## 2 Method
### Optimization Framework
#### 2.3.1 Image Recovery and Regularization  
The error reduction (ER) algorithm [31] and hybrid input-output (HIO) algorithm [32] are commonly used phase retrieval methods for imaging correlography. They use a similar method, iterating between frequency and spatial domains with a set of constraints in both. The main difference between them is that the HIO algorithm applies a feedback function outside the object's support. Figure 5: (**a**) The original image. (**b**) The far-field diffraction pattern of the red circle area under ideal conditions. (**c**) The far-field diffraction pattern of the red circle area with phase perturbation. (**d**) The estimated far-field diffraction pattern via correlography. (**e**) The ptychographic reconstruction results of the first scan position with estimated far-field diffraction patterns. In fact, the conventional ptychographic iterative engine (PIE) for ptychographic imaging can be seen as using the ER algorithm for each probe position sequentially if the probe is assumed as a finite support. The main difference between PIE and ER is that PIE updates the subobject sequentially in each iteration, while the ER updates the entire object in each iteration. In this work, we introduce the amplitude flow optimization framework [33] for ptychographic reconstruction and incorporate a regularizer named as regularization by denoising (RED) [34] to suppress noise. This method was first proposed by [PERSON] in the research [35]. It is suitable for various system models and suppresses noise well. As with the relationship between PIE and ER, our work updates the subobject sequentially in each iteration, while the research [35] updates the entire object in each iteration. The amplitude flow algorithm is a recently reported method used to recover the solution x to a system based on the magnitude measurements \(y^{j}=\left\{\left|\left(a^{j},x\right)\right|\right\}_{j=1}^{m}\), where \(a^{j}\) is the feature or sensing vector, x is the wanted unknown signal, and \(y^{j}\) is the given observation. Previous research has demonstrated that this lower-order loss function based on absolute magnitude has great advantages in statistical and computational efficiency. Additionally, the gradient of the amplitude loss function is very similar to that of the Poisson log-likelihood [36]. In our case, the physical process of data detection corresponds to counting the number of electrons and photons hitting the sensor. Measurements in such a process are known to contain the Poisson distribution of noise [37]. Therefore, the amplitude-based reconstruction framework is reasonable for the proposed imaging process. To extend this framework to our reconstruction, the formulation of the proposed imaging model is rewritten as  
\[y^{j}=\left|A^{j}z\right| \tag{10}\]  
where \(y^{j}\) denotes the filtered Fourier magnitude \(\phi^{j}(u,v)\) of the subobject in Equation (8), and \(z\) denotes the desired recovery object. \(A^{j}\) denotes the relation between the reconstruction value \(z\) and the estimated Fourier magnitude \(y^{j}\), which corresponds to two sequential operations: (i) downsampling of the object caused by aperture function; (ii) estimation of the Fourier magnitude of the subobject via imaging correlography. Following the least-squares criterion, the problem of solving Equation (10) is recast as a minimization problem  
\[\text{min}\ \ \ L(z)=\frac{1}{2m}\sum_{j=1}^{m}\left(\left|A^{j}z\right|-y^{j} \right)^{2} \tag{11}\]  
As mentioned above, imaging objects in this method results in noisy reconstructions. To further suppress noise, the RED framework is applied during the reconstruction. RED is an efficient and flexible approach used to solve inverse imaging problems. It can incorporate an arbitrary denoiser to regularize an arbitrary inverse imaging problem. Another advantage of RED is that it employs the denoising engine in defining the regularization of the inverse problem to make the overall reconstruction function clearer and better defined.