A new method to reconstruct data acquired in a local tomography setup is proposed. This method uses an initial reconstruction and refines it by correcting the low-frequency artifacts, known as the cupping effect. A basis of Gaussian functions is used to correct the initial reconstruction. The coefficients of this basis are found by optimizing iteratively a fidelity term under the constraint of a known sub-region. Using a coarse basis reduces the degrees of freedom of the problem while actually correcting the cupping effect. Simulations show that the known region constraint yields an unbiased reconstruction, in accordance with uniqueness theorems stated in local tomography.