## 2 Method
### Imaging Correlography with Ptychography  
The main problem with imaging correlography is that the estimated power spectrum has a very low signal-to-noise ratio (SNR) for high spatial frequencies, making the phase recovery inaccurate. Therefore, the recovered object suffers from low resolution. The ptychographic framework is used to solve the problem of inaccurate phase recovery when using imaging correlography. 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. In this subsection, we show how the ptychographic framework can be adapted to imaging correlography. Similar to ptychography, an aperture \(P(x,y)\) is introduced to measure the object part-by-part. Here, the aperture is assumed to be a low pass filter with \(P(x,y)=1\) within a finite diameter, and is recentered at m different locations \(\left(c^{j}(x),c^{j}(y)\right)\), \(\left(j=1,2,3\ldots m\right)\) in the object domain. The \(n\)-th speckle intensity pattern of the \(j\)-th section of the object can be described as  
Figure 3: (**a**) Original image, (**b**) actual autocorrelation function of the original image, and (**c**) actual Fourier magnitude of the original image. (**d–f**) Estimated autocorrelation function of the original image from 200, 500, and 1000 speckle patterns, respectively. (**g-i**) Estimated Fourier magnitude of the original image from 200, 500, and 1000 speckle patterns, respectively. \[I_{n}^{j}(u,v)=\left|F\big{[}u_{n}(x,y)\cdot P\big{(}x-c^{j}(x),y-c^{j}(y)\big{)} \big{]}\right|^{2} \tag{5}\]  
According to Equation (3), the autocorrelation function \(\Gamma^{j}(x,y)\) of the \(j\)-th subobject can be estimated from the average spectrum of \(N\) speckle intensity patterns. \[\Gamma^{j}(x,y)=\frac{1}{N}\sum_{n=1}^{N}|F^{-1}|I_{n}^{j}(u,v)-\overline{I_{ n}^{j}}|^{2} \tag{6}\]  
where \(\overline{I_{n}^{j}}\) denotes the ensemble (mean) intensity of the \(n\)-th observed speckle pattern of the \(j\)-th subobject. Since the Fourier transform of the autocorrelation of the object's brightness function is equivalent to the squared modulus of the Fourier transform of the brightness function [4], the Fourier magnitude of the \(j\)-th subobject can be described by  
\[U^{j}(u,v)=\sqrt{F\big{[}\Gamma^{j}(x,y)\big{]}} \tag{7}\]  
Note that the estimated \(U^{j}(u,v)\) can be treated as the measured intensity pattern in the far field plane for ptychography. Then, people can reconstruct the object using ptychographic phase retrieval algorithms. Figure 4 is the block diagram of the imaging correlography with ptychography. Compared with Figures 2 and 4, the overlap of adjacent sections in the original object is crucial to provide redundant constraints. It can effectively solve the multiple solution problems and avoid converging to local optimal solutions for the conventional imaging correlography. Figure 5a is an original image, and the red circle area is the first scan position \((j=1)\). Figure 5b is the far-field diffraction pattern of the red circle area. It can be regarded as the measured data from conventional ptychography under ideal conditions. Then, the rapidly changed random numbers \((-\pi,\pi)\) are used for the phase of the optical field to simulate the phase perturbation. Figure 5c shows the recorded far-field diffraction pattern in this case. Compared with Figure 5b, the measured data in Figure 5c is very noisy. However, we can re-estimate the far-field diffraction pattern (Figure 5d) from a series of noisy measurements (Figure 5c) using the imaging correlography method. Figure 5d is the estimated far-field diffraction pattern via correlography. Figure 5e is the ptychographic reconstruction results of the first scan position with estimated far-field diffraction patterns. Therefore, one can re-estimate the far-field diffraction pattern using imaging correlography if the directly measured data are noisy.