# High Bandwidth-Utilization Digital Holographic Reconstruction Using an Untrained Neural Network
## 3 Method  
When the supervised network is used for image restoration, the cost function is defined to penalize the error between the network's output and the ground truth. Unlike this network, DIP [31] (an unsupervised blind image restoration method) can fit a randomly initialized convolutional neural network (CNN) to a single corrupted image. This method does not need a massive training dataset and ground truth and can naturally learn the uncorrupted and realistic parts using an untrained network in an iteratively convergent manner. Here, inspired by the DIP, we designed a novel learning scheme using the physical model in the training process's cost function for low-carrier frequency fringe demodulation in the slightly off-axis DH system. In the process of fringe demodulation, we recognize the artifacts-free complex amplitude reconstruction of the hologram as a nonlinear optimization problem.  
Essentially, the proposed method migrates the physical model to the unsupervised training process, in which the final artifacts-free recovered results can be regarded as the solution to the nonlinear inverse problem. After the network generates the main body of the object, it continues to generate the delicate phase details of the object instead of the phase artifacts.  
As depicted in Figure 2, the complex amplitude can be reconstructed from the captured hologram (shown in Figure 2a) based on the FT method [6], and then the real and imaginary parts of the reconstructed complex analytic signal are fed into the network as a two-channel input. A deep convolutional auto-encoding network with randomly initialized weights is trained to obtain the real and imaginary parts of the output.  
The reconstructed output (i.e., the reconstructed hologram, as shown in Figure 2g) can be expressed as \(f(O,R)\), where \(O\) is the object wave and \(R\) is the reference wave. The cost function can be defined using the reconstructed hologram and the captured hologram, which will converge to a minimum by updating the reconstructed complex amplitude as follows:  
\[w=\text{argmin}\left\|H-f(O,R)\right\|_{2}^{2} \tag{3}\]  
where \(\omega\) is the weight of the network, and \(H\) is the captured hologram. When the cost function based on the gradient descent method is minimized, the network will search for the optimal results by tuning the model parameters within the parameter space. Specifically, the network first generates the primary instance, which is the rough shape estimate of the reconstructed object. Furthermore, the network gradually recovers the artifact-free phase details of the object from coarse to fine.  
In our method, we use an auto-encoding network with an \"hourglass\" architecture [33,34], as shown in Figure 2d, which encodes a high-dimensional network input into lower-dimensional space and reconstructs a high-dimensional output object from the latent variables. The network architecture uses batch normalization [38] and ReLU activation [39] to accelerate the model convergence. Figure 2d shows the hyper-parameters, the tensor dimensions, and the channels of each layer of the network. Notably, in this network, the wavelet transform and its inverse transform are used to realize down-sampling and up-sampling, which replace the pooling, stridden convolution, or interpolation. According to [40], using a wavelet transform could impose sparsity in the reconstruction process and result in more precise reconstruction with lower distortion, which is more suitable for artifacts-free phase recovery in the slightly off-axis system.