# High Bandwidth-Utilization Digital Holographic Reconstruction Using an Untrained Neural Network
## 1 Introduction  
Quantitative phase imaging (QPI) [1; 2; 3] emerges as a powerful, label-free approach to studying transparent cell structure, which makes a significant contribution to biological research and medical diagnosis. It uses the refractive index as the endogenous contrast agent to generate subcellular-specific quantitative maps of analyzed live bio-structure [4; 5]. Digital Holographic Microscopy (DHM) [6; 7; 8; 9; 10] records complete wavefront information by the principle of interferometry and can complete real-time, highly robust QPI with a single-frame hologram. DHM has emerged as a valuable means in the biomedical field, such as for living cell analysis [11], drug release monitoring in vitro [12], and the optical metrology of nanostructures [13]. According to whether there is a titled angle between the object light and the reference light, the holography is categorized into in-line and off-axis digital holography (DH). In-line DH can achieve high-bandwidth phase reconstruction, but the reconstruction result of the object is far from the ground truth due to the influence of twin images. So, it requires multiple holographic imaging at different distances from the object plane along with iterative algorithms [14; 15], which degrades the temporal resolution. The off-axis configuration [16; 17] introduces an additional coherent reference beam that is tilted and superimposed with the object beam, which separates the autocorrelation and cross-correlation terms in the spatial frequency domain (SFD). However, this configuration requires a sufficiently high carrier frequency hologram at the expense of the imaging system's space bandwidth product (SBP). Therefore, the slightly off-axis DH configuration [18; 19], as a high-SBP single-frame demodulation solution, is proposed. It separates the conjugated spectrum while leaving the autocorrelation term partially overlapped with information-carrying, cross-correlation terms. This novel DH configuration makes full use of the sensor's spatial bandwidth, which can significantly improve the detector's spatial bandwidth utilization (SBU) [20]. Notably, in the slightly off-axis DH system, the inevitable overlapping of spectrum produces phase artifacts, which significantly degrades the imaging quality and impairs the system's practicality. To address this problem and achieve high-resolution, artifact-free imaging, many solutions have been proposed. Based on the theory of \"cepstrum\" and homomorphic filtering, the nonlinear filtering approach [21] is proposed to achieve zero-order suppression. It takes the logarithm of the spatial spectrum as the filtering space and employs the additive property to separate the \(\pm 1\)-order terms and the zero-order term. However, this method needs to guarantee that one of the diffraction terms must be confined to one quadrant of the SFD. Furthermore, the Kramers-Kronig (KK) relation [22; 23] is exploited for phase demodulation, which introduces a complex analytic function and utilizes the half-space bandwidth of the sensor to achieve high-SBP imaging. However, it requires separation of the cross-correlation terms of the interferogram in the extended SFD and needs to guarantee that the intensity of the reference wave is much stronger than the object wave. Of note, an exquisite low-carrier frequency fringe demodulation approach has been presented, namely, variational Hilbert quantitative phase imaging (VHQPI) [24]. It adaptively alleviates the overlapped-spectrum problem and improves the system's reliability, demonstrating the excellent practicality in biological applications. Recently, a high-throughput, artifact-free, slightly off-axis holographic reconstruction method based on Fourier ptychographic microscopy (FPM) [25] has been proposed. In this method, the complex object field can be iteratively reconstructed from the recorded hologram by solving a nonlinear optimization problem. With the development of artificial intelligence (AI), deep-learning-based (DL-based) imaging schemes provide a new paradigm for QPI [26; 27; 28; 29].