## 2 Analysis of Imaging Properties of HDOE in Wide-Bandwidth Spectrum
### Multimodal Loss Function  
According to the analysis in Section 2.2, it can be concluded that the degradation of HDOE imaging is a complex process, including blurring and edge artifacts caused by chromatic aberration, and stray light pollution caused by non-design order diffraction of light waves. Consquently, four loss function components are tailored to supervise learning various aspects of HDOE image degradation.  
The learning function is \(G:L\to H\), where \(L\) is the low-quality image captured by the detector and \(H\) is the ground truth. Therefore, this process can be expressed as \(\hat{\text{H}}=\text{G}(L)\), where \(\hat{\text{H}}\) is the high-quality image generated by the DDZMR. The loss function of the DDZMR network is formulated as follows:  
\[\text{L}_{DDZMR}=\lambda_{i}\text{L}_{\text{information}}+\lambda_{c1}\text{L}_{ color}+\lambda_{c2}\text{L}_{\text{chromatic}}+\lambda_{s}\text{L}_{stray-light} \tag{12}\]  
The image information and detail loss function was composed of the standard cross entropy function [29], which quantifies the difference in pixel intensity distribution between the reconstructed image \(\hat{\text{H}}\) and the ground truth \(H\). For a total of \(N_{H}\) pixels in \(\hat{\text{H}}\), \(\text{L}_{\text{information}}\) is calculated as:  
\[\text{L}_{\text{information}}=\frac{1}{N_{H}}\sum_{i=1}^{N_{H}}\left[-H_{i}log \hat{\text{H}}_{i}-(1-H_{i})log(1-\hat{\text{H}}_{i})\right] \tag{13}\]  
The contrast evaluation loss function evaluates and calculates the overall similarity of the real measurement values in the RGB space of the reconstruction \(\hat{\text{H}}\) and \(H\), respectively [30]. The images captured by SHDCIS have contrast differences between the three RGB channels and the real image. This difference is quantified by expressing \(\Delta r\), \(\Delta g\), and \(\Delta b\) as the per-channel numerical difference between \(\hat{\text{H}}\) and \(L\). Thus, the contrast evaluation loss function is expressed as:  
\[\text{L}_{\text{color}}=\left\|4(\Delta r-\Delta g)^{2}+(\Delta r+\Delta g-2 \Delta b)^{2}\right\|_{2} \tag{14}\]  
The chromatic aberration loss function designed to evaluate artifacts and blurring at the edges of images, which are mainly caused by the chromatic aberration of HDOE. Local image gradients get used to measure the effect of chromatic aberration correction in DDZMR network output images. The standard \(3\times 3\) Sobel operator to calculate the spatial gradient of image \(I\) is expressed as: \(\
abla I=\sqrt{I_{x}^{2}+I_{y}^{2}}\)[31]. Thus, the chromatic aberration loss functions are defined as [32]:  
\[\mathrm{L_{chromatic}}=\left\|\
abla H|^{2}-\left|\
abla\mathrm{\hat{H}}\right| ^{2}\right\|_{1} \tag{15}\]  
The purpose of the stray light loss function is to force the image generated by the network to restore the feature content similar to the ground truth, and remove the haze-like noise pollution, which is caused by the stray light pollution formed by the non-design order diffraction of the light wave on the HDOE. The haze-like noise pollution measurement function is defined as the high-level features extracted by the last convolutional layer of the pre-trained VGG-16 network [33]. Consequently, the stray light loss function is expressed as:  
\[\mathrm{L_{stray-light}}=\left\|\Phi_{VGG}(H)-\Phi_{VGG}(\mathrm{\hat{H}}) \right\|_{2} \tag{16}\]