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This paper is concerned with the detection of objects immersed in anisotropicmedia from boundary measurements. We propose an accurate approach based on theKohn-Vogelius formulation and the topological sensitivity analysis method. Theinverse problem is formulated as a topology optimization one minimizing anenergy like functional. A topological asymptotic expansion is derived for theanisotropic Laplace operator. The unknown object is reconstructed using alevel-set curve of the topological gradient. The efficiency and accuracy of theproposed algorithm are illustrated by some numerical results. MOTS-CL\'ES :Probl\`eme inverse g\'eom\'etrique, Laplace anisotrope, formulation deKohn-Vogelius, analyse de sensibilit\'e, optimisation topologique.

A fast reconstruction algorithm for geometric inverse problems using topological sensitivity analysis and Dirichlet-Neumann cost functional approach