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Algorithm

Our algorithm consists of inserting small heterogeneities (or cracks) in regions where the topological gradient is smaller than a given threshold. There regions are the edges of the image. The algorithm is as follows:

$ \bullet$
Initialization: $ c=c_0$ (constant value everywhere).
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Calculation of $ u_0$ and $ p_0$ , respectively solutions of the direct (2.21) and adjoint (2.27) unperturbed problems.
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Computation of the $ 2\times 2$ matrix $ M(x)$ defined by (2.28), and of its lowest eigenvalue $ \lambda_{min}(M(x))$ at each point of the domain.
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Set the new conductivity:

$\displaystyle c_1 = \left\{ \begin{array}{l} \varepsilon \textrm{ if } x\in\Ome... ...lambda_{min}(M(x)) < \alpha < 0,\\ c_0 \textrm{ elsewhere,} \end{array} \right.$ (2.29)

where $ \varepsilon>0$ is assumed to be small, and $ \alpha$ is a negative threshold.
$ \bullet$
Calculation of $ u_1$ , the solution of the perturbed direct problem (2.21) using $ c=c_1$ .

The image $ u_1$ is the restored image.


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