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This work considers properties of the Neumann-to-Dirichlet map for theconductivity equation under the assumption that the conductivity is identicallyone close to the boundary of the examined smooth, bounded and simply connecteddomain. It is demonstrated that the so-called bisweep data, i.e., the(relative) potential differences between two boundary points when deltacurrents of opposite signs are applied at the very same points, uniquelydetermine the whole Neumann-to-Dirichlet map. In two dimensions, the bisweepdata extend as a holomorphic function of two variables to some (interior)neighborhood of the product boundary. It follows that the wholeNeumann-to-Dirichlet map is characterized by the derivatives of the bisweepdata at an arbitrary point. On the diagonal of the product boundary, thesederivatives can be given with the help of the derivatives of the (relative)boundary potentials at some fixed point caused by the distributional currentdensities supported at the same point, and thus such point measurementsuniquely define the Neumann-to-Dirichlet map. This observation also leads to anew, truly local uniqueness result for the so-called Calder\'on inverseconductivity problem.

Point measurements for a Neumann-to-Dirichlet map and the Calder\'on problem in the plane