# Physics-Based Noise Modeling and Deep Learning for Denoising Permanently Shadowed Lunar Images
## 2 Datasets
### Simulated Data of Lunar PSRs  
The core challenge in obtaining high-quality permanent shadow region images lies in the scarcity of photon reflections from these low-light areas, where photon noise and sensor noise dominate [3]. Therefore, in this study, we adopted the physical noise model proposed by [PERSON] et al. (2021) [29] to train the deep learning model. This model, based on the imaging principles of the LROC NAC charge-coupled device (CCD), is used to simulate images of the permanent shadow regions. The specific noise synthesis formula is shown in Equation (1), and the flowchart of the process is illustrated in Figure 1.  
\[I=N\times\left(F\times\left(S+N_{p}\right)\right)+\left(N_{d}+T\right)+N_{b}+N_{ r}+N_{c} \tag{1}\]  
In the above equation, the mean photon signal \(S\) represents the ideal image without interference, while \(I\) denotes the simulated low-light image, and \(N_{p}\) represents photon noise. The flat-field response is described by \(F\), and \(N\) captures the nonlinear response of the camera. The expression (\(N_{d}+T\)) represents dark current noise, where \(T\) is the temperature of the CCD in the NAC camera. Additionally, \(N_{b}\) stands for dark bias noise, \(N_{r}\) for Read noise, and \(N_{c}\) corresponds to companding noise. Specifically, photon noise \(N_{p}\) arises from the random arrival of photons at the CCD sensor and follows a Poisson distribution. When recording digital noise (DN) values, the LROC NAC responds linearly to the photon signal \(S\). However, when DN values fall below 600, the response becomes nonlinear, necessitating the application of a nonlinear response correction function. The flat-field correction \(F\) addresses the issue of pixel sensitivity differences in the NACs, achieved by multiplying the forward response by the flat-field response. In the noise superposition process, only noiseless sunlight images are used, which contain only the forward response and must be multiplied by the flat-field response \(F\). This flat-field response is generated from a specialized image, ensuring that the mean value of the central region is 1.0 [36]. Read noise \(N_{r}\) is the random system noise generated when carriers are converted into analog voltage signals. Dark noise consists of two components: dark current noise (\(N_{d}+T\)) and dark bias noise \(N_{b}\). Dark current noise originates from thermally excited carriers in the CCD, manifesting as horizontal and vertical stripes. Dark bias noise is deterministic, primarily appearing as vertical stripes. Therefore, dark noise can be considered a form of striped noise. To simulate dark noise, we randomly selected dark noise frames containing dark current noise, dark bias noise, and read noise. Companding  
Figure 1: The flowchart of data synthesis. noise \(N_{c}\) was not introduced, as these frames used a one-to-one mapping from 12-bit images to 8-bit format. Companding noise \(N_{c}\) is the noise generated when the NAC image undergoes a lossy compression process, reducing the 16-bit image to an 8-bit image, reflecting the core limitation of DN resolution. For the NACs, a compression scheme specifically designed for images with DN values lower than 500 was used. The noise generation process involves applying a compression function to the synthetic noise image, followed by reverse compression. Figure 2 illustrates the specific process by which the compression function reduces 12-bit values to 8-bit values [29]. The main difference between sunlight-exposed images and PSR images lies in the lighting environment. Sunlight-exposed images capture areas directly exposed to sunlight, while PSR images record areas illuminated only by indirect light. To better simulate the lighting conditions of the PSRs in the training data, we selected sunlight-exposed images whose secondary incident light angles are similar to the expected angles in the PSRs. These angles were determined by analyzing secondary incident light angles across several craters, which allowed for the specification of the lighting range for the training data.