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We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $\Omega\subset\mathbb{R}^{n}$ when the so-called Neumann-to-Dirichlet map is locally given on a non empty curved portion $\Sigma$ of the boundary $\partial\Omega$. We prove that anisotropic conductivities that are \textit{a-priori} known to be piecewise constant matrices on a given partition of $\Omega$ with curved interfaces can be uniquely determined in the interior from the knowledge of the local Neumann-to-Dirichlet map. This is joint work with Giovanni Alessandrini and Maarten V. de Hoop.