## 3 The Method  
### Presentation of Some Models with a Diffusion Term  
First of all, we must consider that an observed digital image \(f_{0}\) is the noisy image. This digital image could be interpreted as a real image \(f\) (the desired one) with one (or several) noise(s) or residual(s) of image r (see [7], p. 6, (18) r = N(0;\(\sigma\)) is considered as a Gaussian noise with mean = 0 and deviation \(\sigma\)). The effects of the degradation (caused by the camera or object motion, atmospheric turbulence optics, etc.) on the piecewise smooth image \(f\) are modelled by a linear or a nonlinear operator \(K\). Usually, the operator \(K\) is named the degradation operator or degradation function, and it is also known as a point spread function (PSF) or kernel.  
This is modelled by the following equation  
\[f_{0}=K(f)+r. \tag{1}\]  
The smoothing process consists of approaching \(f\) from \(f_{0}\) by means of a minimisation problem of the energy function \(E(f;f_{0};\lambda)\) as defined below  
\[E(f,f_{0},\lambda)=E_{reg}(f)+\lambda E_{approx}(f,f_{0}) \tag{2}\]  
where  
\(E(f;f_{0};\lambda)\) stands for the energy that allows one to go from \(f_{0}\) to the desired image;  
\(E_{reg}(f)\) denotes the regularisation energy to the real image \(f\);  
\(E_{approx}(f,f_{0})\) interprets the energy of approximation between the real image \(f\) and the observed image \(f_{0}\), also named the fitting term or the fidelity term;  
\(\lambda\) is the fidelity parameter or the penalty parameter.  
The main goal is to recover an image corrupted by the noise while preserving edges and some details on the observed image and small scale structures such as textures and other details on the observed image.  
Among examples of energies functions, we can cite the following continuous energies functions.  
\[E(f,f_{0},\lambda)=\int_{\Omega}\lvert\
abla f\rvert dx+\frac{\lambda}{2}\int_ {\Omega}\lvert f-f_{0}\rvert^{2}dx, \tag{3}\]  
\[E(f,f_{0},\lambda)=\int_{\Omega}\lvert\
abla f\rvert^{2}dx+\frac{\lambda}{2} \int_{\Omega}\lvert f-f_{0}\rvert^{2}dx. \tag{4}\]  
Models (3) and (4) were introduced by [PERSON] et al. in [3].  
We can also cite the model associated with the following energy function  
\[E(f,f_{0},\lambda)=\int_{\Omega}\lvert\
abla f\rvert^{p(x)}dx+\frac{\lambda}{ 2}\int_{\Omega}\lvert f-f_{0}\rvert^{2}dx. \tag{5}\]  
here, the variable exponent \(p\) is defined as follows  
\[p(x)=\left\{\begin{array}{ll}1&\text{if }\lvert x\rvert>M,\\ 2&\text{if }x=0\end{array}\right.\]  
where \(M>0\) is a positive constant. Precisely, \(M>0\) is either the index of the preservation (i.e., \(\lvert x\rvert>M\)) or the index of degradation (i.e., \(\lvert x\rvert\leq M\)). For more details, a reader interested in the field can refer to [1].  
A variant of Model (5) is the following  
\[E(f,f_{0},\epsilon)=\frac{1}{\epsilon}\int_{|\
abla f(x)|<\epsilon} \lvert\
abla f\rvert^{2}dx+\frac{1}{2\epsilon}\int_{|\
abla f|\geq\epsilon} \Bigl{(}\lvert\
abla f\rvert-\frac{\epsilon}{2}\Bigr{)}dx+\frac{\lambda}{2} \int_{\Omega}\lvert f-f_{0}\rvert^{2}dx. \tag{6}\]  
This energy function was introduced by [PERSON] [2]. In this model, we can observe that \(p=2\) for \(\lvert\
abla f(x)\rvert<\epsilon\), while \(p=1\) for \(\lvert\
abla f(x)\rvert\geq\epsilon\).