## 3 Experiments and Results
### Results
#### 3.2.3 Effect of the Reconstruction Algorithm  
At last, we compare the proposed optimization framework (AF-RED) with three other reconstruction algorithms, including error reduction (ER) [31], Wirtinger flow (WF) [40], and amplitude flow (AF) [33]. ER is a baseline algorithm for ptychographic reconstruction. WF and AF are both new phase retrieval algorithms. The Wirtinger flow optimization is based on intensity measurements, while the AF optimization is based on amplitude measurements. Because the gradient of the Poisson log-likelihood function is very similar to amplitude-based loss function, PWF [44] is not included in the comparison algorithms. Regarding the proposed reconstruction algorithm (AF-RED), the wavelet filter is employed as the denoiser in the regularization.  
Figure 12a shows the ptychographic measurements for the original object. The circles describe different aperture positions, and the red circle is the central position. Figure 12b is one of the measured speckle patterns for the central probe. Figure 12c-f are the reconstruction results with ER, WF, AF, and AF-RED, respectively. All comparison algorithms use the same initial  
Figure 11: Visual reconstruction results of ptychographic imaging correlography with various speckle frames.  
Figure 10: Quantitative reconstruction results of ptychographic imaging correlography with various speckle frames.  
The Fourier modulus estimated by imaging correlography is not as accurate as that directly measured by ptychographic imaging in microscopy. It contains noise and loses a lot of high-frequency information. Therefore, the recovered data with ER suffer from apparent noise because they are sensitive to input noise. The WF phase retrieval algorithm is based on the Gaussian assumption, which is suitable for microscopic imaging. Regarding the long-distance ptychographic imaging correlography, the main noises are Poisson noise and speckle noise. Thus, there are obvious cloudy features surrounding the recovered object in Figure 12d. The AF reconstruction algorithm constructed by lower-order loss function has great advantages in statistical and computation efficiency compared to Wirtinger flow optimization. Therefore, the noise in Figure 12e is weaker than Figure 12d. The proposed reconstruction method outperforms the other three algorithms significantly by incorporating the advantages of AF and RED frameworks. It not only adopts the lower-order loss function to reconstruct the true solution with high probability, but also recognizes and filters the outliers using the RED framework. Figure 13 is the quantitative comparison of the reconstruction results using different algorithms. We can see that the convergence curve of the proposed algorithm is faster and smoother.  
Figure 12: (**a**) Ptychographic measurements for the original object. (**b**) One of the measured speckle patterns for the center aperture (red circle area (**a**)). (**c**) The reconstruction results for the error reduction (ER) algorithm. (**d**) The reconstruction results for the Wirtinger flow (WF) algorithm. (**e**) The reconstruction results for the amplitude flow (AF) algorithm. (**f**) The reconstruction results for the proposed amplitude flow regularization by denoising (AF-RED) algorithm.  
Figure 13: Quantitative comparison of the reconstruction results using different algorithms.