Motivated by the interior tomography problem, we propose a method for exact reconstruction of a region of interest of a function from its local Radon transform in any number of dimensions. Our aim is to verify the feasibility of a one-dimensional reconstruction procedure that can provide the foundation for an efficient algorithm. For a broad class of functions, including piecewise polynomials and generalized splines, we prove that an exact reconstruction is possible by minimizing a generalized total variation seminorm along lines. The main difference with previous works is that our approach is inherently one-dimensional and that it imposes less constraints on the class of admissible signals. Within this formulation, we derive unique reconstruction results using properties of the Hilbert transform, and we present numerical examples of the reconstruction.