Dirichlet and Neumann formulations for the inpainting problem

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Dirichlet and Neumann formulations for the inpainting problem

In our approach, we now denote by $\Omega$ the image and $\Gamma$ its boundary, $\omega\subset\Omega$ the missing part of the image and $\gamma$ its boundary. Let be the image that we want to restore. We assume that is known in $\Omega\backslash\omega$ , and unknown in $\omega$ .

The idea is to adapt the crack localization method to inpainting: crack detection first allows us to identify the cracks (or edges) $\sigma$ of the hidden part $\omega$ of the image, and then we will impose that the Laplacian of the restored image is equal to zero in $\omega\backslash\sigma$ . For a given crack $\sigma\subset\omega$ , as (Dirichlet condition) and $\partial_n v$ (Neumann condition) are known on the boundary $\gamma$ of $\omega$ , we can solve two different problems inside $\omega$ .

For a given crack $\sigma$ , we denote by $u_D\in H^1(\Omega\backslash\sigma)$ the solution of the following Dirichlet problem:

$\displaystyle \left\{ \begin{array}{l} \Delta u_D = 0 \quad in \ \omega\backsla... ...uad on\ \sigma,\\ u_D = v\quad in \ \Omega\backslash\omega. \end{array} \right.$

(2.11)

Outside $\omega$ , the solution is equal to the original image, and inside $\omega$ , we use equation (2.8).

In the same way, if we assume to be enough regular, we can consider the solution $u_N\in H^1(\Omega\backslash\sigma)$ of the following Neumann problem:

$\displaystyle \left\{ \begin{array}{l} \Delta u_N = 0 \quad in \ \omega\backsla... ...ad on\ \sigma, \\ u_N = v\quad in \ \Omega\backslash\omega. \end{array} \right.$

(2.12)

Note that from the numerical point of view, it is much more easy to solve an approximated Neumann problem:

$\displaystyle \left\{ \begin{array}{l} \Delta u_N = 0 \quad in \ \omega\backsla... ...lpha \Delta u_N + u_N = v\quad in \ \Omega\backslash\omega, \end{array} \right.$

(2.13)

where $\alpha$ is a small positive number.

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