## 2 Method  
### Image Formation Model for Imaging Correlography  
Here, we assume that the rough-surfaced object is flood-illuminated with the laser beam. Each realization of the field emerging from the object's surface is randomly and coherently dephased, which can be given by the product of  
\[u_{n}(x,y)=\big{|}u_{o}(x,y)\big{|}exp\left[j\varphi_{n}(x,y)\right] \tag{1}\]  
where \(n=1,2,\ldots N\) represent the \(N\) independent speckle pattern realizations, \(\big{|}u_{o}(x,y)\big{|}\) is the object's field amplitude reflectivity, \(\varphi_{n}(x,y)\) is the phase of the reflected field associated with height profile of the object's random rough surface, and \((x,y)\) are the lateral coordinates in the object plane. The \(n\)-th  
Figure 1: Schematic diagram of the ptychographic imaging correlography technique.  
intensity measurement \(I_{n}(u,v)\) of a fully developed laser speckle pattern is recorded by the detectors in the observation plane.  
\[I_{n}(u,v)=\left|\mathcal{F}(u_{n}(x,y))\right|^{2} \tag{2}\]  
where \(\mathcal{F}\) denotes the Fourier transform operator and \((u,v)\) denotes the frequency coordinates in the observation plane. The independent intensity measurements can be realized by rotating the object slightly. From the previous research [30], the autocorrelation function of the object's brightness distribution function \(\Gamma(x,y)\) can be described as  
\[\Gamma(x,y)=\frac{1}{N}\sum_{n=1}^{N}\left|\mathcal{F}^{-1}\left\{I_{n}(u,v)- \overline{I_{n}}\right\}\right|^{2} \tag{3}\]  
where \(\overline{I_{n}}\) is the ensemble (mean) intensity of the \(n\)-th speckle intensity patterns, \(\mathcal{F}^{-1}\) is the inverse Fourier transform operator, and \(N\) is the total number of speckle patterns. Then, a measure of the object's Fourier modulus can be estimated by Fourier transform of the autocorrelation function \(\Gamma(x,y)\), followed by taking the square root.  
\[U(u,v)=\sqrt{\mathcal{F}[\Gamma(x,y)]} \tag{4}\]  
where \(U(u,v)\) denotes the Fourier magnitude of the object's intensity distribution. Then, a phase retrieval algorithm is applied to recover the phase associated with the \(U(u,v)\). By performing the inverse Fourier transform on the synthesized Fourier data, people can reconstruct the image of the object. Figure 2 is the block diagram of the imaging correlography.  
For practical imaging correlography geometries, a single speckle pattern realization (\(N=1\)) yields a rather noisy estimate of the object's Fourier magnitude. Therefore, in most cases, a series of speckle pattern images are required to form a Fourier magnitude estimate that is acceptable for reconstructing an image of the object. Figure 3a is applied to generate the speckle intensity patterns. Figure 3b shows the actual autocorrelation function of the original image, and Figure 3c shows the  
Figure 2: Block diagram of the imaging correlography.  
actual Fourier magnitude of the original image. Figure 3d-f are estimated autocorrelation functions \(\Gamma(x,y)\) from 200, 500, and 1000 speckle frames, respectively, and Figure 3g-i are estimated Fourier magnitudes \(U(u,v)\) from 200, 500, and 1000 speckle frames, respectively. Compared with the actual data and estimated data, it is clear that the autocorrelation function and Fourier magnitude estimated in this manner are noisy. Especially at the higher spatial frequencies, the noise is pronounced and the Fourier modulus is weak. With the number of speckle patterns increasing, the apparent noise at higher spatial frequencies is reduced. It is quite difficult for the iterative phase retrieval algorithm to converge to the global optimal solution with these noisy estimated Fourier modulus data.