# High Bandwidth-Utilization Digital Holographic Reconstruction Using an Untrained Neural Network
## 4 Simulation  
In this section, the simulation experiment is designed to demonstrate the validity and performance of the proposed method. To discuss the properties of the method quantitatively, we design the comparison experiment with two single-frame fringe demodulation algorithms based on the physical model: the conventional FT method and VHQPI [24]. We implement our method using the PyTorch Framework [41] in a GPU workstation with an NVIDIA GeForce RTX2080 Ti graphics card. A fixed learning rate of 0.005 for simulation experiment is adopted for the Adam optimizer [42]. A simulated hologram of size \(160\times 160\) obtained excellent training results after 5000 iterations. Specifically, we select two images from MATLAB as intensity and a phase of the complex amplitude of the object wave as \(O(x,y)=A_{O}(x,y)\exp[iq_{O}(x,y)]\). Likewise, in the reference light arm, the plane wave field \(R(x,y)=A_{R}(x,y)\exp[iq_{R}(x,y)]\) is propagated. However, it is difficult to guarantee that the reference light intensity is a constant due to the system's instability and environmental disturbances, so we simulate the reference wave intensity using a two-dimensional standard Gaussian distribution as \(A_{R}(x,y)=\frac{1}{2\pi^{2}}\exp(-(x^{2}+y^{2})/2)\). According to Equation (1), we can naturally obtain the simulated hologram (as shown in Figure 3e). It is also worth noting that we add a pupil function  
Figure 2: The overall block diagram of the proposed learning approach. (**a**) The captured hologram. (**b**,**c**) The real and imaginary parts of the complex amplitudes recovered by the FT method as the input of the network. (**d**) The deep convolutional auto-encoding network with “hourglass” architecture. (**e**,**f**) The corresponding results reconstructed by the network. (**g**) The reconstructed hologram. constraint to the iterative update of the object light in the method. This is because, unlike the lens-free in-line DH system, the slightly off-axis DH system is the imaging system with an objective lens. Adding pupil function constraints during the reconstruction process better complies with the physical model's iterative structure. Moreover, the learning process of the object wave's complex amplitude is often accompanied by high-frequency artifacts, and such a constraint can cleverly realize iterative correction of the artifacts. Figure 3 demonstrates the performance of our method compared with the conventional FT method and the VHQPI. Figure 3b,c show the phase recovery results through the FT method and the VHQPI, respectively, whose enlarged views are shown in Figure 3(b1,c1). In the FT-based phase reconstruction, we implement SFD filtering using the window calculated by the simulated numerical aperture (NA). The imaging result is severely disturbed by artifacts due to the overlapping-spectrum problem. Because of the restrictions of the limited physical model, VHQPI performs a limited capability of zero-order spectrum suppression, and the reconstructed phase results still have certain phase artifacts. In contrast, the result of our method can suppress phase artifacts efficiently and achieve high-accuracy phase imaging, as shown in Figure 3(d,d1). Figure 3a is the ground truth gained by phase recovery after theoretical background removal, and its magnified view is shown in Figure 3(a1). Intuitively, we can easily find that our results are more similar to the ground truth than the other two methods. To discuss the performance of methods quantitatively, the Root Mean Square Error (RMSE) is calculated to measure the average of the squares of the pixel-wise errors between the ground truth image and the reconstructed image. Furthermore, the difference distribution between the recovered results and the ground truth are shown in Figure 3(b2,c2,d2). We can conclude that our method has higher reconstruction accuracy and better artifact-suppression ability. In the real DH system, the captured hologram is always affected by noise due to environmental factors and imperfect detectors, etc.