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Variational formulation

Let $ \Omega\subset \mathbb{R}^2$ be an open bounded domain, and $ v\in L^2(\Omega)$ be the noisy image. The enhancement of $ v$ is based on the resolution of the following problem:

$\displaystyle \textrm{find } u\in H^1(\Omega) \textrm{ such that } \left\{ \beg... ...v & in & \Omega,\\ \partial_n u = 0 & on & \partial \Omega, \end{array} \right.$ (2.21)

where $ n$ is the outward unit normal to $ \partial\Omega$ , and $ c$ is the conductivity, to be defined in the following. Several choices can be made for the conductivity, mainly $ c$ equal to a constant value (linear diffusion method: it is fast, but it blurs important structures), or $ c$ defined by a nonlinear function of $ \nabla u$ (nonlinear diffusion method, edge-preserving [114,13]). In the topological gradient approach, $ c$ takes only two values: a constant value $ c_0$ (close to $ 1$ ) in the smooth part of the image, and a very small values $ \varepsilon$ (close to 0 ) on the edges or cracks in order to preserve them.

Setting $ c=0$ on a part of the image is equivalent to perturbing the domain by the insertion of cracks. For a given point $ x_0\in\Omega$ and for a given small parameter $ \rho>0$ , we consider $ \Omega_\rho = \Omega\backslash {\sigma_\rho}$ the perturbed domain by the insertion of a crack $ \sigma_\rho = x_0+\rho \sigma(n)$ , where $ \sigma(n)$ is a straight crack and $ n$ is a unit vector normal to the crack. The variational formulation of the perturbed problem is the following:

$\displaystyle \textrm{find } u_\rho\in H^1(\Omega_\rho) \textrm{ such that } a_\rho(u_\rho,w) = l_\rho(w),\ \forall w\in H^1(\Omega_\rho),$ (2.22)

where $ a_\rho$ (resp. $ l_\rho$ ) is the following bilinear (resp. linear) form defined on $ H^1(\Omega_\rho)$ (resp. $ L^2(\Omega_\rho)$ ) by

$\displaystyle a_\rho(u,w) = \int_{\Omega_\rho} (c\nabla u\nabla w+uw)\, dx, \qquad l_\rho(w) = \int_{\Omega_\rho} vw\, dx.$ (2.23)

Edge detection if equivalent to looking for a subdomain of $ \Omega$ where the energy is small. So our goal is to minimize the energy norm outside edges:

$\displaystyle j(\rho)=J(\Omega_\rho,u_\rho) = \int_{\Omega_\rho} \Vert\nabla u_\rho\Vert^2.$ (2.24)

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