \([0.025,\ 0.5]\). The contrast for the horizontal and vertical bars belonging to each group is computed to quantify the reconstruction performance. The contrast \(C\) is given as  
\[C=\frac{\overline{w}-\overline{b}}{\overline{w}+\overline{b}} \tag{15}\]  
where \(\overline{b}\) and \(\overline{w}\) are the average intensity of the black and white bars, respectively. To aid our discussion, the limit of resolvability is defined to be when the contrast of a group drops below 20% (MTF 20%). Figure 7: The data processing flow chart for ptychographic imaging correlography. Note: AF = amplitude flow; RED = regularization by denoising. MTF is the modulation transfer function of the imaging system. In the first simulation, we capture a \(8\times 8\) grid of images with 66.7% overlap between the neighboring positions. Figure 8b shows the ptychographic measurements for the resolution chart. The recovered image for conventional imaging correlography is shown in Figure 8c, and the recovered image for ptychographic imaging correlography is shown in Figure 8d. The plot in Figure 8e shows the contrast of the groups in the recovered images. The recovered results for conventional imaging correlography cannot resolve the features that have widths of 20 pixels (0.025 line pairs/pixel) before the contrast drops below 20%. However, we are able to recover elements that are only 5 pixels wide (0.100 line pairs/pixel) using ptychographic imaging correlography. Compared with conventional imaging correlography, ptychographic imaging correlography improves the resolution of the object four-fold via the ptychographic measurements and the proposed optimization framework. Figure 8: (**a**) Ground truth resolution chart. (**b**) Ptychographic measurements for the resolution chart. (**c**) Recovered image for conventional imaging correlography. (**d**) Recovered image for ptychographic imaging correlography. (**e**) The contrast of the groups in the recovered images. The four images in Figure 6 are employed as the original test images to analyze the recovery performance again. Here, the size of the images is \(256\times 256\) pixels, the overlap ratio between two adjacent positions is \(69\%\), the number of vertical and horizontal measurements is \(\mathrm{m}=5\times 5\), and the measured speckle pattern for each subobject is \(\mathrm{N}=150\). In Figure 9, we visually compare the reconstruction results using the conventional method and ptychographic method, respectively. The previous research demonstrated that the Fourier transform of a function with finite support could be uniquely estimated using analytic continuation. However, it is difficult for the conventional method to recover the original image, since a small error in the estimated Fourier component could produce a large error in the recovered values. Many estimated extrapolations are consistent with the real data, with results apparently being superimposed, upside down, or showing twin images during the reconstruction. From the reconstruction results, we can see that the ptychographic imaging correlography outperforms the conventional methods. This method successfully recovers all images. The redundant data provided by adjacent measurements means the algorithm converges with the global optimum with a high probability. The overlapping region of subapertures can constrain the solution space of the reconstruction procedure. As the ratio of intensity measurements to unknowns is increased, the likelihood of ambiguous solutions existing reduces. Therefore, the resolution of the recovered objects is improved.