## 2 Basis and Fundamentals
### Nonlinear Diffusion  
[PERSON] and [PERSON] proposed a classical nonlinear diffusion model based on the linear diffusion (or heat equation) [6], and the diffusion process can be represented by a partial differential equation:  
\[\frac{\partial I}{\partial t}(x,y,t)=\text{div}\bigg{(}\frac{1}{1+|\
abla I|^{ 2}/k^{2}}\cdot\
abla I\bigg{)} \tag{1}\]  
where \(I(x,y,t)\) refers to the image at time \(t\), div means the divergence operator, \(\
abla I(x,y,t)\) means the spatial gradient of the image at time \(t\), and \(k\) denotes a threshold linked to the features strength. This partial differential equation can be derived by minimizing a related energy functional:  
\[E_{PM}(I)=\frac{k^{2}}{2}log\Big{(}1+|\
abla I|^{2}/k^{2}\Big{)} \tag{2}\]  
Here \(\frac{1}{1+|\
abla I|^{2}/k^{2}}\) means a diffusion velocity, and it can be replaced by other function \(C(|\
abla I|)\) as necessary. In the PM model, the diffusion velocity is determined by a fixed pattern at each iteration, which reduces the diffusion as the gradient module increases to protect the edges present in the image. Some degraded images can be successfully recovered by solving the above equation with a proper initial condition and boundary condition, even if the artifacts still occur in the result due to the instability of the diffusion process. The total variation model (TV) is another representative one of nonlinear diffusion models, and it was first proposed by [PERSON], [PERSON] and [PERSON] [1]. The bounded variation (BV) function is introduced to denote the ideal image and then the discontinuities edge can be well-preserved when removing the noise. This model has been widely applied to many other fields as a regularization technique [26]. It can be denoted as  
\[\frac{\partial I}{\partial t}=\text{div}\bigg{(}\frac{\
abla I}{|\
abla I|} \bigg{)}-\lambda(I-f). \tag{3}\]  
Here, \(f\) denotes the observed degraded image and \(\lambda>0\) is a Lagrangian multiplier which balances different terms. It can be obtained by minimizing a combined energy functional  
\[E_{TV}(u)=\int_{\Omega}|\
abla I|dxdy+\frac{\lambda}{2}\int_{\Omega}|I-f|^{2}dxdy \tag{4}\]  
Differing from the diffusion controlling mechanism of the PM model, the TV model prevents smoothing across the edge everywhere, even if in an almost plain area. For preserving the discontinuous edge and smoothing the approximately flat area, it is often necessary to carefully design numerical schemes with a small time step. Based on the TV model and the PM model, lots of research has been presented to improve the results at different views [3; 27; 28; 29]. The CTAD model [30] was presented to treat the restoration tasks on astronomical images. The local variance of the image gray is embedded in the diffusion velocity of the model for an adaptive process. Specifically, the advanced diffusion velocity is denoted as  
\[C_{CTAD}(|\
abla I|,\sigma^{2})=\frac{1}{1+|\
abla I|^{2}/(k^{2}\cdot\sigma^{ 2})} \tag{5}\]  
where \(\sigma^{2}\) is the local variance of gray levels in a proper neighborhood. A weighted Laplacian was incorporated the modified Perona-Malik model, where the directional Laplacian diffuses the image along the edge direction of the original image [16]. In this model, the diffusion is designed along a unit directional vector \(\overrightarrow{\mathbf{n}}=\left(-f_{y},f_{x}\right)/|\
abla f|\), and the noise can be suppressed by the weighted Laplace operator. The related partial differential equation can be denoted as  
\[\frac{\partial I}{\partial t}=\overrightarrow{\mathbf{n}}\
abla^{T}[C(| \
abla I|)\
abla I]\overrightarrow{\mathbf{n}}^{T}+\alpha\cdot C(|\
abla I|)\Delta I \tag{6}\]  
where \(C(|\
abla I|)=1/\sqrt{1+(|\
abla I|)^{2}}\), \(\alpha\) denotes a small positive constant and \(f\) is the observed degraded image. [PERSON] et al. proposed a modified model that introduces both a probability value and gradient of local image gray to reduce the noise and smooth the noise-free part adaptively [20].