A class of inverse problems for the identification of an unknown geometric object from given measurements is considered. A concept for object imaging based on optimality conditions and level sets is introduced which provides high resolution properties of the identification problem and stability to discretization and noise errors. As a specific case, the identification of the center of a test object of arbitrary shape and unknown boundary conditions from d boundary measurements in d spatial dimensions in the context of the Helmholtz equation is described in detail. For analysis and numerical realization, methods from topology optimization, generalized singular perturbations endowed with variational techniques, and a Petrov-Galerkin enrichment within generalized FEM are used.