## 2 Partial Differential Equations in Image Compression
### Pde-Based Interpolation  
The main goal of PDE-based interpolation is to reconstruct the original image from the known pixels without losing their primary information. With PDE-based interpolation, the image can be reconstructed from sparse data with relative accuracy. The concept of diffusion is known mainly from the physical context. It is a process that compensates for concentration differences without creating or destroying mass. This idea can be applied to image processing tasks, and we will formulate it in a continuous framework. Let \(\Omega\subset\mathbb{R}^{n}\) be an \(n\)-dimensional image domain. An unknown scalar-valued function \(v:\Omega\to\mathbb{R}\) must be recovered, of which only its values on a subset \(\Omega_{1}\subset\Omega\) are known. The goal is to find an interpolating function \(u:\Omega\to\mathbb{R}\) that is smooth and close to \(v\) in \(\Omega\setminus\Omega_{1}\) and identical to \(v\) in \(\Omega_{1}\). This problem can be embedded in an evolution environment with an evolution parameter \(t\geq 0\). Its solution \(u(x,t)\) yields the desired interpolating function as its steady state (\(t\to\infty\)). The evolution is initialized with a function \(f:\Omega\to\mathbb{R}\) initialized to \(v\) identical to \(\Omega_{1}\) and set to \(\Omega\setminus\Omega_{1}\) to any value:\[f(x):=\left\{ \begin{array}{ll}v(x)&\quad\text{if }x\in\Omega_{1}\\ 0&\quad\text{else.}\end{array} \right. \tag{1}\]  
The evolution is considered  
\[\partial_{t}u\ =\ (1-c(x))\,Lu\ -\ c(x)\,(u-f) \tag{2}\]  
with \(f\) as initial value,  
\[u(x,0)\ =\ f(x), \tag{3}\]  
and reflecting (homogeneous Neumann) boundary conditions on the image boundary \(\partial\Omega\). The function \(c:\Omega\to\mathbb{R}\) is the characteristic function on \(\Omega_{1}\), i.e.,  
\[c(x):=\left\{ \begin{array}{ll}1&\quad\text{if }x\in\Omega_{1}\\ 0&\quad\text{else,}\end{array} \right. \tag{4}\]  
and \(L\) is some elliptic differential operator. The idea is to solve the steady state equation  
\[(1-c(x))\,Lu\ -\ c(x)\,(u-f)=0 \tag{5}\]  
with reflecting boundary conditions. In \(\Omega_{1}\), \(c(x)=1\) such that the interpolation condition \(u(x)=f(x)=v(x)\) is fulfilled. In \(\Omega\setminus\Omega_{1}\), it follows from \(c(x)=0\) that the solution has to satisfy \(Lu=0\). This elliptic PDE can be regarded as the steady state of the evolution equation  
\[\partial_{t}u=Lu \tag{6}\]  
with the Dirichlet boundary conditions provided by the interpolation data on \(\Omega_{1}\). **Specific Smoothing Operators. ** There are many possibilities for the elliptic differential operator \(L\). The simplest and best studied uses the Laplace operator \(Lu:=\Delta u\), which leads to _linear isotropic diffusion_[35]:  
\[\partial_{t}u=\Delta u, \tag{7}\]  
also called _homogeneous diffusion_ (_HD_). A prototype for a higher-order differential operator is the biharmonic operator \(Lu:=-\Delta^{2}u\) providing the _biharmonic smoothing (BS)_ evolution  
\[\partial_{t}u=-\Delta^{2}u. \tag{8}\]  
Using it for interpolation comes down to thin plate spline interpolation [36], rotationally invariant multidimensional generalizations of the cubic spline interpolation. The higher-order nonlinear diffusion considered in this paper is related to the differential operator based on Laplacian \(Lu:=-\Delta(g(|\Delta u|^{2})\,\Delta u)\) proposed by [37]. _The fourth-order nonlinear diffusion_ equation is  
\[\partial_{t}u=-\Delta(g(|\Delta u|^{2})\,\Delta u) \tag{9}\]  
where the diffusivity function \(g\) is  
\[g(|\Delta u|^{2})=\frac{1}{\sqrt{1+|\Delta u|^{2}/\lambda^{2}}}. \tag{10}\]  
with some contrast parameter \(\lambda>0\). Since the highest derivative operator is 4, it will be denoted as _fourth-order Charbonnier diffusion (4 ChD)_.