## 3 The Proposed Method
### A Novel Energy Functional  
The existence of an inconsistent background (which different in parts) is disadvantageous for treating the polluted and low-contrast image. More exactly, many geometric terms (such as gradient, Laplacian, etc.) are unavoidable to be polluted by not only noise, but also the inconsistent background. Accurate estimation of the background can help to remove the noise and enhance the image quality more efficiently. The observation can be generally denoted as \(f(x)=I(x)+v(x)+n(x)\), and a simple example is shown in the first row of Figure 1. Here, \(f\) is the observed image polluted by Gaussian white noise \(n\) with variance \(\sigma^{2}\) and inconsistent background \(v\), and \(I\) is the clear version of \(f\). Assume that \(v\) is almost smooth everywhere, then a novel energy functional is presented as  
\[E(I,v)=\int_{\Omega}|\
abla I|dxdy+\alpha\int_{\Omega}|\
abla v|^{2}dxdy+\beta \int_{\Omega}|\
abla(I+v)|dxdy \tag{12}\]  
It can be found that the first term means the total variation of the ideal image \(I\). The second term is the \(L_{2}\) energy of the gradient field of background \(v\) and the minimization can be linked to solve a smoothing version of \(v\). The third term is the total variation of \(I+v\), and the minimization means an edge-preserved process. \(\alpha\) and \(\beta\) are both weight factors to balance these terms. Based on the observation model, the variance prior of the noise can be related to a fitting functional \(E_{0}(I,v)=\int_{\Omega}(I+v-f)^{2}dxdy\). In a dynamic minimization process, it can limit the solved \(u,v\) not far away from the observation \(f\). Then an optimization problem to determine \(I,v\) can be denoted as  
\[\min_{I,v}\ E(I,v)\ \ s.t.\ \ E_{0}(I,v)=\sigma^{2}.\]  
Apply the Lagrange method and then the above problem is equivalent to an unconstrained optimization problem, that is \[ \begin{split}\min_{I,v}E_{new}(I,v)&=E(I,v)+\lambda E_{0} (I,v)\\ &=\int_{\Omega}|\
abla I|+\alpha|\
abla v|^{2}+\beta|\
abla(I+v)|+ \lambda(I+v-f)^{2}dxdy\\ &\triangleq\int_{\Omega}F(I,v)dxdy\end{split}  \tag{13}\]  
Here, \(\lambda\) means a Lagrange multiplier to combine the geometric energy and the fitting error. Apply the following Euler-Lagrange equations to compute the gradient flow of (13), and introduce the artificial term \(\frac{\partial I}{\partial t},\frac{\partial v}{\partial t}\), the final evolution equations can be obtained as (14)  
\[\left\{ \begin{array}{l}\frac{\partial F}{\partial I}-\frac{ \partial}{\partial x}(\frac{\partial F}{\partial I_{x}})-\frac{\partial}{ \partial y}(\frac{\partial F}{\partial I_{y}})=0,\\ \frac{\partial F}{\partial v}-\frac{\partial}{\partial x}(\frac{ \partial F}{\partial v_{x}})-\frac{\partial}{\partial y}(\frac{\partial F}{ \partial v_{y}})=0.\end{array} \right.\]  
\[\left\{ \begin{array}{l}\frac{\partial I(x,y,t)}{\partial t}=\
abla\cdot \left(\frac{\
abla I}{|\
abla I|}\right)+\beta\
abla\cdot\left[\frac{\
abla( I+v)}{|\
abla(I+v)|}\right]-\lambda(I+v-f),\\ \frac{\partial v(x,y,t)}{\partial t}=\alpha\Delta v+\beta\
abla\cdot \left[\frac{\
abla(I+v)}{|\
abla(I+v)|}\right]-\lambda(I+v-f).\end{array} \right. \tag{14}\]  
In the evolution, the first equation will preserve the features, including defects to be detected due to the two total variation diffusion terms. The second equation is mainly to approximate the ideal background \(v\) while features are well-protected. Set the homogeneous Neumann boundary condition on \(\partial\Omega\), that is  
\[\frac{\partial I(x,y,t)}{\partial\mathbf{n}}=0,\;\frac{\partial v(x,y,t)}{ \partial\mathbf{n}}=0, \tag{15}\]  
where \(\mathbf{n}\) means the normal vector on the boundary \(\partial\Omega\). Set the initial condition in \(\Omega\)  
\[I(x,y,0)=f(x,y)-v_{0}(x,y),\;v(x,y,0)=v_{0}(x,y), \tag{16}\]  
then Equations (14) can be solved by some numerical methods, such as finite difference. Here, \(v_{0}(x,y)\) means an initial estimation of the background. Figure 1 shows the traditional evolution and the proposed evolution on an artificial image with two suspect areas.