We recall the functionnal to minimize : $$ \begin{align} \bar{I}_{\gamma}(\varphi,V,\tilde{T})=&{\mbox{var}}_g\,\tilde{T}+\dfrac{\nu}{2}\,\|T(\varphi)-R\|_{L^2(\Omega)}^2+\displaystyle{\int_{\Omega}}\,QW(V)\,dx\nonumber \\ &+\dfrac{\gamma}{2}\,\|V-\nabla \varphi\|_{L^2(\Omega,M_2)}^2+\gamma\,\|\tilde{T}-T \circ \varphi\|_{L^1(\Omega)}. \label{le-guyader-equation8} \end{align} $$ with \(QW(\xi)=\left\{\begin{array}{ll} W(\xi)\,\,\,{\mbox{if $\|\xi\|^2 \geq 2\dfrac{\lambda+\mu}{\lambda+2\mu}$}},\\ \Psi({\mbox{det}}\,\xi)\,\,\,{\mbox{if $\|\xi\|^2 < 2\dfrac{\lambda+\mu}{\lambda+2\mu}$}}, \end{array}\right.\) We set \(\widehat{\mathcal{W}}={\mbox{Id}}+W_0^{1,2}(\Omega,\mathbb{R}^2)\) and \(\widehat{\mathcal{\chi}}=\left\{V \in L^{4}(\Omega,M_2(\mathbb{R}))\right\}\). The decoupled problem consists in minimizing \(\bar{I}_{\gamma} \) on \(\widehat{\mathcal{W}} \times \widehat{\mathcal{\chi}}\times BV(\Omega,g)\)

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min det \(\nabla \varphi \)=0.002, max det \(\nabla \varphi \)=2.32.

\( \Delta MI = 12\% \).

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min det \(\nabla \varphi \)=0.37, max det \(\nabla \varphi \)=2.25.

\( \Delta MI = 15\% \).

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min det \(\nabla \varphi \)=0.63, max det \(\nabla \varphi \)=1.28.

\( \Delta MI = 6\% \).

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min det \(\nabla \varphi \)=0.68, max det \(\nabla \varphi \)=1.42.

\( \Delta MI = 14\% \).

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min det \(\nabla \varphi \)=0.46, max det \(\nabla \varphi \)=1.58.

\( \Delta MI = 35\% \).

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min det \(\nabla \varphi \)=0.17, max det \(\nabla \varphi \)=1.45.

\( \Delta MI = 24\% \).

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min det \(\nabla \varphi \)=0.052, max det \(\nabla \varphi \)=2.31.

\( \Delta MI = 41\% \).

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min det \(\nabla \varphi \)=0.006, max det \(\nabla \varphi \)=14.8.

\( \Delta MI = 50\% \).

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min det \(\nabla \varphi \)=0.01, max det \(\nabla \varphi \)=3.83.

\( \Delta MI = 33\% \).

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min det \(\nabla \varphi \)=0.005, max det \(\nabla \varphi \)=2.31.

\( \Delta MI =27\% \).

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min det \(\nabla \varphi \)=0.18, max det \(\nabla \varphi \)=2.92.

\( \Delta MI = 27\% \).

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min det \(\nabla \varphi \)=0.11, max det \(\nabla \varphi \)=3.06.

\( \Delta MI = 24\% \).

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min det \(\nabla \varphi \)=0.65, max det \(\nabla \varphi \)=1.89.

\( \Delta MI = 12\% \).

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The result is better adding the weighted total variation in particular on the boundary, but the hole is not reproduced either.

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- G. E. Christensen, Deformable shape models for anatomy, PhD thesis, Washing- ton University, Sever Institute of technology, USA, 1994.
- J. Modersitzki, Numerical Methods for Image Registration, Oxford University Press, 2004
- T. Lin, C. Le Guyader, I. Dinov, P. Thompson, A. Toga, and L. Vese, Gene Expression Data to Mouse Atlas Registration Using a Nonlinear Elasticity Smoother and Landmark Points Constraints, Washington, J. Sci. Comput. 50 (2012), pp. 586– 609.